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newlib/libm/ld/e_remainderl.c | 40 + newlib/libm/ld/e_sinhl.c | 134 +++ newlib/libm/ld/fpmath.h | 82 ++ newlib/libm/ld/math_private.h | 924 ++++++++++++++++++ newlib/libm/ld/s_asinhl.c | 91 ++ newlib/libm/ld/s_atanl.c | 85 ++ newlib/libm/ld/s_cbrtl.c | 143 +++ newlib/libm/ld/s_ceill.c | 101 ++ newlib/libm/ld/s_copysignl.c | 44 + newlib/libm/ld/s_cosl.c | 102 ++ newlib/libm/ld/s_fabsl.c | 45 + newlib/libm/ld/s_fdim.c | 48 + newlib/libm/ld/s_floorl.c | 101 ++ newlib/libm/ld/s_fmal.c | 274 ++++++ newlib/libm/ld/s_fmaxl.c | 57 ++ newlib/libm/ld/s_fminl.c | 57 ++ newlib/libm/ld/s_frexpl.c | 64 ++ newlib/libm/ld/s_ilogbl.c | 53 + newlib/libm/ld/s_llrintl.c | 9 + newlib/libm/ld/s_llroundl.c | 11 + newlib/libm/ld/s_logbl.c | 54 + newlib/libm/ld/s_lrint.c | 60 ++ newlib/libm/ld/s_lrintl.c | 9 + newlib/libm/ld/s_lround.c | 70 ++ newlib/libm/ld/s_lroundl.c | 11 + newlib/libm/ld/s_modfl.c | 103 ++ newlib/libm/ld/s_nearbyint.c | 61 ++ newlib/libm/ld/s_nextafterl.c | 80 ++ 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+++ newlib/libm/ld80/k_cospil.h | 42 + newlib/libm/ld80/k_expl.h | 301 ++++++ newlib/libm/ld80/k_sinpil.h | 42 + newlib/libm/ld80/s_cospil.c | 129 +++ newlib/libm/ld80/s_erfl.c | 337 +++++++ newlib/libm/ld80/s_exp2l.c | 290 ++++++ newlib/libm/ld80/s_expl.c | 279 ++++++ newlib/libm/ld80/s_logl.c | 722 ++++++++++++++ newlib/libm/ld80/s_sinpil.c | 140 +++ 75 files changed, 12021 insertions(+) create mode 100644 newlib/libc/include/sys/endian.h create mode 100644 newlib/libc/machine/aarch64/machine/_fpmath.h create mode 100644 newlib/libc/machine/i386/machine/_fpmath.h create mode 100644 newlib/libc/machine/x86_64/machine/_fpmath.h create mode 100644 newlib/libm/ld/e_acoshl.c create mode 100644 newlib/libm/ld/e_acosl.c create mode 100644 newlib/libm/ld/e_asinl.c create mode 100644 newlib/libm/ld/e_atan2l.c create mode 100644 newlib/libm/ld/e_atanhl.c create mode 100644 newlib/libm/ld/e_coshl.c create mode 100644 newlib/libm/ld/e_fmodl.c create mode 100644 newlib/libm/ld/e_lgammal.c create mode 100644 newlib/libm/ld/e_remainderl.c create mode 100644 newlib/libm/ld/e_sinhl.c create mode 100644 newlib/libm/ld/fpmath.h create mode 100644 newlib/libm/ld/math_private.h create mode 100644 newlib/libm/ld/s_asinhl.c create mode 100644 newlib/libm/ld/s_atanl.c create mode 100644 newlib/libm/ld/s_cbrtl.c create mode 100644 newlib/libm/ld/s_ceill.c create mode 100644 newlib/libm/ld/s_copysignl.c create mode 100644 newlib/libm/ld/s_cosl.c create mode 100644 newlib/libm/ld/s_fabsl.c create mode 100644 newlib/libm/ld/s_fdim.c create mode 100644 newlib/libm/ld/s_floorl.c create mode 100644 newlib/libm/ld/s_fmal.c create mode 100644 newlib/libm/ld/s_fmaxl.c create mode 100644 newlib/libm/ld/s_fminl.c create mode 100644 newlib/libm/ld/s_frexpl.c create mode 100644 newlib/libm/ld/s_ilogbl.c create mode 100644 newlib/libm/ld/s_llrintl.c create mode 100644 newlib/libm/ld/s_llroundl.c create mode 100644 newlib/libm/ld/s_logbl.c create mode 100644 newlib/libm/ld/s_lrint.c create mode 100644 newlib/libm/ld/s_lrintl.c create mode 100644 newlib/libm/ld/s_lround.c create mode 100644 newlib/libm/ld/s_lroundl.c create mode 100644 newlib/libm/ld/s_modfl.c create mode 100644 newlib/libm/ld/s_nearbyint.c create mode 100644 newlib/libm/ld/s_nextafterl.c create mode 100644 newlib/libm/ld/s_nexttoward.c create mode 100644 newlib/libm/ld/s_nexttowardf.c create mode 100644 newlib/libm/ld/s_remquol.c create mode 100644 newlib/libm/ld/s_rintl.c create mode 100644 newlib/libm/ld/s_roundl.c create mode 100644 newlib/libm/ld/s_scalbln.c create mode 100644 newlib/libm/ld/s_scalbnl.c create mode 100644 newlib/libm/ld/s_sinl.c create mode 100644 newlib/libm/ld/s_tanhl.c create mode 100644 newlib/libm/ld/s_tanl.c create mode 100644 newlib/libm/ld/s_truncl.c create mode 100644 newlib/libm/ld128/b_tgammal.c create mode 100644 newlib/libm/ld128/e_powl.c create mode 100644 newlib/libm/ld128/e_rem_pio2l.h create mode 100644 newlib/libm/ld128/invtrig.h create mode 100644 newlib/libm/ld128/k_expl.h create mode 100644 newlib/libm/ld128/s_erfl.c create mode 100644 newlib/libm/ld128/s_exp2l.c create mode 100644 newlib/libm/ld128/s_expl.c create mode 100644 newlib/libm/ld128/s_logl.c create mode 100644 newlib/libm/ld80/b_expl.c create mode 100644 newlib/libm/ld80/b_logl.c create mode 100644 newlib/libm/ld80/b_tgammal.c create mode 100644 newlib/libm/ld80/e_powl.c create mode 100644 newlib/libm/ld80/e_rem_pio2l.h create mode 100644 newlib/libm/ld80/invtrig.h create mode 100644 newlib/libm/ld80/k_cospil.h create mode 100644 newlib/libm/ld80/k_expl.h create mode 100644 newlib/libm/ld80/k_sinpil.h create mode 100644 newlib/libm/ld80/s_cospil.c create mode 100644 newlib/libm/ld80/s_erfl.c create mode 100644 newlib/libm/ld80/s_exp2l.c create mode 100644 newlib/libm/ld80/s_expl.c create mode 100644 newlib/libm/ld80/s_logl.c create mode 100644 newlib/libm/ld80/s_sinpil.c diff --git a/newlib/libc/include/sys/endian.h b/newlib/libc/include/sys/endian.h new file mode 100644 index 000000000..d91c442e9 --- /dev/null +++ b/newlib/libc/include/sys/endian.h @@ -0,0 +1,177 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2002 Thomas Moestl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#ifndef _SYS_ENDIAN_H_ +#define _SYS_ENDIAN_H_ + +#include +#include +#include + +#ifndef _UINT8_T_DECLARED +typedef __uint8_t uint8_t; +#define _UINT8_T_DECLARED +#endif + +#ifndef _UINT16_T_DECLARED +typedef __uint16_t uint16_t; +#define _UINT16_T_DECLARED +#endif + +#ifndef _UINT32_T_DECLARED +typedef __uint32_t uint32_t; +#define _UINT32_T_DECLARED +#endif + +#ifndef _UINT64_T_DECLARED +typedef __uint64_t uint64_t; +#define _UINT64_T_DECLARED +#endif + +/* + * Note: While tempting to try to avoid namespace pollution from this file, + * several software packages assume these marcos are defined, even when it + * defines _POSIX_C_SOURCE to request an unpolluted namespace. + */ + +/* + * General byte order swapping functions. + */ +#define bswap16(x) __bswap16(x) +#define bswap32(x) __bswap32(x) +#define bswap64(x) __bswap64(x) + +/* Alignment-agnostic encode/decode bytestream to/from little/big endian. */ +static __inline uint16_t +be16dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return ((p[0] << 8) | p[1]); +} + +static __inline uint32_t +be32dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return (((unsigned)p[0] << 24) | (p[1] << 16) | (p[2] << 8) | p[3]); +} + +static __inline uint64_t +be64dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return (((uint64_t)be32dec(p) << 32) | be32dec(p + 4)); +} + +static __inline uint16_t +le16dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return ((p[1] << 8) | p[0]); +} + +static __inline uint32_t +le32dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return (((unsigned)p[3] << 24) | (p[2] << 16) | (p[1] << 8) | p[0]); +} + +static __inline uint64_t +le64dec(const void *pp) +{ + uint8_t const *p = (uint8_t const *)pp; + + return (((uint64_t)le32dec(p + 4) << 32) | le32dec(p)); +} + +static __inline void +be16enc(void *pp, uint16_t u) +{ + uint8_t *p = (uint8_t *)pp; + + p[0] = (u >> 8) & 0xff; + p[1] = u & 0xff; +} + +static __inline void +be32enc(void *pp, uint32_t u) +{ + uint8_t *p = (uint8_t *)pp; + + p[0] = (u >> 24) & 0xff; + p[1] = (u >> 16) & 0xff; + p[2] = (u >> 8) & 0xff; + p[3] = u & 0xff; +} + +static __inline void +be64enc(void *pp, uint64_t u) +{ + uint8_t *p = (uint8_t *)pp; + + be32enc(p, (uint32_t)(u >> 32)); + be32enc(p + 4, (uint32_t)(u & 0xffffffffU)); +} + +static __inline void +le16enc(void *pp, uint16_t u) +{ + uint8_t *p = (uint8_t *)pp; + + p[0] = u & 0xff; + p[1] = (u >> 8) & 0xff; +} + +static __inline void +le32enc(void *pp, uint32_t u) +{ + uint8_t *p = (uint8_t *)pp; + + p[0] = u & 0xff; + p[1] = (u >> 8) & 0xff; + p[2] = (u >> 16) & 0xff; + p[3] = (u >> 24) & 0xff; +} + +static __inline void +le64enc(void *pp, uint64_t u) +{ + uint8_t *p = (uint8_t *)pp; + + le32enc(p, (uint32_t)(u & 0xffffffffU)); + le32enc(p + 4, (uint32_t)(u >> 32)); +} +#endif /* _SYS_ENDIAN_H_ */ diff --git a/newlib/libc/machine/aarch64/machine/_fpmath.h b/newlib/libc/machine/aarch64/machine/_fpmath.h new file mode 100644 index 000000000..71d0a7152 --- /dev/null +++ b/newlib/libc/machine/aarch64/machine/_fpmath.h @@ -0,0 +1,58 @@ +/*- + * Copyright (c) 2002, 2003 David Schultz + * Copyright (2) 2014 The FreeBSD Foundation + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +union IEEEl2bits { + long double e; + struct { + unsigned long manl :64; + unsigned long manh :48; + unsigned int exp :15; + unsigned int sign :1; + } bits; + /* TODO andrew: Check the packing here */ + struct { + unsigned long manl :64; + unsigned long manh :48; + unsigned int expsign :16; + } xbits; +}; + +#define LDBL_NBIT 0 +#define LDBL_IMPLICIT_NBIT +#define mask_nbit_l(u) ((void)0) + +#define LDBL_MANH_SIZE 48 +#define LDBL_MANL_SIZE 64 + +#define LDBL_TO_ARRAY32(u, a) do { \ + (a)[0] = (uint32_t)(u).bits.manl; \ + (a)[1] = (uint32_t)((u).bits.manl >> 32); \ + (a)[2] = (uint32_t)(u).bits.manh; \ + (a)[3] = (uint32_t)((u).bits.manh >> 32); \ +} while(0) diff --git a/newlib/libc/machine/i386/machine/_fpmath.h b/newlib/libc/machine/i386/machine/_fpmath.h new file mode 100644 index 000000000..874439c34 --- /dev/null +++ b/newlib/libc/machine/i386/machine/_fpmath.h @@ -0,0 +1,56 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2002, 2003 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +union IEEEl2bits { + long double e; + struct { + unsigned int manl :32; + unsigned int manh :32; + unsigned int exp :15; + unsigned int sign :1; + unsigned int junk :16; + } bits; + struct { + unsigned long long man :64; + unsigned int expsign :16; + unsigned int junk :16; + } xbits; +}; + +#define LDBL_NBIT 0x80000000 +#define mask_nbit_l(u) ((u).bits.manh &= ~LDBL_NBIT) + +#define LDBL_MANH_SIZE 32 +#define LDBL_MANL_SIZE 32 + +#define LDBL_TO_ARRAY32(u, a) do { \ + (a)[0] = (uint32_t)(u).bits.manl; \ + (a)[1] = (uint32_t)(u).bits.manh; \ +} while (0) diff --git a/newlib/libc/machine/x86_64/machine/_fpmath.h b/newlib/libc/machine/x86_64/machine/_fpmath.h new file mode 100644 index 000000000..8be7b7dba --- /dev/null +++ b/newlib/libc/machine/x86_64/machine/_fpmath.h @@ -0,0 +1,57 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2002, 2003 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +union IEEEl2bits { + long double e; + struct { + unsigned int manl :32; + unsigned int manh :32; + unsigned int exp :15; + unsigned int sign :1; + unsigned int junkl :16; + unsigned int junkh :32; + } bits; + struct { + unsigned long man :64; + unsigned int expsign :16; + unsigned long junk :48; + } xbits; +}; + +#define LDBL_NBIT 0x80000000 +#define mask_nbit_l(u) ((u).bits.manh &= ~LDBL_NBIT) + +#define LDBL_MANH_SIZE 32 +#define LDBL_MANL_SIZE 32 + +#define LDBL_TO_ARRAY32(u, a) do { \ + (a)[0] = (uint32_t)(u).bits.manl; \ + (a)[1] = (uint32_t)(u).bits.manh; \ +} while (0) diff --git a/newlib/libm/ld/e_acoshl.c b/newlib/libm/ld/e_acoshl.c new file mode 100644 index 000000000..b9f3aed67 --- /dev/null +++ b/newlib/libm/ld/e_acoshl.c @@ -0,0 +1,89 @@ +/* from: FreeBSD: head/lib/msun/src/e_acosh.c 176451 2008-02-22 02:30:36Z das */ + +/* @(#)e_acosh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See e_acosh.c for complete comments. + * + * Converted to long double by David Schultz and + * Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* EXP_LARGE is the threshold above which we use acosh(x) ~= log(2x). */ +#if LDBL_MANT_DIG == 64 +#define EXP_LARGE 34 +#elif LDBL_MANT_DIG == 113 +#define EXP_LARGE 58 +#else +#error "Unsupported long double format" +#endif + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +one = 1.0; + +#if LDBL_MANT_DIG == 64 +static const union IEEEl2bits +u_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L); +#define ln2 u_ln2.e +#elif LDBL_MANT_DIG == 113 +static const long double +ln2 = 6.93147180559945309417232121458176568e-1L; /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ +#else +#error "Unsupported long double format" +#endif + +long double +acoshl(long double x) +{ + long double t; + int16_t hx; + + ENTERI(); + GET_LDBL_EXPSIGN(hx, x); + if (hx < 0x3fff) { /* x < 1, or misnormal */ + RETURNI((x-x)/(x-x)); + } else if (hx >= BIAS + EXP_LARGE) { /* x >= LARGE */ + if (hx >= 0x7fff) { /* x is inf, NaN or misnormal */ + RETURNI(x+x); + } else + RETURNI(logl(x)+ln2); /* acosh(huge)=log(2x), or misnormal */ + } else if (hx == 0x3fff && x == 1) { + RETURNI(0.0); /* acosh(1) = 0 */ + } else if (hx >= 0x4000) { /* LARGE > x >= 2, or misnormal */ + t=x*x; + RETURNI(logl(2.0*x-one/(x+sqrtl(t-one)))); + } else { /* 1 +__FBSDID("$FreeBSD$"); + +/* + * See comments in e_acos.c. + * Converted to long double by David Schultz . + */ + +#include + +#include "invtrig.h" +#include "math.h" +#include "math_private.h" + +static const long double +one= 1.00000000000000000000e+00; + +#ifdef __i386__ +/* XXX Work around the fact that gcc truncates long double constants on i386 */ +static volatile double +pi1 = 3.14159265358979311600e+00, /* 0x1.921fb54442d18p+1 */ +pi2 = 1.22514845490862001043e-16; /* 0x1.1a80000000000p-53 */ +#define pi ((long double)pi1 + pi2) +#else +static const long double +pi = 3.14159265358979323846264338327950280e+00L; +#endif + +long double +acosl(long double x) +{ + union IEEEl2bits u; + long double z,p,q,r,w,s,c,df; + int16_t expsign, expt; + u.e = x; + expsign = u.xbits.expsign; + expt = expsign & 0x7fff; + if(expt >= BIAS) { /* |x| >= 1 */ + if(expt==BIAS && ((u.bits.manh&~LDBL_NBIT)|u.bits.manl)==0) { + if (expsign>0) return 0.0; /* acos(1) = 0 */ + else return pi+2.0*pio2_lo; /* acos(-1)= pi */ + } + return (x-x)/(x-x); /* acos(|x|>1) is NaN */ + } + if(expt 0.5 */ + z = (one-x)*0.5; + s = sqrtl(z); + u.e = s; + u.bits.manl = 0; + df = u.e; + c = (z-df*df)/(s+df); + p = P(z); + q = Q(z); + r = p/q; + w = r*s+c; + return 2.0*(df+w); + } +} diff --git a/newlib/libm/ld/e_asinl.c b/newlib/libm/ld/e_asinl.c new file mode 100644 index 000000000..a85765f1b --- /dev/null +++ b/newlib/libm/ld/e_asinl.c @@ -0,0 +1,77 @@ + +/* @(#)e_asin.c 1.3 95/01/18 */ +/* FreeBSD: head/lib/msun/src/e_asin.c 176451 2008-02-22 02:30:36Z das */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See comments in e_asin.c. + * Converted to long double by David Schultz . + */ + +#include + +#include "invtrig.h" +#include "math.h" +#include "math_private.h" + +static const long double +one = 1.00000000000000000000e+00, +huge = 1.000e+300; + +long double +asinl(long double x) +{ + union IEEEl2bits u; + long double t=0.0,w,p,q,c,r,s; + int16_t expsign, expt; + u.e = x; + expsign = u.xbits.expsign; + expt = expsign & 0x7fff; + if(expt >= BIAS) { /* |x|>= 1 */ + if(expt==BIAS && ((u.bits.manh&~LDBL_NBIT)|u.bits.manl)==0) + /* asin(1)=+-pi/2 with inexact */ + return x*pio2_hi+x*pio2_lo; + return (x-x)/(x-x); /* asin(|x|>1) is NaN */ + } else if (exptone) return x;/* return x with inexact if x!=0*/ + } + t = x*x; + p = P(t); + q = Q(t); + w = p/q; + return x+x*w; + } + /* 1> |x|>= 0.5 */ + w = one-fabsl(x); + t = w*0.5; + p = P(t); + q = Q(t); + s = sqrtl(t); + if(u.bits.manh>=THRESH) { /* if |x| is close to 1 */ + w = p/q; + t = pio2_hi-(2.0*(s+s*w)-pio2_lo); + } else { + u.e = s; + u.bits.manl = 0; + w = u.e; + c = (t-w*w)/(s+w); + r = p/q; + p = 2.0*s*r-(pio2_lo-2.0*c); + q = pio4_hi-2.0*w; + t = pio4_hi-(p-q); + } + if(expsign>0) return t; else return -t; +} diff --git a/newlib/libm/ld/e_atan2l.c b/newlib/libm/ld/e_atan2l.c new file mode 100644 index 000000000..94ebdec54 --- /dev/null +++ b/newlib/libm/ld/e_atan2l.c @@ -0,0 +1,120 @@ + +/* @(#)e_atan2.c 1.3 95/01/18 */ +/* FreeBSD: head/lib/msun/src/e_atan2.c 176451 2008-02-22 02:30:36Z das */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See comments in e_atan2.c. + * Converted to long double by David Schultz . + */ + +#include + +#include "invtrig.h" +#include "math.h" +#include "math_private.h" + +static volatile long double +tiny = 1.0e-300; +static const long double +zero = 0.0; + +#ifdef __i386__ +/* XXX Work around the fact that gcc truncates long double constants on i386 */ +static volatile double +pi1 = 3.14159265358979311600e+00, /* 0x1.921fb54442d18p+1 */ +pi2 = 1.22514845490862001043e-16; /* 0x1.1a80000000000p-53 */ +#define pi ((long double)pi1 + pi2) +#else +static const long double +pi = 3.14159265358979323846264338327950280e+00L; +#endif + +long double +atan2l(long double y, long double x) +{ + union IEEEl2bits ux, uy; + long double z; + int32_t k,m; + int16_t exptx, expsignx, expty, expsigny; + + uy.e = y; + expsigny = uy.xbits.expsign; + expty = expsigny & 0x7fff; + ux.e = x; + expsignx = ux.xbits.expsign; + exptx = expsignx & 0x7fff; + + if ((exptx==BIAS+LDBL_MAX_EXP && + ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)!=0) || /* x is NaN */ + (expty==BIAS+LDBL_MAX_EXP && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)!=0)) /* y is NaN */ + return nan_mix(x, y); + if (expsignx==BIAS && ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)==0) + return atanl(y); /* x=1.0 */ + m = ((expsigny>>15)&1)|((expsignx>>14)&2); /* 2*sign(x)+sign(y) */ + + /* when y = 0 */ + if(expty==0 && ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)==0) { + switch(m) { + case 0: + case 1: return y; /* atan(+-0,+anything)=+-0 */ + case 2: return pi+tiny;/* atan(+0,-anything) = pi */ + case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */ + } + } + /* when x = 0 */ + if(exptx==0 && ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)==0) + return (expsigny<0)? -pio2_hi-tiny: pio2_hi+tiny; + + /* when x is INF */ + if(exptx==BIAS+LDBL_MAX_EXP) { + if(expty==BIAS+LDBL_MAX_EXP) { + switch(m) { + case 0: return pio2_hi*0.5+tiny;/* atan(+INF,+INF) */ + case 1: return -pio2_hi*0.5-tiny;/* atan(-INF,+INF) */ + case 2: return 1.5*pio2_hi+tiny;/*atan(+INF,-INF)*/ + case 3: return -1.5*pio2_hi-tiny;/*atan(-INF,-INF)*/ + } + } else { + switch(m) { + case 0: return zero ; /* atan(+...,+INF) */ + case 1: return -zero ; /* atan(-...,+INF) */ + case 2: return pi+tiny ; /* atan(+...,-INF) */ + case 3: return -pi-tiny ; /* atan(-...,-INF) */ + } + } + } + /* when y is INF */ + if(expty==BIAS+LDBL_MAX_EXP) + return (expsigny<0)? -pio2_hi-tiny: pio2_hi+tiny; + + /* compute y/x */ + k = expty-exptx; + if(k > LDBL_MANT_DIG+2) { /* |y/x| huge */ + z=pio2_hi+pio2_lo; + m&=1; + } + else if(expsignx<0&&k<-LDBL_MANT_DIG-2) z=0.0; /* |y/x| tiny, x<0 */ + else z=atanl(fabsl(y/x)); /* safe to do y/x */ + switch (m) { + case 0: return z ; /* atan(+,+) */ + case 1: return -z ; /* atan(-,+) */ + case 2: return pi-(z-pi_lo);/* atan(+,-) */ + default: /* case 3 */ + return (z-pi_lo)-pi;/* atan(-,-) */ + } +} diff --git a/newlib/libm/ld/e_atanhl.c b/newlib/libm/ld/e_atanhl.c new file mode 100644 index 000000000..11d56ea52 --- /dev/null +++ b/newlib/libm/ld/e_atanhl.c @@ -0,0 +1,74 @@ +/* from: FreeBSD: head/lib/msun/src/e_atanh.c 176451 2008-02-22 02:30:36Z das */ + +/* @(#)e_atanh.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See e_atanh.c for complete comments. + * + * Converted to long double by David Schultz and + * Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* EXP_TINY is the threshold below which we use atanh(x) ~= x. */ +#if LDBL_MANT_DIG == 64 +#define EXP_TINY -34 +#elif LDBL_MANT_DIG == 113 +#define EXP_TINY -58 +#else +#error "Unsupported long double format" +#endif + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const double one = 1.0, huge = 1e300; +static const double zero = 0.0; + +long double +atanhl(long double x) +{ + long double t; + uint16_t hx, ix; + + ENTERI(); + GET_LDBL_EXPSIGN(hx, x); + ix = hx & 0x7fff; + if (ix >= 0x3fff) /* |x| >= 1, or NaN or misnormal */ + RETURNI(fabsl(x) == 1 ? x / zero : (x - x) / (x - x)); + if (ix < BIAS + EXP_TINY && (huge + x) > zero) + RETURNI(x); /* x is tiny */ + SET_LDBL_EXPSIGN(x, ix); + if (ix < 0x3ffe) { /* |x| < 0.5, or misnormal */ + t = x+x; + t = 0.5*log1pl(t+t*x/(one-x)); + } else + t = 0.5*log1pl((x+x)/(one-x)); + RETURNI((hx & 0x8000) == 0 ? t : -t); +} diff --git a/newlib/libm/ld/e_coshl.c b/newlib/libm/ld/e_coshl.c new file mode 100644 index 000000000..4e3b28311 --- /dev/null +++ b/newlib/libm/ld/e_coshl.c @@ -0,0 +1,132 @@ +/* from: FreeBSD: head/lib/msun/src/e_coshl.c XXX */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See e_cosh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const volatile long double huge = 0x1p10000L, tiny = 0x1p-10000L; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-1, 1], range ~[-1.8211e-21, 1.8211e-21]: + * |cosh(x) - c(x)| < 2**-68.8 + */ +static const union IEEEl2bits +C4u = LD80C(0xaaaaaaaaaaaaac78, -5, 4.16666666666666682297e-2L); +#define C4 C4u.e +static const double +C2 = 0.5, +C6 = 1.3888888888888616e-3, /* 0x16c16c16c16b99.0p-62 */ +C8 = 2.4801587301767953e-5, /* 0x1a01a01a027061.0p-68 */ +C10 = 2.7557319163300398e-7, /* 0x127e4fb6c9b55f.0p-74 */ +C12 = 2.0876768371393075e-9, /* 0x11eed99406a3f4.0p-81 */ +C14 = 1.1469537039374480e-11, /* 0x1938c67cd18c48.0p-89 */ +C16 = 4.8473490896852041e-14; /* 0x1b49c429701e45.0p-97 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-1, 1], range ~[-2.3194e-37, 2.3194e-37]: + * |cosh(x) - c(x)| < 2**-121.69 + */ +static const long double +C4 = 4.16666666666666666666666666666666225e-2L, /* 0x1555555555555555555555555554e.0p-117L */ +C6 = 1.38888888888888888888888888889434831e-3L, /* 0x16c16c16c16c16c16c16c16c1dd7a.0p-122L */ +C8 = 2.48015873015873015873015871870962089e-5L, /* 0x1a01a01a01a01a01a01a017af2756.0p-128L */ +C10 = 2.75573192239858906525574318600800201e-7L, /* 0x127e4fb7789f5c72ef01c8a040640.0p-134L */ +C12 = 2.08767569878680989791444691755468269e-9L, /* 0x11eed8eff8d897b543d0679607399.0p-141L */ +C14= 1.14707455977297247387801189650495351e-11L, /* 0x193974a8c07c9d24ae169a7fa9b54.0p-149L */ +C16 = 4.77947733238737883626416876486279985e-14L; /* 0x1ae7f3e733b814d4e1b90f5727fe4.0p-157L */ +static const double +C2 = 0.5, +C18 = 1.5619206968597871e-16, /* 0x16827863b9900b.0p-105 */ +C20 = 4.1103176218528049e-19, /* 0x1e542ba3d3c269.0p-114 */ +C22 = 8.8967926401641701e-22, /* 0x10ce399542a014.0p-122 */ +C24 = 1.6116681626523904e-24, /* 0x1f2c981d1f0cb7.0p-132 */ +C26 = 2.5022374732804632e-27; /* 0x18c7ecf8b2c4a0.0p-141 */ +#else +#error "Unsupported long double format" +#endif /* LDBL_MANT_DIG == 64 */ + +/* log(2**16385 - 0.5) rounded up: */ +static const float +o_threshold = 1.13572168e4; /* 0xb174de.0p-10 */ + +long double +coshl(long double x) +{ + long double hi,lo,x2,x4; +#if LDBL_MANT_DIG == 113 + double dx2; +#endif + uint16_t ix; + + GET_LDBL_EXPSIGN(ix,x); + ix &= 0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) return x*x; + + ENTERI(); + + /* |x| < 1, return 1 or c(x) */ + if(ix<0x3fff) { + if (ix o_threshold, cosh(x) overflow */ + RETURNI(huge*huge); +} diff --git a/newlib/libm/ld/e_fmodl.c b/newlib/libm/ld/e_fmodl.c new file mode 100644 index 000000000..ad3bcc34c --- /dev/null +++ b/newlib/libm/ld/e_fmodl.c @@ -0,0 +1,149 @@ +/* @(#)e_fmod.c 1.3 95/01/18 */ +/*- + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#define BIAS (LDBL_MAX_EXP - 1) + +#if LDBL_MANL_SIZE > 32 +typedef uint64_t manl_t; +#else +typedef uint32_t manl_t; +#endif + +#if LDBL_MANH_SIZE > 32 +typedef uint64_t manh_t; +#else +typedef uint32_t manh_t; +#endif + +/* + * These macros add and remove an explicit integer bit in front of the + * fractional mantissa, if the architecture doesn't have such a bit by + * default already. + */ +#ifdef LDBL_IMPLICIT_NBIT +#define SET_NBIT(hx) ((hx) | (1ULL << LDBL_MANH_SIZE)) +#define HFRAC_BITS LDBL_MANH_SIZE +#else +#define SET_NBIT(hx) (hx) +#define HFRAC_BITS (LDBL_MANH_SIZE - 1) +#endif + +#define MANL_SHIFT (LDBL_MANL_SIZE - 1) + +static const long double one = 1.0, Zero[] = {0.0, -0.0,}; + +/* + * fmodl(x,y) + * Return x mod y in exact arithmetic + * Method: shift and subtract + * + * Assumptions: + * - The low part of the mantissa fits in a manl_t exactly. + * - The high part of the mantissa fits in an int64_t with enough room + * for an explicit integer bit in front of the fractional bits. + */ +long double +fmodl(long double x, long double y) +{ + union IEEEl2bits ux, uy; + int64_t hx,hz; /* We need a carry bit even if LDBL_MANH_SIZE is 32. */ + manh_t hy; + manl_t lx,ly,lz; + int ix,iy,n,sx; + + ux.e = x; + uy.e = y; + sx = ux.bits.sign; + + /* purge off exception values */ + if((uy.bits.exp|uy.bits.manh|uy.bits.manl)==0 || /* y=0 */ + (ux.bits.exp == BIAS + LDBL_MAX_EXP) || /* or x not finite */ + (uy.bits.exp == BIAS + LDBL_MAX_EXP && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)!=0)) /* or y is NaN */ + return nan_mix_op(x, y, *)/nan_mix_op(x, y, *); + if(ux.bits.exp<=uy.bits.exp) { + if((ux.bits.exp>MANL_SHIFT); lx = lx+lx;} + else { + if ((hz|lz)==0) /* return sign(x)*0 */ + return Zero[sx]; + hx = hz+hz+(lz>>MANL_SHIFT); lx = lz+lz; + } + } + hz=hx-hy;lz=lx-ly; if(lx=0) {hx=hz;lx=lz;} + + /* convert back to floating value and restore the sign */ + if((hx|lx)==0) /* return sign(x)*0 */ + return Zero[sx]; + while(hx<(1ULL<>MANL_SHIFT); lx = lx+lx; + iy -= 1; + } + ux.bits.manh = hx; /* The mantissa is truncated here if needed. */ + ux.bits.manl = lx; + if (iy < LDBL_MIN_EXP) { + ux.bits.exp = iy + (BIAS + 512); + ux.e *= 0x1p-512; + } else { + ux.bits.exp = iy + BIAS; + } + x = ux.e * one; /* create necessary signal */ + return x; /* exact output */ +} diff --git a/newlib/libm/ld/e_lgammal.c b/newlib/libm/ld/e_lgammal.c new file mode 100644 index 000000000..ebc2fc78c --- /dev/null +++ b/newlib/libm/ld/e_lgammal.c @@ -0,0 +1,25 @@ +/* @(#)e_lgamma.c 1.3 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include "math.h" +#include "math_private.h" + +extern int signgam; + +long double +lgammal(long double x) +{ + return lgammal_r(x,&signgam); +} diff --git a/newlib/libm/ld/e_remainderl.c b/newlib/libm/ld/e_remainderl.c new file mode 100644 index 000000000..4a6786309 --- /dev/null +++ b/newlib/libm/ld/e_remainderl.c @@ -0,0 +1,40 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +long double +remainderl(long double x, long double y) +{ + int quo; + + return (remquol(x, y, &quo)); +} diff --git a/newlib/libm/ld/e_sinhl.c b/newlib/libm/ld/e_sinhl.c new file mode 100644 index 000000000..38d3df195 --- /dev/null +++ b/newlib/libm/ld/e_sinhl.c @@ -0,0 +1,134 @@ +/* from: FreeBSD: head/lib/msun/src/e_sinhl.c XXX */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See e_sinh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const long double shuge = 0x1p16383L; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-1, 1], range ~[-6.6749e-22, 6.6749e-22]: + * |sinh(x)/x - s(x)| < 2**-70.3 + */ +static const union IEEEl2bits +S3u = LD80C(0xaaaaaaaaaaaaaaaa, -3, 1.66666666666666666658e-1L); +#define S3 S3u.e +static const double +S5 = 8.3333333333333332e-3, /* 0x11111111111111.0p-59 */ +S7 = 1.9841269841270074e-4, /* 0x1a01a01a01a070.0p-65 */ +S9 = 2.7557319223873889e-6, /* 0x171de3a5565fe6.0p-71 */ +S11 = 2.5052108406704084e-8, /* 0x1ae6456857530f.0p-78 */ +S13 = 1.6059042748655297e-10, /* 0x161245fa910697.0p-85 */ +S15 = 7.6470006914396920e-13, /* 0x1ae7ce4eff2792.0p-93 */ +S17 = 2.8346142308424267e-15; /* 0x19882ce789ffc6.0p-101 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-1, 1], range ~[-2.9673e-36, 2.9673e-36]: + * |sinh(x)/x - s(x)| < 2**-118.0 + */ +static const long double +S3 = 1.66666666666666666666666666666666033e-1L, /* 0x1555555555555555555555555553b.0p-115L */ +S5 = 8.33333333333333333333333333337643193e-3L, /* 0x111111111111111111111111180f5.0p-119L */ +S7 = 1.98412698412698412698412697391263199e-4L, /* 0x1a01a01a01a01a01a01a0176aad11.0p-125L */ +S9 = 2.75573192239858906525574406205464218e-6L, /* 0x171de3a556c7338faac243aaa9592.0p-131L */ +S11 = 2.50521083854417187749675637460977997e-8L, /* 0x1ae64567f544e38fe59b3380d7413.0p-138L */ +S13 = 1.60590438368216146368737762431552702e-10L, /* 0x16124613a86d098059c7620850fc2.0p-145L */ +S15 = 7.64716373181980539786802470969096440e-13L, /* 0x1ae7f3e733b814193af09ce723043.0p-153L */ +S17 = 2.81145725434775409870584280722701574e-15L; /* 0x1952c77030c36898c3fd0b6dfc562.0p-161L */ +static const double +S19= 8.2206352435411005e-18, /* 0x12f49b4662b86d.0p-109 */ +S21= 1.9572943931418891e-20, /* 0x171b8f2fab9628.0p-118 */ +S23 = 3.8679983530666939e-23, /* 0x17617002b73afc.0p-127 */ +S25 = 6.5067867911512749e-26; /* 0x1423352626048a.0p-136 */ +#else +#error "Unsupported long double format" +#endif /* LDBL_MANT_DIG == 64 */ + +/* log(2**16385 - 0.5) rounded up: */ +static const float +o_threshold = 1.13572168e4; /* 0xb174de.0p-10 */ + +long double +sinhl(long double x) +{ + long double hi,lo,x2,x4; +#if LDBL_MANT_DIG == 113 + double dx2; +#endif + double s; + int16_t ix,jx; + + GET_LDBL_EXPSIGN(jx,x); + ix = jx&0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) return x+x; + + ENTERI(); + + s = 1; + if (jx<0) s = -1; + + /* |x| < 64, return x, s(x), or accurate s*(exp(|x|)/2-1/exp(|x|)/2) */ + if (ix<0x4005) { /* |x|<64 */ + if (ix1) RETURNI(x); /* sinh(tiny) = tiny with inexact */ + if (ix<0x3fff) { /* |x|<1 */ + x2 = x*x; +#if LDBL_MANT_DIG == 64 + x4 = x2*x2; + RETURNI(((S17*x2 + S15)*x4 + (S13*x2 + S11))*(x2*x*x4*x4) + + ((S9*x2 + S7)*x2 + S5)*(x2*x*x2) + S3*(x2*x) + x); +#elif LDBL_MANT_DIG == 113 + dx2 = x2; + RETURNI(((((((((((S25*dx2 + S23)*dx2 + + S21)*x2 + S19)*x2 + + S17)*x2 + S15)*x2 + S13)*x2 + S11)*x2 + S9)*x2 + S7)*x2 + + S5)* (x2*x*x2) + + S3*(x2*x) + x); +#endif + } + k_hexpl(fabsl(x), &hi, &lo); + RETURNI(s*(lo - 0.25/(hi + lo) + hi)); + } + + /* |x| in [64, o_threshold], return correctly-overflowing s*exp(|x|)/2 */ + if (fabsl(x) <= o_threshold) + RETURNI(s*hexpl(fabsl(x))); + + /* |x| > o_threshold, sinh(x) overflow */ + return x*shuge; +} diff --git a/newlib/libm/ld/fpmath.h b/newlib/libm/ld/fpmath.h new file mode 100644 index 000000000..ce935eb40 --- /dev/null +++ b/newlib/libm/ld/fpmath.h @@ -0,0 +1,82 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2003 Mike Barcroft + * Copyright (c) 2002 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#ifndef _FPMATH_H_ +#define _FPMATH_H_ + +#include +#include "_fpmath.h" + +#ifndef _IEEE_WORD_ORDER +#define _IEEE_WORD_ORDER _BYTE_ORDER +#endif + +union IEEEf2bits { + float f; + struct { +#if _BYTE_ORDER == _LITTLE_ENDIAN + unsigned int man :23; + unsigned int exp :8; + unsigned int sign :1; +#else /* _BIG_ENDIAN */ + unsigned int sign :1; + unsigned int exp :8; + unsigned int man :23; +#endif + } bits; +}; + +#define DBL_MANH_SIZE 20 +#define DBL_MANL_SIZE 32 + +union IEEEd2bits { + double d; + struct { +#if _BYTE_ORDER == _LITTLE_ENDIAN +#if _IEEE_WORD_ORDER == _LITTLE_ENDIAN + unsigned int manl :32; +#endif + unsigned int manh :20; + unsigned int exp :11; + unsigned int sign :1; +#if _IEEE_WORD_ORDER == _BIG_ENDIAN + unsigned int manl :32; +#endif +#else /* _BIG_ENDIAN */ + unsigned int sign :1; + unsigned int exp :11; + unsigned int manh :20; + unsigned int manl :32; +#endif + } bits; +}; + +#endif /* !_FPMATH_H */ diff --git a/newlib/libm/ld/math_private.h b/newlib/libm/ld/math_private.h new file mode 100644 index 000000000..df526e71e --- /dev/null +++ b/newlib/libm/ld/math_private.h @@ -0,0 +1,924 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * from: @(#)fdlibm.h 5.1 93/09/24 + * $FreeBSD$ + */ + +#ifndef _MATH_PRIVATE_H_ +#define _MATH_PRIVATE_H_ + +#include +#include + +/* + * The original fdlibm code used statements like: + * n0 = ((*(int*)&one)>>29)^1; * index of high word * + * ix0 = *(n0+(int*)&x); * high word of x * + * ix1 = *((1-n0)+(int*)&x); * low word of x * + * to dig two 32 bit words out of the 64 bit IEEE floating point + * value. That is non-ANSI, and, moreover, the gcc instruction + * scheduler gets it wrong. We instead use the following macros. + * Unlike the original code, we determine the endianness at compile + * time, not at run time; I don't see much benefit to selecting + * endianness at run time. + */ + +/* + * A union which permits us to convert between a double and two 32 bit + * ints. + */ + +#ifdef __arm__ +#if defined(__VFP_FP__) || defined(__ARM_EABI__) +#define IEEE_WORD_ORDER BYTE_ORDER +#else +#define IEEE_WORD_ORDER BIG_ENDIAN +#endif +#else /* __arm__ */ +#define IEEE_WORD_ORDER BYTE_ORDER +#endif + +/* A union which permits us to convert between a long double and + four 32 bit ints. */ + +#if IEEE_WORD_ORDER == BIG_ENDIAN + +typedef union +{ + long double value; + struct { + u_int32_t mswhi; + u_int32_t mswlo; + u_int32_t lswhi; + u_int32_t lswlo; + } parts32; + struct { + u_int64_t msw; + u_int64_t lsw; + } parts64; +} ieee_quad_shape_type; + +#endif + +#if IEEE_WORD_ORDER == LITTLE_ENDIAN + +typedef union +{ + long double value; + struct { + u_int32_t lswlo; + u_int32_t lswhi; + u_int32_t mswlo; + u_int32_t mswhi; + } parts32; + struct { + u_int64_t lsw; + u_int64_t msw; + } parts64; +} ieee_quad_shape_type; + +#endif + +#if IEEE_WORD_ORDER == BIG_ENDIAN + +typedef union +{ + double value; + struct + { + u_int32_t msw; + u_int32_t lsw; + } parts; + struct + { + u_int64_t w; + } xparts; +} ieee_double_shape_type; + +#endif + +#if IEEE_WORD_ORDER == LITTLE_ENDIAN + +typedef union +{ + double value; + struct + { + u_int32_t lsw; + u_int32_t msw; + } parts; + struct + { + u_int64_t w; + } xparts; +} ieee_double_shape_type; + +#endif + +/* Get two 32 bit ints from a double. */ + +#define EXTRACT_WORDS(ix0,ix1,d) \ +do { \ + ieee_double_shape_type ew_u; \ + ew_u.value = (d); \ + (ix0) = ew_u.parts.msw; \ + (ix1) = ew_u.parts.lsw; \ +} while (0) + +/* Get a 64-bit int from a double. */ +#define EXTRACT_WORD64(ix,d) \ +do { \ + ieee_double_shape_type ew_u; \ + ew_u.value = (d); \ + (ix) = ew_u.xparts.w; \ +} while (0) + +/* Get the more significant 32 bit int from a double. */ + +#define GET_HIGH_WORD(i,d) \ +do { \ + ieee_double_shape_type gh_u; \ + gh_u.value = (d); \ + (i) = gh_u.parts.msw; \ +} while (0) + +/* Get the less significant 32 bit int from a double. */ + +#define GET_LOW_WORD(i,d) \ +do { \ + ieee_double_shape_type gl_u; \ + gl_u.value = (d); \ + (i) = gl_u.parts.lsw; \ +} while (0) + +/* Set a double from two 32 bit ints. */ + +#define INSERT_WORDS(d,ix0,ix1) \ +do { \ + ieee_double_shape_type iw_u; \ + iw_u.parts.msw = (ix0); \ + iw_u.parts.lsw = (ix1); \ + (d) = iw_u.value; \ +} while (0) + +/* Set a double from a 64-bit int. */ +#define INSERT_WORD64(d,ix) \ +do { \ + ieee_double_shape_type iw_u; \ + iw_u.xparts.w = (ix); \ + (d) = iw_u.value; \ +} while (0) + +/* Set the more significant 32 bits of a double from an int. */ + +#define SET_HIGH_WORD(d,v) \ +do { \ + ieee_double_shape_type sh_u; \ + sh_u.value = (d); \ + sh_u.parts.msw = (v); \ + (d) = sh_u.value; \ +} while (0) + +/* Set the less significant 32 bits of a double from an int. */ + +#define SET_LOW_WORD(d,v) \ +do { \ + ieee_double_shape_type sl_u; \ + sl_u.value = (d); \ + sl_u.parts.lsw = (v); \ + (d) = sl_u.value; \ +} while (0) + +/* + * A union which permits us to convert between a float and a 32 bit + * int. + */ + +typedef union +{ + float value; + /* FIXME: Assumes 32 bit int. */ + unsigned int word; +} ieee_float_shape_type; + +/* Get a 32 bit int from a float. */ + +#define GET_FLOAT_WORD(i,d) \ +do { \ + ieee_float_shape_type gf_u; \ + gf_u.value = (d); \ + (i) = gf_u.word; \ +} while (0) + +/* Set a float from a 32 bit int. */ + +#define SET_FLOAT_WORD(d,i) \ +do { \ + ieee_float_shape_type sf_u; \ + sf_u.word = (i); \ + (d) = sf_u.value; \ +} while (0) + +/* + * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long + * double. + */ + +#define EXTRACT_LDBL80_WORDS(ix0,ix1,d) \ +do { \ + union IEEEl2bits ew_u; \ + ew_u.e = (d); \ + (ix0) = ew_u.xbits.expsign; \ + (ix1) = ew_u.xbits.man; \ +} while (0) + +/* + * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit + * long double. + */ + +#define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d) \ +do { \ + union IEEEl2bits ew_u; \ + ew_u.e = (d); \ + (ix0) = ew_u.xbits.expsign; \ + (ix1) = ew_u.xbits.manh; \ + (ix2) = ew_u.xbits.manl; \ +} while (0) + +/* Get expsign as a 16 bit int from a long double. */ + +#define GET_LDBL_EXPSIGN(i,d) \ +do { \ + union IEEEl2bits ge_u; \ + ge_u.e = (d); \ + (i) = ge_u.xbits.expsign; \ +} while (0) + +/* + * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int + * mantissa. + */ + +#define INSERT_LDBL80_WORDS(d,ix0,ix1) \ +do { \ + union IEEEl2bits iw_u; \ + iw_u.xbits.expsign = (ix0); \ + iw_u.xbits.man = (ix1); \ + (d) = iw_u.e; \ +} while (0) + +/* + * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints + * comprising the mantissa. + */ + +#define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2) \ +do { \ + union IEEEl2bits iw_u; \ + iw_u.xbits.expsign = (ix0); \ + iw_u.xbits.manh = (ix1); \ + iw_u.xbits.manl = (ix2); \ + (d) = iw_u.e; \ +} while (0) + +/* Set expsign of a long double from a 16 bit int. */ + +#define SET_LDBL_EXPSIGN(d,v) \ +do { \ + union IEEEl2bits se_u; \ + se_u.e = (d); \ + se_u.xbits.expsign = (v); \ + (d) = se_u.e; \ +} while (0) + +#ifdef __i386__ +/* Long double constants are broken on i386. */ +#define LD80C(m, ex, v) { \ + .xbits.man = __CONCAT(m, ULL), \ + .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0), \ +} +#else +/* The above works on non-i386 too, but we use this to check v. */ +#define LD80C(m, ex, v) { .e = (v), } +#endif + +#ifdef FLT_EVAL_METHOD +/* + * Attempt to get strict C99 semantics for assignment with non-C99 compilers. + */ +#if FLT_EVAL_METHOD == 0 || __GNUC__ == 0 +#define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) +#else +#define STRICT_ASSIGN(type, lval, rval) do { \ + volatile type __lval; \ + \ + if (sizeof(type) >= sizeof(long double)) \ + (lval) = (rval); \ + else { \ + __lval = (rval); \ + (lval) = __lval; \ + } \ +} while (0) +#endif +#endif /* FLT_EVAL_METHOD */ + +/* Support switching the mode to FP_PE if necessary. */ +#if defined(__i386__) && !defined(NO_FPSETPREC) +#define ENTERI() ENTERIT(long double) +#define ENTERIT(returntype) \ + returntype __retval; \ + fp_prec_t __oprec; \ + \ + if ((__oprec = fpgetprec()) != FP_PE) \ + fpsetprec(FP_PE) +#define RETURNI(x) do { \ + __retval = (x); \ + if (__oprec != FP_PE) \ + fpsetprec(__oprec); \ + RETURNF(__retval); \ +} while (0) +#define ENTERV() \ + fp_prec_t __oprec; \ + \ + if ((__oprec = fpgetprec()) != FP_PE) \ + fpsetprec(FP_PE) +#define RETURNV() do { \ + if (__oprec != FP_PE) \ + fpsetprec(__oprec); \ + return; \ +} while (0) +#else +#define ENTERI() +#define ENTERIT(x) +#define RETURNI(x) RETURNF(x) +#define ENTERV() +#define RETURNV() return +#endif + +/* Default return statement if hack*_t() is not used. */ +#define RETURNF(v) return (v) + +/* + * 2sum gives the same result as 2sumF without requiring |a| >= |b| or + * a == 0, but is slower. + */ +#define _2sum(a, b) do { \ + __typeof(a) __s, __w; \ + \ + __w = (a) + (b); \ + __s = __w - (a); \ + (b) = ((a) - (__w - __s)) + ((b) - __s); \ + (a) = __w; \ +} while (0) + +/* + * 2sumF algorithm. + * + * "Normalize" the terms in the infinite-precision expression a + b for + * the sum of 2 floating point values so that b is as small as possible + * relative to 'a'. (The resulting 'a' is the value of the expression in + * the same precision as 'a' and the resulting b is the rounding error.) + * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and + * exponent overflow or underflow must not occur. This uses a Theorem of + * Dekker (1971). See Knuth (1981) 4.2.2 Theorem C. The name "TwoSum" + * is apparently due to Skewchuk (1997). + * + * For this to always work, assignment of a + b to 'a' must not retain any + * extra precision in a + b. This is required by C standards but broken + * in many compilers. The brokenness cannot be worked around using + * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this + * algorithm would be destroyed by non-null strict assignments. (The + * compilers are correct to be broken -- the efficiency of all floating + * point code calculations would be destroyed similarly if they forced the + * conversions.) + * + * Fortunately, a case that works well can usually be arranged by building + * any extra precision into the type of 'a' -- 'a' should have type float_t, + * double_t or long double. b's type should be no larger than 'a's type. + * Callers should use these types with scopes as large as possible, to + * reduce their own extra-precision and efficiciency problems. In + * particular, they shouldn't convert back and forth just to call here. + */ +#ifdef DEBUG +#define _2sumF(a, b) do { \ + __typeof(a) __w; \ + volatile __typeof(a) __ia, __ib, __r, __vw; \ + \ + __ia = (a); \ + __ib = (b); \ + assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib)); \ + \ + __w = (a) + (b); \ + (b) = ((a) - __w) + (b); \ + (a) = __w; \ + \ + /* The next 2 assertions are weak if (a) is already long double. */ \ + assert((long double)__ia + __ib == (long double)(a) + (b)); \ + __vw = __ia + __ib; \ + __r = __ia - __vw; \ + __r += __ib; \ + assert(__vw == (a) && __r == (b)); \ +} while (0) +#else /* !DEBUG */ +#define _2sumF(a, b) do { \ + __typeof(a) __w; \ + \ + __w = (a) + (b); \ + (b) = ((a) - __w) + (b); \ + (a) = __w; \ +} while (0) +#endif /* DEBUG */ + +/* + * Set x += c, where x is represented in extra precision as a + b. + * x must be sufficiently normalized and sufficiently larger than c, + * and the result is then sufficiently normalized. + * + * The details of ordering are that |a| must be >= |c| (so that (a, c) + * can be normalized without extra work to swap 'a' with c). The details of + * the normalization are that b must be small relative to the normalized 'a'. + * Normalization of (a, c) makes the normalized c tiny relative to the + * normalized a, so b remains small relative to 'a' in the result. However, + * b need not ever be tiny relative to 'a'. For example, b might be about + * 2**20 times smaller than 'a' to give about 20 extra bits of precision. + * That is usually enough, and adding c (which by normalization is about + * 2**53 times smaller than a) cannot change b significantly. However, + * cancellation of 'a' with c in normalization of (a, c) may reduce 'a' + * significantly relative to b. The caller must ensure that significant + * cancellation doesn't occur, either by having c of the same sign as 'a', + * or by having |c| a few percent smaller than |a|. Pre-normalization of + * (a, b) may help. + * + * This is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2 + * exercise 19). We gain considerable efficiency by requiring the terms to + * be sufficiently normalized and sufficiently increasing. + */ +#define _3sumF(a, b, c) do { \ + __typeof(a) __tmp; \ + \ + __tmp = (c); \ + _2sumF(__tmp, (a)); \ + (b) += (a); \ + (a) = __tmp; \ +} while (0) + +/* + * Common routine to process the arguments to nan(), nanf(), and nanl(). + */ +void _scan_nan(uint32_t *__words, int __num_words, const char *__s); + +/* + * Mix 0, 1 or 2 NaNs. First add 0 to each arg. This normally just turns + * signaling NaNs into quiet NaNs by setting a quiet bit. We do this + * because we want to never return a signaling NaN, and also because we + * don't want the quiet bit to affect the result. Then mix the converted + * args using the specified operation. + * + * When one arg is NaN, the result is typically that arg quieted. When both + * args are NaNs, the result is typically the quietening of the arg whose + * mantissa is largest after quietening. When neither arg is NaN, the + * result may be NaN because it is indeterminate, or finite for subsequent + * construction of a NaN as the indeterminate 0.0L/0.0L. + * + * Technical complications: the result in bits after rounding to the final + * precision might depend on the runtime precision and/or on compiler + * optimizations, especially when different register sets are used for + * different precisions. Try to make the result not depend on at least the + * runtime precision by always doing the main mixing step in long double + * precision. Try to reduce dependencies on optimizations by adding the + * the 0's in different precisions (unless everything is in long double + * precision). + */ +#define nan_mix(x, y) (nan_mix_op((x), (y), +)) +#define nan_mix_op(x, y, op) (((x) + 0.0L) op ((y) + 0)) + +#ifdef _COMPLEX_H + +/* + * C99 specifies that complex numbers have the same representation as + * an array of two elements, where the first element is the real part + * and the second element is the imaginary part. + */ +typedef union { + float complex f; + float a[2]; +} float_complex; +typedef union { + double complex f; + double a[2]; +} double_complex; +typedef union { + long double complex f; + long double a[2]; +} long_double_complex; +#define REALPART(z) ((z).a[0]) +#define IMAGPART(z) ((z).a[1]) + +/* + * Inline functions that can be used to construct complex values. + * + * The C99 standard intends x+I*y to be used for this, but x+I*y is + * currently unusable in general since gcc introduces many overflow, + * underflow, sign and efficiency bugs by rewriting I*y as + * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product. + * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted + * to -0.0+I*0.0. + * + * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL() + * to construct complex values. Compilers that conform to the C99 + * standard require the following functions to avoid the above issues. + */ + +#ifndef CMPLXF +static __inline float complex +CMPLXF(float x, float y) +{ + float_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#ifndef CMPLX +static __inline double complex +CMPLX(double x, double y) +{ + double_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#ifndef CMPLXL +static __inline long double complex +CMPLXL(long double x, long double y) +{ + long_double_complex z; + + REALPART(z) = x; + IMAGPART(z) = y; + return (z.f); +} +#endif + +#endif /* _COMPLEX_H */ + +/* + * The rnint() family rounds to the nearest integer for a restricted range + * range of args (up to about 2**MANT_DIG). We assume that the current + * rounding mode is FE_TONEAREST so that this can be done efficiently. + * Extra precision causes more problems in practice, and we only centralize + * this here to reduce those problems, and have not solved the efficiency + * problems. The exp2() family uses a more delicate version of this that + * requires extracting bits from the intermediate value, so it is not + * centralized here and should copy any solution of the efficiency problems. + */ + +static inline double +rnint(__double_t x) +{ + /* + * This casts to double to kill any extra precision. This depends + * on the cast being applied to a double_t to avoid compiler bugs + * (this is a cleaner version of STRICT_ASSIGN()). This is + * inefficient if there actually is extra precision, but is hard + * to improve on. We use double_t in the API to minimise conversions + * for just calling here. Note that we cannot easily change the + * magic number to the one that works directly with double_t, since + * the rounding precision is variable at runtime on x86 so the + * magic number would need to be variable. Assuming that the + * rounding precision is always the default is too fragile. This + * and many other complications will move when the default is + * changed to FP_PE. + */ + return ((double)(x + 0x1.8p52) - 0x1.8p52); +} + +static inline float +rnintf(__float_t x) +{ + /* + * As for rnint(), except we could just call that to handle the + * extra precision case, usually without losing efficiency. + */ + return ((float)(x + 0x1.8p23F) - 0x1.8p23F); +} + +#ifdef LDBL_MANT_DIG +/* + * The complications for extra precision are smaller for rnintl() since it + * can safely assume that the rounding precision has been increased from + * its default to FP_PE on x86. We don't exploit that here to get small + * optimizations from limiting the rangle to double. We just need it for + * the magic number to work with long doubles. ld128 callers should use + * rnint() instead of this if possible. ld80 callers should prefer + * rnintl() since for amd64 this avoids swapping the register set, while + * for i386 it makes no difference (assuming FP_PE), and for other arches + * it makes little difference. + */ +static inline long double +rnintl(long double x) +{ + return (x + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2 - + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2); +} +#endif /* LDBL_MANT_DIG */ + +/* + * irint() and i64rint() give the same result as casting to their integer + * return type provided their arg is a floating point integer. They can + * sometimes be more efficient because no rounding is required. + */ +#if defined(amd64) || defined(__i386__) +#define irint(x) \ + (sizeof(x) == sizeof(float) && \ + sizeof(__float_t) == sizeof(long double) ? irintf(x) : \ + sizeof(x) == sizeof(double) && \ + sizeof(__double_t) == sizeof(long double) ? irintd(x) : \ + sizeof(x) == sizeof(long double) ? irintl(x) : (int)(x)) +#else +#define irint(x) ((int)(x)) +#endif + +#define i64rint(x) ((int64_t)(x)) /* only needed for ld128 so not opt. */ + +#if defined(__i386__) +static __inline int +irintf(float x) +{ + int n; + + __asm("fistl %0" : "=m" (n) : "t" (x)); + return (n); +} + +static __inline int +irintd(double x) +{ + int n; + + __asm("fistl %0" : "=m" (n) : "t" (x)); + return (n); +} +#endif + +#if defined(__amd64__) || defined(__i386__) +static __inline int +irintl(long double x) +{ + int n; + + __asm("fistl %0" : "=m" (n) : "t" (x)); + return (n); +} +#endif + +#ifdef DEBUG +#if defined(__amd64__) || defined(__i386__) +#define breakpoint() asm("int $3") +#else +#include + +#define breakpoint() raise(SIGTRAP) +#endif +#endif + +/* Write a pari script to test things externally. */ +#ifdef DOPRINT +#include + +#ifndef DOPRINT_SWIZZLE +#define DOPRINT_SWIZZLE 0 +#endif + +#ifdef DOPRINT_LD80 + +#define DOPRINT_START(xp) do { \ + uint64_t __lx; \ + uint16_t __hx; \ + \ + /* Hack to give more-problematic args. */ \ + EXTRACT_LDBL80_WORDS(__hx, __lx, *xp); \ + __lx ^= DOPRINT_SWIZZLE; \ + INSERT_LDBL80_WORDS(*xp, __hx, __lx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#elif defined(DOPRINT_D64) + +#define DOPRINT_START(xp) do { \ + uint32_t __hx, __lx; \ + \ + EXTRACT_WORDS(__hx, __lx, *xp); \ + __lx ^= DOPRINT_SWIZZLE; \ + INSERT_WORDS(*xp, __hx, __lx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#elif defined(DOPRINT_F32) + +#define DOPRINT_START(xp) do { \ + uint32_t __hx; \ + \ + GET_FLOAT_WORD(__hx, *xp); \ + __hx ^= DOPRINT_SWIZZLE; \ + SET_FLOAT_WORD(*xp, __hx); \ + printf("x = %.21Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */ + +#ifndef DOPRINT_SWIZZLE_HIGH +#define DOPRINT_SWIZZLE_HIGH 0 +#endif + +#define DOPRINT_START(xp) do { \ + uint64_t __lx, __llx; \ + uint16_t __hx; \ + \ + EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp); \ + __llx ^= DOPRINT_SWIZZLE; \ + __lx ^= DOPRINT_SWIZZLE_HIGH; \ + INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx); \ + printf("x = %.36Lg; ", (long double)*xp); \ +} while (0) +#define DOPRINT_END1(v) \ + printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v)) +#define DOPRINT_END2(hi, lo) \ + printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n", \ + (long double)(hi), (long double)(lo)) + +#endif /* DOPRINT_LD80 */ + +#else /* !DOPRINT */ +#define DOPRINT_START(xp) +#define DOPRINT_END1(v) +#define DOPRINT_END2(hi, lo) +#endif /* DOPRINT */ + +#define RETURNP(x) do { \ + DOPRINT_END1(x); \ + RETURNF(x); \ +} while (0) +#define RETURNPI(x) do { \ + DOPRINT_END1(x); \ + RETURNI(x); \ +} while (0) +#define RETURN2P(x, y) do { \ + DOPRINT_END2((x), (y)); \ + RETURNF((x) + (y)); \ +} while (0) +#define RETURN2PI(x, y) do { \ + DOPRINT_END2((x), (y)); \ + RETURNI((x) + (y)); \ +} while (0) +#ifdef STRUCT_RETURN +#define RETURNSP(rp) do { \ + if (!(rp)->lo_set) \ + RETURNP((rp)->hi); \ + RETURN2P((rp)->hi, (rp)->lo); \ +} while (0) +#define RETURNSPI(rp) do { \ + if (!(rp)->lo_set) \ + RETURNPI((rp)->hi); \ + RETURN2PI((rp)->hi, (rp)->lo); \ +} while (0) +#endif +#define SUM2P(x, y) ({ \ + const __typeof (x) __x = (x); \ + const __typeof (y) __y = (y); \ + \ + DOPRINT_END2(__x, __y); \ + __x + __y; \ +}) + +/* + * ieee style elementary functions + * + * We rename functions here to improve other sources' diffability + * against fdlibm. + */ +#define __ieee754_sqrt sqrt +#define __ieee754_acos acos +#define __ieee754_acosh acosh +#define __ieee754_log log +#define __ieee754_log2 log2 +#define __ieee754_atanh atanh +#define __ieee754_asin asin +#define __ieee754_atan2 atan2 +#define __ieee754_exp exp +#define __ieee754_cosh cosh +#define __ieee754_fmod fmod +#define __ieee754_pow pow +#define __ieee754_lgamma lgamma +#define __ieee754_gamma gamma +#define __ieee754_lgamma_r lgamma_r +#define __ieee754_gamma_r gamma_r +#define __ieee754_log10 log10 +#define __ieee754_sinh sinh +#define __ieee754_hypot hypot +#define __ieee754_j0 j0 +#define __ieee754_j1 j1 +#define __ieee754_y0 y0 +#define __ieee754_y1 y1 +#define __ieee754_jn jn +#define __ieee754_yn yn +#define __ieee754_remainder remainder +#define __ieee754_scalb scalb +#define __ieee754_sqrtf sqrtf +#define __ieee754_acosf acosf +#define __ieee754_acoshf acoshf +#define __ieee754_logf logf +#define __ieee754_atanhf atanhf +#define __ieee754_asinf asinf +#define __ieee754_atan2f atan2f +#define __ieee754_expf expf +#define __ieee754_coshf coshf +#define __ieee754_fmodf fmodf +#define __ieee754_powf powf +#define __ieee754_lgammaf lgammaf +#define __ieee754_gammaf gammaf +#define __ieee754_lgammaf_r lgammaf_r +#define __ieee754_gammaf_r gammaf_r +#define __ieee754_log10f log10f +#define __ieee754_log2f log2f +#define __ieee754_sinhf sinhf +#define __ieee754_hypotf hypotf +#define __ieee754_j0f j0f +#define __ieee754_j1f j1f +#define __ieee754_y0f y0f +#define __ieee754_y1f y1f +#define __ieee754_jnf jnf +#define __ieee754_ynf ynf +#define __ieee754_remainderf remainderf +#define __ieee754_scalbf scalbf + +/* fdlibm kernel function */ +int __kernel_rem_pio2(double*,double*,int,int,int); + +/* double precision kernel functions */ +#ifndef INLINE_REM_PIO2 +int __ieee754_rem_pio2(double,double*); +#endif +double __kernel_sin(double,double,int); +double __kernel_cos(double,double); +double __kernel_tan(double,double,int); +double __ldexp_exp(double,int); +#ifdef _COMPLEX_H +double complex __ldexp_cexp(double complex,int); +#endif + +/* float precision kernel functions */ +#ifndef INLINE_REM_PIO2F +int __ieee754_rem_pio2f(float,double*); +#endif +#ifndef INLINE_KERNEL_SINDF +float __kernel_sindf(double); +#endif +#ifndef INLINE_KERNEL_COSDF +float __kernel_cosdf(double); +#endif +#ifndef INLINE_KERNEL_TANDF +float __kernel_tandf(double,int); +#endif +float __ldexp_expf(float,int); +#ifdef _COMPLEX_H +float complex __ldexp_cexpf(float complex,int); +#endif + +/* long double precision kernel functions */ +long double __kernel_sinl(long double, long double, int); +long double __kernel_cosl(long double, long double); +long double __kernel_tanl(long double, long double, int); + +#endif /* !_MATH_PRIVATE_H_ */ diff --git a/newlib/libm/ld/s_asinhl.c b/newlib/libm/ld/s_asinhl.c new file mode 100644 index 000000000..ba28f599c --- /dev/null +++ b/newlib/libm/ld/s_asinhl.c @@ -0,0 +1,91 @@ +/* from: FreeBSD: head/lib/msun/src/e_acosh.c 176451 2008-02-22 02:30:36Z das */ + +/* @(#)s_asinh.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See s_asinh.c for complete comments. + * + * Converted to long double by David Schultz and + * Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* EXP_LARGE is the threshold above which we use asinh(x) ~= log(2x). */ +/* EXP_TINY is the threshold below which we use asinh(x) ~= x. */ +#if LDBL_MANT_DIG == 64 +#define EXP_LARGE 34 +#define EXP_TINY -34 +#elif LDBL_MANT_DIG == 113 +#define EXP_LARGE 58 +#define EXP_TINY -58 +#else +#error "Unsupported long double format" +#endif + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ +huge= 1.00000000000000000000e+300; + +#if LDBL_MANT_DIG == 64 +static const union IEEEl2bits +u_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L); +#define ln2 u_ln2.e +#elif LDBL_MANT_DIG == 113 +static const long double +ln2 = 6.93147180559945309417232121458176568e-1L; /* 0x162e42fefa39ef35793c7673007e6.0p-113 */ +#else +#error "Unsupported long double format" +#endif + +long double +asinhl(long double x) +{ + long double t, w; + uint16_t hx, ix; + + ENTERI(); + GET_LDBL_EXPSIGN(hx, x); + ix = hx & 0x7fff; + if (ix >= 0x7fff) RETURNI(x+x); /* x is inf, NaN or misnormal */ + if (ix < BIAS + EXP_TINY) { /* |x| < TINY, or misnormal */ + if (huge + x > one) RETURNI(x); /* return x inexact except 0 */ + } + if (ix >= BIAS + EXP_LARGE) { /* |x| >= LARGE, or misnormal */ + w = logl(fabsl(x))+ln2; + } else if (ix >= 0x4000) { /* LARGE > |x| >= 2.0, or misnormal */ + t = fabsl(x); + w = logl(2.0*t+one/(sqrtl(x*x+one)+t)); + } else { /* 2.0 > |x| >= TINY, or misnormal */ + t = x*x; + w =log1pl(fabsl(x)+t/(one+sqrtl(one+t))); + } + RETURNI((hx & 0x8000) == 0 ? w : -w); +} diff --git a/newlib/libm/ld/s_atanl.c b/newlib/libm/ld/s_atanl.c new file mode 100644 index 000000000..ff29c3ce8 --- /dev/null +++ b/newlib/libm/ld/s_atanl.c @@ -0,0 +1,85 @@ +/* @(#)s_atan.c 5.1 93/09/24 */ +/* FreeBSD: head/lib/msun/src/s_atan.c 176451 2008-02-22 02:30:36Z das */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See comments in s_atan.c. + * Converted to long double by David Schultz . + */ + +#include + +#include "invtrig.h" +#include "math.h" +#include "math_private.h" + +static const long double +one = 1.0, +huge = 1.0e300; + +long double +atanl(long double x) +{ + union IEEEl2bits u; + long double w,s1,s2,z; + int id; + int16_t expsign, expt; + int32_t expman; + + u.e = x; + expsign = u.xbits.expsign; + expt = expsign & 0x7fff; + if(expt >= ATAN_CONST) { /* if |x| is large, atan(x)~=pi/2 */ + if(expt == BIAS + LDBL_MAX_EXP && + ((u.bits.manh&~LDBL_NBIT)|u.bits.manl)!=0) + return x+x; /* NaN */ + if(expsign>0) return atanhi[3]+atanlo[3]; + else return -atanhi[3]-atanlo[3]; + } + /* Extract the exponent and the first few bits of the mantissa. */ + /* XXX There should be a more convenient way to do this. */ + expman = (expt << 8) | ((u.bits.manh >> (MANH_SIZE - 9)) & 0xff); + if (expman < ((BIAS - 2) << 8) + 0xc0) { /* |x| < 0.4375 */ + if (expt < ATAN_LINEAR) { /* if |x| is small, atanl(x)~=x */ + if(huge+x>one) return x; /* raise inexact */ + } + id = -1; + } else { + x = fabsl(x); + if (expman < (BIAS << 8) + 0x30) { /* |x| < 1.1875 */ + if (expman < ((BIAS - 1) << 8) + 0x60) { /* 7/16 <=|x|<11/16 */ + id = 0; x = (2.0*x-one)/(2.0+x); + } else { /* 11/16<=|x|< 19/16 */ + id = 1; x = (x-one)/(x+one); + } + } else { + if (expman < ((BIAS + 1) << 8) + 0x38) { /* |x| < 2.4375 */ + id = 2; x = (x-1.5)/(one+1.5*x); + } else { /* 2.4375 <= |x| < 2^ATAN_CONST */ + id = 3; x = -1.0/x; + } + }} + /* end of argument reduction */ + z = x*x; + w = z*z; + /* break sum aT[i]z**(i+1) into odd and even poly */ + s1 = z*T_even(w); + s2 = w*T_odd(w); + if (id<0) return x - x*(s1+s2); + else { + z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x); + return (expsign<0)? -z:z; + } +} diff --git a/newlib/libm/ld/s_cbrtl.c b/newlib/libm/ld/s_cbrtl.c new file mode 100644 index 000000000..b15c96ecd --- /dev/null +++ b/newlib/libm/ld/s_cbrtl.c @@ -0,0 +1,143 @@ +/*- + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * The argument reduction and testing for exceptional cases was + * written by Steven G. Kargl with input from Bruce D. Evans + * and David A. Schultz. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#define BIAS (LDBL_MAX_EXP - 1) + +static const unsigned + B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ + +long double +cbrtl(long double x) +{ + union IEEEl2bits u, v; + long double r, s, t, w; + double dr, dt, dx; + float ft, fx; + uint32_t hx; + uint16_t expsign; + int k; + + u.e = x; + expsign = u.xbits.expsign; + k = expsign & 0x7fff; + + /* + * If x = +-Inf, then cbrt(x) = +-Inf. + * If x = NaN, then cbrt(x) = NaN. + */ + if (k == BIAS + LDBL_MAX_EXP) + return (x + x); + + ENTERI(); + if (k == 0) { + /* If x = +-0, then cbrt(x) = +-0. */ + if ((u.bits.manh | u.bits.manl) == 0) + RETURNI(x); + /* Adjust subnormal numbers. */ + u.e *= 0x1.0p514; + k = u.bits.exp; + k -= BIAS + 514; + } else + k -= BIAS; + u.xbits.expsign = BIAS; + v.e = 1; + + x = u.e; + switch (k % 3) { + case 1: + case -2: + x = 2*x; + k--; + break; + case 2: + case -1: + x = 4*x; + k -= 2; + break; + } + v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3); + + /* + * The following is the guts of s_cbrtf, with the handling of + * special values removed and extra care for accuracy not taken, + * but with most of the extra accuracy not discarded. + */ + + /* ~5-bit estimate: */ + fx = x; + GET_FLOAT_WORD(hx, fx); + SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1)); + + /* ~16-bit estimate: */ + dx = x; + dt = ft; + dr = dt * dt * dt; + dt = dt * (dx + dx + dr) / (dx + dr + dr); + + /* ~47-bit estimate: */ + dr = dt * dt * dt; + dt = dt * (dx + dx + dr) / (dx + dr + dr); + +#if LDBL_MANT_DIG == 64 + /* + * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8). + * Round it away from zero to 32 bits (32 so that t*t is exact, and + * away from zero for technical reasons). + */ + volatile double vd2 = 0x1.0p32; + volatile double vd1 = 0x1.0p-31; + #define vd ((long double)vd2 + vd1) + + t = dt + vd - 0x1.0p32; +#elif LDBL_MANT_DIG == 113 + /* + * Round dt away from zero to 47 bits. Since we don't trust the 47, + * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and + * might be avoidable in this case, since on most machines dt will + * have been evaluated in 53-bit precision and the technical reasons + * for rounding up might not apply to either case in cbrtl() since + * dt is much more accurate than needed. + */ + t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60; +#else +#error "Unsupported long double format" +#endif + + /* + * Final step Newton iteration to 64 or 113 bits with + * error < 0.667 ulps + */ + s=t*t; /* t*t is exact */ + r=x/s; /* error <= 0.5 ulps; |r| < |t| */ + w=t+t; /* t+t is exact */ + r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ + t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */ + + t *= v.e; + RETURNI(t); +} diff --git a/newlib/libm/ld/s_ceill.c b/newlib/libm/ld/s_ceill.c new file mode 100644 index 000000000..2d1045fe6 --- /dev/null +++ b/newlib/libm/ld/s_ceill.c @@ -0,0 +1,101 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * From: @(#)s_ceil.c 5.1 93/09/24 + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * ceill(x) + * Return x rounded toward -inf to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to ceill(x). + */ + +#include +#include +#include + +#include "fpmath.h" + +#ifdef LDBL_IMPLICIT_NBIT +#define MANH_SIZE (LDBL_MANH_SIZE + 1) +#define INC_MANH(u, c) do { \ + uint64_t o = u.bits.manh; \ + u.bits.manh += (c); \ + if (u.bits.manh < o) \ + u.bits.exp++; \ +} while (0) +#else +#define MANH_SIZE LDBL_MANH_SIZE +#define INC_MANH(u, c) do { \ + uint64_t o = u.bits.manh; \ + u.bits.manh += (c); \ + if (u.bits.manh < o) { \ + u.bits.exp++; \ + u.bits.manh |= 1llu << (LDBL_MANH_SIZE - 1); \ + } \ +} while (0) +#endif + +static const long double huge = 1.0e300; + +long double +ceill(long double x) +{ + union IEEEl2bits u = { .e = x }; + int e = u.bits.exp - LDBL_MAX_EXP + 1; + + if (e < MANH_SIZE - 1) { + if (e < 0) { /* raise inexact if x != 0 */ + if (huge + x > 0.0) + if (u.bits.exp > 0 || + (u.bits.manh | u.bits.manl) != 0) + u.e = u.bits.sign ? -0.0 : 1.0; + } else { + uint64_t m = ((1llu << MANH_SIZE) - 1) >> (e + 1); + if (((u.bits.manh & m) | u.bits.manl) == 0) + return (x); /* x is integral */ + if (!u.bits.sign) { +#ifdef LDBL_IMPLICIT_NBIT + if (e == 0) + u.bits.exp++; + else +#endif + INC_MANH(u, 1llu << (MANH_SIZE - e - 1)); + } + if (huge + x > 0.0) { /* raise inexact flag */ + u.bits.manh &= ~m; + u.bits.manl = 0; + } + } + } else if (e < LDBL_MANT_DIG - 1) { + uint64_t m = (uint64_t)-1 >> (64 - LDBL_MANT_DIG + e + 1); + if ((u.bits.manl & m) == 0) + return (x); /* x is integral */ + if (!u.bits.sign) { + if (e == MANH_SIZE - 1) + INC_MANH(u, 1); + else { + uint64_t o = u.bits.manl; + u.bits.manl += 1llu << (LDBL_MANT_DIG - e - 1); + if (u.bits.manl < o) /* got a carry */ + INC_MANH(u, 1); + } + } + if (huge + x > 0.0) /* raise inexact flag */ + u.bits.manl &= ~m; + } + return (u.e); +} diff --git a/newlib/libm/ld/s_copysignl.c b/newlib/libm/ld/s_copysignl.c new file mode 100644 index 000000000..bd6744705 --- /dev/null +++ b/newlib/libm/ld/s_copysignl.c @@ -0,0 +1,44 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 Stefan Farfeleder + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include + +#include "fpmath.h" + +long double +copysignl(long double x, long double y) +{ + union IEEEl2bits ux, uy; + + ux.e = x; + uy.e = y; + ux.bits.sign = uy.bits.sign; + return (ux.e); +} diff --git a/newlib/libm/ld/s_cosl.c b/newlib/libm/ld/s_cosl.c new file mode 100644 index 000000000..3d066483f --- /dev/null +++ b/newlib/libm/ld/s_cosl.c @@ -0,0 +1,102 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * Limited testing on pseudorandom numbers drawn within [-2e8:4e8] shows + * an accuracy of <= 0.7412 ULP. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#if LDBL_MANT_DIG == 64 +#include "../ld80/e_rem_pio2l.h" +static const union IEEEl2bits +pio4u = LD80C(0xc90fdaa22168c235, -00001, 7.85398163397448309628e-01L); +#define pio4 (pio4u.e) +#elif LDBL_MANT_DIG == 113 +#include "../ld128/e_rem_pio2l.h" +long double pio4 = 7.85398163397448309615660845819875721e-1L; +#else +#error "Unsupported long double format" +#endif + +long double +cosl(long double x) +{ + union IEEEl2bits z; + int e0; + long double y[2]; + long double hi, lo; + + z.e = x; + z.bits.sign = 0; + + /* If x = +-0 or x is a subnormal number, then cos(x) = 1 */ + if (z.bits.exp == 0) + return (1.0); + + /* If x = NaN or Inf, then cos(x) = NaN. */ + if (z.bits.exp == 32767) + return ((x - x) / (x - x)); + + ENTERI(); + + /* Optimize the case where x is already within range. */ + if (z.e < pio4) + RETURNI(__kernel_cosl(z.e, 0)); + + e0 = __ieee754_rem_pio2l(x, y); + hi = y[0]; + lo = y[1]; + + switch (e0 & 3) { + case 0: + hi = __kernel_cosl(hi, lo); + break; + case 1: + hi = - __kernel_sinl(hi, lo, 1); + break; + case 2: + hi = - __kernel_cosl(hi, lo); + break; + case 3: + hi = __kernel_sinl(hi, lo, 1); + break; + } + + RETURNI(hi); +} diff --git a/newlib/libm/ld/s_fabsl.c b/newlib/libm/ld/s_fabsl.c new file mode 100644 index 000000000..5076d8a9b --- /dev/null +++ b/newlib/libm/ld/s_fabsl.c @@ -0,0 +1,45 @@ +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * Copyright (c) 2003 Dag-Erling Smørgrav + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer + * in this position and unchanged. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * 3. The name of the author may not be used to endorse or promote products + * derived from this software without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include + +#include "fpmath.h" + +long double +fabsl(long double x) +{ + union IEEEl2bits u; + + u.e = x; + u.bits.sign = 0; + return (u.e); +} diff --git a/newlib/libm/ld/s_fdim.c b/newlib/libm/ld/s_fdim.c new file mode 100644 index 000000000..c40c3e9d3 --- /dev/null +++ b/newlib/libm/ld/s_fdim.c @@ -0,0 +1,48 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#define DECL(type, fn) \ +type \ +fn(type x, type y) \ +{ \ + \ + if (isnan(x)) \ + return (x); \ + if (isnan(y)) \ + return (y); \ + return (x > y ? x - y : 0.0); \ +} + +DECL(double, fdim) +DECL(float, fdimf) +DECL(long double, fdiml) diff --git a/newlib/libm/ld/s_floorl.c b/newlib/libm/ld/s_floorl.c new file mode 100644 index 000000000..6cec3e781 --- /dev/null +++ b/newlib/libm/ld/s_floorl.c @@ -0,0 +1,101 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * From: @(#)s_floor.c 5.1 93/09/24 + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * floorl(x) + * Return x rounded toward -inf to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to floorl(x). + */ + +#include +#include +#include + +#include "fpmath.h" + +#ifdef LDBL_IMPLICIT_NBIT +#define MANH_SIZE (LDBL_MANH_SIZE + 1) +#define INC_MANH(u, c) do { \ + uint64_t o = u.bits.manh; \ + u.bits.manh += (c); \ + if (u.bits.manh < o) \ + u.bits.exp++; \ +} while (0) +#else +#define MANH_SIZE LDBL_MANH_SIZE +#define INC_MANH(u, c) do { \ + uint64_t o = u.bits.manh; \ + u.bits.manh += (c); \ + if (u.bits.manh < o) { \ + u.bits.exp++; \ + u.bits.manh |= 1llu << (LDBL_MANH_SIZE - 1); \ + } \ +} while (0) +#endif + +static const long double huge = 1.0e300; + +long double +floorl(long double x) +{ + union IEEEl2bits u = { .e = x }; + int e = u.bits.exp - LDBL_MAX_EXP + 1; + + if (e < MANH_SIZE - 1) { + if (e < 0) { /* raise inexact if x != 0 */ + if (huge + x > 0.0) + if (u.bits.exp > 0 || + (u.bits.manh | u.bits.manl) != 0) + u.e = u.bits.sign ? -1.0 : 0.0; + } else { + uint64_t m = ((1llu << MANH_SIZE) - 1) >> (e + 1); + if (((u.bits.manh & m) | u.bits.manl) == 0) + return (x); /* x is integral */ + if (u.bits.sign) { +#ifdef LDBL_IMPLICIT_NBIT + if (e == 0) + u.bits.exp++; + else +#endif + INC_MANH(u, 1llu << (MANH_SIZE - e - 1)); + } + if (huge + x > 0.0) { /* raise inexact flag */ + u.bits.manh &= ~m; + u.bits.manl = 0; + } + } + } else if (e < LDBL_MANT_DIG - 1) { + uint64_t m = (uint64_t)-1 >> (64 - LDBL_MANT_DIG + e + 1); + if ((u.bits.manl & m) == 0) + return (x); /* x is integral */ + if (u.bits.sign) { + if (e == MANH_SIZE - 1) + INC_MANH(u, 1); + else { + uint64_t o = u.bits.manl; + u.bits.manl += 1llu << (LDBL_MANT_DIG - e - 1); + if (u.bits.manl < o) /* got a carry */ + INC_MANH(u, 1); + } + } + if (huge + x > 0.0) /* raise inexact flag */ + u.bits.manl &= ~m; + } + return (u.e); +} diff --git a/newlib/libm/ld/s_fmal.c b/newlib/libm/ld/s_fmal.c new file mode 100644 index 000000000..a379346c1 --- /dev/null +++ b/newlib/libm/ld/s_fmal.c @@ -0,0 +1,274 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2005-2011 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include +#include + +#include "fpmath.h" + +/* + * A struct dd represents a floating-point number with twice the precision + * of a long double. We maintain the invariant that "hi" stores the high-order + * bits of the result. + */ +struct dd { + long double hi; + long double lo; +}; + +/* + * Compute a+b exactly, returning the exact result in a struct dd. We assume + * that both a and b are finite, but make no assumptions about their relative + * magnitudes. + */ +static inline struct dd +dd_add(long double a, long double b) +{ + struct dd ret; + long double s; + + ret.hi = a + b; + s = ret.hi - a; + ret.lo = (a - (ret.hi - s)) + (b - s); + return (ret); +} + +/* + * Compute a+b, with a small tweak: The least significant bit of the + * result is adjusted into a sticky bit summarizing all the bits that + * were lost to rounding. This adjustment negates the effects of double + * rounding when the result is added to another number with a higher + * exponent. For an explanation of round and sticky bits, see any reference + * on FPU design, e.g., + * + * J. Coonen. An Implementation Guide to a Proposed Standard for + * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980. + */ +static inline long double +add_adjusted(long double a, long double b) +{ + struct dd sum; + union IEEEl2bits u; + + sum = dd_add(a, b); + if (sum.lo != 0) { + u.e = sum.hi; + if ((u.bits.manl & 1) == 0) + sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); + } + return (sum.hi); +} + +/* + * Compute ldexp(a+b, scale) with a single rounding error. It is assumed + * that the result will be subnormal, and care is taken to ensure that + * double rounding does not occur. + */ +static inline long double +add_and_denormalize(long double a, long double b, int scale) +{ + struct dd sum; + int bits_lost; + union IEEEl2bits u; + + sum = dd_add(a, b); + + /* + * If we are losing at least two bits of accuracy to denormalization, + * then the first lost bit becomes a round bit, and we adjust the + * lowest bit of sum.hi to make it a sticky bit summarizing all the + * bits in sum.lo. With the sticky bit adjusted, the hardware will + * break any ties in the correct direction. + * + * If we are losing only one bit to denormalization, however, we must + * break the ties manually. + */ + if (sum.lo != 0) { + u.e = sum.hi; + bits_lost = -u.bits.exp - scale + 1; + if ((bits_lost != 1) ^ (int)(u.bits.manl & 1)) + sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); + } + return (ldexp(sum.hi, scale)); +} + +/* + * Compute a*b exactly, returning the exact result in a struct dd. We assume + * that both a and b are normalized, so no underflow or overflow will occur. + * The current rounding mode must be round-to-nearest. + */ +static inline struct dd +dd_mul(long double a, long double b) +{ +#if LDBL_MANT_DIG == 64 + static const long double split = 0x1p32L + 1.0; +#elif LDBL_MANT_DIG == 113 + static const long double split = 0x1p57L + 1.0; +#endif + struct dd ret; + long double ha, hb, la, lb, p, q; + + p = a * split; + ha = a - p; + ha += p; + la = a - ha; + + p = b * split; + hb = b - p; + hb += p; + lb = b - hb; + + p = ha * hb; + q = ha * lb + la * hb; + + ret.hi = p + q; + ret.lo = p - ret.hi + q + la * lb; + return (ret); +} + +/* + * Fused multiply-add: Compute x * y + z with a single rounding error. + * + * We use scaling to avoid overflow/underflow, along with the + * canonical precision-doubling technique adapted from: + * + * Dekker, T. A Floating-Point Technique for Extending the + * Available Precision. Numer. Math. 18, 224-242 (1971). + */ +long double +fmal(long double x, long double y, long double z) +{ + long double xs, ys, zs, adj; + struct dd xy, r; + int oround; + int ex, ey, ez; + int spread; + + /* + * Handle special cases. The order of operations and the particular + * return values here are crucial in handling special cases involving + * infinities, NaNs, overflows, and signed zeroes correctly. + */ + if (x == 0.0 || y == 0.0) + return (x * y + z); + if (z == 0.0) + return (x * y); + if (!isfinite(x) || !isfinite(y)) + return (x * y + z); + if (!isfinite(z)) + return (z); + + xs = frexpl(x, &ex); + ys = frexpl(y, &ey); + zs = frexpl(z, &ez); + oround = fegetround(); + spread = ex + ey - ez; + + /* + * If x * y and z are many orders of magnitude apart, the scaling + * will overflow, so we handle these cases specially. Rounding + * modes other than FE_TONEAREST are painful. + */ + if (spread < -LDBL_MANT_DIG) { + feraiseexcept(FE_INEXACT); + if (!isnormal(z)) + feraiseexcept(FE_UNDERFLOW); + switch (oround) { + case FE_TONEAREST: + return (z); + case FE_TOWARDZERO: + if (x > 0.0 ^ y < 0.0 ^ z < 0.0) + return (z); + else + return (nextafterl(z, 0)); + case FE_DOWNWARD: + if (x > 0.0 ^ y < 0.0) + return (z); + else + return (nextafterl(z, -INFINITY)); + default: /* FE_UPWARD */ + if (x > 0.0 ^ y < 0.0) + return (nextafterl(z, INFINITY)); + else + return (z); + } + } + if (spread <= LDBL_MANT_DIG * 2) + zs = ldexpl(zs, -spread); + else + zs = copysignl(LDBL_MIN, zs); + + fesetround(FE_TONEAREST); + /* work around clang bug 8100 */ + volatile long double vxs = xs; + + /* + * Basic approach for round-to-nearest: + * + * (xy.hi, xy.lo) = x * y (exact) + * (r.hi, r.lo) = xy.hi + z (exact) + * adj = xy.lo + r.lo (inexact; low bit is sticky) + * result = r.hi + adj (correctly rounded) + */ + xy = dd_mul(vxs, ys); + r = dd_add(xy.hi, zs); + + spread = ex + ey; + + if (r.hi == 0.0) { + /* + * When the addends cancel to 0, ensure that the result has + * the correct sign. + */ + fesetround(oround); + volatile long double vzs = zs; /* XXX gcc CSE bug workaround */ + return (xy.hi + vzs + ldexpl(xy.lo, spread)); + } + + if (oround != FE_TONEAREST) { + /* + * There is no need to worry about double rounding in directed + * rounding modes. + */ + fesetround(oround); + /* work around clang bug 8100 */ + volatile long double vrlo = r.lo; + adj = vrlo + xy.lo; + return (ldexpl(r.hi + adj, spread)); + } + + adj = add_adjusted(r.lo, xy.lo); + if (spread + ilogbl(r.hi) > -16383) + return (ldexpl(r.hi + adj, spread)); + else + return (add_and_denormalize(r.hi, adj, spread)); +} diff --git a/newlib/libm/ld/s_fmaxl.c b/newlib/libm/ld/s_fmaxl.c new file mode 100644 index 000000000..c0d7c8832 --- /dev/null +++ b/newlib/libm/ld/s_fmaxl.c @@ -0,0 +1,57 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#include "fpmath.h" + +long double +fmaxl(long double x, long double y) +{ + union IEEEl2bits u[2]; + + u[0].e = x; + mask_nbit_l(u[0]); + u[1].e = y; + mask_nbit_l(u[1]); + + /* Check for NaNs to avoid raising spurious exceptions. */ + if (u[0].bits.exp == 32767 && (u[0].bits.manh | u[0].bits.manl) != 0) + return (y); + if (u[1].bits.exp == 32767 && (u[1].bits.manh | u[1].bits.manl) != 0) + return (x); + + /* Handle comparisons of signed zeroes. */ + if (u[0].bits.sign != u[1].bits.sign) + return (u[0].bits.sign ? y : x); + + return (x > y ? x : y); +} diff --git a/newlib/libm/ld/s_fminl.c b/newlib/libm/ld/s_fminl.c new file mode 100644 index 000000000..97604b38b --- /dev/null +++ b/newlib/libm/ld/s_fminl.c @@ -0,0 +1,57 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#include "fpmath.h" + +long double +fminl(long double x, long double y) +{ + union IEEEl2bits u[2]; + + u[0].e = x; + mask_nbit_l(u[0]); + u[1].e = y; + mask_nbit_l(u[1]); + + /* Check for NaNs to avoid raising spurious exceptions. */ + if (u[0].bits.exp == 32767 && (u[0].bits.manh | u[0].bits.manl) != 0) + return (y); + if (u[1].bits.exp == 32767 && (u[1].bits.manh | u[1].bits.manl) != 0) + return (x); + + /* Handle comparisons of signed zeroes. */ + if (u[0].bits.sign != u[1].bits.sign) + return (u[1].bits.sign ? y : x); + + return (x < y ? x : y); +} diff --git a/newlib/libm/ld/s_frexpl.c b/newlib/libm/ld/s_frexpl.c new file mode 100644 index 000000000..66e284fdd --- /dev/null +++ b/newlib/libm/ld/s_frexpl.c @@ -0,0 +1,64 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004-2005 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include +#include + +#include "fpmath.h" + +#if LDBL_MAX_EXP != 0x4000 +#error "Unsupported long double format" +#endif + +long double +frexpl(long double x, int *ex) +{ + union IEEEl2bits u; + + u.e = x; + switch (u.bits.exp) { + case 0: /* 0 or subnormal */ + if ((u.bits.manl | u.bits.manh) == 0) { + *ex = 0; + } else { + u.e *= 0x1.0p514; + *ex = u.bits.exp - 0x4200; + u.bits.exp = 0x3ffe; + } + break; + case 0x7fff: /* infinity or NaN; value of *ex is unspecified */ + break; + default: /* normal */ + *ex = u.bits.exp - 0x3ffe; + u.bits.exp = 0x3ffe; + break; + } + return (u.e); +} diff --git a/newlib/libm/ld/s_ilogbl.c b/newlib/libm/ld/s_ilogbl.c new file mode 100644 index 000000000..3211f4409 --- /dev/null +++ b/newlib/libm/ld/s_ilogbl.c @@ -0,0 +1,53 @@ +/* + * From: @(#)s_ilogb.c 5.1 93/09/24 + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include +#include + +#include "fpmath.h" + +int +ilogbl(long double x) +{ + union IEEEl2bits u; + unsigned long m; + int b; + + u.e = x; + if (u.bits.exp == 0) { + if ((u.bits.manl | u.bits.manh) == 0) + return (FP_ILOGB0); + /* denormalized */ + if (u.bits.manh == 0) { + m = 1lu << (LDBL_MANL_SIZE - 1); + for (b = LDBL_MANH_SIZE; !(u.bits.manl & m); m >>= 1) + b++; + } else { + m = 1lu << (LDBL_MANH_SIZE - 1); + for (b = 0; !(u.bits.manh & m); m >>= 1) + b++; + } +#ifdef LDBL_IMPLICIT_NBIT + b++; +#endif + return (LDBL_MIN_EXP - b - 1); + } else if (u.bits.exp < (LDBL_MAX_EXP << 1) - 1) + return (u.bits.exp - LDBL_MAX_EXP + 1); + else if (u.bits.manl != 0 || u.bits.manh != 0) + return (FP_ILOGBNAN); + else + return (INT_MAX); +} diff --git a/newlib/libm/ld/s_llrintl.c b/newlib/libm/ld/s_llrintl.c new file mode 100644 index 000000000..6ef83759c --- /dev/null +++ b/newlib/libm/ld/s_llrintl.c @@ -0,0 +1,9 @@ +#include +__FBSDID("$FreeBSD$"); + +#define type long double +#define roundit rintl +#define dtype long long +#define fn llrintl + +#include "s_lrint.c" diff --git a/newlib/libm/ld/s_llroundl.c b/newlib/libm/ld/s_llroundl.c new file mode 100644 index 000000000..02c44eb4b --- /dev/null +++ b/newlib/libm/ld/s_llroundl.c @@ -0,0 +1,11 @@ +#include +__FBSDID("$FreeBSD$"); + +#define type long double +#define roundit roundl +#define dtype long long +#define DTYPE_MIN LLONG_MIN +#define DTYPE_MAX LLONG_MAX +#define fn llroundl + +#include "s_lround.c" diff --git a/newlib/libm/ld/s_logbl.c b/newlib/libm/ld/s_logbl.c new file mode 100644 index 000000000..ee1a91fd8 --- /dev/null +++ b/newlib/libm/ld/s_logbl.c @@ -0,0 +1,54 @@ +/* + * From: @(#)s_ilogb.c 5.1 93/09/24 + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include +#include + +#include "fpmath.h" + +long double +logbl(long double x) +{ + union IEEEl2bits u; + unsigned long m; + int b; + + u.e = x; + if (u.bits.exp == 0) { + if ((u.bits.manl | u.bits.manh) == 0) { /* x == 0 */ + u.bits.sign = 1; + return (1.0L / u.e); + } + /* denormalized */ + if (u.bits.manh == 0) { + m = 1lu << (LDBL_MANL_SIZE - 1); + for (b = LDBL_MANH_SIZE; !(u.bits.manl & m); m >>= 1) + b++; + } else { + m = 1lu << (LDBL_MANH_SIZE - 1); + for (b = 0; !(u.bits.manh & m); m >>= 1) + b++; + } +#ifdef LDBL_IMPLICIT_NBIT + b++; +#endif + return ((long double)(LDBL_MIN_EXP - b - 1)); + } + if (u.bits.exp < (LDBL_MAX_EXP << 1) - 1) /* normal */ + return ((long double)(u.bits.exp - LDBL_MAX_EXP + 1)); + else /* +/- inf or nan */ + return (x * x); +} diff --git a/newlib/libm/ld/s_lrint.c b/newlib/libm/ld/s_lrint.c new file mode 100644 index 000000000..ad9b978fa --- /dev/null +++ b/newlib/libm/ld/s_lrint.c @@ -0,0 +1,60 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2005 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +#include +#include + +#ifndef type +__FBSDID("$FreeBSD$"); +#define type double +#define roundit rint +#define dtype long +#define fn lrint +#endif + +/* + * C99 says we should not raise a spurious inexact exception when an + * invalid exception is raised. Unfortunately, the set of inputs + * that overflows depends on the rounding mode when 'dtype' has more + * significant bits than 'type'. Hence, we bend over backwards for the + * sake of correctness; an MD implementation could be more efficient. + */ +dtype +fn(type x) +{ + fenv_t env; + dtype d; + + feholdexcept(&env); + d = (dtype)roundit(x); + if (fetestexcept(FE_INVALID)) + feclearexcept(FE_INEXACT); + feupdateenv(&env); + return (d); +} diff --git a/newlib/libm/ld/s_lrintl.c b/newlib/libm/ld/s_lrintl.c new file mode 100644 index 000000000..497b442f3 --- /dev/null +++ b/newlib/libm/ld/s_lrintl.c @@ -0,0 +1,9 @@ +#include +__FBSDID("$FreeBSD$"); + +#define type long double +#define roundit rintl +#define dtype long +#define fn lrintl + +#include "s_lrint.c" diff --git a/newlib/libm/ld/s_lround.c b/newlib/libm/ld/s_lround.c new file mode 100644 index 000000000..1dd8e697f --- /dev/null +++ b/newlib/libm/ld/s_lround.c @@ -0,0 +1,70 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2005 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +#include +#include +#include + +#ifndef type +__FBSDID("$FreeBSD$"); +#define type double +#define roundit round +#define dtype long +#define DTYPE_MIN LONG_MIN +#define DTYPE_MAX LONG_MAX +#define fn lround +#endif + +/* + * If type has more precision than dtype, the endpoints dtype_(min|max) are + * of the form xxx.5; they are "out of range" because lround() rounds away + * from 0. On the other hand, if type has less precision than dtype, then + * all values that are out of range are integral, so we might as well assume + * that everything is in range. At compile time, INRANGE(x) should reduce to + * two floating-point comparisons in the former case, or TRUE otherwise. + */ +static const type type_min = (type)DTYPE_MIN; +static const type type_max = (type)DTYPE_MAX; +static const type dtype_min = (type)DTYPE_MIN - 0.5; +static const type dtype_max = (type)DTYPE_MAX + 0.5; +#define INRANGE(x) (dtype_max - type_max != 0.5 || \ + ((x) > dtype_min && (x) < dtype_max)) + +dtype +fn(type x) +{ + + if (INRANGE(x)) { + x = roundit(x); + return ((dtype)x); + } else { + feraiseexcept(FE_INVALID); + return (DTYPE_MAX); + } +} diff --git a/newlib/libm/ld/s_lroundl.c b/newlib/libm/ld/s_lroundl.c new file mode 100644 index 000000000..e410827e2 --- /dev/null +++ b/newlib/libm/ld/s_lroundl.c @@ -0,0 +1,11 @@ +#include +__FBSDID("$FreeBSD$"); + +#define type long double +#define roundit roundl +#define dtype long +#define DTYPE_MIN LONG_MIN +#define DTYPE_MAX LONG_MAX +#define fn lroundl + +#include "s_lround.c" diff --git a/newlib/libm/ld/s_modfl.c b/newlib/libm/ld/s_modfl.c new file mode 100644 index 000000000..2d83bbe33 --- /dev/null +++ b/newlib/libm/ld/s_modfl.c @@ -0,0 +1,103 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * Derived from s_modf.c, which has the following Copyright: + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * $FreeBSD$ + */ + +#include +#include +#include + +#include "fpmath.h" + +#if LDBL_MANL_SIZE > 32 +#define MASK ((uint64_t)-1) +#else +#define MASK ((uint32_t)-1) +#endif +/* Return the last n bits of a word, representing the fractional part. */ +#define GETFRAC(bits, n) ((bits) & ~(MASK << (n))) +/* The number of fraction bits in manh, not counting the integer bit */ +#define HIBITS (LDBL_MANT_DIG - LDBL_MANL_SIZE) + +static const long double zero[] = { 0.0L, -0.0L }; + +long double +modfl(long double x, long double *iptr) +{ + union IEEEl2bits ux; + int e; + + ux.e = x; + e = ux.bits.exp - LDBL_MAX_EXP + 1; + if (e < HIBITS) { /* Integer part is in manh. */ + if (e < 0) { /* |x|<1 */ + *iptr = zero[ux.bits.sign]; + return (x); + } else { + if ((GETFRAC(ux.bits.manh, HIBITS - 1 - e) | + ux.bits.manl) == 0) { /* X is an integer. */ + *iptr = x; + return (zero[ux.bits.sign]); + } else { + /* Clear all but the top e+1 bits. */ + ux.bits.manh >>= HIBITS - 1 - e; + ux.bits.manh <<= HIBITS - 1 - e; + ux.bits.manl = 0; + *iptr = ux.e; + return (x - ux.e); + } + } + } else if (e >= LDBL_MANT_DIG - 1) { /* x has no fraction part. */ + *iptr = x; + if (x != x) /* Handle NaNs. */ + return (x); + return (zero[ux.bits.sign]); + } else { /* Fraction part is in manl. */ + if (GETFRAC(ux.bits.manl, LDBL_MANT_DIG - 1 - e) == 0) { + /* x is integral. */ + *iptr = x; + return (zero[ux.bits.sign]); + } else { + /* Clear all but the top e+1 bits. */ + ux.bits.manl >>= LDBL_MANT_DIG - 1 - e; + ux.bits.manl <<= LDBL_MANT_DIG - 1 - e; + *iptr = ux.e; + return (x - ux.e); + } + } +} diff --git a/newlib/libm/ld/s_nearbyint.c b/newlib/libm/ld/s_nearbyint.c new file mode 100644 index 000000000..796dbaf98 --- /dev/null +++ b/newlib/libm/ld/s_nearbyint.c @@ -0,0 +1,61 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +/* + * We save and restore the floating-point environment to avoid raising + * an inexact exception. We can get away with using fesetenv() + * instead of feclearexcept()/feupdateenv() to restore the environment + * because the only exception defined for rint() is overflow, and + * rounding can't overflow as long as emax >= p. + * + * The volatile keyword is needed below because clang incorrectly assumes + * that rint won't raise any floating-point exceptions. Declaring ret volatile + * is sufficient to trick the compiler into doing the right thing. + */ +#define DECL(type, fn, rint) \ +type \ +fn(type x) \ +{ \ + volatile type ret; \ + fenv_t env; \ + \ + fegetenv(&env); \ + ret = rint(x); \ + fesetenv(&env); \ + return (ret); \ +} + +DECL(double, nearbyint, rint) +DECL(float, nearbyintf, rintf) +DECL(long double, nearbyintl, rintl) diff --git a/newlib/libm/ld/s_nextafterl.c b/newlib/libm/ld/s_nextafterl.c new file mode 100644 index 000000000..9c61a436a --- /dev/null +++ b/newlib/libm/ld/s_nextafterl.c @@ -0,0 +1,80 @@ +/* @(#)s_nextafter.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* IEEE functions + * nextafter(x,y) + * return the next machine floating-point number of x in the + * direction toward y. + * Special cases: + */ + +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#if LDBL_MAX_EXP != 0x4000 +#error "Unsupported long double format" +#endif + +long double +nextafterl(long double x, long double y) +{ + volatile long double t; + union IEEEl2bits ux, uy; + + ux.e = x; + uy.e = y; + + if ((ux.bits.exp == 0x7fff && + ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl) != 0) || + (uy.bits.exp == 0x7fff && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl) != 0)) + return x+y; /* x or y is nan */ + if(x==y) return y; /* x=y, return y */ + if(x==0.0) { + ux.bits.manh = 0; /* return +-minsubnormal */ + ux.bits.manl = 1; + ux.bits.sign = uy.bits.sign; + t = ux.e*ux.e; + if(t==ux.e) return t; else return ux.e; /* raise underflow flag */ + } + if(x>0.0 ^ x +__FBSDID("$FreeBSD$"); + +/* + * We assume that a long double has a 15-bit exponent. On systems + * where long double is the same as double, nexttoward() is an alias + * for nextafter(), so we don't use this routine. + */ + +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#if LDBL_MAX_EXP != 0x4000 +#error "Unsupported long double format" +#endif + +double +nexttoward(double x, long double y) +{ + union IEEEl2bits uy; + volatile double t; + int32_t hx,ix; + u_int32_t lx; + + EXTRACT_WORDS(hx,lx,x); + ix = hx&0x7fffffff; /* |x| */ + uy.e = y; + + if(((ix>=0x7ff00000)&&((ix-0x7ff00000)|lx)!=0) || + (uy.bits.exp == 0x7fff && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl) != 0)) + return x+y; /* x or y is nan */ + if(x==y) return (double)y; /* x=y, return y */ + if(x==0.0) { + INSERT_WORDS(x,uy.bits.sign<<31,1); /* return +-minsubnormal */ + t = x*x; + if(t==x) return t; else return x; /* raise underflow flag */ + } + if(hx>0.0 ^ x < y) { /* x -= ulp */ + if(lx==0) hx -= 1; + lx -= 1; + } else { /* x += ulp */ + lx += 1; + if(lx==0) hx += 1; + } + ix = hx&0x7ff00000; + if(ix>=0x7ff00000) return x+x; /* overflow */ + if(ix<0x00100000) { /* underflow */ + t = x*x; + if(t!=x) { /* raise underflow flag */ + INSERT_WORDS(x,hx,lx); + return x; + } + } + INSERT_WORDS(x,hx,lx); + return x; +} diff --git a/newlib/libm/ld/s_nexttowardf.c b/newlib/libm/ld/s_nexttowardf.c new file mode 100644 index 000000000..9ddfff961 --- /dev/null +++ b/newlib/libm/ld/s_nexttowardf.c @@ -0,0 +1,59 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#define LDBL_INFNAN_EXP (LDBL_MAX_EXP * 2 - 1) + +float +nexttowardf(float x, long double y) +{ + union IEEEl2bits uy; + volatile float t; + int32_t hx,ix; + + GET_FLOAT_WORD(hx,x); + ix = hx&0x7fffffff; /* |x| */ + uy.e = y; + + if((ix>0x7f800000) || + (uy.bits.exp == LDBL_INFNAN_EXP && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl) != 0)) + return x+y; /* x or y is nan */ + if(x==y) return (float)y; /* x=y, return y */ + if(ix==0) { /* x == 0 */ + SET_FLOAT_WORD(x,(uy.bits.sign<<31)|1);/* return +-minsubnormal */ + t = x*x; + if(t==x) return t; else return x; /* raise underflow flag */ + } + if(hx>=0 ^ x < y) /* x -= ulp */ + hx -= 1; + else /* x += ulp */ + hx += 1; + ix = hx&0x7f800000; + if(ix>=0x7f800000) return x+x; /* overflow */ + if(ix<0x00800000) { /* underflow */ + t = x*x; + if(t!=x) { /* raise underflow flag */ + SET_FLOAT_WORD(x,hx); + return x; + } + } + SET_FLOAT_WORD(x,hx); + return x; +} diff --git a/newlib/libm/ld/s_remquol.c b/newlib/libm/ld/s_remquol.c new file mode 100644 index 000000000..a9f5813d0 --- /dev/null +++ b/newlib/libm/ld/s_remquol.c @@ -0,0 +1,173 @@ +/* @(#)e_fmod.c 1.3 95/01/18 */ +/*- + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#define BIAS (LDBL_MAX_EXP - 1) + +#if LDBL_MANL_SIZE > 32 +typedef uint64_t manl_t; +#else +typedef uint32_t manl_t; +#endif + +#if LDBL_MANH_SIZE > 32 +typedef uint64_t manh_t; +#else +typedef uint32_t manh_t; +#endif + +/* + * These macros add and remove an explicit integer bit in front of the + * fractional mantissa, if the architecture doesn't have such a bit by + * default already. + */ +#ifdef LDBL_IMPLICIT_NBIT +#define SET_NBIT(hx) ((hx) | (1ULL << LDBL_MANH_SIZE)) +#define HFRAC_BITS LDBL_MANH_SIZE +#else +#define SET_NBIT(hx) (hx) +#define HFRAC_BITS (LDBL_MANH_SIZE - 1) +#endif + +#define MANL_SHIFT (LDBL_MANL_SIZE - 1) + +static const long double Zero[] = {0.0L, -0.0L}; + +/* + * Return the IEEE remainder and set *quo to the last n bits of the + * quotient, rounded to the nearest integer. We choose n=31 because + * we wind up computing all the integer bits of the quotient anyway as + * a side-effect of computing the remainder by the shift and subtract + * method. In practice, this is far more bits than are needed to use + * remquo in reduction algorithms. + * + * Assumptions: + * - The low part of the mantissa fits in a manl_t exactly. + * - The high part of the mantissa fits in an int64_t with enough room + * for an explicit integer bit in front of the fractional bits. + */ +long double +remquol(long double x, long double y, int *quo) +{ + union IEEEl2bits ux, uy; + int64_t hx,hz; /* We need a carry bit even if LDBL_MANH_SIZE is 32. */ + manh_t hy; + manl_t lx,ly,lz; + int ix,iy,n,q,sx,sxy; + + ux.e = x; + uy.e = y; + sx = ux.bits.sign; + sxy = sx ^ uy.bits.sign; + ux.bits.sign = 0; /* |x| */ + uy.bits.sign = 0; /* |y| */ + + /* purge off exception values */ + if((uy.bits.exp|uy.bits.manh|uy.bits.manl)==0 || /* y=0 */ + (ux.bits.exp == BIAS + LDBL_MAX_EXP) || /* or x not finite */ + (uy.bits.exp == BIAS + LDBL_MAX_EXP && + ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)!=0)) /* or y is NaN */ + return nan_mix_op(x, y, *)/nan_mix_op(x, y, *); + if(ux.bits.exp<=uy.bits.exp) { + if((ux.bits.exp>MANL_SHIFT); lx = lx+lx;} + else {hx = hz+hz+(lz>>MANL_SHIFT); lx = lz+lz; q++;} + q <<= 1; + } + hz=hx-hy;lz=lx-ly; if(lx=0) {hx=hz;lx=lz;q++;} + + /* convert back to floating value and restore the sign */ + if((hx|lx)==0) { /* return sign(x)*0 */ + q &= 0x7fffffff; + *quo = (sxy ? -q : q); + return Zero[sx]; + } + while(hx<(1ULL<>MANL_SHIFT); lx = lx+lx; + iy -= 1; + } + ux.bits.manh = hx; /* The integer bit is truncated here if needed. */ + ux.bits.manl = lx; + if (iy < LDBL_MIN_EXP) { + ux.bits.exp = iy + (BIAS + 512); + ux.e *= 0x1p-512; + } else { + ux.bits.exp = iy + BIAS; + } +fixup: + x = ux.e; /* |x| */ + y = fabsl(y); + if (y < LDBL_MIN * 2) { + if (x+x>y || (x+x==y && (q & 1))) { + q++; + x-=y; + } + } else if (x>0.5*y || (x==0.5*y && (q & 1))) { + q++; + x-=y; + } + ux.e = x; + ux.bits.sign ^= sx; + x = ux.e; + q &= 0x7fffffff; + *quo = (sxy ? -q : q); + return x; +} diff --git a/newlib/libm/ld/s_rintl.c b/newlib/libm/ld/s_rintl.c new file mode 100644 index 000000000..1e9824d83 --- /dev/null +++ b/newlib/libm/ld/s_rintl.c @@ -0,0 +1,92 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "fpmath.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual bias, min exp and expsign packing. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const float +shift[2] = { +#if LDBL_MANT_DIG == 64 + 0x1.0p63, -0x1.0p63 +#elif LDBL_MANT_DIG == 113 + 0x1.0p112, -0x1.0p112 +#else +#error "Unsupported long double format" +#endif +}; +static const float zero[2] = { 0.0, -0.0 }; + +long double +rintl(long double x) +{ + union IEEEl2bits u; + uint32_t expsign; + int ex, sign; + + u.e = x; + expsign = u.xbits.expsign; + ex = expsign & 0x7fff; + + if (ex >= BIAS + LDBL_MANT_DIG - 1) { + if (ex == BIAS + LDBL_MAX_EXP) + return (x + x); /* Inf, NaN, or unsupported format */ + return (x); /* finite and already an integer */ + } + sign = expsign >> 15; + + /* + * The following code assumes that intermediate results are + * evaluated in long double precision. If they are evaluated in + * greater precision, double rounding may occur, and if they are + * evaluated in less precision (as on i386), results will be + * wildly incorrect. + */ + x += shift[sign]; + x -= shift[sign]; + + /* + * If the result is +-0, then it must have the same sign as x, but + * the above calculation doesn't always give this. Fix up the sign. + */ + if (ex < BIAS && x == 0.0L) + return (zero[sign]); + + return (x); +} diff --git a/newlib/libm/ld/s_roundl.c b/newlib/libm/ld/s_roundl.c new file mode 100644 index 000000000..8d1c02a7f --- /dev/null +++ b/newlib/libm/ld/s_roundl.c @@ -0,0 +1,64 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2003, Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +long double +roundl(long double x) +{ + long double t; + uint16_t hx; + + GET_LDBL_EXPSIGN(hx, x); + if ((hx & 0x7fff) == 0x7fff) + return (x + x); + + ENTERI(); + + if (!(hx & 0x8000)) { + t = floorl(x); + if (t - x <= -0.5L) + t += 1; + RETURNI(t); + } else { + t = floorl(-x); + if (t + x <= -0.5L) + t += 1; + RETURNI(-t); + } +} diff --git a/newlib/libm/ld/s_scalbln.c b/newlib/libm/ld/s_scalbln.c new file mode 100644 index 000000000..c27420c92 --- /dev/null +++ b/newlib/libm/ld/s_scalbln.c @@ -0,0 +1,56 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2004 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#define NMAX 65536 +#define NMIN -65536 + +double +scalbln(double x, long n) +{ + + return (scalbn(x, (n > NMAX) ? NMAX : (n < NMIN) ? NMIN : (int)n)); +} + +float +scalblnf(float x, long n) +{ + + return (scalbnf(x, (n > NMAX) ? NMAX : (n < NMIN) ? NMIN : (int)n)); +} + +long double +scalblnl(long double x, long n) +{ + + return (scalbnl(x, (n > NMAX) ? NMAX : (n < NMIN) ? NMIN : (int)n)); +} diff --git a/newlib/libm/ld/s_scalbnl.c b/newlib/libm/ld/s_scalbnl.c new file mode 100644 index 000000000..6044c1b1d --- /dev/null +++ b/newlib/libm/ld/s_scalbnl.c @@ -0,0 +1,49 @@ +/* + * Copyright (c) 2005-2020 Rich Felker, et al. + * + * SPDX-License-Identifier: MIT + * + * Please see https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT + * for all contributors to musl. + */ +#include +#include +#include "math_private.h" +#include "fpmath.h" +/* + * scalbnl (long double x, int n) + * scalbnl(x,n) returns x* 2**n computed by exponent + * manipulation rather than by actually performing an + * exponentiation or a multiplication. + */ +#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384 +long double scalbnl(long double x, int n) +{ + union IEEEl2bits u; + + if (n > 16383) { + x *= 0x1p16383L; + n -= 16383; + if (n > 16383) { + x *= 0x1p16383L; + n -= 16383; + if (n > 16383) + n = 16383; + } + } else if (n < -16382) { + x *= 0x1p-16382L * 0x1p113L; + n += 16382 - 113; + if (n < -16382) { + x *= 0x1p-16382L * 0x1p113L; + n += 16382 - 113; + if (n < -16382) + n = -16382; + } + } + u.e = 1.0; + u.xbits.expsign = 0x3fff + n; + return x * u.e; +} +__strong_reference(scalbnl, ldexpl); +#endif + diff --git a/newlib/libm/ld/s_sinl.c b/newlib/libm/ld/s_sinl.c new file mode 100644 index 000000000..f1ef84c8c --- /dev/null +++ b/newlib/libm/ld/s_sinl.c @@ -0,0 +1,95 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#ifdef __i386__ +#include +#endif + +#include "math.h" +#include "math_private.h" +#if LDBL_MANT_DIG == 64 +#include "../ld80/e_rem_pio2l.h" +#elif LDBL_MANT_DIG == 113 +#include "../ld128/e_rem_pio2l.h" +#else +#error "Unsupported long double format" +#endif + +long double +sinl(long double x) +{ + union IEEEl2bits z; + int e0, s; + long double y[2]; + long double hi, lo; + + z.e = x; + s = z.bits.sign; + z.bits.sign = 0; + + /* If x = +-0 or x is a subnormal number, then sin(x) = x */ + if (z.bits.exp == 0) + return (x); + + /* If x = NaN or Inf, then sin(x) = NaN. */ + if (z.bits.exp == 32767) + return ((x - x) / (x - x)); + + ENTERI(); + + /* Optimize the case where x is already within range. */ + if (z.e < M_PI_4) { + hi = __kernel_sinl(z.e, 0, 0); + RETURNI(s ? -hi : hi); + } + + e0 = __ieee754_rem_pio2l(x, y); + hi = y[0]; + lo = y[1]; + + switch (e0 & 3) { + case 0: + hi = __kernel_sinl(hi, lo, 1); + break; + case 1: + hi = __kernel_cosl(hi, lo); + break; + case 2: + hi = - __kernel_sinl(hi, lo, 1); + break; + case 3: + hi = - __kernel_cosl(hi, lo); + break; + } + + RETURNI(hi); +} diff --git a/newlib/libm/ld/s_tanhl.c b/newlib/libm/ld/s_tanhl.c new file mode 100644 index 000000000..b7531866d --- /dev/null +++ b/newlib/libm/ld/s_tanhl.c @@ -0,0 +1,174 @@ +/* from: FreeBSD: head/lib/msun/src/s_tanhl.c XXX */ + +/* @(#)s_tanh.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See s_tanh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "math.h" +#include "math_private.h" +#include "fpmath.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const volatile double tiny = 1.0e-300; +static const double one = 1.0; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-0.25, 0.25], range ~[-1.6304e-22, 1.6304e-22]: + * |tanh(x)/x - t(x)| < 2**-72.3 + */ +static const union IEEEl2bits +T3u = LD80C(0xaaaaaaaaaaaaaa9f, -2, -3.33333333333333333017e-1L); +#define T3 T3u.e +static const double +T5 = 1.3333333333333314e-1, /* 0x1111111111110a.0p-55 */ +T7 = -5.3968253968210485e-2, /* -0x1ba1ba1ba1a1a1.0p-57 */ +T9 = 2.1869488531393817e-2, /* 0x1664f488172022.0p-58 */ +T11 = -8.8632352345964591e-3, /* -0x1226e34bc138d5.0p-59 */ +T13 = 3.5921169709993771e-3, /* 0x1d6d371d3e400f.0p-61 */ +T15 = -1.4555786415756001e-3, /* -0x17d923aa63814d.0p-62 */ +T17 = 5.8645267876296793e-4, /* 0x13378589b85aa7.0p-63 */ +T19 = -2.1121033571392224e-4; /* -0x1baf0af80c4090.0p-65 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-0.25, 0.25], range ~[-2.4211e-37, 2.4211e-37]: + * |tanh(x)/x - t(x)| < 2**121.6 + */ +static const long double +T3 = -3.33333333333333333333333333333332980e-1L, /* -0x1555555555555555555555555554e.0p-114L */ +T5 = 1.33333333333333333333333333332707260e-1L, /* 0x1111111111111111111111110ab7b.0p-115L */ +T7 = -5.39682539682539682539682535723482314e-2L, /* -0x1ba1ba1ba1ba1ba1ba1ba17b5fc98.0p-117L */ +T9 = 2.18694885361552028218693591149061717e-2L, /* 0x1664f4882c10f9f32d6b1a12a25e5.0p-118L */ +T11 = -8.86323552990219656883762347736381851e-3L, /* -0x1226e355e6c23c8f5a5a0f386cb4d.0p-119L */ +T13 = 3.59212803657248101358314398220822722e-3L, /* 0x1d6d3d0e157ddfb403ad3637442c6.0p-121L */ +T15 = -1.45583438705131796512568010348874662e-3L; /* -0x17da36452b75e150c44cc34253b34.0p-122L */ +static const double +T17 = 5.9002744094556621e-4, /* 0x1355824803668e.0p-63 */ +T19 = -2.3912911424260516e-4, /* -0x1f57d7734c8dde.0p-65 */ +T21 = 9.6915379535512898e-5, /* 0x1967e18ad6a6ca.0p-66 */ +T23 = -3.9278322983156353e-5, /* -0x1497d8e6b75729.0p-67 */ +T25 = 1.5918887220143869e-5, /* 0x10b1319998cafa.0p-68 */ +T27 = -6.4514295231630956e-6, /* -0x1b0f2b71b218eb.0p-70 */ +T29 = 2.6120754043964365e-6, /* 0x15e963a3cf3a39.0p-71 */ +T31 = -1.0407567231003314e-6, /* -0x1176041e656869.0p-72 */ +T33 = 3.4744117554063574e-7; /* 0x1750fe732cab9c.0p-74 */ +#endif /* LDBL_MANT_DIG == 64 */ + +static inline long double +divl(long double a, long double b, long double c, long double d, + long double e, long double f) +{ + long double inv, r; + float fr, fw; + + _2sumF(a, c); + b = b + c; + _2sumF(d, f); + e = e + f; + + inv = 1 / (d + e); + + r = (a + b) * inv; + fr = r; + r = fr; + + fw = d + e; + e = d - fw + e; + d = fw; + + r = r + (a - d * r + b - e * r) * inv; + + return r; +} + +long double +tanhl(long double x) +{ + long double hi,lo,s,x2,x4,z; +#if LDBL_MANT_DIG == 113 + double dx2; +#endif + int16_t jx,ix; + + GET_LDBL_EXPSIGN(jx,x); + ix = jx&0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) { + if (jx>=0) return one/x+one; /* tanh(+-inf)=+-1 */ + else return one/x-one; /* tanh(NaN) = NaN */ + } + + ENTERI(); + + /* |x| < 40 */ + if (ix < 0x4004 || fabsl(x) < 40) { /* |x|<40 */ + if (__predict_false(ix= 40, return +-1 */ + } else { + z = one - tiny; /* raise inexact flag */ + } + s = 1; + if (jx<0) s = -1; + RETURNI(s*z); +} diff --git a/newlib/libm/ld/s_tanl.c b/newlib/libm/ld/s_tanl.c new file mode 100644 index 000000000..0c5228ec8 --- /dev/null +++ b/newlib/libm/ld/s_tanl.c @@ -0,0 +1,97 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * Limited testing on pseudorandom numbers drawn within [0:4e8] shows + * an accuracy of <= 1.5 ULP where 247024 values of x out of 40 million + * possibles resulted in tan(x) that exceeded 0.5 ULP (ie., 0.6%). + */ + +#include +#ifdef __i386__ +#include +#endif + +#include "math.h" +#include "math_private.h" +#if LDBL_MANT_DIG == 64 +#include "../ld80/e_rem_pio2l.h" +#elif LDBL_MANT_DIG == 113 +#include "../ld128/e_rem_pio2l.h" +#else +#error "Unsupported long double format" +#endif + +long double +tanl(long double x) +{ + union IEEEl2bits z; + int e0, s; + long double y[2]; + long double hi, lo; + + z.e = x; + s = z.bits.sign; + z.bits.sign = 0; + + /* If x = +-0 or x is subnormal, then tan(x) = x. */ + if (z.bits.exp == 0) + return (x); + + /* If x = NaN or Inf, then tan(x) = NaN. */ + if (z.bits.exp == 32767) + return ((x - x) / (x - x)); + + ENTERI(); + + /* Optimize the case where x is already within range. */ + if (z.e < M_PI_4) { + hi = __kernel_tanl(z.e, 0, 0); + RETURNI(s ? -hi : hi); + } + + e0 = __ieee754_rem_pio2l(x, y); + hi = y[0]; + lo = y[1]; + + switch (e0 & 3) { + case 0: + case 2: + hi = __kernel_tanl(hi, lo, 0); + break; + case 1: + case 3: + hi = __kernel_tanl(hi, lo, 1); + break; + } + + RETURNI(hi); +} diff --git a/newlib/libm/ld/s_truncl.c b/newlib/libm/ld/s_truncl.c new file mode 100644 index 000000000..9e2b51132 --- /dev/null +++ b/newlib/libm/ld/s_truncl.c @@ -0,0 +1,68 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * From: @(#)s_floor.c 5.1 93/09/24 + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * truncl(x) + * Return x rounded toward 0 to integral value + * Method: + * Bit twiddling. + * Exception: + * Inexact flag raised if x not equal to truncl(x). + */ + +#include +#include +#include + +#include "fpmath.h" + +#ifdef LDBL_IMPLICIT_NBIT +#define MANH_SIZE (LDBL_MANH_SIZE + 1) +#else +#define MANH_SIZE LDBL_MANH_SIZE +#endif + +static const long double huge = 1.0e300; +static const float zero[] = { 0.0, -0.0 }; + +long double +truncl(long double x) +{ + union IEEEl2bits u = { .e = x }; + int e = u.bits.exp - LDBL_MAX_EXP + 1; + + if (e < MANH_SIZE - 1) { + if (e < 0) { /* raise inexact if x != 0 */ + if (huge + x > 0.0) + u.e = zero[u.bits.sign]; + } else { + uint64_t m = ((1llu << MANH_SIZE) - 1) >> (e + 1); + if (((u.bits.manh & m) | u.bits.manl) == 0) + return (x); /* x is integral */ + if (huge + x > 0.0) { /* raise inexact flag */ + u.bits.manh &= ~m; + u.bits.manl = 0; + } + } + } else if (e < LDBL_MANT_DIG - 1) { + uint64_t m = (uint64_t)-1 >> (64 - LDBL_MANT_DIG + e + 1); + if ((u.bits.manl & m) == 0) + return (x); /* x is integral */ + if (huge + x > 0.0) /* raise inexact flag */ + u.bits.manl &= ~m; + } + return (u.e); +} diff --git a/newlib/libm/ld128/b_tgammal.c b/newlib/libm/ld128/b_tgammal.c new file mode 100644 index 000000000..1c995ab6d --- /dev/null +++ b/newlib/libm/ld128/b_tgammal.c @@ -0,0 +1,57 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2013 David Chisnall + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include +#include + +/* + * If long double is not the same size as double, then these will lose + * precision and we should emit a warning whenever something links against + * them. + */ +#if (LDBL_MANT_DIG > 53) +#define WARN_IMPRECISE(x) \ + __warn_references(x, # x " has lower than advertised precision"); +#else +#define WARN_IMPRECISE(x) +#endif +/* + * Declare the functions as weak variants so that other libraries providing + * real versions can override them. + */ +#define DECLARE_WEAK(x)\ + __weak_reference(imprecise_## x, x);\ + WARN_IMPRECISE(x) + +#define DECLARE_IMPRECISE(f) \ + long double imprecise_ ## f ## l(long double v) { return f(v); }\ + DECLARE_WEAK(f ## l) + +DECLARE_IMPRECISE(tgamma); diff --git a/newlib/libm/ld128/e_powl.c b/newlib/libm/ld128/e_powl.c new file mode 100644 index 000000000..12b92a1a9 --- /dev/null +++ b/newlib/libm/ld128/e_powl.c @@ -0,0 +1,443 @@ +/*- + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * Copyright (c) 2008 Stephen L. Moshier + * + * Permission to use, copy, modify, and distribute this software for any + * purpose with or without fee is hereby granted, provided that the above + * copyright notice and this permission notice appear in all copies. + * + * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES + * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR + * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES + * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN + * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF + * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. + */ + +/* powl(x,y) return x**y + * + * n + * Method: Let x = 2 * (1+f) + * 1. Compute and return log2(x) in two pieces: + * log2(x) = w1 + w2, + * where w1 has 113-53 = 60 bit trailing zeros. + * 2. Perform y*log2(x) = n+y' by simulating multi-precision + * arithmetic, where |y'|<=0.5. + * 3. Return x**y = 2**n*exp(y'*log2) + * + * Special cases: + * 1. (anything) ** 0 is 1 + * 2. (anything) ** 1 is itself + * 3. (anything) ** NAN is NAN + * 4. NAN ** (anything except 0) is NAN + * 5. +-(|x| > 1) ** +INF is +INF + * 6. +-(|x| > 1) ** -INF is +0 + * 7. +-(|x| < 1) ** +INF is +0 + * 8. +-(|x| < 1) ** -INF is +INF + * 9. +-1 ** +-INF is NAN + * 10. +0 ** (+anything except 0, NAN) is +0 + * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 + * 12. +0 ** (-anything except 0, NAN) is +INF + * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF + * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) + * 15. +INF ** (+anything except 0,NAN) is +INF + * 16. +INF ** (-anything except 0,NAN) is +0 + * 17. -INF ** (anything) = -0 ** (-anything) + * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) + * 19. (-anything except 0 and inf) ** (non-integer) is NAN + * + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "math_private.h" + +static const long double bp[] = { + 1.0L, + 1.5L, +}; + +/* log_2(1.5) */ +static const long double dp_h[] = { + 0.0, + 5.8496250072115607565592654282227158546448E-1L +}; + +/* Low part of log_2(1.5) */ +static const long double dp_l[] = { + 0.0, + 1.0579781240112554492329533686862998106046E-16L +}; + +static const long double zero = 0.0L, + one = 1.0L, + two = 2.0L, + two113 = 1.0384593717069655257060992658440192E34L, + huge = 1.0e3000L, + tiny = 1.0e-3000L; + +/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) + z = (x-1)/(x+1) + 1 <= x <= 1.25 + Peak relative error 2.3e-37 */ +static const long double LN[] = +{ + -3.0779177200290054398792536829702930623200E1L, + 6.5135778082209159921251824580292116201640E1L, + -4.6312921812152436921591152809994014413540E1L, + 1.2510208195629420304615674658258363295208E1L, + -9.9266909031921425609179910128531667336670E-1L +}; +static const long double LD[] = +{ + -5.129862866715009066465422805058933131960E1L, + 1.452015077564081884387441590064272782044E2L, + -1.524043275549860505277434040464085593165E2L, + 7.236063513651544224319663428634139768808E1L, + -1.494198912340228235853027849917095580053E1L + /* 1.0E0 */ +}; + +/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) + 0 <= x <= 0.5 + Peak relative error 5.7e-38 */ +static const long double PN[] = +{ + 5.081801691915377692446852383385968225675E8L, + 9.360895299872484512023336636427675327355E6L, + 4.213701282274196030811629773097579432957E4L, + 5.201006511142748908655720086041570288182E1L, + 9.088368420359444263703202925095675982530E-3L, +}; +static const long double PD[] = +{ + 3.049081015149226615468111430031590411682E9L, + 1.069833887183886839966085436512368982758E8L, + 8.259257717868875207333991924545445705394E5L, + 1.872583833284143212651746812884298360922E3L, + /* 1.0E0 */ +}; + +static const long double + /* ln 2 */ + lg2 = 6.9314718055994530941723212145817656807550E-1L, + lg2_h = 6.9314718055994528622676398299518041312695E-1L, + lg2_l = 2.3190468138462996154948554638754786504121E-17L, + ovt = 8.0085662595372944372e-0017L, + /* 2/(3*log(2)) */ + cp = 9.6179669392597560490661645400126142495110E-1L, + cp_h = 9.6179669392597555432899980587535537779331E-1L, + cp_l = 5.0577616648125906047157785230014751039424E-17L; + +long double +powl(long double x, long double y) +{ + long double z, ax, z_h, z_l, p_h, p_l; + long double yy1, t1, t2, r, s, t, u, v, w; + long double s2, s_h, s_l, t_h, t_l; + int32_t i, j, k, yisint, n; + u_int32_t ix, iy; + int32_t hx, hy; + ieee_quad_shape_type o, p, q; + + p.value = x; + hx = p.parts32.mswhi; + ix = hx & 0x7fffffff; + + q.value = y; + hy = q.parts32.mswhi; + iy = hy & 0x7fffffff; + + + /* y==zero: x**0 = 1 */ + if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) + return one; + + /* 1.0**y = 1; -1.0**+-Inf = 1 */ + if (x == one) + return one; + if (x == -1.0L && iy == 0x7fff0000 + && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) + return one; + + /* +-NaN return x+y */ + if ((ix > 0x7fff0000) + || ((ix == 0x7fff0000) + && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0)) + || (iy > 0x7fff0000) + || ((iy == 0x7fff0000) + && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0))) + return nan_mix(x, y); + + /* determine if y is an odd int when x < 0 + * yisint = 0 ... y is not an integer + * yisint = 1 ... y is an odd int + * yisint = 2 ... y is an even int + */ + yisint = 0; + if (hx < 0) + { + if (iy >= 0x40700000) /* 2^113 */ + yisint = 2; /* even integer y */ + else if (iy >= 0x3fff0000) /* 1.0 */ + { + if (floorl (y) == y) + { + z = 0.5 * y; + if (floorl (z) == z) + yisint = 2; + else + yisint = 1; + } + } + } + + /* special value of y */ + if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) + { + if (iy == 0x7fff0000) /* y is +-inf */ + { + if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi | + p.parts32.lswlo) == 0) + return y - y; /* +-1**inf is NaN */ + else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ + return (hy >= 0) ? y : zero; + else /* (|x|<1)**-,+inf = inf,0 */ + return (hy < 0) ? -y : zero; + } + if (iy == 0x3fff0000) + { /* y is +-1 */ + if (hy < 0) + return one / x; + else + return x; + } + if (hy == 0x40000000) + return x * x; /* y is 2 */ + if (hy == 0x3ffe0000) + { /* y is 0.5 */ + if (hx >= 0) /* x >= +0 */ + return sqrtl (x); + } + } + + ax = fabsl (x); + /* special value of x */ + if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0) + { + if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) + { + z = ax; /*x is +-0,+-inf,+-1 */ + if (hy < 0) + z = one / z; /* z = (1/|x|) */ + if (hx < 0) + { + if (((ix - 0x3fff0000) | yisint) == 0) + { + z = (z - z) / (z - z); /* (-1)**non-int is NaN */ + } + else if (yisint == 1) + z = -z; /* (x<0)**odd = -(|x|**odd) */ + } + return z; + } + } + + /* (x<0)**(non-int) is NaN */ + if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) + return (x - x) / (x - x); + + /* |y| is huge. + 2^-16495 = 1/2 of smallest representable value. + If (1 - 1/131072)^y underflows, y > 1.4986e9 */ + if (iy > 0x401d654b) + { + /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ + if (iy > 0x407d654b) + { + if (ix <= 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix >= 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + /* over/underflow if x is not close to one */ + if (ix < 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix > 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + + n = 0; + /* take care subnormal number */ + if (ix < 0x00010000) + { + ax *= two113; + n -= 113; + o.value = ax; + ix = o.parts32.mswhi; + } + n += ((ix) >> 16) - 0x3fff; + j = ix & 0x0000ffff; + /* determine interval */ + ix = j | 0x3fff0000; /* normalize ix */ + if (j <= 0x3988) + k = 0; /* |x|> 31) - 1) | (yisint - 1)) == 0) + s = -one; /* (-ve)**(odd int) */ + + /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ + yy1 = y; + o.value = yy1; + o.parts32.lswlo = 0; + o.parts32.lswhi &= 0xf8000000; + yy1 = o.value; + p_l = (y - yy1) * t1 + y * t2; + p_h = yy1 * t1; + z = p_l + p_h; + o.value = z; + j = o.parts32.mswhi; + if (j >= 0x400d0000) /* z >= 16384 */ + { + /* if z > 16384 */ + if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi | + o.parts32.lswlo) != 0) + return s * huge * huge; /* overflow */ + else + { + if (p_l + ovt > z - p_h) + return s * huge * huge; /* overflow */ + } + } + else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ + { + /* z < -16495 */ + if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi | + o.parts32.lswlo) + != 0) + return s * tiny * tiny; /* underflow */ + else + { + if (p_l <= z - p_h) + return s * tiny * tiny; /* underflow */ + } + } + /* compute 2**(p_h+p_l) */ + i = j & 0x7fffffff; + k = (i >> 16) - 0x3fff; + n = 0; + if (i > 0x3ffe0000) + { /* if |z| > 0.5, set n = [z+0.5] */ + n = floorl (z + 0.5L); + t = n; + p_h -= t; + } + t = p_l + p_h; + o.value = t; + o.parts32.lswlo = 0; + o.parts32.lswhi &= 0xf8000000; + t = o.value; + u = t * lg2_h; + v = (p_l - (t - p_h)) * lg2 + t * lg2_l; + z = u + v; + w = v - (z - u); + /* exp(z) */ + t = z * z; + u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); + v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); + t1 = z - t * u / v; + r = (z * t1) / (t1 - two) - (w + z * w); + z = one - (r - z); + o.value = z; + j = o.parts32.mswhi; + j += (n << 16); + if ((j >> 16) <= 0) + z = scalbnl (z, n); /* subnormal output */ + else + { + o.parts32.mswhi = j; + z = o.value; + } + return s * z; +} diff --git a/newlib/libm/ld128/e_rem_pio2l.h b/newlib/libm/ld128/e_rem_pio2l.h new file mode 100644 index 000000000..7516ced49 --- /dev/null +++ b/newlib/libm/ld128/e_rem_pio2l.h @@ -0,0 +1,135 @@ +/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* ld128 version of __ieee754_rem_pio2l(x,y) + * + * return the remainder of x rem pi/2 in y[0]+y[1] + * use __kernel_rem_pio2() + */ + +#include + +#include "math.h" +#include "math_private.h" +#include "../ld/fpmath.h" + +#define BIAS (LDBL_MAX_EXP - 1) + +/* + * XXX need to verify that nonzero integer multiples of pi/2 within the + * range get no closer to a long double than 2**-140, or that + * ilogb(x) + ilogb(min_delta) < 45 - -140. + */ +/* + * invpio2: 113 bits of 2/pi + * pio2_1: first 68 bits of pi/2 + * pio2_1t: pi/2 - pio2_1 + * pio2_2: second 68 bits of pi/2 + * pio2_2t: pi/2 - (pio2_1+pio2_2) + * pio2_3: third 68 bits of pi/2 + * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) + */ + +static const double +zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ +two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */ + +static const long double +invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ +pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ +pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ +pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ +pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ +pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ +pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ + +static inline __always_inline int +__ieee754_rem_pio2l(long double x, long double *y) +{ + union IEEEl2bits u,u1; + long double z,w,t,r,fn; + double tx[5],ty[3]; + int64_t n; + int e0,ex,i,j,nx; + int16_t expsign; + + u.e = x; + expsign = u.xbits.expsign; + ex = expsign & 0x7fff; + if (ex < BIAS + 45 || ex == BIAS + 45 && + u.bits.manh < 0x921fb54442d1LL) { + /* |x| ~< 2^45*(pi/2), medium size */ + /* TODO: use only double precision for fn, as in expl(). */ + fn = rnintl(x * invpio2); + n = i64rint(fn); + r = x-fn*pio2_1; + w = fn*pio2_1t; /* 1st round good to 180 bit */ + { + union IEEEl2bits u2; + int ex1; + j = ex; + y[0] = r-w; + u2.e = y[0]; + ex1 = u2.xbits.expsign & 0x7fff; + i = j-ex1; + if(i>51) { /* 2nd iteration needed, good to 248 */ + t = r; + w = fn*pio2_2; + r = t-w; + w = fn*pio2_2t-((t-r)-w); + y[0] = r-w; + u2.e = y[0]; + ex1 = u2.xbits.expsign & 0x7fff; + i = j-ex1; + if(i>119) { /* 3rd iteration need, 316 bits acc */ + t = r; /* will cover all possible cases */ + w = fn*pio2_3; + r = t-w; + w = fn*pio2_3t-((t-r)-w); + y[0] = r-w; + } + } + } + y[1] = (r-y[0])-w; + return n; + } + /* + * all other (large) arguments + */ + if(ex==0x7fff) { /* x is inf or NaN */ + y[0]=y[1]=x-x; return 0; + } + /* set z = scalbn(|x|,ilogb(x)-23) */ + u1.e = x; + e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ + u1.xbits.expsign = ex - e0; + z = u1.e; + for(i=0;i<4;i++) { + tx[i] = (double)((int32_t)(z)); + z = (z-tx[i])*two24; + } + tx[4] = z; + nx = 5; + while(tx[nx-1]==zero) nx--; /* skip zero term */ + n = __kernel_rem_pio2(tx,ty,e0,nx,3); + t = (long double)ty[2] + ty[1]; + r = t + ty[0]; + w = ty[0] - (r - t); + if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} + y[0] = r; y[1] = w; return n; +} diff --git a/newlib/libm/ld128/invtrig.h b/newlib/libm/ld128/invtrig.h new file mode 100644 index 000000000..423b56847 --- /dev/null +++ b/newlib/libm/ld128/invtrig.h @@ -0,0 +1,115 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include + +#include "fpmath.h" + +#define BIAS (LDBL_MAX_EXP - 1) +#define MANH_SIZE (LDBL_MANH_SIZE + 1) + +/* Approximation thresholds. */ +#define ASIN_LINEAR (BIAS - 56) /* 2**-56 */ +#define ACOS_CONST (BIAS - 113) /* 2**-113 */ +#define ATAN_CONST (BIAS + 113) /* 2**113 */ +#define ATAN_LINEAR (BIAS - 56) /* 2**-56 */ + +/* 0.95 */ +#define THRESH ((0xe666666666666666ULL>>(64-(MANH_SIZE-1)))|LDBL_NBIT) + +/* Constants shared by the long double inverse trig functions. */ +#define pS0 _ItL_pS0 +#define pS1 _ItL_pS1 +#define pS2 _ItL_pS2 +#define pS3 _ItL_pS3 +#define pS4 _ItL_pS4 +#define pS5 _ItL_pS5 +#define pS6 _ItL_pS6 +#define pS7 _ItL_pS7 +#define pS8 _ItL_pS8 +#define pS9 _ItL_pS9 +#define qS1 _ItL_qS1 +#define qS2 _ItL_qS2 +#define qS3 _ItL_qS3 +#define qS4 _ItL_qS4 +#define qS5 _ItL_qS5 +#define qS6 _ItL_qS6 +#define qS7 _ItL_qS7 +#define qS8 _ItL_qS8 +#define qS9 _ItL_qS9 +#define atanhi _ItL_atanhi +#define atanlo _ItL_atanlo +#define aT _ItL_aT +#define pi_lo _ItL_pi_lo + +#define pio2_hi atanhi[3] +#define pio2_lo atanlo[3] +#define pio4_hi atanhi[1] + +/* Constants shared by the long double inverse trig functions. */ +extern const long double pS0, pS1, pS2, pS3, pS4, pS5, pS6, pS7, pS8, pS9; +extern const long double qS1, qS2, qS3, qS4, qS5, qS6, qS7, qS8, qS9; +extern const long double atanhi[], atanlo[], aT[]; +extern const long double pi_lo; + +static inline long double +P(long double x) +{ + + return (x * (pS0 + x * (pS1 + x * (pS2 + x * (pS3 + x * \ + (pS4 + x * (pS5 + x * (pS6 + x * (pS7 + x * (pS8 + x * \ + pS9)))))))))); +} + +static inline long double +Q(long double x) +{ + + return (1.0 + x * (qS1 + x * (qS2 + x * (qS3 + x * (qS4 + x * \ + (qS5 + x * (qS6 + x * (qS7 + x * (qS8 + x * qS9))))))))); +} + +static inline long double +T_even(long double x) +{ + + return (aT[0] + x * (aT[2] + x * (aT[4] + x * (aT[6] + x * \ + (aT[8] + x * (aT[10] + x * (aT[12] + x * (aT[14] + x * \ + (aT[16] + x * (aT[18] + x * (aT[20] + x * aT[22]))))))))))); +} + +static inline long double +T_odd(long double x) +{ + + return (aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] + x * \ + (aT[9] + x * (aT[11] + x * (aT[13] + x * (aT[15] + x * \ + (aT[17] + x * (aT[19] + x * (aT[21] + x * aT[23]))))))))))); +} diff --git a/newlib/libm/ld128/k_expl.h b/newlib/libm/ld128/k_expl.h new file mode 100644 index 000000000..159338fe3 --- /dev/null +++ b/newlib/libm/ld128/k_expl.h @@ -0,0 +1,324 @@ +/* from: FreeBSD: head/lib/msun/ld128/s_expl.c 251345 2013-06-03 20:09:22Z kargl */ + +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * ld128 version of k_expl.h. See ../ld80/s_expl.c for most comments. + * + * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments + * about the secondary kernels. + */ + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +/* + * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must + * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest + * bits zero so that multiplication of it by n is exact. + */ +INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ +L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */ +static const long double +/* 0x1.62e42fefa39ef35793c768000000p-8 */ +L1 = 5.41521234812457272982212595914567508e-3L; + +/* + * XXX values in hex in comments have been lost (or were never present) + * from here. + */ +static const long double +/* + * Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]: + * |exp(x) - p(x)| < 2**-124.9 + * (0.002708 is ln2/(2*INTERVALS) rounded up a little). + * + * XXX the coeffs aren't very carefully rounded, and I get 3.6 more bits. + */ +A2 = 0.5, +A3 = 1.66666666666666666666666666651085500e-1L, +A4 = 4.16666666666666666666666666425885320e-2L, +A5 = 8.33333333333333333334522877160175842e-3L, +A6 = 1.38888888888888888889971139751596836e-3L; + +static const double +A7 = 1.9841269841269470e-4, /* 0x1.a01a01a019f91p-13 */ +A8 = 2.4801587301585286e-5, /* 0x1.71de3ec75a967p-19 */ +A9 = 2.7557324277411235e-6, /* 0x1.71de3ec75a967p-19 */ +A10 = 2.7557333722375069e-7; /* 0x1.27e505ab56259p-22 */ + +static const struct { + /* + * hi must be rounded to at most 106 bits so that multiplication + * by r1 in expm1l() is exact, but it is rounded to 88 bits due to + * historical accidents. + * + * XXX it is wasteful to use long double for both hi and lo. ld128 + * exp2l() uses only float for lo (in a very differently organized + * table; ld80 exp2l() is different again. It uses 2 doubles in a + * table organized like this one. 1 double and 1 float would + * suffice). There are different packing/locality/alignment/caching + * problems with these methods. + * + * XXX C's bad %a format makes the bits unreadable. They happen + * to all line up for the hi values 1 before the point and 88 + * in 22 nybbles, but for the low values the nybbles are shifted + * randomly. + */ + long double hi; + long double lo; +} tbl[INTERVALS] = { + 0x1p0L, 0x0p0L, + 0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L, + 0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L, + 0x1.04315e86e7f84bd738f9a2p0L, 0xd.a47e6ed040bb4bfc05af6455e9b8p-96L, + 0x1.059b0d31585743ae7c548ep0L, 0xb.68ca417fe53e3495f7df4baf84a0p-92L, + 0x1.0706b29ddf6ddc6dc403a8p0L, 0x1.d87b27ed07cb8b092ac75e311753p-88L, + 0x1.0874518759bc808c35f25cp0L, 0x1.9427fa2b041b2d6829d8993a0d01p-88L, + 0x1.09e3ecac6f3834521e060cp0L, 0x5.84d6b74ba2e023da730e7fccb758p-92L, + 0x1.0b5586cf9890f6298b92b6p0L, 0x1.1842a98364291408b3ceb0a2a2bbp-88L, + 0x1.0cc922b7247f7407b705b8p0L, 0x9.3dc5e8aac564e6fe2ef1d431fd98p-92L, + 0x1.0e3ec32d3d1a2020742e4ep0L, 0x1.8af6a552ac4b358b1129e9f966a4p-88L, + 0x1.0fb66affed31af232091dcp0L, 0x1.8a1426514e0b627bda694a400a27p-88L, + 0x1.11301d0125b50a4ebbf1aep0L, 0xd.9318ceac5cc47ab166ee57427178p-92L, + 0x1.12abdc06c31cbfb92bad32p0L, 0x4.d68e2f7270bdf7cedf94eb1cb818p-92L, + 0x1.1429aaea92ddfb34101942p0L, 0x1.b2586d01844b389bea7aedd221d4p-88L, + 0x1.15a98c8a58e512480d573cp0L, 0x1.d5613bf92a2b618ee31b376c2689p-88L, + 0x1.172b83c7d517adcdf7c8c4p0L, 0x1.0eb14a792035509ff7d758693f24p-88L, + 0x1.18af9388c8de9bbbf70b9ap0L, 0x3.c2505c97c0102e5f1211941d2840p-92L, + 0x1.1a35beb6fcb753cb698f68p0L, 0x1.2d1c835a6c30724d5cfae31b84e5p-88L, + 0x1.1bbe084045cd39ab1e72b4p0L, 0x4.27e35f9acb57e473915519a1b448p-92L, + 0x1.1d4873168b9aa7805b8028p0L, 0x9.90f07a98b42206e46166cf051d70p-92L, + 0x1.1ed5022fcd91cb8819ff60p0L, 0x1.121d1e504d36c47474c9b7de6067p-88L, + 0x1.2063b88628cd63b8eeb028p0L, 0x1.50929d0fc487d21c2b84004264dep-88L, + 0x1.21f49917ddc962552fd292p0L, 0x9.4bdb4b61ea62477caa1dce823ba0p-92L, + 0x1.2387a6e75623866c1fadb0p0L, 0x1.c15cb593b0328566902df69e4de2p-88L, + 0x1.251ce4fb2a63f3582ab7dep0L, 0x9.e94811a9c8afdcf796934bc652d0p-92L, + 0x1.26b4565e27cdd257a67328p0L, 0x1.d3b249dce4e9186ddd5ff44e6b08p-92L, + 0x1.284dfe1f5638096cf15cf0p0L, 0x3.ca0967fdaa2e52d7c8106f2e262cp-92L, + 0x1.29e9df51fdee12c25d15f4p0L, 0x1.a24aa3bca890ac08d203fed80a07p-88L, + 0x1.2b87fd0dad98ffddea4652p0L, 0x1.8fcab88442fdc3cb6de4519165edp-88L, + 0x1.2d285a6e4030b40091d536p0L, 0xd.075384589c1cd1b3e4018a6b1348p-92L, + 0x1.2ecafa93e2f5611ca0f45cp0L, 0x1.523833af611bdcda253c554cf278p-88L, + 0x1.306fe0a31b7152de8d5a46p0L, 0x3.05c85edecbc27343629f502f1af2p-92L, + 0x1.32170fc4cd8313539cf1c2p0L, 0x1.008f86dde3220ae17a005b6412bep-88L, + 0x1.33c08b26416ff4c9c8610cp0L, 0x1.96696bf95d1593039539d94d662bp-88L, + 0x1.356c55f929ff0c94623476p0L, 0x3.73af38d6d8d6f9506c9bbc93cbc0p-92L, + 0x1.371a7373aa9caa7145502ep0L, 0x1.4547987e3e12516bf9c699be432fp-88L, + 0x1.38cae6d05d86585a9cb0d8p0L, 0x1.bed0c853bd30a02790931eb2e8f0p-88L, + 0x1.3a7db34e59ff6ea1bc9298p0L, 0x1.e0a1d336163fe2f852ceeb134067p-88L, + 0x1.3c32dc313a8e484001f228p0L, 0xb.58f3775e06ab66353001fae9fca0p-92L, + 0x1.3dea64c12342235b41223ep0L, 0x1.3d773fba2cb82b8244267c54443fp-92L, + 0x1.3fa4504ac801ba0bf701aap0L, 0x4.1832fb8c1c8dbdff2c49909e6c60p-92L, + 0x1.4160a21f72e29f84325b8ep0L, 0x1.3db61fb352f0540e6ba05634413ep-88L, + 0x1.431f5d950a896dc7044394p0L, 0x1.0ccec81e24b0caff7581ef4127f7p-92L, + 0x1.44e086061892d03136f408p0L, 0x1.df019fbd4f3b48709b78591d5cb5p-88L, + 0x1.46a41ed1d005772512f458p0L, 0x1.229d97df404ff21f39c1b594d3a8p-88L, + 0x1.486a2b5c13cd013c1a3b68p0L, 0x1.062f03c3dd75ce8757f780e6ec99p-88L, + 0x1.4a32af0d7d3de672d8bcf4p0L, 0x6.f9586461db1d878b1d148bd3ccb8p-92L, + 0x1.4bfdad5362a271d4397afep0L, 0xc.42e20e0363ba2e159c579f82e4b0p-92L, + 0x1.4dcb299fddd0d63b36ef1ap0L, 0x9.e0cc484b25a5566d0bd5f58ad238p-92L, + 0x1.4f9b2769d2ca6ad33d8b68p0L, 0x1.aa073ee55e028497a329a7333dbap-88L, + 0x1.516daa2cf6641c112f52c8p0L, 0x4.d822190e718226177d7608d20038p-92L, + 0x1.5342b569d4f81df0a83c48p0L, 0x1.d86a63f4e672a3e429805b049465p-88L, + 0x1.551a4ca5d920ec52ec6202p0L, 0x4.34ca672645dc6c124d6619a87574p-92L, + 0x1.56f4736b527da66ecb0046p0L, 0x1.64eb3c00f2f5ab3d801d7cc7272dp-88L, + 0x1.58d12d497c7fd252bc2b72p0L, 0x1.43bcf2ec936a970d9cc266f0072fp-88L, + 0x1.5ab07dd48542958c930150p0L, 0x1.91eb345d88d7c81280e069fbdb63p-88L, + 0x1.5c9268a5946b701c4b1b80p0L, 0x1.6986a203d84e6a4a92f179e71889p-88L, + 0x1.5e76f15ad21486e9be4c20p0L, 0x3.99766a06548a05829e853bdb2b52p-92L, + 0x1.605e1b976dc08b076f592ap0L, 0x4.86e3b34ead1b4769df867b9c89ccp-92L, + 0x1.6247eb03a5584b1f0fa06ep0L, 0x1.d2da42bb1ceaf9f732275b8aef30p-88L, + 0x1.6434634ccc31fc76f8714cp0L, 0x4.ed9a4e41000307103a18cf7a6e08p-92L, + 0x1.66238825522249127d9e28p0L, 0x1.b8f314a337f4dc0a3adf1787ff74p-88L, + 0x1.68155d44ca973081c57226p0L, 0x1.b9f32706bfe4e627d809a85dcc66p-88L, + 0x1.6a09e667f3bcc908b2fb12p0L, 0x1.66ea957d3e3adec17512775099dap-88L, + 0x1.6c012750bdabeed76a9980p0L, 0xf.4f33fdeb8b0ecd831106f57b3d00p-96L, + 0x1.6dfb23c651a2ef220e2cbep0L, 0x1.bbaa834b3f11577ceefbe6c1c411p-92L, + 0x1.6ff7df9519483cf87e1b4ep0L, 0x1.3e213bff9b702d5aa477c12523cep-88L, + 0x1.71f75e8ec5f73dd2370f2ep0L, 0xf.0acd6cb434b562d9e8a20adda648p-92L, + 0x1.73f9a48a58173bd5c9a4e6p0L, 0x8.ab1182ae217f3a7681759553e840p-92L, + 0x1.75feb564267c8bf6e9aa32p0L, 0x1.a48b27071805e61a17b954a2dad8p-88L, + 0x1.780694fde5d3f619ae0280p0L, 0x8.58b2bb2bdcf86cd08e35fb04c0f0p-92L, + 0x1.7a11473eb0186d7d51023ep0L, 0x1.6cda1f5ef42b66977960531e821bp-88L, + 0x1.7c1ed0130c1327c4933444p0L, 0x1.937562b2dc933d44fc828efd4c9cp-88L, + 0x1.7e2f336cf4e62105d02ba0p0L, 0x1.5797e170a1427f8fcdf5f3906108p-88L, + 0x1.80427543e1a11b60de6764p0L, 0x9.a354ea706b8e4d8b718a672bf7c8p-92L, + 0x1.82589994cce128acf88afap0L, 0xb.34a010f6ad65cbbac0f532d39be0p-92L, + 0x1.8471a4623c7acce52f6b96p0L, 0x1.c64095370f51f48817914dd78665p-88L, + 0x1.868d99b4492ec80e41d90ap0L, 0xc.251707484d73f136fb5779656b70p-92L, + 0x1.88ac7d98a669966530bcdep0L, 0x1.2d4e9d61283ef385de170ab20f96p-88L, + 0x1.8ace5422aa0db5ba7c55a0p0L, 0x1.92c9bb3e6ed61f2733304a346d8fp-88L, + 0x1.8cf3216b5448bef2aa1cd0p0L, 0x1.61c55d84a9848f8c453b3ca8c946p-88L, + 0x1.8f1ae991577362b982745cp0L, 0x7.2ed804efc9b4ae1458ae946099d4p-92L, + 0x1.9145b0b91ffc588a61b468p0L, 0x1.f6b70e01c2a90229a4c4309ea719p-88L, + 0x1.93737b0cdc5e4f4501c3f2p0L, 0x5.40a22d2fc4af581b63e8326efe9cp-92L, + 0x1.95a44cbc8520ee9b483694p0L, 0x1.a0fc6f7c7d61b2b3a22a0eab2cadp-88L, + 0x1.97d829fde4e4f8b9e920f8p0L, 0x1.1e8bd7edb9d7144b6f6818084cc7p-88L, + 0x1.9a0f170ca07b9ba3109b8cp0L, 0x4.6737beb19e1eada6825d3c557428p-92L, + 0x1.9c49182a3f0901c7c46b06p0L, 0x1.1f2be58ddade50c217186c90b457p-88L, + 0x1.9e86319e323231824ca78ep0L, 0x6.4c6e010f92c082bbadfaf605cfd4p-92L, + 0x1.a0c667b5de564b29ada8b8p0L, 0xc.ab349aa0422a8da7d4512edac548p-92L, + 0x1.a309bec4a2d3358c171f76p0L, 0x1.0daad547fa22c26d168ea762d854p-88L, + 0x1.a5503b23e255c8b424491cp0L, 0xa.f87bc8050a405381703ef7caff50p-92L, + 0x1.a799e1330b3586f2dfb2b0p0L, 0x1.58f1a98796ce8908ae852236ca94p-88L, + 0x1.a9e6b5579fdbf43eb243bcp0L, 0x1.ff4c4c58b571cf465caf07b4b9f5p-88L, + 0x1.ac36bbfd3f379c0db966a2p0L, 0x1.1265fc73e480712d20f8597a8e7bp-88L, + 0x1.ae89f995ad3ad5e8734d16p0L, 0x1.73205a7fbc3ae675ea440b162d6cp-88L, + 0x1.b0e07298db66590842acdep0L, 0x1.c6f6ca0e5dcae2aafffa7a0554cbp-88L, + 0x1.b33a2b84f15faf6bfd0e7ap0L, 0x1.d947c2575781dbb49b1237c87b6ep-88L, + 0x1.b59728de559398e3881110p0L, 0x1.64873c7171fefc410416be0a6525p-88L, + 0x1.b7f76f2fb5e46eaa7b081ap0L, 0xb.53c5354c8903c356e4b625aacc28p-92L, + 0x1.ba5b030a10649840cb3c6ap0L, 0xf.5b47f297203757e1cc6eadc8bad0p-92L, + 0x1.bcc1e904bc1d2247ba0f44p0L, 0x1.b3d08cd0b20287092bd59be4ad98p-88L, + 0x1.bf2c25bd71e088408d7024p0L, 0x1.18e3449fa073b356766dfb568ff4p-88L, + 0x1.c199bdd85529c2220cb12ap0L, 0x9.1ba6679444964a36661240043970p-96L, + 0x1.c40ab5fffd07a6d14df820p0L, 0xf.1828a5366fd387a7bdd54cdf7300p-92L, + 0x1.c67f12e57d14b4a2137fd2p0L, 0xf.2b301dd9e6b151a6d1f9d5d5f520p-96L, + 0x1.c8f6d9406e7b511acbc488p0L, 0x5.c442ddb55820171f319d9e5076a8p-96L, + 0x1.cb720dcef90691503cbd1ep0L, 0x9.49db761d9559ac0cb6dd3ed599e0p-92L, + 0x1.cdf0b555dc3f9c44f8958ep0L, 0x1.ac51be515f8c58bdfb6f5740a3a4p-88L, + 0x1.d072d4a07897b8d0f22f20p0L, 0x1.a158e18fbbfc625f09f4cca40874p-88L, + 0x1.d2f87080d89f18ade12398p0L, 0x9.ea2025b4c56553f5cdee4c924728p-92L, + 0x1.d5818dcfba48725da05aeap0L, 0x1.66e0dca9f589f559c0876ff23830p-88L, + 0x1.d80e316c98397bb84f9d04p0L, 0x8.805f84bec614de269900ddf98d28p-92L, + 0x1.da9e603db3285708c01a5ap0L, 0x1.6d4c97f6246f0ec614ec95c99392p-88L, + 0x1.dd321f301b4604b695de3cp0L, 0x6.30a393215299e30d4fb73503c348p-96L, + 0x1.dfc97337b9b5eb968cac38p0L, 0x1.ed291b7225a944efd5bb5524b927p-88L, + 0x1.e264614f5a128a12761fa0p0L, 0x1.7ada6467e77f73bf65e04c95e29dp-88L, + 0x1.e502ee78b3ff6273d13014p0L, 0x1.3991e8f49659e1693be17ae1d2f9p-88L, + 0x1.e7a51fbc74c834b548b282p0L, 0x1.23786758a84f4956354634a416cep-88L, + 0x1.ea4afa2a490d9858f73a18p0L, 0xf.5db301f86dea20610ceee13eb7b8p-92L, + 0x1.ecf482d8e67f08db0312fap0L, 0x1.949cef462010bb4bc4ce72a900dfp-88L, + 0x1.efa1bee615a27771fd21a8p0L, 0x1.2dac1f6dd5d229ff68e46f27e3dfp-88L, + 0x1.f252b376bba974e8696fc2p0L, 0x1.6390d4c6ad5476b5162f40e1d9a9p-88L, + 0x1.f50765b6e4540674f84b76p0L, 0x2.862baff99000dfc4352ba29b8908p-92L, + 0x1.f7bfdad9cbe138913b4bfep0L, 0x7.2bd95c5ce7280fa4d2344a3f5618p-92L, + 0x1.fa7c1819e90d82e90a7e74p0L, 0xb.263c1dc060c36f7650b4c0f233a8p-92L, + 0x1.fd3c22b8f71f10975ba4b2p0L, 0x1.2bcf3a5e12d269d8ad7c1a4a8875p-88L +}; + +/* + * Kernel for expl(x). x must be finite and not tiny or huge. + * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). + * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). + */ +static inline void +__k_expl(long double x, long double *hip, long double *lop, int *kp) +{ + long double q, r, r1, t; + double dr, fn, r2; + int n, n2; + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + fn = rnint((double)x * INV_L); + n = irint(fn); + n2 = (unsigned)n % INTERVALS; + /* Depend on the sign bit being propagated: */ + *kp = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ + dr = r; + q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + + dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); + t = tbl[n2].lo + tbl[n2].hi; + *hip = tbl[n2].hi; + *lop = tbl[n2].lo + t * (q + r1); +} + +/* + * XXX: the rest of the functions are identical for ld80 and ld128. + * However, we should use scalbnl() for ld128, since long double + * multiplication was very slow on sparc64 and no new evaluation has + * been made for aarch64 and/or riscv. + */ + +static inline void +k_hexpl(long double x, long double *hip, long double *lop) +{ + float twopkm1; + int k; + + __k_expl(x, hip, lop, &k); + SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); + *hip *= twopkm1; + *lop *= twopkm1; +} + +static inline long double +hexpl(long double x) +{ + long double hi, lo, twopkm2; + int k; + + twopkm2 = 1; + __k_expl(x, &hi, &lo, &k); + SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); + return (lo + hi) * 2 * twopkm2; +} + +#ifdef _COMPLEX_H +/* + * See ../src/k_exp.c for details. + */ +static inline long double complex +__ldexp_cexpl(long double complex z, int expt) +{ + long double c, exp_x, hi, lo, s; + long double x, y, scale1, scale2; + int half_expt, k; + + x = creall(z); + y = cimagl(z); + __k_expl(x, &hi, &lo, &k); + + exp_x = (lo + hi) * 0x1p16382L; + expt += k - 16382; + + scale1 = 1; + half_expt = expt / 2; + SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); + scale2 = 1; + SET_LDBL_EXPSIGN(scale2, BIAS + expt - half_expt); + + sincosl(y, &s, &c); + return (CMPLXL(c * exp_x * scale1 * scale2, + s * exp_x * scale1 * scale2)); +} +#endif /* _COMPLEX_H */ diff --git a/newlib/libm/ld128/s_erfl.c b/newlib/libm/ld128/s_erfl.c new file mode 100644 index 000000000..e29c9691f --- /dev/null +++ b/newlib/libm/ld128/s_erfl.c @@ -0,0 +1,329 @@ +/* @(#)s_erf.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See s_erf.c for complete comments. + * + * Converted to long double by Steven G. Kargl. + */ +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double tiny = 0x1p-10000L; + +static const double +half= 0.5, +one = 1, +two = 2; +/* + * In the domain [0, 2**-40], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-120. + */ +static const long double +efx = 1.28379167095512573896158903121545167e-01L, /* 0xecbff6a7, 0x481dd788, 0xb64d21a8, 0xeb06fc3f */ +efx8 = 1.02703333676410059116927122497236133e+00L, /* 0xecbff6a7, 0x481dd788, 0xb64d21a8, 0xeb06ff3f */ +/* + * Domain [0, 0.84375], range ~[-1.919e-38, 1.919e-38]: + * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-125.29 + */ +pp0 = 1.28379167095512573896158903121545167e-01L, /* 0x3ffc06eb, 0xa8214db6, 0x88d71d48, 0xa7f6bfec */ +pp1 = -3.14931554396568573802046931159683404e-01L, /* 0xbffd427d, 0x6ada7263, 0x547eb096, 0x95f37463 */ +pp2 = -5.27514920282183487103576956956725309e-02L, /* 0xbffab023, 0xe5a271e3, 0xb0e79b01, 0x2f7ac962 */ +pp3 = -1.13202828509005281355609495523452713e-02L, /* 0xbff872f1, 0x6a5023a1, 0xe08b3884, 0x326af20f */ +pp4 = -9.18626155872522453865998391206048506e-04L, /* 0xbff4e19f, 0xea5fb024, 0x43247a37, 0xe430b06c */ +pp5 = -7.87518862406176274922506447157284230e-05L, /* 0xbff14a4f, 0x31a85fe0, 0x7fff2204, 0x09c49b37 */ +pp6 = -3.42357944472240436548115331090560881e-06L, /* 0xbfeccb81, 0x4b43c336, 0xcd2eb6c2, 0x903f2d87 */ +pp7 = -1.37317432573890412634717890726745428e-07L, /* 0xbfe826e3, 0x0e915eb6, 0x42aee414, 0xf7e36805 */ +pp8 = -2.71115170113861755855049008732113726e-09L, /* 0xbfe2749e, 0x2b94fd00, 0xecb4d166, 0x0efb91f8 */ +pp9 = -3.37925756196555959454018189718117864e-11L, /* 0xbfdc293e, 0x1d9060cb, 0xd043204a, 0x314cd7f0 */ +qq1 = 4.76672625471551170489978555182449450e-01L, /* 0x3ffde81c, 0xde6531f0, 0x76803bee, 0x526e29e9 */ +qq2 = 1.06713144672281502058807525850732240e-01L, /* 0x3ffbb518, 0xd7a6bb74, 0xcd9bdd33, 0x7601eee5 */ +qq3 = 1.47747613127513761102189201923147490e-02L, /* 0x3ff8e423, 0xae527e18, 0xf12cb447, 0x723b4749 */ +qq4 = 1.39939377672028671891148770908874816e-03L, /* 0x3ff56ed7, 0xba055d84, 0xc21b45c4, 0x388d1812 */ +qq5 = 9.44302939359455241271983309378738276e-05L, /* 0x3ff18c11, 0xc18c99a4, 0x86d0fe09, 0x46387b4c */ +qq6 = 4.56199342312522842161301671745365650e-06L, /* 0x3fed3226, 0x73421d05, 0x08875300, 0x32fa1432 */ +qq7 = 1.53019260483764773845294600092361197e-07L, /* 0x3fe8489b, 0x3a63f627, 0x2b9ad2ce, 0x26516e57 */ +qq8 = 3.25542691121324805094777901250005508e-09L, /* 0x3fe2bf6c, 0x26d93a29, 0x9142be7c, 0x9f1dd043 */ +qq9 = 3.37405581964478060434410167262684979e-11L; /* 0x3fdc28c8, 0xfb8fa1be, 0x10e57eec, 0xaa19e49f */ + +static const long double +erx = 8.42700792949714894142232424201210961e-01L, /* 0x3ffeaf76, 0x7a741088, 0xb0000000, 0x00000000 */ +/* + * Domain [0.84375, 1.25], range ~[-2.521e-36, 2.523e-36]: + * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-120.15 + */ +pa0 = -2.48010117891186017024438233323795897e-17L, /* 0xbfc7c97f, 0x77812279, 0x6c877f22, 0xef4bfb2e */ +pa1 = 4.15107497420594680894327969504526489e-01L, /* 0x3ffda911, 0xf096fbc2, 0x55662005, 0x2337fa64 */ +pa2 = -3.94180628087084846724448515851892609e-02L, /* 0xbffa42e9, 0xab54528c, 0xad529da1, 0x6efc2af3 */ +pa3 = 4.48897599625192107295954790681677462e-02L, /* 0x3ffa6fbc, 0xa65edba1, 0x0e4cbcea, 0x73ef9a31 */ +pa4 = 8.02069252143016600110972019232995528e-02L, /* 0x3ffb4887, 0x0e8b548e, 0x3230b417, 0x11b553b3 */ +pa5 = -1.02729816533435279443621120242391295e-02L, /* 0xbff850a0, 0x041de3ee, 0xd5bca6c9, 0x4ef5f9f2 */ +pa6 = 5.70777694530755634864821094419982095e-03L, /* 0x3ff77610, 0x9b501e10, 0x4c978382, 0x742df68f */ +pa7 = 1.22635150233075521018231779267077071e-03L, /* 0x3ff5417b, 0x0e623682, 0x60327da0, 0x96b9219e */ +pa8 = 5.36100234820204569428412542856666503e-04L, /* 0x3ff41912, 0x27ceb4c1, 0x1d3298ec, 0x84ced627 */ +pa9 = -1.97753571846365167177187858667583165e-04L, /* 0xbff29eb8, 0x23f5bcf3, 0x15c83c46, 0xe4fda98b */ +pa10 = 6.19333039900846970674794789568415105e-05L, /* 0x3ff103c4, 0x60f88e46, 0xc0c9fb02, 0x13cc7fc1 */ +pa11 = -5.40531400436645861492290270311751349e-06L, /* 0xbfed6abe, 0x9665f8a8, 0xdd0ad3ba, 0xe5dc0ee3 */ +qa1 = 9.05041313265490487793231810291907851e-01L, /* 0x3ffecf61, 0x93340222, 0xe9930620, 0xc4e61168 */ +qa2 = 6.79848064708886864767240880834868092e-01L, /* 0x3ffe5c15, 0x0ba858dc, 0xf7900ae9, 0xfea1e09a */ +qa3 = 4.04720609926471677581066689316516445e-01L, /* 0x3ffd9e6f, 0x145e9b00, 0x6d8c1749, 0xd2928623 */ +qa4 = 1.69183273898369996364661075664302225e-01L, /* 0x3ffc5a7c, 0xc2a363c1, 0xd6c19097, 0xef9b4063 */ +qa5 = 7.44476185988067992342479750486764248e-02L, /* 0x3ffb30ef, 0xfc7259ef, 0x1bcbb089, 0x686dd62d */ +qa6 = 2.02981172725892407200420389604788573e-02L, /* 0x3ff94c90, 0x7976cb0e, 0x21e1d36b, 0x0f09ca2b */ +qa7 = 6.94281866271607668268269403102277234e-03L, /* 0x3ff7c701, 0x2b193250, 0xc5d46ecc, 0x374843d8 */ +qa8 = 1.12952275469171559611651594706820034e-03L, /* 0x3ff52818, 0xfd2a7c06, 0xd13e38fd, 0xda4b34f5 */ +qa9 = 3.13736683241992737197226578597710179e-04L, /* 0x3ff348fa, 0x0cb48d18, 0x051f849b, 0x135ccf74 */ +qa10 = 1.17037675204033225470121134087771410e-05L, /* 0x3fee88b6, 0x98f47704, 0xa5d8f8f2, 0xc6422e11 */ +qa11 = 4.61312518293853991439362806880973592e-06L, /* 0x3fed3594, 0xe31db94f, 0x3592b693, 0xed4386b4 */ +qa12 = -1.02158572037456893687737553657431771e-06L; /* 0xbfeb123a, 0xd60d9b1e, 0x1f6fdeb9, 0x7dc8410a */ +/* + * Domain [1.25,2.85715], range ~[-2.922e-37,2.922e-37]: + * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-121.36 + */ +static const long double +ra0 = -9.86494292470069009555706994426014461e-03L, /* 0xbff84341, 0x239e8709, 0xe941b06a, 0xcb4b6ec5 */ +ra1 = -1.13580436992565640457579040117568870e+00L, /* 0xbfff22c4, 0x133f7c0d, 0x72d5e231, 0x2eb1ee3f */ +ra2 = -4.89744330295291950661185707066921755e+01L, /* 0xc00487cb, 0xa38b4fc2, 0xc136695b, 0xc1df8047 */ +ra3 = -1.10766149300215937173768072715352140e+03L, /* 0xc00914ea, 0x55e6beb3, 0xabc50e07, 0xb6e5664d */ +ra4 = -1.49991031232170934967642795601952100e+04L, /* 0xc00cd4b8, 0xd33243e6, 0xffbf6545, 0x3c57ef6e */ +ra5 = -1.29805749738318462882524181556996692e+05L, /* 0xc00ffb0d, 0xbfeed9b6, 0x5b2a3ff4, 0xe245bd3c */ +ra6 = -7.42828497044940065828871976644647850e+05L, /* 0xc0126ab5, 0x8fe7caca, 0x473352d9, 0xcd4e0c90 */ +ra7 = -2.85637299581890734287995171242421106e+06L, /* 0xc0145cad, 0xa7f76fe7, 0x3e358051, 0x1799f927 */ +ra8 = -7.40674797129824999383748865571026084e+06L, /* 0xc015c412, 0x6fe29c02, 0x298ad158, 0x7d24e45c */ +ra9 = -1.28653420911930973914078724204151759e+07L, /* 0xc016889e, 0x7c2eb0dc, 0x95d5863b, 0x0aa34dc3 */ +ra10 = -1.47198163599330179552932489109452638e+07L, /* 0xc016c136, 0x90b84923, 0xf9bcb497, 0x19bbd0f5 */ +ra11 = -1.07812992258382800318665248311522624e+07L, /* 0xc0164904, 0xe673a113, 0x35d7f079, 0xe13701f3 */ +ra12 = -4.83545565681708642630419905537756076e+06L, /* 0xc0152721, 0xfea094a8, 0x869eb39d, 0x413d6f13 */ +ra13 = -1.23956521201673964822976917356685286e+06L, /* 0xc0132ea0, 0xd3646baa, 0x2fe62b0d, 0xbae5ce85 */ +ra14 = -1.62289333553652417591275333240371812e+05L, /* 0xc0103cf8, 0xaab1e2d6, 0x4c25e014, 0x248d76ab */ +ra15 = -8.82890392601176969729168894389833110e+03L, /* 0xc00c13e7, 0x3b3d8f94, 0x6fbda6f6, 0xe7049a82 */ +ra16 = -1.22591866337261720023681535568334619e+02L, /* 0xc005ea5e, 0x12358891, 0xcfa712c5, 0x77f050d4 */ +sa1 = 6.44508918884710829371852723353794047e+01L, /* 0x400501cd, 0xb69a6c0f, 0x5716de14, 0x47161af6 */ +sa2 = 1.76118475473171481523704824327358534e+03L, /* 0x4009b84b, 0xd305829f, 0xc4c771b0, 0xbf1f7f9b */ +sa3 = 2.69448346969488374857087646131950188e+04L, /* 0x400da503, 0x56bacc05, 0x4fdba68d, 0x2cca27e6 */ +sa4 = 2.56826633369941456778326497384543763e+05L, /* 0x4010f59d, 0x51124428, 0x69c41de6, 0xbd0d5753 */ +sa5 = 1.60647413092257206847700054645905859e+06L, /* 0x40138834, 0xa2184244, 0x557a1bed, 0x68c9d556 */ +sa6 = 6.76963075165099718574753447122393797e+06L, /* 0x40159d2f, 0x7b01b0cc, 0x8bac9e95, 0x5d35d56e */ +sa7 = 1.94295690905361884290986932493647741e+07L, /* 0x40172878, 0xc1172d61, 0x3068501e, 0x2f3c71da */ +sa8 = 3.79774781017759149060839255547073541e+07L, /* 0x401821be, 0xc30d06fe, 0x410563d7, 0x032111fd */ +sa9 = 5.00659831846029484248302236457727397e+07L, /* 0x40187df9, 0x1f97a111, 0xc51d6ac2, 0x4b389793 */ +sa10 = 4.36486287620506484276130525941972541e+07L, /* 0x40184d03, 0x3a618ae0, 0x2a723357, 0xfa45c60a */ +sa11 = 2.43779678791333894255510508253951934e+07L, /* 0x401773fa, 0x6fe10ee2, 0xc467850d, 0xc6b7ff30 */ +sa12 = 8.30732360384443202039372372212966542e+06L, /* 0x4015fb09, 0xee6a5631, 0xdd98de7e, 0x8b00461a */ +sa13 = 1.60160846942050515734192397495105693e+06L, /* 0x40138704, 0x8782bf13, 0x5b8fb315, 0xa898abe5 */ +sa14 = 1.54255505242533291014555153757001825e+05L, /* 0x40102d47, 0xc0abc98e, 0x843c9490, 0xb4352440 */ +sa15 = 5.87949220002375547561467275493888824e+03L, /* 0x400b6f77, 0xe00d21d1, 0xec4d41e8, 0x2f8e1673 */ +sa16 = 4.97272976346793193860385983372237710e+01L; /* 0x40048dd1, 0x816c1b3f, 0x24f540a6, 0x4cfe03cc */ +/* + * Domain [2.85715,9], range ~[-7.886e-37,7.918e-37]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-120 + */ +static const long double +rb0 = -9.86494292470008707171371994479162369e-3L, /* 0xbff84341, 0x239e86f4, 0x2f57e561, 0xf4469360 */ +rb1 = -1.57047326624110727986326503729442830L, /* 0xbfff920a, 0x8935bf73, 0x8803b894, 0x4656482d */ +rb2 = -1.03228196364885474342132255440317065e2L, /* 0xc0059ce9, 0xac4ed0ff, 0x2cff0ff7, 0x5e70d1ab */ +rb3 = -3.74000570653418227179358710865224376e3L, /* 0xc00ad380, 0x2ebf7835, 0xf6b07ed2, 0x861242f7 */ +rb4 = -8.35435477739098044190860390632813956e4L, /* 0xc00f4657, 0x8c3ae934, 0x3647d7b3, 0x80e76fb7 */ +rb5 = -1.21398672055223642118716640216747152e6L, /* 0xc0132862, 0x2b8761c8, 0x27d18c0f, 0x137c9463 */ +rb6 = -1.17669175877248796101665344873273970e7L, /* 0xc0166719, 0x0b2cea46, 0x81f14174, 0x11602ea5 */ +rb7 = -7.66108006086998253606773064264599615e7L, /* 0xc019243f, 0x3c26f4f0, 0x1cc05241, 0x3b953728 */ +rb8 = -3.32547117558141845968704725353130804e8L, /* 0xc01b3d24, 0x42d8ee26, 0x24ef6f3b, 0x604a8c65 */ +rb9 = -9.41561252426350696802167711221739746e8L, /* 0xc01cc0f8, 0xad23692a, 0x8ddb2310, 0xe9937145 */ +rb10 = -1.67157110805390944549427329626281063e9L, /* 0xc01d8e88, 0x9a903734, 0x09a55fa3, 0xd205c903 */ +rb11 = -1.74339631004410841337645931421427373e9L, /* 0xc01d9fa8, 0x77582d2a, 0xc183b8ab, 0x7e00cb05 */ +rb12 = -9.57655233596934915727573141357471703e8L, /* 0xc01cc8a5, 0x460cc685, 0xd0271fa0, 0x6a70e3da */ +rb13 = -2.26320062731339353035254704082495066e8L, /* 0xc01aafab, 0xd7d76721, 0xc9720e11, 0x6a8bd489 */ +rb14 = -1.42777302996263256686002973851837039e7L, /* 0xc016b3b8, 0xc499689f, 0x2b88d965, 0xc32414f9 */ +sb1 = 1.08512869705594540211033733976348506e2L, /* 0x4005b20d, 0x2db7528d, 0x00d20dcb, 0x858f6191 */ +sb2 = 5.02757713761390460534494530537572834e3L, /* 0x400b3a39, 0x3bf4a690, 0x3025d28d, 0xfd40a891 */ +sb3 = 1.31019107205412870059331647078328430e5L, /* 0x400fffcb, 0x1b71d05e, 0x3b28361d, 0x2a3c3690 */ +sb4 = 2.13021555152296846166736757455018030e6L, /* 0x40140409, 0x3c6984df, 0xc4491d7c, 0xb04aa08d */ +sb5 = 2.26649105281820861953868568619768286e7L, /* 0x401759d6, 0xce8736f0, 0xf28ad037, 0x2a901e0c */ +sb6 = 1.61071939490875921812318684143076081e8L, /* 0x401a3338, 0x686fb541, 0x6bd27d06, 0x4f95c9ac */ +sb7 = 7.66895673844301852676056750497991966e8L, /* 0x401c6daf, 0x31cec121, 0x54699126, 0x4bd9bf9e */ +sb8 = 2.41884450436101936436023058196042526e9L, /* 0x401e2059, 0x46b0b8d7, 0x87b64cbf, 0x78bc296d */ +sb9 = 4.92403055884071695093305291535107666e9L, /* 0x401f257e, 0xbe5ed739, 0x39e17346, 0xcadd2e55 */ +sb10 = 6.18627786365587486459633615573786416e9L, /* 0x401f70bb, 0x1be7a7e7, 0x6a45b5ae, 0x607c70f0 */ +sb11 = 4.45898013426501378097430226324743199e9L, /* 0x401f09c6, 0xa32643d7, 0xf1724620, 0x9ea46c32 */ +sb12 = 1.63006115763329848117160344854224975e9L, /* 0x401d84a3, 0x0996887f, 0x65a4f43b, 0x978c1d74 */ +sb13 = 2.39216717012421697446304015847567721e8L, /* 0x401ac845, 0x09a065c2, 0x30095da7, 0x9d72d6ae */ +sb14 = 7.84837329009278694937250358810225609e6L; /* 0x4015df06, 0xd5290e15, 0x63031fac, 0x4d9c894c */ +/* + * Domain [9,108], range ~[-5.324e-38,5.340e-38]: + * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-124 + */ +static const long double +rc0 = -9.86494292470008707171367567652935673e-3L, /* 0xbff84341, 0x239e86f4, 0x2f57e55b, 0x1aa10fd3 */ +rc1 = -1.26229447747315096406518846411562266L, /* 0xbfff4325, 0xbb1aab28, 0xda395cd9, 0xfb861c15 */ +rc2 = -6.13742634438922591780742637728666162e1L, /* 0xc004eafe, 0x7dd51cd8, 0x3c7c5928, 0x751e50cf */ +rc3 = -1.50455835478908280402912854338421517e3L, /* 0xc0097823, 0xbc15b9ab, 0x3d60745c, 0x523e80a5 */ +rc4 = -2.04415631865861549920184039902945685e4L, /* 0xc00d3f66, 0x40b3fc04, 0x5388f2ec, 0xb009e1f0 */ +rc5 = -1.57625662981714582753490610560037638e5L, /* 0xc01033dc, 0xd4dc95b6, 0xfd4da93b, 0xf355b4a9 */ +rc6 = -6.73473451616752528402917538033283794e5L, /* 0xc01248d8, 0x2e73a4f9, 0xcded49c5, 0xfa3bfeb7 */ +rc7 = -1.47433165421387483167186683764364857e6L, /* 0xc01367f1, 0xba77a8f7, 0xcfdd0dbb, 0x25d554b3 */ +rc8 = -1.38811981807868828563794929997744139e6L, /* 0xc01352e5, 0x7d16d9ad, 0xbbdcbf38, 0x38fbc5ea */ +rc9 = -3.59659700530831825640766479698155060e5L, /* 0xc0115f3a, 0xecd57f45, 0x21f8ad6c, 0x910a5958 */ +sc1 = 7.72730753022908298637508998072635696e1L, /* 0x40053517, 0xa10d52bc, 0xdabb55b6, 0xbd0328cd */ +sc2 = 2.36825757341694050500333261769082182e3L, /* 0x400a2808, 0x3e0a9b42, 0x82977842, 0x9c5de29e */ +sc3 = 3.72210540173034735352888847134073099e4L, /* 0x400e22ca, 0x1ba827ef, 0xac8390d7, 0x1fc39a41 */ +sc4 = 3.24136032646418336712461033591393412e5L, /* 0x40113c8a, 0x0216e100, 0xc59d1e44, 0xf0e68d9d */ +sc5 = 1.57836135851134393802505823370009175e6L, /* 0x40138157, 0x95bc7664, 0x17575961, 0xdbe58eeb */ +sc6 = 4.12881981392063738026679089714182355e6L, /* 0x4014f801, 0x9e82e8d2, 0xb8b3a70e, 0xfd84185d */ +sc7 = 5.24438427289213488410596395361544142e6L, /* 0x40154017, 0x81177109, 0x2aa6c3b0, 0x1f106625 */ +sc8 = 2.59909544563616121735963429710382149e6L, /* 0x40143d45, 0xbb90a9b1, 0x12bf9390, 0xa827a700 */ +sc9 = 2.80930665169282501639651995082335693e5L; /* 0x40111258, 0xaa92222e, 0xa97e3216, 0xa237fa6c */ + +long double +erfl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx, llx; + int32_t i; + uint16_t hx; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfl(nan)=nan */ + i = (hx>>15)<<1; + return (1-i)+one/x; /* erfl(+-inf)=+-1 */ + } + + ax = fabsl(x); + if(ax < 0.84375) { + if(ax < 0x1p-40L) { + if(ax < 0x1p-16373L) + return (8*x+efx8*x)/8; /* avoid spurious underflow */ + return x + efx*x; + } + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*(pp5+z*(pp6+z*(pp7+ + z*(pp8+z*pp9)))))))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*(qq6+z*(qq7+ + z*(qq8+z*qq9)))))))); + y = r/s; + return x + x*y; + } + if(ax < 1.25) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*(pa7+ + s*(pa8+s*(pa9+s*(pa10+s*pa11)))))))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*(qa7+ + s*(qa8+s*(qa9+s*(qa10+s*(qa11+s*qa12))))))))))); + if(x>=0) return (erx + P/Q); else return (-erx - P/Q); + } + if (ax >= 9) { /* inf>|x|>= 9 */ + if(x>=0) return (one-tiny); else return (tiny-one); + } + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*(ra9+s*(ra10+s*(ra11+s*(ra12+s*(ra13+s*(ra14+ + s*(ra15+s*ra16))))))))))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*(sa9+s*(sa10+s*(sa11+s*(sa12+s*(sa13+s*(sa14+ + s*(sa15+s*sa16))))))))))))))); + } else { /* |x| >= 2.85715 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*(rb7+ + s*(rb8+s*(rb9+s*(rb10+s*(rb11+s*(rb12+s*(rb13+ + s*rb14))))))))))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*(sb7+ + s*(sb8+s*(sb9+s*(sb10+s*(sb11+s*(sb12+s*(sb13+ + s*sb14))))))))))))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>=0) return (one-r/ax); else return (r/ax-one); +} + +long double +erfcl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx, llx; + uint16_t hx; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfcl(nan)=nan */ + /* erfcl(+-inf)=0,2 */ + return ((hx>>15)<<1)+one/x; + } + + ax = fabsl(x); + if(ax < 0.84375L) { + if(ax < 0x1p-34L) + return one-x; + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*(pp5+z*(pp6+z*(pp7+ + z*(pp8+z*pp9)))))))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*(qq6+z*(qq7+ + z*(qq8+z*qq9)))))))); + y = r/s; + if(ax < 0.25L) { /* x<1/4 */ + return one-(x+x*y); + } else { + r = x*y; + r += (x-half); + return half - r; + } + } + if(ax < 1.25L) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*(pa7+ + s*(pa8+s*(pa9+s*(pa10+s*pa11)))))))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*(qa7+ + s*(qa8+s*(qa9+s*(qa10+s*(qa11+s*qa12))))))))))); + if(x>=0) { + z = one-erx; return z - P/Q; + } else { + z = erx+P/Q; return one+z; + } + } + + if(ax < 108) { /* |x| < 108 */ + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*(ra9+s*(ra10+s*(ra11+s*(ra12+s*(ra13+s*(ra14+ + s*(ra15+s*ra16))))))))))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*(sa9+s*(sa10+s*(sa11+s*(sa12+s*(sa13+s*(sa14+ + s*(sa15+s*sa16))))))))))))))); + } else if(ax < 9) { + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*(rb7+ + s*(rb8+s*(rb9+s*(rb10+s*(rb11+s*(rb12+s*(rb13+ + s*rb14))))))))))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*(sb7+ + s*(sb8+s*(sb9+s*(sb10+s*(sb11+s*(sb12+s*(sb13+ + s*sb14))))))))))))); + } else { + if(x < -9) return two-tiny; /* x < -9 */ + R=rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+s*(rc6+s*(rc7+ + s*(rc8+s*rc9)))))))); + S=one+s*(sc1+s*(sc2+s*(sc3+s*(sc4+s*(sc5+s*(sc6+s*(sc7+ + s*(sc8+s*sc9)))))))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>0) return r/ax; else return two-r/ax; + } else { + if(x>0) return tiny*tiny; else return two-tiny; + } +} diff --git a/newlib/libm/ld128/s_exp2l.c b/newlib/libm/ld128/s_exp2l.c new file mode 100644 index 000000000..ee3d2c782 --- /dev/null +++ b/newlib/libm/ld128/s_exp2l.c @@ -0,0 +1,429 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2005-2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "fpmath.h" +#include "math.h" + +#define TBLBITS 7 +#define TBLSIZE (1 << TBLBITS) + +#define BIAS (LDBL_MAX_EXP - 1) +#define EXPMASK (BIAS + LDBL_MAX_EXP) + +static volatile long double + huge = 0x1p10000L, + twom10000 = 0x1p-10000L; + +static const long double + P1 = 0x1.62e42fefa39ef35793c7673007e6p-1L, + P2 = 0x1.ebfbdff82c58ea86f16b06ec9736p-3L, + P3 = 0x1.c6b08d704a0bf8b33a762bad3459p-5L, + P4 = 0x1.3b2ab6fba4e7729ccbbe0b4f3fc2p-7L, + P5 = 0x1.5d87fe78a67311071dee13fd11d9p-10L, + P6 = 0x1.430912f86c7876f4b663b23c5fe5p-13L; + +static const double + P7 = 0x1.ffcbfc588b041p-17, + P8 = 0x1.62c0223a5c7c7p-20, + P9 = 0x1.b52541ff59713p-24, + P10 = 0x1.e4cf56a391e22p-28, + redux = 0x1.8p112 / TBLSIZE; + +static const long double tbl[TBLSIZE] = { + 0x1.6a09e667f3bcc908b2fb1366dfeap-1L, + 0x1.6c012750bdabeed76a99800f4edep-1L, + 0x1.6dfb23c651a2ef220e2cbe1bc0d4p-1L, + 0x1.6ff7df9519483cf87e1b4f3e1e98p-1L, + 0x1.71f75e8ec5f73dd2370f2ef0b148p-1L, + 0x1.73f9a48a58173bd5c9a4e68ab074p-1L, + 0x1.75feb564267c8bf6e9aa33a489a8p-1L, + 0x1.780694fde5d3f619ae02808592a4p-1L, + 0x1.7a11473eb0186d7d51023f6ccb1ap-1L, + 0x1.7c1ed0130c1327c49334459378dep-1L, + 0x1.7e2f336cf4e62105d02ba1579756p-1L, + 0x1.80427543e1a11b60de67649a3842p-1L, + 0x1.82589994cce128acf88afab34928p-1L, + 0x1.8471a4623c7acce52f6b97c6444cp-1L, + 0x1.868d99b4492ec80e41d90ac2556ap-1L, + 0x1.88ac7d98a669966530bcdf2d4cc0p-1L, + 0x1.8ace5422aa0db5ba7c55a192c648p-1L, + 0x1.8cf3216b5448bef2aa1cd161c57ap-1L, + 0x1.8f1ae991577362b982745c72eddap-1L, + 0x1.9145b0b91ffc588a61b469f6b6a0p-1L, + 0x1.93737b0cdc5e4f4501c3f2540ae8p-1L, + 0x1.95a44cbc8520ee9b483695a0e7fep-1L, + 0x1.97d829fde4e4f8b9e920f91e8eb6p-1L, + 0x1.9a0f170ca07b9ba3109b8c467844p-1L, + 0x1.9c49182a3f0901c7c46b071f28dep-1L, + 0x1.9e86319e323231824ca78e64c462p-1L, + 0x1.a0c667b5de564b29ada8b8cabbacp-1L, + 0x1.a309bec4a2d3358c171f770db1f4p-1L, + 0x1.a5503b23e255c8b424491caf88ccp-1L, + 0x1.a799e1330b3586f2dfb2b158f31ep-1L, + 0x1.a9e6b5579fdbf43eb243bdff53a2p-1L, + 0x1.ac36bbfd3f379c0db966a3126988p-1L, + 0x1.ae89f995ad3ad5e8734d17731c80p-1L, + 0x1.b0e07298db66590842acdfc6fb4ep-1L, + 0x1.b33a2b84f15faf6bfd0e7bd941b0p-1L, + 0x1.b59728de559398e3881111648738p-1L, + 0x1.b7f76f2fb5e46eaa7b081ab53ff6p-1L, + 0x1.ba5b030a10649840cb3c6af5b74cp-1L, + 0x1.bcc1e904bc1d2247ba0f45b3d06cp-1L, + 0x1.bf2c25bd71e088408d7025190cd0p-1L, + 0x1.c199bdd85529c2220cb12a0916bap-1L, + 0x1.c40ab5fffd07a6d14df820f17deap-1L, + 0x1.c67f12e57d14b4a2137fd20f2a26p-1L, + 0x1.c8f6d9406e7b511acbc48805c3f6p-1L, + 0x1.cb720dcef90691503cbd1e949d0ap-1L, + 0x1.cdf0b555dc3f9c44f8958fac4f12p-1L, + 0x1.d072d4a07897b8d0f22f21a13792p-1L, + 0x1.d2f87080d89f18ade123989ea50ep-1L, + 0x1.d5818dcfba48725da05aeb66dff8p-1L, + 0x1.d80e316c98397bb84f9d048807a0p-1L, + 0x1.da9e603db3285708c01a5b6d480cp-1L, + 0x1.dd321f301b4604b695de3c0630c0p-1L, + 0x1.dfc97337b9b5eb968cac39ed284cp-1L, + 0x1.e264614f5a128a12761fa17adc74p-1L, + 0x1.e502ee78b3ff6273d130153992d0p-1L, + 0x1.e7a51fbc74c834b548b2832378a4p-1L, + 0x1.ea4afa2a490d9858f73a18f5dab4p-1L, + 0x1.ecf482d8e67f08db0312fb949d50p-1L, + 0x1.efa1bee615a27771fd21a92dabb6p-1L, + 0x1.f252b376bba974e8696fc3638f24p-1L, + 0x1.f50765b6e4540674f84b762861a6p-1L, + 0x1.f7bfdad9cbe138913b4bfe72bd78p-1L, + 0x1.fa7c1819e90d82e90a7e74b26360p-1L, + 0x1.fd3c22b8f71f10975ba4b32bd006p-1L, + 0x1.0000000000000000000000000000p+0L, + 0x1.0163da9fb33356d84a66ae336e98p+0L, + 0x1.02c9a3e778060ee6f7caca4f7a18p+0L, + 0x1.04315e86e7f84bd738f9a20da442p+0L, + 0x1.059b0d31585743ae7c548eb68c6ap+0L, + 0x1.0706b29ddf6ddc6dc403a9d87b1ep+0L, + 0x1.0874518759bc808c35f25d942856p+0L, + 0x1.09e3ecac6f3834521e060c584d5cp+0L, + 0x1.0b5586cf9890f6298b92b7184200p+0L, + 0x1.0cc922b7247f7407b705b893dbdep+0L, + 0x1.0e3ec32d3d1a2020742e4f8af794p+0L, + 0x1.0fb66affed31af232091dd8a169ep+0L, + 0x1.11301d0125b50a4ebbf1aed9321cp+0L, + 0x1.12abdc06c31cbfb92bad324d6f84p+0L, + 0x1.1429aaea92ddfb34101943b2588ep+0L, + 0x1.15a98c8a58e512480d573dd562aep+0L, + 0x1.172b83c7d517adcdf7c8c50eb162p+0L, + 0x1.18af9388c8de9bbbf70b9a3c269cp+0L, + 0x1.1a35beb6fcb753cb698f692d2038p+0L, + 0x1.1bbe084045cd39ab1e72b442810ep+0L, + 0x1.1d4873168b9aa7805b8028990be8p+0L, + 0x1.1ed5022fcd91cb8819ff61121fbep+0L, + 0x1.2063b88628cd63b8eeb0295093f6p+0L, + 0x1.21f49917ddc962552fd29294bc20p+0L, + 0x1.2387a6e75623866c1fadb1c159c0p+0L, + 0x1.251ce4fb2a63f3582ab7de9e9562p+0L, + 0x1.26b4565e27cdd257a673281d3068p+0L, + 0x1.284dfe1f5638096cf15cf03c9fa0p+0L, + 0x1.29e9df51fdee12c25d15f5a25022p+0L, + 0x1.2b87fd0dad98ffddea46538fca24p+0L, + 0x1.2d285a6e4030b40091d536d0733ep+0L, + 0x1.2ecafa93e2f5611ca0f45d5239a4p+0L, + 0x1.306fe0a31b7152de8d5a463063bep+0L, + 0x1.32170fc4cd8313539cf1c3009330p+0L, + 0x1.33c08b26416ff4c9c8610d96680ep+0L, + 0x1.356c55f929ff0c94623476373be4p+0L, + 0x1.371a7373aa9caa7145502f45452ap+0L, + 0x1.38cae6d05d86585a9cb0d9bed530p+0L, + 0x1.3a7db34e59ff6ea1bc9299e0a1fep+0L, + 0x1.3c32dc313a8e484001f228b58cf0p+0L, + 0x1.3dea64c12342235b41223e13d7eep+0L, + 0x1.3fa4504ac801ba0bf701aa417b9cp+0L, + 0x1.4160a21f72e29f84325b8f3dbacap+0L, + 0x1.431f5d950a896dc704439410b628p+0L, + 0x1.44e086061892d03136f409df0724p+0L, + 0x1.46a41ed1d005772512f459229f0ap+0L, + 0x1.486a2b5c13cd013c1a3b69062f26p+0L, + 0x1.4a32af0d7d3de672d8bcf46f99b4p+0L, + 0x1.4bfdad5362a271d4397afec42e36p+0L, + 0x1.4dcb299fddd0d63b36ef1a9e19dep+0L, + 0x1.4f9b2769d2ca6ad33d8b69aa0b8cp+0L, + 0x1.516daa2cf6641c112f52c84d6066p+0L, + 0x1.5342b569d4f81df0a83c49d86bf4p+0L, + 0x1.551a4ca5d920ec52ec620243540cp+0L, + 0x1.56f4736b527da66ecb004764e61ep+0L, + 0x1.58d12d497c7fd252bc2b7343d554p+0L, + 0x1.5ab07dd48542958c93015191e9a8p+0L, + 0x1.5c9268a5946b701c4b1b81697ed4p+0L, + 0x1.5e76f15ad21486e9be4c20399d12p+0L, + 0x1.605e1b976dc08b076f592a487066p+0L, + 0x1.6247eb03a5584b1f0fa06fd2d9eap+0L, + 0x1.6434634ccc31fc76f8714c4ee122p+0L, + 0x1.66238825522249127d9e29b92ea2p+0L, + 0x1.68155d44ca973081c57227b9f69ep+0L, +}; + +static const float eps[TBLSIZE] = { + -0x1.5c50p-101, + -0x1.5d00p-106, + 0x1.8e90p-102, + -0x1.5340p-103, + 0x1.1bd0p-102, + -0x1.4600p-105, + -0x1.7a40p-104, + 0x1.d590p-102, + -0x1.d590p-101, + 0x1.b100p-103, + -0x1.0d80p-105, + 0x1.6b00p-103, + -0x1.9f00p-105, + 0x1.c400p-103, + 0x1.e120p-103, + -0x1.c100p-104, + -0x1.9d20p-103, + 0x1.a800p-108, + 0x1.4c00p-106, + -0x1.9500p-106, + 0x1.6900p-105, + -0x1.29d0p-100, + 0x1.4c60p-103, + 0x1.13a0p-102, + -0x1.5b60p-103, + -0x1.1c40p-103, + 0x1.db80p-102, + 0x1.91a0p-102, + 0x1.dc00p-105, + 0x1.44c0p-104, + 0x1.9710p-102, + 0x1.8760p-103, + -0x1.a720p-103, + 0x1.ed20p-103, + -0x1.49c0p-102, + -0x1.e000p-111, + 0x1.86a0p-103, + 0x1.2b40p-103, + -0x1.b400p-108, + 0x1.1280p-99, + -0x1.02d8p-102, + -0x1.e3d0p-103, + -0x1.b080p-105, + -0x1.f100p-107, + -0x1.16c0p-105, + -0x1.1190p-103, + -0x1.a7d2p-100, + 0x1.3450p-103, + -0x1.67c0p-105, + 0x1.4b80p-104, + -0x1.c4e0p-103, + 0x1.6000p-108, + -0x1.3f60p-105, + 0x1.93f0p-104, + 0x1.5fe0p-105, + 0x1.6f80p-107, + -0x1.7600p-106, + 0x1.21e0p-106, + -0x1.3a40p-106, + -0x1.40c0p-104, + -0x1.9860p-105, + -0x1.5d40p-108, + -0x1.1d70p-106, + 0x1.2760p-105, + 0x0.0000p+0, + 0x1.21e2p-104, + -0x1.9520p-108, + -0x1.5720p-106, + -0x1.4810p-106, + -0x1.be00p-109, + 0x1.0080p-105, + -0x1.5780p-108, + -0x1.d460p-105, + -0x1.6140p-105, + 0x1.4630p-104, + 0x1.ad50p-103, + 0x1.82e0p-105, + 0x1.1d3cp-101, + 0x1.6100p-107, + 0x1.ec30p-104, + 0x1.f200p-108, + 0x1.0b40p-103, + 0x1.3660p-102, + 0x1.d9d0p-103, + -0x1.02d0p-102, + 0x1.b070p-103, + 0x1.b9c0p-104, + -0x1.01c0p-103, + -0x1.dfe0p-103, + 0x1.1b60p-104, + -0x1.ae94p-101, + -0x1.3340p-104, + 0x1.b3d8p-102, + -0x1.6e40p-105, + -0x1.3670p-103, + 0x1.c140p-104, + 0x1.1840p-101, + 0x1.1ab0p-102, + -0x1.a400p-104, + 0x1.1f00p-104, + -0x1.7180p-103, + 0x1.4ce0p-102, + 0x1.9200p-107, + -0x1.54c0p-103, + 0x1.1b80p-105, + -0x1.1828p-101, + 0x1.5720p-102, + -0x1.a060p-100, + 0x1.9160p-102, + 0x1.a280p-104, + 0x1.3400p-107, + 0x1.2b20p-102, + 0x1.7800p-108, + 0x1.cfd0p-101, + 0x1.2ef0p-102, + -0x1.2760p-99, + 0x1.b380p-104, + 0x1.0048p-101, + -0x1.60b0p-102, + 0x1.a1ccp-100, + -0x1.a640p-104, + -0x1.08a0p-101, + 0x1.7e60p-102, + 0x1.22c0p-103, + -0x1.7200p-106, + 0x1.f0f0p-102, + 0x1.eb4ep-99, + 0x1.c6e0p-103, +}; + +/* + * exp2l(x): compute the base 2 exponential of x + * + * Accuracy: Peak error < 0.502 ulp. + * + * Method: (accurate tables) + * + * Reduce x: + * x = 2**k + y, for integer k and |y| <= 1/2. + * Thus we have exp2(x) = 2**k * exp2(y). + * + * Reduce y: + * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE. + * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]), + * with |z - eps[i]| <= 2**-8 + 2**-98 for the table used. + * + * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via + * a degree-10 minimax polynomial with maximum error under 2**-120. + * The values in exp2t[] and eps[] are chosen such that + * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such + * that exp2t[i] is accurate to 2**-122. + * + * Note that the range of i is +-TBLSIZE/2, so we actually index the tables + * by i0 = i + TBLSIZE/2. + * + * This method is due to Gal, with many details due to Gal and Bachelis: + * + * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library + * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991). + */ +long double +exp2l(long double x) +{ + union IEEEl2bits u, v; + long double r, t, twopk, twopkp10000, z; + uint32_t hx, ix, i0; + int k; + + u.e = x; + + /* Filter out exceptional cases. */ + hx = u.xbits.expsign; + ix = hx & EXPMASK; + if (ix >= BIAS + 14) { /* |x| >= 16384 */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (u.xbits.manh != 0 + || u.xbits.manl != 0 + || (hx & 0x8000) == 0) + return (x + x); /* x is NaN or +Inf */ + else + return (0.0); /* x is -Inf */ + } + if (x >= 16384) + return (huge * huge); /* overflow */ + if (x <= -16495) + return (twom10000 * twom10000); /* underflow */ + } else if (ix <= BIAS - 115) { /* |x| < 0x1p-115 */ + return (1.0 + x); + } + + /* + * Reduce x, computing z, i0, and k. The low bits of x + redux + * contain the 16-bit integer part of the exponent (k) followed by + * TBLBITS fractional bits (i0). We use bit tricks to extract these + * as integers, then set z to the remainder. + * + * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8. + * Then the low-order word of x + redux is 0x000abc12, + * We split this into k = 0xabc and i0 = 0x12 (adjusted to + * index into the table), then we compute z = 0x0.003456p0. + * + * XXX If the exponent is negative, the computation of k depends on + * '>>' doing sign extension. + */ + u.e = x + redux; + i0 = (u.bits.manl & 0xffffffff) + TBLSIZE / 2; + k = (int)i0 >> TBLBITS; + i0 = i0 & (TBLSIZE - 1); + u.e -= redux; + z = x - u.e; + v.xbits.manh = 0; + v.xbits.manl = 0; + if (k >= LDBL_MIN_EXP) { + v.xbits.expsign = LDBL_MAX_EXP - 1 + k; + twopk = v.e; + } else { + v.xbits.expsign = LDBL_MAX_EXP - 1 + k + 10000; + twopkp10000 = v.e; + } + + /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */ + t = tbl[i0]; /* exp2t[i0] */ + z -= eps[i0]; /* eps[i0] */ + r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + z * (P6 + + z * (P7 + z * (P8 + z * (P9 + z * P10))))))))); + + /* Scale by 2**k. */ + if(k >= LDBL_MIN_EXP) { + if (k == LDBL_MAX_EXP) + return (r * 2.0 * 0x1p16383L); + return (r * twopk); + } else { + return (r * twopkp10000 * twom10000); + } +} diff --git a/newlib/libm/ld128/s_expl.c b/newlib/libm/ld128/s_expl.c new file mode 100644 index 000000000..5b786af49 --- /dev/null +++ b/newlib/libm/ld128/s_expl.c @@ -0,0 +1,326 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. + */ + +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double +huge = 0x1p10000L, +tiny = 0x1p-10000L; + +static const long double +twom10000 = 0x1p-10000L; + +static const long double +/* log(2**16384 - 0.5) rounded towards zero: */ +/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ +o_threshold = 11356.523406294143949491931077970763428L, +/* log(2**(-16381-64-1)) rounded towards zero: */ +u_threshold = -11433.462743336297878837243843452621503L; + +long double +expl(long double x) +{ + union IEEEl2bits u; + long double hi, lo, t, twopk; + int k; + uint16_t hx, ix; + + DOPRINT_START(&x); + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf or -NaN */ + RETURNP(-1 / x); + RETURNP(x + x); /* x is +Inf or +NaN */ + } + if (x > o_threshold) + RETURNP(huge * huge); + if (x < u_threshold) + RETURNP(tiny * tiny); + } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ + RETURN2P(1, x); /* 1 with inexact iff x != 0 */ + } + + ENTERI(); + + twopk = 1; + __k_expl(x, &hi, &lo, &k); + t = SUM2P(hi, lo); + + /* Scale by 2**k. */ + /* + * XXX sparc64 multiplication was so slow that scalbnl() is faster, + * but performance on aarch64 and riscv hasn't yet been quantified. + */ + if (k >= LDBL_MIN_EXP) { + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L); + SET_LDBL_EXPSIGN(twopk, BIAS + k); + RETURNI(t * twopk); + } else { + SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); + RETURNI(t * twopk * twom10000); + } +} + +/* + * Our T1 and T2 are chosen to be approximately the points where method + * A and method B have the same accuracy. Tang's T1 and T2 are the + * points where method A's accuracy changes by a full bit. For Tang, + * this drop in accuracy makes method A immediately less accurate than + * method B, but our larger INTERVALS makes method A 2 bits more + * accurate so it remains the most accurate method significantly + * closer to the origin despite losing the full bit in our extended + * range for it. + * + * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. + * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear + * in both subintervals, so set T3 = 2**-5, which places the condition + * into the [T1, T3] interval. + * + * XXX we now do this more to (partially) balance the number of terms + * in the C and D polys than to avoid checking the condition in both + * intervals. + * + * XXX these micro-optimizations are excessive. + */ +static const double +T1 = -0.1659, /* ~-30.625/128 * log(2) */ +T2 = 0.1659, /* ~30.625/128 * log(2) */ +T3 = 0.03125; + +/* + * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 + * + * XXX none of the long double C or D coeffs except C10 is correctly printed. + * If you re-print their values in %.35Le format, the result is always + * different. For example, the last 2 digits in C3 should be 59, not 67. + * 67 is apparently from rounding an extra-precision value to 36 decimal + * places. + */ +static const long double +C3 = 1.66666666666666666666666666666666667e-1L, +C4 = 4.16666666666666666666666666666666645e-2L, +C5 = 8.33333333333333333333333333333371638e-3L, +C6 = 1.38888888888888888888888888891188658e-3L, +C7 = 1.98412698412698412698412697235950394e-4L, +C8 = 2.48015873015873015873015112487849040e-5L, +C9 = 2.75573192239858906525606685484412005e-6L, +C10 = 2.75573192239858906612966093057020362e-7L, +C11 = 2.50521083854417203619031960151253944e-8L, +C12 = 2.08767569878679576457272282566520649e-9L, +C13 = 1.60590438367252471783548748824255707e-10L; + +/* + * XXX this has 1 more coeff than needed. + * XXX can start the double coeffs but not the double mults at C10. + * With my coeffs (C10-C17 double; s = best_s): + * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 + */ +static const double +C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ +C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ +C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ +C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ +C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ + +/* + * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 + */ +static const long double +D3 = 1.66666666666666666666666666666682245e-1L, +D4 = 4.16666666666666666666666666634228324e-2L, +D5 = 8.33333333333333333333333364022244481e-3L, +D6 = 1.38888888888888888888887138722762072e-3L, +D7 = 1.98412698412698412699085805424661471e-4L, +D8 = 2.48015873015873015687993712101479612e-5L, +D9 = 2.75573192239858944101036288338208042e-6L, +D10 = 2.75573192239853161148064676533754048e-7L, +D11 = 2.50521083855084570046480450935267433e-8L, +D12 = 2.08767569819738524488686318024854942e-9L, +D13 = 1.60590442297008495301927448122499313e-10L; + +/* + * XXX this has 1 more coeff than needed. + * XXX can start the double coeffs but not the double mults at D11. + * With my coeffs (D11-D16 double): + * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 + */ +static const double +D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ +D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ +D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ +D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ + +long double +expm1l(long double x) +{ + union IEEEl2bits u, v; + long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; + long double x_lo, x2; + double dr, dx, fn, r2; + int k, n, n2; + uint16_t hx, ix; + + DOPRINT_START(&x); + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf or -NaN */ + RETURNP(-1 / x - 1); + RETURNP(x + x); /* x is +Inf or +NaN */ + } + if (x > o_threshold) + RETURNP(huge * huge); + /* + * expm1l() never underflows, but it must avoid + * unrepresentable large negative exponents. We used a + * much smaller threshold for large |x| above than in + * expl() so as to handle not so large negative exponents + * in the same way as large ones here. + */ + if (hx & 0x8000) /* x <= -128 */ + RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */ + } + + ENTERI(); + + if (T1 < x && x < T2) { + x2 = x * x; + dx = x; + + if (x < T3) { + if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ + /* x (rounded) with inexact if x != 0: */ + RETURNPI(x == 0 ? x : + (0x1p200 * x + fabsl(x)) * 0x1p-200); + } + q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + + x * (C7 + x * (C8 + x * (C9 + x * (C10 + + x * (C11 + x * (C12 + x * (C13 + + dx * (C14 + dx * (C15 + dx * (C16 + + dx * (C17 + dx * C18)))))))))))))); + } else { + q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + + x * (D7 + x * (D8 + x * (D9 + x * (D10 + + x * (D11 + x * (D12 + x * (D13 + + dx * (D14 + dx * (D15 + dx * (D16 + + dx * D17))))))))))))); + } + + x_hi = (float)x; + x_lo = x - x_hi; + hx2_hi = x_hi * x_hi / 2; + hx2_lo = x_lo * (x + x_hi) / 2; + if (ix >= BIAS - 7) + RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); + else + RETURN2PI(x, hx2_lo + q + hx2_hi); + } + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + fn = rnint((double)x * INV_L); + n = irint(fn); + n2 = (unsigned)n % INTERVALS; + k = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Prepare scale factor. */ + v.e = 1; + v.xbits.expsign = BIAS + k; + twopk = v.e; + + /* + * Evaluate lower terms of + * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). + */ + dr = r; + q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + + dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); + + t = tbl[n2].lo + tbl[n2].hi; + + if (k == 0) { + t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); + RETURNI(t); + } + if (k == -1) { + t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); + RETURNI(t / 2); + } + if (k < -7) { + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); + RETURNI(t * twopk - 1); + } + if (k > 2 * LDBL_MANT_DIG - 1) { + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L - 1); + RETURNI(t * twopk - 1); + } + + v.xbits.expsign = BIAS - k; + twomk = v.e; + + if (k > LDBL_MANT_DIG - 1) + t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); + else + t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); + RETURNI(t * twopk); +} diff --git a/newlib/libm/ld128/s_logl.c b/newlib/libm/ld128/s_logl.c new file mode 100644 index 000000000..4774a271e --- /dev/null +++ b/newlib/libm/ld128/s_logl.c @@ -0,0 +1,740 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007-2013 Bruce D. Evans + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +/** + * Implementation of the natural logarithm of x for 128-bit format. + * + * First decompose x into its base 2 representation: + * + * log(x) = log(X * 2**k), where X is in [1, 2) + * = log(X) + k * log(2). + * + * Let X = X_i + e, where X_i is the center of one of the intervals + * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256) + * and X is in this interval. Then + * + * log(X) = log(X_i + e) + * = log(X_i * (1 + e / X_i)) + * = log(X_i) + log(1 + e / X_i). + * + * The values log(X_i) are tabulated below. Let d = e / X_i and use + * + * log(1 + d) = p(d) + * + * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of + * suitably high degree. + * + * To get sufficiently small roundoff errors, k * log(2), log(X_i), and + * sometimes (if |k| is not large) the first term in p(d) must be evaluated + * and added up in extra precision. Extra precision is not needed for the + * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final + * error is controlled mainly by the error in the second term in p(d). The + * error in this term itself is at most 0.5 ulps from the d*d operation in + * it. The error in this term relative to the first term is thus at most + * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of + * at most twice this at the point of the final rounding step. Thus the + * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive + * testing of a float variant of this function showed a maximum final error + * of 0.5008 ulps. Non-exhaustive testing of a double variant of this + * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256). + * + * We made the maximum of |d| (and thus the total relative error and the + * degree of p(d)) small by using a large number of intervals. Using + * centers of intervals instead of endpoints reduces this maximum by a + * factor of 2 for a given number of intervals. p(d) is special only + * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen + * naturally. The most accurate minimax polynomial of a given degree might + * be different, but then we wouldn't want it since we would have to do + * extra work to avoid roundoff error (especially for P0*d instead of d). + */ + +#ifdef DEBUG +#include +#include +#endif + +#include "fpmath.h" +#include "math.h" +#ifndef NO_STRUCT_RETURN +#define STRUCT_RETURN +#endif +#include "math_private.h" + +#if !defined(NO_UTAB) && !defined(NO_UTABL) +#define USE_UTAB +#endif + +/* + * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]: + * |log(1 + d)/d - p(d)| < 2**-122.7 + */ +static const long double +P2 = -0.5L, +P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */ +P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */ +P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */ +P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */ +P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */ +P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */ +/* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */ +static const double +P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */ +P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */ +P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */ +P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */ +P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */ +P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */ + +static volatile const double zero = 0; + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define TSIZE (INTERVALS + 1) +#define G(i) (T[(i)].G) +#define F_hi(i) (T[(i)].F_hi) +#define F_lo(i) (T[(i)].F_lo) +#define ln2_hi F_hi(TSIZE - 1) +#define ln2_lo F_lo(TSIZE - 1) +#define E(i) (U[(i)].E) +#define H(i) (U[(i)].H) + +static const struct { + float G; /* 1/(1 + i/128) rounded to 8/9 bits */ + float F_hi; /* log(1 / G_i) rounded (see below) */ + /* The compiler will insert 8 bytes of padding here. */ + long double F_lo; /* next 113 bits for log(1 / G_i) */ +} T[TSIZE] = { + /* + * ln2_hi and each F_hi(i) are rounded to a number of bits that + * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. + * + * The last entry (for X just below 2) is used to define ln2_hi + * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly + * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. + * This is needed for accuracy when x is just below 1. (To avoid + * special cases, such x are "reduced" strangely to X just below + * 2 and dk = -1, and then the exact cancellation is needed + * because any the error from any non-exactness would be too + * large). + * + * The relevant range of dk is [-16445, 16383]. The maximum number + * of bits in F_hi(i) that works is very dependent on i but has + * a minimum of 93. We only need about 12 bits in F_hi(i) for + * it to provide enough extra precision. + * + * We round F_hi(i) to 24 bits so that it can have type float, + * mainly to minimize the size of the table. Using all 24 bits + * in a float for it automatically satisfies the above constraints. + */ + 0x800000.0p-23, 0, 0, + 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L, + 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L, + 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L, + 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L, + 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L, + 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L, + 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L, + 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L, + 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L, + 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L, + 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L, + 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L, + 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L, + 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L, + 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L, + 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L, + 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L, + 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L, + 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L, + 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L, + 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L, + 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L, + 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L, + 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L, + 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L, + 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L, + 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L, + 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L, + 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L, + 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L, + 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L, + 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L, + 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L, + 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L, + 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L, + 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L, + 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L, + 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L, + 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L, + 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L, + 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L, + 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L, + 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L, + 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L, + 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L, + 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L, + 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L, + 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L, + 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L, + 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L, + 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L, + 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L, + 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L, + 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L, + 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L, + 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L, + 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L, + 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L, + 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L, + 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L, + 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L, + 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L, + 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L, + 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L, + 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L, + 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L, + 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L, + 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L, + 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L, + 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L, + 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L, + 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L, + 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L, + 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L, + 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L, + 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L, + 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L, + 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L, + 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L, + 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L, + 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L, + 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L, + 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L, + 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L, + 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L, + 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L, + 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L, + 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L, + 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L, + 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L, + 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L, + 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L, + 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L, + 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L, + 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L, + 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L, + 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L, + 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L, + 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L, + 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L, + 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L, + 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L, + 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L, + 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L, + 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L, + 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L, + 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L, + 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L, + 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L, + 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L, + 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L, + 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L, + 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L, + 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L, + 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L, + 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L, + 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L, + 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L, + 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L, + 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L, + 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L, + 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L, + 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L, + 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L, + 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L, + 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L, + 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L, + 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L, +}; + +#ifdef USE_UTAB +static const struct { + float H; /* 1 + i/INTERVALS (exact) */ + float E; /* H(i) * G(i) - 1 (exact) */ +} U[TSIZE] = { + 0x800000.0p-23, 0, + 0x810000.0p-23, -0x800000.0p-37, + 0x820000.0p-23, -0x800000.0p-35, + 0x830000.0p-23, -0x900000.0p-34, + 0x840000.0p-23, -0x800000.0p-33, + 0x850000.0p-23, -0xc80000.0p-33, + 0x860000.0p-23, -0xa00000.0p-36, + 0x870000.0p-23, 0x940000.0p-33, + 0x880000.0p-23, 0x800000.0p-35, + 0x890000.0p-23, -0xc80000.0p-34, + 0x8a0000.0p-23, 0xe00000.0p-36, + 0x8b0000.0p-23, 0x900000.0p-33, + 0x8c0000.0p-23, -0x800000.0p-35, + 0x8d0000.0p-23, -0xe00000.0p-33, + 0x8e0000.0p-23, 0x880000.0p-33, + 0x8f0000.0p-23, -0xa80000.0p-34, + 0x900000.0p-23, -0x800000.0p-35, + 0x910000.0p-23, 0x800000.0p-37, + 0x920000.0p-23, 0x900000.0p-35, + 0x930000.0p-23, 0xd00000.0p-35, + 0x940000.0p-23, 0xe00000.0p-35, + 0x950000.0p-23, 0xc00000.0p-35, + 0x960000.0p-23, 0xe00000.0p-36, + 0x970000.0p-23, -0x800000.0p-38, + 0x980000.0p-23, -0xc00000.0p-35, + 0x990000.0p-23, -0xd00000.0p-34, + 0x9a0000.0p-23, 0x880000.0p-33, + 0x9b0000.0p-23, 0xe80000.0p-35, + 0x9c0000.0p-23, -0x800000.0p-35, + 0x9d0000.0p-23, 0xb40000.0p-33, + 0x9e0000.0p-23, 0x880000.0p-34, + 0x9f0000.0p-23, -0xe00000.0p-35, + 0xa00000.0p-23, 0x800000.0p-33, + 0xa10000.0p-23, -0x900000.0p-36, + 0xa20000.0p-23, -0xb00000.0p-33, + 0xa30000.0p-23, -0xa00000.0p-36, + 0xa40000.0p-23, 0x800000.0p-33, + 0xa50000.0p-23, -0xf80000.0p-35, + 0xa60000.0p-23, 0x880000.0p-34, + 0xa70000.0p-23, -0x900000.0p-33, + 0xa80000.0p-23, -0x800000.0p-35, + 0xa90000.0p-23, 0x900000.0p-34, + 0xaa0000.0p-23, 0xa80000.0p-33, + 0xab0000.0p-23, -0xac0000.0p-34, + 0xac0000.0p-23, -0x800000.0p-37, + 0xad0000.0p-23, 0xf80000.0p-35, + 0xae0000.0p-23, 0xf80000.0p-34, + 0xaf0000.0p-23, -0xac0000.0p-33, + 0xb00000.0p-23, -0x800000.0p-33, + 0xb10000.0p-23, -0xb80000.0p-34, + 0xb20000.0p-23, -0x800000.0p-34, + 0xb30000.0p-23, -0xb00000.0p-35, + 0xb40000.0p-23, -0x800000.0p-35, + 0xb50000.0p-23, -0xe00000.0p-36, + 0xb60000.0p-23, -0x800000.0p-35, + 0xb70000.0p-23, -0xb00000.0p-35, + 0xb80000.0p-23, -0x800000.0p-34, + 0xb90000.0p-23, -0xb80000.0p-34, + 0xba0000.0p-23, -0x800000.0p-33, + 0xbb0000.0p-23, -0xac0000.0p-33, + 0xbc0000.0p-23, 0x980000.0p-33, + 0xbd0000.0p-23, 0xbc0000.0p-34, + 0xbe0000.0p-23, 0xe00000.0p-36, + 0xbf0000.0p-23, -0xb80000.0p-35, + 0xc00000.0p-23, -0x800000.0p-33, + 0xc10000.0p-23, 0xa80000.0p-33, + 0xc20000.0p-23, 0x900000.0p-34, + 0xc30000.0p-23, -0x800000.0p-35, + 0xc40000.0p-23, -0x900000.0p-33, + 0xc50000.0p-23, 0x820000.0p-33, + 0xc60000.0p-23, 0x800000.0p-38, + 0xc70000.0p-23, -0x820000.0p-33, + 0xc80000.0p-23, 0x800000.0p-33, + 0xc90000.0p-23, -0xa00000.0p-36, + 0xca0000.0p-23, -0xb00000.0p-33, + 0xcb0000.0p-23, 0x840000.0p-34, + 0xcc0000.0p-23, -0xd00000.0p-34, + 0xcd0000.0p-23, 0x800000.0p-33, + 0xce0000.0p-23, -0xe00000.0p-35, + 0xcf0000.0p-23, 0xa60000.0p-33, + 0xd00000.0p-23, -0x800000.0p-35, + 0xd10000.0p-23, 0xb40000.0p-33, + 0xd20000.0p-23, -0x800000.0p-35, + 0xd30000.0p-23, 0xaa0000.0p-33, + 0xd40000.0p-23, -0xe00000.0p-35, + 0xd50000.0p-23, 0x880000.0p-33, + 0xd60000.0p-23, -0xd00000.0p-34, + 0xd70000.0p-23, 0x9c0000.0p-34, + 0xd80000.0p-23, -0xb00000.0p-33, + 0xd90000.0p-23, -0x800000.0p-38, + 0xda0000.0p-23, 0xa40000.0p-33, + 0xdb0000.0p-23, -0xdc0000.0p-34, + 0xdc0000.0p-23, 0xc00000.0p-35, + 0xdd0000.0p-23, 0xca0000.0p-33, + 0xde0000.0p-23, -0xb80000.0p-34, + 0xdf0000.0p-23, 0xd00000.0p-35, + 0xe00000.0p-23, 0xc00000.0p-33, + 0xe10000.0p-23, -0xf40000.0p-34, + 0xe20000.0p-23, 0x800000.0p-37, + 0xe30000.0p-23, 0x860000.0p-33, + 0xe40000.0p-23, -0xc80000.0p-33, + 0xe50000.0p-23, -0xa80000.0p-34, + 0xe60000.0p-23, 0xe00000.0p-36, + 0xe70000.0p-23, 0x880000.0p-33, + 0xe80000.0p-23, -0xe00000.0p-33, + 0xe90000.0p-23, -0xfc0000.0p-34, + 0xea0000.0p-23, -0x800000.0p-35, + 0xeb0000.0p-23, 0xe80000.0p-35, + 0xec0000.0p-23, 0x900000.0p-33, + 0xed0000.0p-23, 0xe20000.0p-33, + 0xee0000.0p-23, -0xac0000.0p-33, + 0xef0000.0p-23, -0xc80000.0p-34, + 0xf00000.0p-23, -0x800000.0p-35, + 0xf10000.0p-23, 0x800000.0p-35, + 0xf20000.0p-23, 0xb80000.0p-34, + 0xf30000.0p-23, 0x940000.0p-33, + 0xf40000.0p-23, 0xc80000.0p-33, + 0xf50000.0p-23, -0xf20000.0p-33, + 0xf60000.0p-23, -0xc80000.0p-33, + 0xf70000.0p-23, -0xa20000.0p-33, + 0xf80000.0p-23, -0x800000.0p-33, + 0xf90000.0p-23, -0xc40000.0p-34, + 0xfa0000.0p-23, -0x900000.0p-34, + 0xfb0000.0p-23, -0xc80000.0p-35, + 0xfc0000.0p-23, -0x800000.0p-35, + 0xfd0000.0p-23, -0x900000.0p-36, + 0xfe0000.0p-23, -0x800000.0p-37, + 0xff0000.0p-23, -0x800000.0p-39, + 0x800000.0p-22, 0, +}; +#endif /* USE_UTAB */ + +#ifdef STRUCT_RETURN +#define RETURN1(rp, v) do { \ + (rp)->hi = (v); \ + (rp)->lo_set = 0; \ + return; \ +} while (0) + +#define RETURN2(rp, h, l) do { \ + (rp)->hi = (h); \ + (rp)->lo = (l); \ + (rp)->lo_set = 1; \ + return; \ +} while (0) + +struct ld { + long double hi; + long double lo; + int lo_set; +}; +#else +#define RETURN1(rp, v) RETURNF(v) +#define RETURN2(rp, h, l) RETURNI((h) + (l)) +#endif + +#ifdef STRUCT_RETURN +static inline __always_inline void +k_logl(long double x, struct ld *rp) +#else +long double +logl(long double x) +#endif +{ + long double d, val_hi, val_lo; + double dd, dk; + uint64_t lx, llx; + int i, k; + uint16_t hx; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + k = -16383; +#if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ + if (x == 1) + RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ +#endif + if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ + if (((hx & 0x7fff) | lx | llx) == 0) + RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ + if (hx != 0) + /* log(neg or NaN) = qNaN: */ + RETURN1(rp, (x - x) / zero); + x *= 0x1.0p113; /* subnormal; scale up x */ + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + k = -16383 - 113; + } else if (hx >= 0x7fff) + RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ +#ifndef STRUCT_RETURN + ENTERI(); +#endif + k += hx; + dk = k; + + /* Scale x to be in [1, 2). */ + SET_LDBL_EXPSIGN(x, 0x3fff); + + /* 0 <= i <= INTERVALS: */ +#define L2I (49 - LOG2_INTERVALS) + i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); + + /* + * -0.005280 < d < 0.004838. In particular, the infinite- + * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits + * ensures that d is representable without extra precision for + * this bound on |d| (since when this calculation is expressed + * as x*G(i)-1, the multiplication needs as many extra bits as + * G(i) has and the subtraction cancels 8 bits). But for + * most i (107 cases out of 129), the infinite-precision |d| + * is <= 2**-8. G(i) is rounded to 9 bits for such i to give + * better accuracy (this works by improving the bound on |d|, + * which in turn allows rounding to 9 bits in more cases). + * This is only important when the original x is near 1 -- it + * lets us avoid using a special method to give the desired + * accuracy for such x. + */ + if (0) + d = x * G(i) - 1; + else { +#ifdef USE_UTAB + d = (x - H(i)) * G(i) + E(i); +#else + long double x_hi; + double x_lo; + + /* + * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. + * G(i) has at most 9 bits, so the splitting point is not + * critical. + */ + INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, + llx & 0xffffffffff000000ULL); + x_lo = x - x_hi; + d = x_hi * G(i) - 1 + x_lo * G(i); +#endif + } + + /* + * Our algorithm depends on exact cancellation of F_lo(i) and + * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is + * at the end of the table. This and other technical complications + * make it difficult to avoid the double scaling in (dk*ln2) * + * log(base) for base != e without losing more accuracy and/or + * efficiency than is gained. + */ + /* + * Use double precision operations wherever possible, since + * long double operations are emulated and were very slow on + * the old sparc64 and unknown on the newer aarch64 and riscv + * machines. Also, don't try to improve parallelism by + * increasing the number of operations, since any parallelism + * on such machines is needed for the emulation. Horner's + * method is good for this, and is also good for accuracy. + * Horner's method doesn't handle the `lo' term well, either + * for efficiency or accuracy. However, for accuracy we + * evaluate d * d * P2 separately to take advantage of by P2 + * being exact, and this gives a good place to sum the 'lo' + * term too. + */ + dd = (double)d; + val_lo = d * d * d * (P3 + + d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + + dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + + dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2; + val_hi = d; +#ifdef DEBUG + if (fetestexcept(FE_UNDERFLOW)) + breakpoint(); +#endif + + _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); + RETURN2(rp, val_hi, val_lo); +} + +long double +log1pl(long double x) +{ + long double d, d_hi, f_lo, val_hi, val_lo; + long double f_hi, twopminusk; + double d_lo, dd, dk; + uint64_t lx, llx; + int i, k; + int16_t ax, hx; + + DOPRINT_START(&x); + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + if (hx < 0x3fff) { /* x < 1, or x neg NaN */ + ax = hx & 0x7fff; + if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ + if (ax == 0x3fff && (lx | llx) == 0) + RETURNP(-1 / zero); /* log1p(-1) = -Inf */ + /* log1p(x < 1, or x NaN) = qNaN: */ + RETURNP((x - x) / (x - x)); + } + if (ax <= 0x3f8d) { /* |x| < 2**-113 */ + if ((int)x == 0) + RETURNP(x); /* x with inexact if x != 0 */ + } + f_hi = 1; + f_lo = x; + } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ + RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ + } else if (hx < 0x40e1) { /* 1 <= x < 2**226 */ + f_hi = x; + f_lo = 1; + } else { /* 2**226 <= x < +Inf */ + f_hi = x; + f_lo = 0; /* avoid underflow of the P3 term */ + } + ENTERI(); + x = f_hi + f_lo; + f_lo = (f_hi - x) + f_lo; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + k = -16383; + + k += hx; + dk = k; + + SET_LDBL_EXPSIGN(x, 0x3fff); + twopminusk = 1; + SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); + f_lo *= twopminusk; + + i = (lx + (1LL << (L2I - 2))) >> (L2I - 1); + + /* + * x*G(i)-1 (with a reduced x) can be represented exactly, as + * above, but now we need to evaluate the polynomial on d = + * (x+f_lo)*G(i)-1 and extra precision is needed for that. + * Since x+x_lo is a hi+lo decomposition and subtracting 1 + * doesn't lose too many bits, an inexact calculation for + * f_lo*G(i) is good enough. + */ + if (0) + d_hi = x * G(i) - 1; + else { +#ifdef USE_UTAB + d_hi = (x - H(i)) * G(i) + E(i); +#else + long double x_hi; + double x_lo; + + INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx, + llx & 0xffffffffff000000ULL); + x_lo = x - x_hi; + d_hi = x_hi * G(i) - 1 + x_lo * G(i); +#endif + } + d_lo = f_lo * G(i); + + /* + * This is _2sumF(d_hi, d_lo) inlined. The condition + * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not + * always satisifed, so it is not clear that this works, but + * it works in practice. It works even if it gives a wrong + * normalized d_lo, since |d_lo| > |d_hi| implies that i is + * nonzero and d is tiny, so the F(i) term dominates d_lo. + * In float precision: + * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. + * And if d is only a little tinier than that, we would have + * another underflow problem for the P3 term; this is also ruled + * out by exhaustive testing.) + */ + d = d_hi + d_lo; + d_lo = d_hi - d + d_lo; + d_hi = d; + + dd = (double)d; + val_lo = d * d * d * (P3 + + d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 + + dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 + + dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2; + val_hi = d_hi; +#ifdef DEBUG + if (fetestexcept(FE_UNDERFLOW)) + breakpoint(); +#endif + + _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); + RETURN2PI(val_hi, val_lo); +} + +#ifdef STRUCT_RETURN + +long double +logl(long double x) +{ + struct ld r; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + RETURNSPI(&r); +} + +/* + * 29+113 bit decompositions. The bits are distributed so that the products + * of the hi terms are exact in double precision. The types are chosen so + * that the products of the hi terms are done in at least double precision, + * without any explicit conversions. More natural choices would require a + * slow long double precision multiplication. + */ +static const double +invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */ +invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */ +static const long double +invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */ +invln2_lo = 6.33178418956604368501892137426645911e-10L, /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */ +invln10_lo_plus_hi = invln10_lo + invln10_hi, +invln2_lo_plus_hi = invln2_lo + invln2_hi; + +long double +log10l(long double x) +{ + struct ld r; + long double hi, lo; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + if (!r.lo_set) + RETURNPI(r.hi); + _2sumF(r.hi, r.lo); + hi = (float)r.hi; + lo = r.lo + (r.hi - hi); + RETURN2PI(invln10_hi * hi, + invln10_lo_plus_hi * lo + invln10_lo * hi); +} + +long double +log2l(long double x) +{ + struct ld r; + long double hi, lo; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + if (!r.lo_set) + RETURNPI(r.hi); + _2sumF(r.hi, r.lo); + hi = (float)r.hi; + lo = r.lo + (r.hi - hi); + RETURN2PI(invln2_hi * hi, + invln2_lo_plus_hi * lo + invln2_lo * hi); +} + +#endif /* STRUCT_RETURN */ diff --git a/newlib/libm/ld80/b_expl.c b/newlib/libm/ld80/b_expl.c new file mode 100644 index 000000000..21bacf449 --- /dev/null +++ b/newlib/libm/ld80/b_expl.c @@ -0,0 +1,113 @@ +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * Copyright (c) 1985, 1993 + * The Regents of the University of California. All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * 3. Neither the name of the University nor the names of its contributors + * may be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +/* + * See bsdsrc/b_exp.c for implementation details. + * + * bsdrc/b_exp.c converted to long double by Steven G. Kargl. + */ + +#include "fpmath.h" +#include "math_private.h" + +static const union IEEEl2bits + p0u = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-01L), + p1u = LD80C(0xb60b60b60b60b59a, -9, -2.77777777777777775377e-03L), + p2u = LD80C(0x8ab355e008a3cfce, -14, 6.61375661375629297465e-05L), + p3u = LD80C(0xddebbc994b0c1376, -20, -1.65343915327882529784e-06L), + p4u = LD80C(0xb354784cb4ef4c41, -25, 4.17535101591534118469e-08L), + p5u = LD80C(0x913e8a718382ce75, -30, -1.05679137034774806475e-09L), + p6u = LD80C(0xe8f0042aa134502e, -36, 2.64819349895429516863e-11L); +#define p1 (p0u.e) +#define p2 (p1u.e) +#define p3 (p2u.e) +#define p4 (p3u.e) +#define p5 (p4u.e) +#define p6 (p5u.e) +#define p7 (p6u.e) + +/* + * lnhuge = (LDBL_MAX_EXP + 9) * log(2.) + * lntiny = (LDBL_MIN_EXP - 64 - 10) * log(2.) + * invln2 = 1 / log(2.) + */ +static const union IEEEl2bits +ln2hiu = LD80C(0xb17217f700000000, -1, 6.93147180369123816490e-01L), +ln2lou = LD80C(0xd1cf79abc9e3b398, -33, 1.90821492927058781614e-10L), +lnhugeu = LD80C(0xb18b0c0330a8fad9, 13, 1.13627617309191834574e+04L), +lntinyu = LD80C(0xb236f28a68bc3bd7, 13, -1.14057368561139000667e+04L), +invln2u = LD80C(0xb8aa3b295c17f0bc, 0, 1.44269504088896340739e+00L); +#define ln2hi (ln2hiu.e) +#define ln2lo (ln2lou.e) +#define lnhuge (lnhugeu.e) +#define lntiny (lntinyu.e) +#define invln2 (invln2u.e) + +/* returns exp(r = x + c) for |c| < |x| with no overlap. */ + +static long double +__exp__D(long double x, long double c) +{ + long double hi, lo, z; + int k; + + if (x != x) /* x is NaN. */ + return(x); + + if (x <= lnhuge) { + if (x >= lntiny) { + /* argument reduction: x --> x - k*ln2 */ + z = invln2 * x; + k = z + copysignl(0.5L, x); + + /* + * Express (x + c) - k * ln2 as hi - lo. + * Let x = hi - lo rounded. + */ + hi = x - k * ln2hi; /* Exact. */ + lo = k * ln2lo - c; + x = hi - lo; + + /* Return 2^k*[1+x+x*c/(2+c)] */ + z = x * x; + c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 + + z * (p5 + z * (p6 + z * p7)))))); + c = (x * c) / (2 - c); + + return (ldexpl(1 + (hi - (lo - c)), k)); + } else { + /* exp(-INF) is 0. exp(-big) underflows to 0. */ + return (isfinite(x) ? ldexpl(1., -5000) : 0); + } + } else + /* exp(INF) is INF, exp(+big#) overflows to INF */ + return (isfinite(x) ? ldexpl(1., 5000) : x); +} diff --git a/newlib/libm/ld80/b_logl.c b/newlib/libm/ld80/b_logl.c new file mode 100644 index 000000000..b11eacbe1 --- /dev/null +++ b/newlib/libm/ld80/b_logl.c @@ -0,0 +1,375 @@ +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * Copyright (c) 1992, 1993 + * The Regents of the University of California. All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * 3. Neither the name of the University nor the names of its contributors + * may be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +/* + * See bsdsrc/b_log.c for implementation details. + * + * bsdrc/b_log.c converted to long double by Steven G. Kargl. + */ + +#define N 128 + +/* + * Coefficients in the polynomial approximation of log(1+f/F). + * Domain of x is [0,1./256] with 2**(-84.48) precision. + */ +static const union IEEEl2bits + a1u = LD80C(0xaaaaaaaaaaaaaaab, -4, 8.33333333333333333356e-02L), + a2u = LD80C(0xcccccccccccccd29, -7, 1.25000000000000000781e-02L), + a3u = LD80C(0x9249249241ed3764, -9, 2.23214285711721994134e-03L), + a4u = LD80C(0xe38e959e1e7e01cf, -12, 4.34030476540000360640e-04L); +#define A1 (a1u.e) +#define A2 (a2u.e) +#define A3 (a3u.e) +#define A4 (a4u.e) + +/* + * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. + * Used for generation of extend precision logarithms. + * The constant 35184372088832 is 2^45, so the divide is exact. + * It ensures correct reading of logF_head, even for inaccurate + * decimal-to-binary conversion routines. (Everybody gets the + * right answer for integers less than 2^53.) + * Values for log(F) were generated using error < 10^-57 absolute + * with the bc -l package. + */ + +static double logF_head[N+1] = { + 0., + .007782140442060381246, + .015504186535963526694, + .023167059281547608406, + .030771658666765233647, + .038318864302141264488, + .045809536031242714670, + .053244514518837604555, + .060624621816486978786, + .067950661908525944454, + .075223421237524235039, + .082443669210988446138, + .089612158689760690322, + .096729626458454731618, + .103796793681567578460, + .110814366340264314203, + .117783035656430001836, + .124703478501032805070, + .131576357788617315236, + .138402322859292326029, + .145182009844575077295, + .151916042025732167530, + .158605030176659056451, + .165249572895390883786, + .171850256926518341060, + .178407657472689606947, + .184922338493834104156, + .191394852999565046047, + .197825743329758552135, + .204215541428766300668, + .210564769107350002741, + .216873938300523150246, + .223143551314024080056, + .229374101064877322642, + .235566071312860003672, + .241719936886966024758, + .247836163904594286577, + .253915209980732470285, + .259957524436686071567, + .265963548496984003577, + .271933715484010463114, + .277868451003087102435, + .283768173130738432519, + .289633292582948342896, + .295464212893421063199, + .301261330578199704177, + .307025035294827830512, + .312755710004239517729, + .318453731118097493890, + .324119468654316733591, + .329753286372579168528, + .335355541920762334484, + .340926586970454081892, + .346466767346100823488, + .351976423156884266063, + .357455888922231679316, + .362905493689140712376, + .368325561158599157352, + .373716409793814818840, + .379078352934811846353, + .384411698910298582632, + .389716751140440464951, + .394993808240542421117, + .400243164127459749579, + .405465108107819105498, + .410659924985338875558, + .415827895143593195825, + .420969294644237379543, + .426084395310681429691, + .431173464818130014464, + .436236766774527495726, + .441274560805140936281, + .446287102628048160113, + .451274644139630254358, + .456237433481874177232, + .461175715122408291790, + .466089729924533457960, + .470979715219073113985, + .475845904869856894947, + .480688529345570714212, + .485507815781602403149, + .490303988045525329653, + .495077266798034543171, + .499827869556611403822, + .504556010751912253908, + .509261901790523552335, + .513945751101346104405, + .518607764208354637958, + .523248143765158602036, + .527867089620485785417, + .532464798869114019908, + .537041465897345915436, + .541597282432121573947, + .546132437597407260909, + .550647117952394182793, + .555141507540611200965, + .559615787935399566777, + .564070138285387656651, + .568504735352689749561, + .572919753562018740922, + .577315365035246941260, + .581691739635061821900, + .586049045003164792433, + .590387446602107957005, + .594707107746216934174, + .599008189645246602594, + .603290851438941899687, + .607555250224322662688, + .611801541106615331955, + .616029877215623855590, + .620240409751204424537, + .624433288012369303032, + .628608659422752680256, + .632766669570628437213, + .636907462236194987781, + .641031179420679109171, + .645137961373620782978, + .649227946625615004450, + .653301272011958644725, + .657358072709030238911, + .661398482245203922502, + .665422632544505177065, + .669430653942981734871, + .673422675212350441142, + .677398823590920073911, + .681359224807238206267, + .685304003098281100392, + .689233281238557538017, + .693147180560117703862 +}; + +static double logF_tail[N+1] = { + 0., + -.00000000000000543229938420049, + .00000000000000172745674997061, + -.00000000000001323017818229233, + -.00000000000001154527628289872, + -.00000000000000466529469958300, + .00000000000005148849572685810, + -.00000000000002532168943117445, + -.00000000000005213620639136504, + -.00000000000001819506003016881, + .00000000000006329065958724544, + .00000000000008614512936087814, + -.00000000000007355770219435028, + .00000000000009638067658552277, + .00000000000007598636597194141, + .00000000000002579999128306990, + -.00000000000004654729747598444, + -.00000000000007556920687451336, + .00000000000010195735223708472, + -.00000000000017319034406422306, + -.00000000000007718001336828098, + .00000000000010980754099855238, + -.00000000000002047235780046195, + -.00000000000008372091099235912, + .00000000000014088127937111135, + .00000000000012869017157588257, + .00000000000017788850778198106, + .00000000000006440856150696891, + .00000000000016132822667240822, + -.00000000000007540916511956188, + -.00000000000000036507188831790, + .00000000000009120937249914984, + .00000000000018567570959796010, + -.00000000000003149265065191483, + -.00000000000009309459495196889, + .00000000000017914338601329117, + -.00000000000001302979717330866, + .00000000000023097385217586939, + .00000000000023999540484211737, + .00000000000015393776174455408, + -.00000000000036870428315837678, + .00000000000036920375082080089, + -.00000000000009383417223663699, + .00000000000009433398189512690, + .00000000000041481318704258568, + -.00000000000003792316480209314, + .00000000000008403156304792424, + -.00000000000034262934348285429, + .00000000000043712191957429145, + -.00000000000010475750058776541, + -.00000000000011118671389559323, + .00000000000037549577257259853, + .00000000000013912841212197565, + .00000000000010775743037572640, + .00000000000029391859187648000, + -.00000000000042790509060060774, + .00000000000022774076114039555, + .00000000000010849569622967912, + -.00000000000023073801945705758, + .00000000000015761203773969435, + .00000000000003345710269544082, + -.00000000000041525158063436123, + .00000000000032655698896907146, + -.00000000000044704265010452446, + .00000000000034527647952039772, + -.00000000000007048962392109746, + .00000000000011776978751369214, + -.00000000000010774341461609578, + .00000000000021863343293215910, + .00000000000024132639491333131, + .00000000000039057462209830700, + -.00000000000026570679203560751, + .00000000000037135141919592021, + -.00000000000017166921336082431, + -.00000000000028658285157914353, + -.00000000000023812542263446809, + .00000000000006576659768580062, + -.00000000000028210143846181267, + .00000000000010701931762114254, + .00000000000018119346366441110, + .00000000000009840465278232627, + -.00000000000033149150282752542, + -.00000000000018302857356041668, + -.00000000000016207400156744949, + .00000000000048303314949553201, + -.00000000000071560553172382115, + .00000000000088821239518571855, + -.00000000000030900580513238244, + -.00000000000061076551972851496, + .00000000000035659969663347830, + .00000000000035782396591276383, + -.00000000000046226087001544578, + .00000000000062279762917225156, + .00000000000072838947272065741, + .00000000000026809646615211673, + -.00000000000010960825046059278, + .00000000000002311949383800537, + -.00000000000058469058005299247, + -.00000000000002103748251144494, + -.00000000000023323182945587408, + -.00000000000042333694288141916, + -.00000000000043933937969737844, + .00000000000041341647073835565, + .00000000000006841763641591466, + .00000000000047585534004430641, + .00000000000083679678674757695, + -.00000000000085763734646658640, + .00000000000021913281229340092, + -.00000000000062242842536431148, + -.00000000000010983594325438430, + .00000000000065310431377633651, + -.00000000000047580199021710769, + -.00000000000037854251265457040, + .00000000000040939233218678664, + .00000000000087424383914858291, + .00000000000025218188456842882, + -.00000000000003608131360422557, + -.00000000000050518555924280902, + .00000000000078699403323355317, + -.00000000000067020876961949060, + .00000000000016108575753932458, + .00000000000058527188436251509, + -.00000000000035246757297904791, + -.00000000000018372084495629058, + .00000000000088606689813494916, + .00000000000066486268071468700, + .00000000000063831615170646519, + .00000000000025144230728376072, + -.00000000000017239444525614834 +}; +/* + * Extra precision variant, returning struct {double a, b;}; + * log(x) = a + b to 63 bits, with 'a' rounded to 24 bits. + */ +static struct Double +__log__D(long double x) +{ + int m, j; + long double F, f, g, q, u, v, u1, u2; + struct Double r; + + /* + * Argument reduction: 1 <= g < 2; x/2^m = g; + * y = F*(1 + f/F) for |f| <= 2^-8 + */ + g = frexpl(x, &m); + g *= 2; + m--; + if (m == DBL_MIN_EXP - 1) { + j = ilogbl(g); + m += j; + g = ldexpl(g, -j); + } + j = N * (g - 1) + 0.5L; + F = (1.L / N) * j + 1; + f = g - F; + + g = 1 / (2 * F + f); + u = 2 * f * g; + v = u * u; + q = u * v * (A1 + v * (A2 + v * (A3 + v * A4))); + if (m | j) { + u1 = u + 513; + u1 -= 513; + } else { + u1 = (float)u; + } + u2 = (2 * (f - F * u1) - u1 * f) * g; + + u1 += m * (long double)logF_head[N] + logF_head[j]; + + u2 += logF_tail[j]; + u2 += q; + u2 += logF_tail[N] * m; + r.a = (float)(u1 + u2); /* Only difference is here. */ + r.b = (u1 - r.a) + u2; + return (r); +} diff --git a/newlib/libm/ld80/b_tgammal.c b/newlib/libm/ld80/b_tgammal.c new file mode 100644 index 000000000..121248adb --- /dev/null +++ b/newlib/libm/ld80/b_tgammal.c @@ -0,0 +1,419 @@ +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * Copyright (c) 1992, 1993 + * The Regents of the University of California. All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * 3. Neither the name of the University nor the names of its contributors + * may be used to endorse or promote products derived from this software + * without specific prior written permission. + * + * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +/* + * The original code, FreeBSD's old svn r93211, contain the following + * attribution: + * + * This code by P. McIlroy, Oct 1992; + * + * The financial support of UUNET Communications Services is greatfully + * acknowledged. + * + * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl. + */ + +/* + * See bsdsrc/t_tgamma.c for implementation details. + */ + +#include + +#if LDBL_MAX_EXP != 0x4000 +#error "Unsupported long double format" +#endif + +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* Used in b_log.c and below. */ +struct Double { + long double a; + long double b; +}; + +#include "b_logl.c" +#include "b_expl.c" + +static const double zero = 0.; +static const volatile double tiny = 1e-300; +/* + * x >= 6 + * + * Use the asymptotic approximation (Stirling's formula) adjusted for + * equal-ripples: + * + * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) + * + * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid + * premature round-off. + * + * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. + */ + +/* + * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the + * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The + * variables are clearly misnamed. + */ +static const union IEEEl2bits +ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L), +ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L); +#define ln2pi_hi (ln2pi_hiu.e) +#define ln2pi_lo (ln2pi_lou.e) + +static const union IEEEl2bits + Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L), + Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L), + Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L), + Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L), + Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L), + Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L), + Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L), + Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L), + Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L), + Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L); +#define Pa0 (Pa0u.e) +#define Pa1 (Pa1u.e) +#define Pa2 (Pa2u.e) +#define Pa3 (Pa3u.e) +#define Pa4 (Pa4u.e) +#define Pa5 (Pa5u.e) +#define Pa6 (Pa6u.e) +#define Pa7 (Pa7u.e) +#define Pa8 (Pa8u.e) +#define Pa9 (Pa9u.e) + +static struct Double +large_gam(long double x) +{ + long double p, z, thi, tlo, xhi, xlo; + long double logx; + struct Double u; + + z = 1 / (x * x); + p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + + z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9)))))))); + p = p / x; + + u = __log__D(x); + u.a -= 1; + + /* Split (x - 0.5) in high and low parts. */ + x -= 0.5L; + xhi = (float)x; + xlo = x - xhi; + + /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ + thi = xhi * u.a; + tlo = xlo * u.a + x * u.b; + + /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ + tlo += ln2pi_lo; + tlo += p; + u.a = ln2pi_hi + tlo; + u.a += thi; + u.b = thi - u.a; + u.b += ln2pi_hi; + u.b += tlo; + return (u); +} +/* + * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval + * [1.066.., 2.066..] accurate to 4.25e-19. + * + * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. + */ +static const union IEEEl2bits + a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L), + a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L); +#define a0_hi (a0_hiu.e) +#define a0_lo (a0_lou.e) + +static const union IEEEl2bits +P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L), +P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L), +P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L), +P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L), +P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L), +P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L), +P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L), +P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L), +P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L), +Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L), +Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L), +Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L), +Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L), +Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L), +Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L), +Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L), +Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L); +#define P0 (P0u.e) +#define P1 (P1u.e) +#define P2 (P2u.e) +#define P3 (P3u.e) +#define P4 (P4u.e) +#define P5 (P5u.e) +#define P6 (P6u.e) +#define P7 (P7u.e) +#define P8 (P8u.e) +#define Q1 (Q1u.e) +#define Q2 (Q2u.e) +#define Q3 (Q3u.e) +#define Q4 (Q4u.e) +#define Q5 (Q5u.e) +#define Q6 (Q6u.e) +#define Q7 (Q7u.e) +#define Q8 (Q8u.e) + +static struct Double +ratfun_gam(long double z, long double c) +{ + long double p, q, thi, tlo; + struct Double r; + + q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + + z * (Q6 + z * (Q7 + z * Q8))))))); + p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + + z * (P6 + z * (P7 + z * P8))))))); + p = p / q; + + /* Split z into high and low parts. */ + thi = (float)z; + tlo = (z - thi) + c; + tlo *= (thi + z); + + /* Split (z+c)^2 into high and low parts. */ + thi *= thi; + q = thi; + thi = (float)thi; + tlo += (q - thi); + + /* Split p/q into high and low parts. */ + r.a = (float)p; + r.b = p - r.a; + + tlo = tlo * p + thi * r.b + a0_lo; + thi *= r.a; /* t = (z+c)^2*(P/Q) */ + r.a = (float)(thi + a0_hi); + r.b = ((a0_hi - r.a) + thi) + tlo; + return (r); /* r = a0 + t */ +} +/* + * x < 6 + * + * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, + * 2.066124]. Use a rational approximation centered at the minimum + * (x0+1) to ensure monotonicity. + * + * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) + * It also has correct monotonicity. + */ +static const union IEEEl2bits + xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L); +#define x0 (xm1u.e) + +static const double + left = -0.3955078125; /* left boundary for rat. approx */ + +static long double +small_gam(long double x) +{ + long double t, y, ym1; + struct Double yy, r; + + y = x - 1; + + if (y <= 1 + (left + x0)) { + yy = ratfun_gam(y - x0, 0); + return (yy.a + yy.b); + } + + r.a = (float)y; + yy.a = r.a - 1; + y = y - 1 ; + r.b = yy.b = y - yy.a; + + /* Argument reduction: G(x+1) = x*G(x) */ + for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { + t = r.a * yy.a; + r.b = r.a * yy.b + y * r.b; + r.a = (float)t; + r.b += (t - r.a); + } + + /* Return r*tgamma(y). */ + yy = ratfun_gam(y - x0, 0); + y = r.b * (yy.a + yy.b) + r.a * yy.b; + y += yy.a * r.a; + return (y); +} +/* + * Good on (0, 1+x0+left]. Accurate to 1 ulp. + */ +static long double +smaller_gam(long double x) +{ + long double d, rhi, rlo, t, xhi, xlo; + struct Double r; + + if (x < x0 + left) { + t = (float)x; + d = (t + x) * (x - t); + t *= t; + xhi = (float)(t + x); + xlo = x - xhi; + xlo += t; + xlo += d; + t = 1 - x0; + t += x; + d = 1 - x0; + d -= t; + d += x; + x = xhi + xlo; + } else { + xhi = (float)x; + xlo = x - xhi; + t = x - x0; + d = - x0 - t; + d += x; + } + + r = ratfun_gam(t, d); + d = (float)(r.a / x); + r.a -= d * xhi; + r.a -= d * xlo; + r.a += r.b; + + return (d + r.a / x); +} +/* + * x < 0 + * + * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). + * At negative integers, return NaN and raise invalid. + */ +static const union IEEEl2bits +piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L); +#define pi (piu.e) + +static long double +neg_gam(long double x) +{ + int sgn = 1; + struct Double lg, lsine; + long double y, z; + + y = ceill(x); + if (y == x) /* Negative integer. */ + return ((x - x) / zero); + + z = y - x; + if (z > 0.5) + z = 1 - z; + + y = y / 2; + if (y == ceill(y)) + sgn = -1; + + if (z < 0.25) + z = sinpil(z); + else + z = cospil(0.5 - z); + + /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ + if (x < -1753) { + + if (x < -1760) + return (sgn * tiny * tiny); + y = expl(lgammal(x) / 2); + y *= y; + return (sgn < 0 ? -y : y); + } + + + y = 1 - x; + if (1 - y == x) + y = tgammal(y); + else /* 1-x is inexact */ + y = - x * tgammal(-x); + + if (sgn < 0) y = -y; + return (pi / (y * z)); +} +/* + * xmax comes from lgamma(xmax) - emax * log(2) = 0. + * static const float xmax = 35.040095f + * static const double xmax = 171.624376956302725; + * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), + * ld128: 1.75554834290446291700388921607020320e+03L, + * + * iota is a sloppy threshold to isolate x = 0. + */ +static const double xmax = 1755.54834290446291689; +static const double iota = 0x1p-116; + +long double +tgammal(long double x) +{ + struct Double u; + + ENTERI(); + + if (x >= 6) { + if (x > xmax) + RETURNI(x / zero); + u = large_gam(x); + RETURNI(__exp__D(u.a, u.b)); + } + + if (x >= 1 + left + x0) + RETURNI(small_gam(x)); + + if (x > iota) + RETURNI(smaller_gam(x)); + + if (x > -iota) { + if (x != 0) + u.a = 1 - tiny; /* raise inexact */ + RETURNI(1 / x); + } + + if (!isfinite(x)) + RETURNI(x - x); /* x is NaN or -Inf */ + + RETURNI(neg_gam(x)); +} diff --git a/newlib/libm/ld80/e_powl.c b/newlib/libm/ld80/e_powl.c new file mode 100644 index 000000000..ea25354c2 --- /dev/null +++ b/newlib/libm/ld80/e_powl.c @@ -0,0 +1,662 @@ +/*- + * Copyright (c) 2008 Stephen L. Moshier + * + * Permission to use, copy, modify, and distribute this software for any + * purpose with or without fee is hereby granted, provided that the above + * copyright notice and this permission notice appear in all copies. + * + * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES + * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR + * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES + * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN + * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF + * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include + +#include "math_private.h" + +/* + * Polynomial evaluator: + * P[0] x^n + P[1] x^(n-1) + ... + P[n] + */ +static inline long double +__polevll(long double x, long double *PP, int n) +{ + long double y; + long double *P; + + P = PP; + y = *P++; + do { + y = y * x + *P++; + } while (--n); + + return (y); +} + +/* + * Polynomial evaluator: + * x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n] + */ +static inline long double +__p1evll(long double x, long double *PP, int n) +{ + long double y; + long double *P; + + P = PP; + n -= 1; + y = x + *P++; + do { + y = y * x + *P++; + } while (--n); + + return (y); +} + +/* powl.c + * + * Power function, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, z, powl(); + * + * z = powl( x, y ); + * + * + * + * DESCRIPTION: + * + * Computes x raised to the yth power. Analytically, + * + * x**y = exp( y log(x) ). + * + * Following Cody and Waite, this program uses a lookup table + * of 2**-i/32 and pseudo extended precision arithmetic to + * obtain several extra bits of accuracy in both the logarithm + * and the exponential. + * + * + * + * ACCURACY: + * + * The relative error of pow(x,y) can be estimated + * by y dl ln(2), where dl is the absolute error of + * the internally computed base 2 logarithm. At the ends + * of the approximation interval the logarithm equal 1/32 + * and its relative error is about 1 lsb = 1.1e-19. Hence + * the predicted relative error in the result is 2.3e-21 y . + * + * Relative error: + * arithmetic domain # trials peak rms + * + * IEEE +-1000 40000 2.8e-18 3.7e-19 + * .001 < x < 1000, with log(x) uniformly distributed. + * -1000 < y < 1000, y uniformly distributed. + * + * IEEE 0,8700 60000 6.5e-18 1.0e-18 + * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. + * + * + * ERROR MESSAGES: + * + * message condition value returned + * pow overflow x**y > MAXNUM INFINITY + * pow underflow x**y < 1/MAXNUM 0.0 + * pow domain x<0 and y noninteger 0.0 + * + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#include "math_private.h" + +/* Table size */ +#define NXT 32 +/* log2(Table size) */ +#define LNXT 5 + +/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) + * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 + */ +static long double P[] = { + 8.3319510773868690346226E-4L, + 4.9000050881978028599627E-1L, + 1.7500123722550302671919E0L, + 1.4000100839971580279335E0L, +}; +static long double Q[] = { +/* 1.0000000000000000000000E0L,*/ + 5.2500282295834889175431E0L, + 8.4000598057587009834666E0L, + 4.2000302519914740834728E0L, +}; +/* A[i] = 2^(-i/32), rounded to IEEE long double precision. + * If i is even, A[i] + B[i/2] gives additional accuracy. + */ +static long double A[33] = { + 1.0000000000000000000000E0L, + 9.7857206208770013448287E-1L, + 9.5760328069857364691013E-1L, + 9.3708381705514995065011E-1L, + 9.1700404320467123175367E-1L, + 8.9735453750155359320742E-1L, + 8.7812608018664974155474E-1L, + 8.5930964906123895780165E-1L, + 8.4089641525371454301892E-1L, + 8.2287773907698242225554E-1L, + 8.0524516597462715409607E-1L, + 7.8799042255394324325455E-1L, + 7.7110541270397041179298E-1L, + 7.5458221379671136985669E-1L, + 7.3841307296974965571198E-1L, + 7.2259040348852331001267E-1L, + 7.0710678118654752438189E-1L, + 6.9195494098191597746178E-1L, + 6.7712777346844636413344E-1L, + 6.6261832157987064729696E-1L, + 6.4841977732550483296079E-1L, + 6.3452547859586661129850E-1L, + 6.2092890603674202431705E-1L, + 6.0762367999023443907803E-1L, + 5.9460355750136053334378E-1L, + 5.8186242938878875689693E-1L, + 5.6939431737834582684856E-1L, + 5.5719337129794626814472E-1L, + 5.4525386633262882960438E-1L, + 5.3357020033841180906486E-1L, + 5.2213689121370692017331E-1L, + 5.1094857432705833910408E-1L, + 5.0000000000000000000000E-1L, +}; +static long double B[17] = { + 0.0000000000000000000000E0L, + 2.6176170809902549338711E-20L, +-1.0126791927256478897086E-20L, + 1.3438228172316276937655E-21L, + 1.2207982955417546912101E-20L, +-6.3084814358060867200133E-21L, + 1.3164426894366316434230E-20L, +-1.8527916071632873716786E-20L, + 1.8950325588932570796551E-20L, + 1.5564775779538780478155E-20L, + 6.0859793637556860974380E-21L, +-2.0208749253662532228949E-20L, + 1.4966292219224761844552E-20L, + 3.3540909728056476875639E-21L, +-8.6987564101742849540743E-22L, +-1.2327176863327626135542E-20L, + 0.0000000000000000000000E0L, +}; + +/* 2^x = 1 + x P(x), + * on the interval -1/32 <= x <= 0 + */ +static long double R[] = { + 1.5089970579127659901157E-5L, + 1.5402715328927013076125E-4L, + 1.3333556028915671091390E-3L, + 9.6181291046036762031786E-3L, + 5.5504108664798463044015E-2L, + 2.4022650695910062854352E-1L, + 6.9314718055994530931447E-1L, +}; + +#define douba(k) A[k] +#define doubb(k) B[k] +#define MEXP (NXT*16384.0L) +/* The following if denormal numbers are supported, else -MEXP: */ +#define MNEXP (-NXT*(16384.0L+64.0L)) +/* log2(e) - 1 */ +#define LOG2EA 0.44269504088896340735992L + +#define F W +#define Fa Wa +#define Fb Wb +#define G W +#define Ga Wa +#define Gb u +#define H W +#define Ha Wb +#define Hb Wb + +static const long double MAXLOGL = 1.1356523406294143949492E4L; +static const long double MINLOGL = -1.13994985314888605586758E4L; +static const long double LOGE2L = 6.9314718055994530941723E-1L; +static volatile long double z; +static long double w, W, Wa, Wb, ya, yb, u; +static const long double huge = 0x1p10000L; +#if 0 /* XXX Prevent gcc from erroneously constant folding this. */ +static const long double twom10000 = 0x1p-10000L; +#else +static volatile long double twom10000 = 0x1p-10000L; +#endif + +static long double reducl( long double ); +static long double powil ( long double, int ); + +long double +powl(long double x, long double y) +{ +/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ +int i, nflg, iyflg, yoddint; +long e; + +if( y == 0.0L ) + return( 1.0L ); + +if( x == 1.0L ) + return( 1.0L ); + +if( isnan(x) ) + return ( nan_mix(x, y) ); +if( isnan(y) ) + return ( nan_mix(x, y) ); + +if( y == 1.0L ) + return( x ); + +if( !isfinite(y) && x == -1.0L ) + return( 1.0L ); + +if( y >= LDBL_MAX ) + { + if( x > 1.0L ) + return( INFINITY ); + if( x > 0.0L && x < 1.0L ) + return( 0.0L ); + if( x < -1.0L ) + return( INFINITY ); + if( x > -1.0L && x < 0.0L ) + return( 0.0L ); + } +if( y <= -LDBL_MAX ) + { + if( x > 1.0L ) + return( 0.0L ); + if( x > 0.0L && x < 1.0L ) + return( INFINITY ); + if( x < -1.0L ) + return( 0.0L ); + if( x > -1.0L && x < 0.0L ) + return( INFINITY ); + } +if( x >= LDBL_MAX ) + { + if( y > 0.0L ) + return( INFINITY ); + return( 0.0L ); + } + +w = floorl(y); +/* Set iyflg to 1 if y is an integer. */ +iyflg = 0; +if( w == y ) + iyflg = 1; + +/* Test for odd integer y. */ +yoddint = 0; +if( iyflg ) + { + ya = fabsl(y); + ya = floorl(0.5L * ya); + yb = 0.5L * fabsl(w); + if( ya != yb ) + yoddint = 1; + } + +if( x <= -LDBL_MAX ) + { + if( y > 0.0L ) + { + if( yoddint ) + return( -INFINITY ); + return( INFINITY ); + } + if( y < 0.0L ) + { + if( yoddint ) + return( -0.0L ); + return( 0.0 ); + } + } + + +nflg = 0; /* flag = 1 if x<0 raised to integer power */ +if( x <= 0.0L ) + { + if( x == 0.0L ) + { + if( y < 0.0 ) + { + if( signbit(x) && yoddint ) + return( -INFINITY ); + return( INFINITY ); + } + if( y > 0.0 ) + { + if( signbit(x) && yoddint ) + return( -0.0L ); + return( 0.0 ); + } + if( y == 0.0L ) + return( 1.0L ); /* 0**0 */ + else + return( 0.0L ); /* 0**y */ + } + else + { + if( iyflg == 0 ) + return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ + nflg = 1; + } + } + +/* Integer power of an integer. */ + +if( iyflg ) + { + i = w; + w = floorl(x); + if( (w == x) && (fabsl(y) < 32768.0) ) + { + w = powil( x, (int) y ); + return( w ); + } + } + + +if( nflg ) + x = fabsl(x); + +/* separate significand from exponent */ +x = frexpl( x, &i ); +e = i; + +/* find significand in antilog table A[] */ +i = 1; +if( x <= douba(17) ) + i = 17; +if( x <= douba(i+8) ) + i += 8; +if( x <= douba(i+4) ) + i += 4; +if( x <= douba(i+2) ) + i += 2; +if( x >= douba(1) ) + i = -1; +i += 1; + + +/* Find (x - A[i])/A[i] + * in order to compute log(x/A[i]): + * + * log(x) = log( a x/a ) = log(a) + log(x/a) + * + * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a + */ +x -= douba(i); +x -= doubb(i/2); +x /= douba(i); + + +/* rational approximation for log(1+v): + * + * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) + */ +z = x*x; +w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) ); +w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ + +/* Convert to base 2 logarithm: + * multiply by log2(e) = 1 + LOG2EA + */ +z = LOG2EA * w; +z += w; +z += LOG2EA * x; +z += x; + +/* Compute exponent term of the base 2 logarithm. */ +w = -i; +w = ldexpl( w, -LNXT ); /* divide by NXT */ +w += e; +/* Now base 2 log of x is w + z. */ + +/* Multiply base 2 log by y, in extended precision. */ + +/* separate y into large part ya + * and small part yb less than 1/NXT + */ +ya = reducl(y); +yb = y - ya; + +/* (w+z)(ya+yb) + * = w*ya + w*yb + z*y + */ +F = z * y + w * yb; +Fa = reducl(F); +Fb = F - Fa; + +G = Fa + w * ya; +Ga = reducl(G); +Gb = G - Ga; + +H = Fb + Gb; +Ha = reducl(H); +w = ldexpl( Ga+Ha, LNXT ); + +/* Test the power of 2 for overflow */ +if( w > MEXP ) + return (huge * huge); /* overflow */ + +if( w < MNEXP ) + return (twom10000 * twom10000); /* underflow */ + +e = w; +Hb = H - Ha; + +if( Hb > 0.0L ) + { + e += 1; + Hb -= (1.0L/NXT); /*0.0625L;*/ + } + +/* Now the product y * log2(x) = Hb + e/NXT. + * + * Compute base 2 exponential of Hb, + * where -0.0625 <= Hb <= 0. + */ +z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ + +/* Express e/NXT as an integer plus a negative number of (1/NXT)ths. + * Find lookup table entry for the fractional power of 2. + */ +if( e < 0 ) + i = 0; +else + i = 1; +i = e/NXT + i; +e = NXT*i - e; +w = douba( e ); +z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ +z = z + w; +z = ldexpl( z, i ); /* multiply by integer power of 2 */ + +if( nflg ) + { +/* For negative x, + * find out if the integer exponent + * is odd or even. + */ + w = ldexpl( y, -1 ); + w = floorl(w); + w = ldexpl( w, 1 ); + if( w != y ) + z = -z; /* odd exponent */ + } + +return( z ); +} + + +/* Find a multiple of 1/NXT that is within 1/NXT of x. */ +static inline long double +reducl(long double x) +{ +long double t; + +t = ldexpl( x, LNXT ); +t = floorl( t ); +t = ldexpl( t, -LNXT ); +return(t); +} + +/* powil.c + * + * Real raised to integer power, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, powil(); + * int n; + * + * y = powil( x, n ); + * + * + * + * DESCRIPTION: + * + * Returns argument x raised to the nth power. + * The routine efficiently decomposes n as a sum of powers of + * two. The desired power is a product of two-to-the-kth + * powers of x. Thus to compute the 32767 power of x requires + * 28 multiplications instead of 32767 multiplications. + * + * + * + * ACCURACY: + * + * + * Relative error: + * arithmetic x domain n domain # trials peak rms + * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 + * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 + * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 + * + * Returns MAXNUM on overflow, zero on underflow. + * + */ + +static long double +powil(long double x, int nn) +{ +long double ww, y; +long double s; +int n, e, sign, asign, lx; + +if( x == 0.0L ) + { + if( nn == 0 ) + return( 1.0L ); + else if( nn < 0 ) + return( LDBL_MAX ); + else + return( 0.0L ); + } + +if( nn == 0 ) + return( 1.0L ); + + +if( x < 0.0L ) + { + asign = -1; + x = -x; + } +else + asign = 0; + + +if( nn < 0 ) + { + sign = -1; + n = -nn; + } +else + { + sign = 1; + n = nn; + } + +/* Overflow detection */ + +/* Calculate approximate logarithm of answer */ +s = x; +s = frexpl( s, &lx ); +e = (lx - 1)*n; +if( (e == 0) || (e > 64) || (e < -64) ) + { + s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); + s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L; + } +else + { + s = LOGE2L * e; + } + +if( s > MAXLOGL ) + return (huge * huge); /* overflow */ + +if( s < MINLOGL ) + return (twom10000 * twom10000); /* underflow */ +/* Handle tiny denormal answer, but with less accuracy + * since roundoff error in 1.0/x will be amplified. + * The precise demarcation should be the gradual underflow threshold. + */ +if( s < (-MAXLOGL+2.0L) ) + { + x = 1.0L/x; + sign = -sign; + } + +/* First bit of the power */ +if( n & 1 ) + y = x; + +else + { + y = 1.0L; + asign = 0; + } + +ww = x; +n >>= 1; +while( n ) + { + ww = ww * ww; /* arg to the 2-to-the-kth power */ + if( n & 1 ) /* if that bit is set, then include in product */ + y *= ww; + n >>= 1; + } + +if( asign ) + y = -y; /* odd power of negative number */ +if( sign < 0 ) + y = 1.0L/y; +return(y); +} diff --git a/newlib/libm/ld80/e_rem_pio2l.h b/newlib/libm/ld80/e_rem_pio2l.h new file mode 100644 index 000000000..b7ef5d983 --- /dev/null +++ b/newlib/libm/ld80/e_rem_pio2l.h @@ -0,0 +1,143 @@ +/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* ld80 version of __ieee754_rem_pio2l(x,y) + * + * return the remainder of x rem pi/2 in y[0]+y[1] + * use __kernel_rem_pio2() + */ + +#include + +#include "math.h" +#include "math_private.h" +#include "../ld/fpmath.h" + +#define BIAS (LDBL_MAX_EXP - 1) + +/* + * invpio2: 64 bits of 2/pi + * pio2_1: first 39 bits of pi/2 + * pio2_1t: pi/2 - pio2_1 + * pio2_2: second 39 bits of pi/2 + * pio2_2t: pi/2 - (pio2_1+pio2_2) + * pio2_3: third 39 bits of pi/2 + * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) + */ + +static const double +zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ +two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ +pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ +pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ +pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ + +#if defined(__amd64__) || defined(__i386__) +/* Long double constants are slow on these arches, and broken on i386. */ +static const volatile double +invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ +invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ +pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ +pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ +pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ +pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ +pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ +pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ +#define invpio2 ((long double)invpio2hi + invpio2lo) +#define pio2_1t ((long double)pio2_1thi + pio2_1tlo) +#define pio2_2t ((long double)pio2_2thi + pio2_2tlo) +#define pio2_3t ((long double)pio2_3thi + pio2_3tlo) +#else +static const long double +invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ +pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ +pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ +pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ +#endif + +static inline __always_inline int +__ieee754_rem_pio2l(long double x, long double *y) +{ + union IEEEl2bits u,u1; + long double z,w,t,r,fn; + double tx[3],ty[2]; + int e0,ex,i,j,nx,n; + int16_t expsign; + + u.e = x; + expsign = u.xbits.expsign; + ex = expsign & 0x7fff; + if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { + /* |x| ~< 2^25*(pi/2), medium size */ + fn = rnintl(x*invpio2); + n = irint(fn); + r = x-fn*pio2_1; + w = fn*pio2_1t; /* 1st round good to 102 bit */ + { + union IEEEl2bits u2; + int ex1; + j = ex; + y[0] = r-w; + u2.e = y[0]; + ex1 = u2.xbits.expsign & 0x7fff; + i = j-ex1; + if(i>22) { /* 2nd iteration needed, good to 141 */ + t = r; + w = fn*pio2_2; + r = t-w; + w = fn*pio2_2t-((t-r)-w); + y[0] = r-w; + u2.e = y[0]; + ex1 = u2.xbits.expsign & 0x7fff; + i = j-ex1; + if(i>61) { /* 3rd iteration need, 180 bits acc */ + t = r; /* will cover all possible cases */ + w = fn*pio2_3; + r = t-w; + w = fn*pio2_3t-((t-r)-w); + y[0] = r-w; + } + } + } + y[1] = (r-y[0])-w; + return n; + } + /* + * all other (large) arguments + */ + if(ex==0x7fff) { /* x is inf or NaN */ + y[0]=y[1]=x-x; return 0; + } + /* set z = scalbn(|x|,ilogb(x)-23) */ + u1.e = x; + e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ + u1.xbits.expsign = ex - e0; + z = u1.e; + for(i=0;i<2;i++) { + tx[i] = (double)((int32_t)(z)); + z = (z-tx[i])*two24; + } + tx[2] = z; + nx = 3; + while(tx[nx-1]==zero) nx--; /* skip zero term */ + n = __kernel_rem_pio2(tx,ty,e0,nx,2); + r = (long double)ty[0] + ty[1]; + w = ty[1] - (r - ty[0]); + if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} + y[0] = r; y[1] = w; return n; +} diff --git a/newlib/libm/ld80/invtrig.h b/newlib/libm/ld80/invtrig.h new file mode 100644 index 000000000..be06a044b --- /dev/null +++ b/newlib/libm/ld80/invtrig.h @@ -0,0 +1,116 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + * + * $FreeBSD$ + */ + +#include + +#include "fpmath.h" + +#define BIAS (LDBL_MAX_EXP - 1) +#define MANH_SIZE LDBL_MANH_SIZE + +/* Approximation thresholds. */ +#define ASIN_LINEAR (BIAS - 32) /* 2**-32 */ +#define ACOS_CONST (BIAS - 65) /* 2**-65 */ +#define ATAN_CONST (BIAS + 65) /* 2**65 */ +#define ATAN_LINEAR (BIAS - 32) /* 2**-32 */ + +/* 0.95 */ +#define THRESH ((0xe666666666666666ULL>>(64-(MANH_SIZE-1)))|LDBL_NBIT) + +/* Constants shared by the long double inverse trig functions. */ +#define pS0 _ItL_pS0 +#define pS1 _ItL_pS1 +#define pS2 _ItL_pS2 +#define pS3 _ItL_pS3 +#define pS4 _ItL_pS4 +#define pS5 _ItL_pS5 +#define pS6 _ItL_pS6 +#define qS1 _ItL_qS1 +#define qS2 _ItL_qS2 +#define qS3 _ItL_qS3 +#define qS4 _ItL_qS4 +#define qS5 _ItL_qS5 +#define atanhi _ItL_atanhi +#define atanlo _ItL_atanlo +#define aT _ItL_aT +#define pi_lo _ItL_pi_lo + +#define pio2_hi atanhi[3] +#define pio2_lo atanlo[3] +#define pio4_hi atanhi[1] + +#ifdef STRUCT_DECLS +typedef struct longdouble { + uint64_t mant; + uint16_t expsign; +} LONGDOUBLE; +#else +typedef long double LONGDOUBLE; +#endif + +extern const LONGDOUBLE pS0, pS1, pS2, pS3, pS4, pS5, pS6; +extern const LONGDOUBLE qS1, qS2, qS3, qS4, qS5; +extern const LONGDOUBLE atanhi[], atanlo[], aT[]; +extern const LONGDOUBLE pi_lo; + +#ifndef STRUCT_DECLS + +static inline long double +P(long double x) +{ + + return (x * (pS0 + x * (pS1 + x * (pS2 + x * (pS3 + x * \ + (pS4 + x * (pS5 + x * pS6))))))); +} + +static inline long double +Q(long double x) +{ + + return (1.0 + x * (qS1 + x * (qS2 + x * (qS3 + x * (qS4 + x * qS5))))); +} + +static inline long double +T_even(long double x) +{ + + return (aT[0] + x * (aT[2] + x * (aT[4] + x * (aT[6] + x * \ + (aT[8] + x * (aT[10] + x * aT[12])))))); +} + +static inline long double +T_odd(long double x) +{ + + return (aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] + x * \ + (aT[9] + x * aT[11]))))); +} + +#endif diff --git a/newlib/libm/ld80/k_cospil.h b/newlib/libm/ld80/k_cospil.h new file mode 100644 index 000000000..6e13ef02a --- /dev/null +++ b/newlib/libm/ld80/k_cospil.h @@ -0,0 +1,42 @@ +/*- + * Copyright (c) 2017 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +/* + * See ../src/k_cospi.c for implementation details. + */ + +static inline long double +__kernel_cospil(long double x) +{ + long double hi, lo; + + hi = (float)x; + lo = x - hi; + lo = lo * (pi_lo + pi_hi) + hi * pi_lo; + hi *= pi_hi; + _2sumF(hi, lo); + return (__kernel_cosl(hi, lo)); +} diff --git a/newlib/libm/ld80/k_expl.h b/newlib/libm/ld80/k_expl.h new file mode 100644 index 000000000..a744d2d38 --- /dev/null +++ b/newlib/libm/ld80/k_expl.h @@ -0,0 +1,301 @@ +/* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ + +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See s_expl.c for more comments about __k_expl(). + * + * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments + * about the secondary kernels. + */ + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +/* + * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must + * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest + * bits zero so that multiplication of it by n is exact. + */ +INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ +L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ +L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ +/* + * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: + * |exp(x) - p(x)| < 2**-77.2 + * (0.002708 is ln2/(2*INTERVALS) rounded up a little). + */ +A2 = 0.5, +A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ +A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ +A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ +A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ + +/* + * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where + * the first 53 bits of the significand are stored in hi and the next 53 + * bits are in lo. Tang's paper states that the trailing 6 bits of hi must + * be zero for his algorithm in both single and double precision, because + * the table is re-used in the implementation of expm1() where a floating + * point addition involving hi must be exact. Here hi is double, so + * converting it to long double gives 11 trailing zero bits. + */ +static const struct { + double hi; + double lo; +} tbl[INTERVALS] = { + { 0x1p+0, 0x0p+0 }, + /* + * XXX hi is rounded down, and the formatting is not quite normal. + * But I rather like both. The 0x1.*p format is good for 4N+1 + * mantissa bits. Rounding down makes the lo terms positive, + * so that the columnar formatting can be simpler. + */ + { 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54 }, + { 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53 }, + { 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53 }, + { 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55 }, + { 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53 }, + { 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57 }, + { 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54 }, + { 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54 }, + { 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54 }, + { 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59 }, + { 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53 }, + { 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53 }, + { 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53 }, + { 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53 }, + { 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55 }, + { 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53 }, + { 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53 }, + { 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55 }, + { 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53 }, + { 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54 }, + { 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53 }, + { 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55 }, + { 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55 }, + { 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54 }, + { 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55 }, + { 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55 }, + { 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53 }, + { 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55 }, + { 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53 }, + { 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54 }, + { 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56 }, + { 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55 }, + { 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55 }, + { 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54 }, + { 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53 }, + { 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53 }, + { 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53 }, + { 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53 }, + { 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53 }, + { 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55 }, + { 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53 }, + { 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53 }, + { 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53 }, + { 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59 }, + { 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54 }, + { 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56 }, + { 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54 }, + { 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56 }, + { 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54 }, + { 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53 }, + { 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53 }, + { 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53 }, + { 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53 }, + { 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54 }, + { 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55 }, + { 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54 }, + { 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60 }, + { 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54 }, + { 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53 }, + { 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53 }, + { 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53 }, + { 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53 }, + { 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57 }, + { 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53 }, + { 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53 }, + { 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53 }, + { 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53 }, + { 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53 }, + { 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53 }, + { 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53 }, + { 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54 }, + { 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53 }, + { 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54 }, + { 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56 }, + { 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53 }, + { 0x1.82589994cce12p+0, 0x1.159f115f56694p-53 }, + { 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53 }, + { 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53 }, + { 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54 }, + { 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54 }, + { 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53 }, + { 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55 }, + { 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53 }, + { 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53 }, + { 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53 }, + { 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53 }, + { 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53 }, + { 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56 }, + { 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56 }, + { 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53 }, + { 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54 }, + { 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53 }, + { 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54 }, + { 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54 }, + { 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53 }, + { 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54 }, + { 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53 }, + { 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53 }, + { 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53 }, + { 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53 }, + { 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53 }, + { 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55 }, + { 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53 }, + { 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55 }, + { 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54 }, + { 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54 }, + { 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56 }, + { 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56 }, + { 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53 }, + { 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53 }, + { 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53 }, + { 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55 }, + { 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53 }, + { 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54 }, + { 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54 }, + { 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53 }, + { 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53 }, + { 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55 }, + { 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54 }, + { 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53 }, + { 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53 }, + { 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54 }, + { 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54 }, + { 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54 }, + { 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53 }, + { 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55 }, + { 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 } +}; + +/* + * Kernel for expl(x). x must be finite and not tiny or huge. + * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). + * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). + */ +static inline void +__k_expl(long double x, long double *hip, long double *lop, int *kp) +{ + long double fn, q, r, r1, r2, t, z; + int n, n2; + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + fn = rnintl(x * INV_L); + r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ + n = irint(fn); + n2 = (unsigned)n % INTERVALS; + /* Depend on the sign bit being propagated: */ + *kp = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + + /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ + z = r * r; +#if 0 + q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; +#else + q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6)); +#endif + t = (long double)tbl[n2].lo + tbl[n2].hi; + *hip = tbl[n2].hi; + *lop = tbl[n2].lo + t * (q + r1); +} + +static inline void +k_hexpl(long double x, long double *hip, long double *lop) +{ + float twopkm1; + int k; + + __k_expl(x, hip, lop, &k); + SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); + *hip *= twopkm1; + *lop *= twopkm1; +} + +static inline long double +hexpl(long double x) +{ + long double hi, lo, twopkm2; + int k; + + twopkm2 = 1; + __k_expl(x, &hi, &lo, &k); + SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); + return (lo + hi) * 2 * twopkm2; +} + +#ifdef _COMPLEX_H +/* + * See ../src/k_exp.c for details. + */ +static inline long double complex +__ldexp_cexpl(long double complex z, int expt) +{ + long double c, exp_x, hi, lo, s; + long double x, y, scale1, scale2; + int half_expt, k; + + x = creall(z); + y = cimagl(z); + __k_expl(x, &hi, &lo, &k); + + exp_x = (lo + hi) * 0x1p16382L; + expt += k - 16382; + + scale1 = 1; + half_expt = expt / 2; + SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); + scale2 = 1; + SET_LDBL_EXPSIGN(scale2, BIAS + expt - half_expt); + + sincosl(y, &s, &c); + return (CMPLXL(c * exp_x * scale1 * scale2, + s * exp_x * scale1 * scale2)); +} +#endif /* _COMPLEX_H */ diff --git a/newlib/libm/ld80/k_sinpil.h b/newlib/libm/ld80/k_sinpil.h new file mode 100644 index 000000000..00241b932 --- /dev/null +++ b/newlib/libm/ld80/k_sinpil.h @@ -0,0 +1,42 @@ +/*- + * Copyright (c) 2017 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +/* + * See ../src/k_sinpi.c for implementation details. + */ + +static inline long double +__kernel_sinpil(long double x) +{ + long double hi, lo; + + hi = (float)x; + lo = x - hi; + lo = lo * (pi_lo + pi_hi) + hi * pi_lo; + hi *= pi_hi; + _2sumF(hi, lo); + return (__kernel_sinl(hi, lo, 1)); +} diff --git a/newlib/libm/ld80/s_cospil.c b/newlib/libm/ld80/s_cospil.c new file mode 100644 index 000000000..199479e9e --- /dev/null +++ b/newlib/libm/ld80/s_cospil.c @@ -0,0 +1,129 @@ +/*- + * Copyright (c) 2017 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +/* + * See ../src/s_cospi.c for implementation details. + */ + +#ifdef __i386__ +#include +#endif +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +static const double +pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */ +pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */ + +#include "k_cospil.h" +#include "k_sinpil.h" + +volatile static const double vzero = 0; + +long double +cospil(long double x) +{ + long double ax, c; + uint64_t lx, m; + uint32_t j0; + uint16_t hx, ix; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + ix = hx & 0x7fff; + INSERT_LDBL80_WORDS(ax, ix, lx); + + ENTERI(); + + if (ix < 0x3fff) { /* |x| < 1 */ + if (ix < 0x3ffd) { /* |x| < 0.25 */ + if (ix < 0x3fdd) { /* |x| < 0x1p-34 */ + if ((int)x == 0) + RETURNI(1); + } + RETURNI(__kernel_cospil(ax)); + } + + if (ix < 0x3ffe) /* |x| < 0.5 */ + c = __kernel_sinpil(0.5 - ax); + else if (lx < 0xc000000000000000ull) { /* |x| < 0.75 */ + if (ax == 0.5) + RETURNI(0); + c = -__kernel_sinpil(ax - 0.5); + } else + c = -__kernel_cospil(1 - ax); + RETURNI(c); + } + + if (ix < 0x403e) { /* 1 <= |x| < 0x1p63 */ + /* Determine integer part of ax. */ + j0 = ix - 0x3fff + 1; + if (j0 < 32) { + lx = (lx >> 32) << 32; + lx &= ~(((lx << 32)-1) >> j0); + } else { + m = (uint64_t)-1 >> (j0 + 1); + if (lx & m) lx &= ~m; + } + INSERT_LDBL80_WORDS(x, ix, lx); + + ax -= x; + EXTRACT_LDBL80_WORDS(ix, lx, ax); + + if (ix < 0x3ffe) { /* |x| < 0.5 */ + if (ix < 0x3ffd) /* |x| < 0.25 */ + c = ix == 0 ? 1 : __kernel_cospil(ax); + else + c = __kernel_sinpil(0.5 - ax); + + } else { + if (lx < 0xc000000000000000ull) { /* |x| < 0.75 */ + if (ax == 0.5) + RETURNI(0); + c = -__kernel_sinpil(ax - 0.5); + } else + c = -__kernel_cospil(1 - ax); + } + + if (j0 > 40) + x -= 0x1p40; + if (j0 > 30) + x -= 0x1p30; + j0 = (uint32_t)x; + + RETURNI(j0 & 1 ? -c : c); + } + + if (ix >= 0x7fff) + RETURNI(vzero / vzero); + + /* + * |x| >= 0x1p63 is always an even integer, so return 1. + */ + RETURNI(1); +} diff --git a/newlib/libm/ld80/s_erfl.c b/newlib/libm/ld80/s_erfl.c new file mode 100644 index 000000000..1ae2f9092 --- /dev/null +++ b/newlib/libm/ld80/s_erfl.c @@ -0,0 +1,337 @@ +/* @(#)s_erf.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include +__FBSDID("$FreeBSD$"); + +/* + * See s_erf.c for complete comments. + * + * Converted to long double by Steven G. Kargl. + */ +#include +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile long double tiny = 0x1p-10000L; + +static const double +half= 0.5, +one = 1, +two = 2; +/* + * In the domain [0, 2**-34], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-102. + */ +static const union IEEEl2bits +efxu = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), +efx8u = LD80C(0x8375d410a6db446c, 0, 1.02703333676410059122e+0L), +/* + * Domain [0, 0.84375], range ~[-1.423e-22, 1.423e-22]: + * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-72.573 + */ +pp0u = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), +pp1u = LD80C(0xa46c7d09ec3d0cec, -2, -3.21140201054840180596e-1L), +pp2u = LD80C(0x9b31e66325576f86, -5, -3.78893851760347812082e-2L), +pp3u = LD80C(0x804ac72c9a0b97dd, -7, -7.83032847030604679616e-3L), +pp4u = LD80C(0x9f42bcbc3d5a601d, -12, -3.03765663857082048459e-4L), +pp5u = LD80C(0x9ec4ad6193470693, -16, -1.89266527398167917502e-5L), +qq1u = LD80C(0xdb4b8eb713188d6b, -2, 4.28310832832310510579e-1L), +qq2u = LD80C(0xa5750835b2459bd1, -4, 8.07896272074540216658e-2L), +qq3u = LD80C(0x8b85d6bd6a90b51c, -7, 8.51579638189385354266e-3L), +qq4u = LD80C(0x87332f82cff4ff96, -11, 5.15746855583604912827e-4L), +qq5u = LD80C(0x83466cb6bf9dca00, -16, 1.56492109706256700009e-5L), +qq6u = LD80C(0xf5bf98c2f996bf63, -24, 1.14435527803073879724e-7L); +#define efx (efxu.e) +#define efx8 (efx8u.e) +#define pp0 (pp0u.e) +#define pp1 (pp1u.e) +#define pp2 (pp2u.e) +#define pp3 (pp3u.e) +#define pp4 (pp4u.e) +#define pp5 (pp5u.e) +#define qq1 (qq1u.e) +#define qq2 (qq2u.e) +#define qq3 (qq3u.e) +#define qq4 (qq4u.e) +#define qq5 (qq5u.e) +#define qq6 (qq6u.e) +static const union IEEEl2bits +erxu = LD80C(0xd7bb3d0000000000, -1, 8.42700779438018798828e-1L), +/* + * Domain [0.84375, 1.25], range ~[-8.132e-22, 8.113e-22]: + * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-71.762 + */ +pa0u = LD80C(0xe8211158da02c692, -27, 1.35116960705131296711e-8L), +pa1u = LD80C(0xd488f89f36988618, -2, 4.15107507167065612570e-1L), +pa2u = LD80C(0xece74f8c63fa3942, -4, -1.15675565215949226989e-1L), +pa3u = LD80C(0xc8d31e020727c006, -4, 9.80589241379624665791e-2L), +pa4u = LD80C(0x985d5d5fafb0551f, -5, 3.71984145558422368847e-2L), +pa5u = LD80C(0xa5b6c4854d2f5452, -8, -5.05718799340957673661e-3L), +pa6u = LD80C(0x85c8d58fe3993a47, -8, 4.08277919612202243721e-3L), +pa7u = LD80C(0xddbfbc23677b35cf, -13, 2.11476292145347530794e-4L), +qa1u = LD80C(0xb8a977896f5eff3f, -1, 7.21335860303380361298e-1L), +qa2u = LD80C(0x9fcd662c3d4eac86, -1, 6.24227891731886593333e-1L), +qa3u = LD80C(0x9d0b618eac67ba07, -2, 3.06727455774491855801e-1L), +qa4u = LD80C(0x881a4293f6d6c92d, -3, 1.32912674218195890535e-1L), +qa5u = LD80C(0xbab144f07dea45bf, -5, 4.55792134233613027584e-2L), +qa6u = LD80C(0xa6c34ba438bdc900, -7, 1.01783980070527682680e-2L), +qa7u = LD80C(0x8fa866dc20717a91, -9, 2.19204436518951438183e-3L); +#define erx (erxu.e) +#define pa0 (pa0u.e) +#define pa1 (pa1u.e) +#define pa2 (pa2u.e) +#define pa3 (pa3u.e) +#define pa4 (pa4u.e) +#define pa5 (pa5u.e) +#define pa6 (pa6u.e) +#define pa7 (pa7u.e) +#define qa1 (qa1u.e) +#define qa2 (qa2u.e) +#define qa3 (qa3u.e) +#define qa4 (qa4u.e) +#define qa5 (qa5u.e) +#define qa6 (qa6u.e) +#define qa7 (qa7u.e) +static const union IEEEl2bits +/* + * Domain [1.25,2.85715], range ~[-2.334e-22,2.334e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-71.860 + */ +ra0u = LD80C(0xa1a091e0fb4f335a, -7, -9.86494298915814308249e-3L), +ra1u = LD80C(0xc2b0d045ae37df6b, -1, -7.60510460864878271275e-1L), +ra2u = LD80C(0xf2cec3ee7da636c5, 3, -1.51754798236892278250e+1L), +ra3u = LD80C(0x813cc205395adc7d, 7, -1.29237335516455333420e+2L), +ra4u = LD80C(0x8737c8b7b4062c2f, 9, -5.40871625829510494776e+2L), +ra5u = LD80C(0x8ffe5383c08d4943, 10, -1.15194769466026108551e+3L), +ra6u = LD80C(0x983573e64d5015a9, 10, -1.21767039790249025544e+3L), +ra7u = LD80C(0x92a794e763a6d4db, 9, -5.86618463370624636688e+2L), +ra8u = LD80C(0xd5ad1fae77c3d9a3, 6, -1.06838132335777049840e+2L), +ra9u = LD80C(0x934c1a247807bb9c, 2, -4.60303980944467334806e+0L), +sa1u = LD80C(0xd342f90012bb1189, 4, 2.64077014928547064865e+1L), +sa2u = LD80C(0x839be13d9d5da883, 8, 2.63217811300123973067e+2L), +sa3u = LD80C(0x9f8cba6d1ae1b24b, 10, 1.27639775710344617587e+3L), +sa4u = LD80C(0xcaa83f403713e33e, 11, 3.24251544209971162003e+3L), +sa5u = LD80C(0x8796aff2f3c47968, 12, 4.33883591261332837874e+3L), +sa6u = LD80C(0xb6ef97f9c753157b, 11, 2.92697460344182158454e+3L), +sa7u = LD80C(0xe02aee5f83773d1c, 9, 8.96670799139389559818e+2L), +sa8u = LD80C(0xc82b83855b88e07e, 6, 1.00084987800048510018e+2L), +sa9u = LD80C(0x92f030aefadf28ad, 1, 2.29591004455459083843e+0L); +#define ra0 (ra0u.e) +#define ra1 (ra1u.e) +#define ra2 (ra2u.e) +#define ra3 (ra3u.e) +#define ra4 (ra4u.e) +#define ra5 (ra5u.e) +#define ra6 (ra6u.e) +#define ra7 (ra7u.e) +#define ra8 (ra8u.e) +#define ra9 (ra9u.e) +#define sa1 (sa1u.e) +#define sa2 (sa2u.e) +#define sa3 (sa3u.e) +#define sa4 (sa4u.e) +#define sa5 (sa5u.e) +#define sa6 (sa6u.e) +#define sa7 (sa7u.e) +#define sa8 (sa8u.e) +#define sa9 (sa9u.e) +/* + * Domain [2.85715,7], range ~[-8.323e-22,8.390e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-70.326 + */ +static const union IEEEl2bits +rb0u = LD80C(0xa1a091cf43abcd26, -7, -9.86494292470284646962e-3L), +rb1u = LD80C(0xd19d2df1cbb8da0a, -1, -8.18804618389296662837e-1L), +rb2u = LD80C(0x9a4dd1383e5daf5b, 4, -1.92879967111618594779e+1L), +rb3u = LD80C(0xbff0ae9fc0751de6, 7, -1.91940164551245394969e+2L), +rb4u = LD80C(0xdde08465310b472b, 9, -8.87508080766577324539e+2L), +rb5u = LD80C(0xe796e1d38c8c70a9, 10, -1.85271506669474503781e+3L), +rb6u = LD80C(0xbaf655a76e0ab3b5, 10, -1.49569795581333675349e+3L), +rb7u = LD80C(0x95d21e3e75503c21, 8, -2.99641547972948019157e+2L), +sb1u = LD80C(0x814487ed823c8cbd, 5, 3.23169247732868256569e+1L), +sb2u = LD80C(0xbe4bfbb1301304be, 8, 3.80593618534539961773e+2L), +sb3u = LD80C(0x809c4ade46b927c7, 11, 2.05776827838541292848e+3L), +sb4u = LD80C(0xa55284359f3395a8, 12, 5.29031455540062116327e+3L), +sb5u = LD80C(0xbcfa72da9b820874, 12, 6.04730608102312640462e+3L), +sb6u = LD80C(0x9d09a35988934631, 11, 2.51260238030767176221e+3L), +sb7u = LD80C(0xd675bbe542c159fa, 7, 2.14459898308561015684e+2L); +#define rb0 (rb0u.e) +#define rb1 (rb1u.e) +#define rb2 (rb2u.e) +#define rb3 (rb3u.e) +#define rb4 (rb4u.e) +#define rb5 (rb5u.e) +#define rb6 (rb6u.e) +#define rb7 (rb7u.e) +#define sb1 (sb1u.e) +#define sb2 (sb2u.e) +#define sb3 (sb3u.e) +#define sb4 (sb4u.e) +#define sb5 (sb5u.e) +#define sb6 (sb6u.e) +#define sb7 (sb7u.e) +/* + * Domain [7,108], range ~[-4.422e-22,4.422e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rc(x)/sc(x)| < 2**-70.938 + */ +static const union IEEEl2bits +/* err = -4.422092275318925082e-22 -70.937689 */ +rc0u = LD80C(0xa1a091cf437a17ad, -7, -9.86494292470008707260e-3L), +rc1u = LD80C(0xbe79c5a978122b00, -1, -7.44045595049165939261e-1L), +rc2u = LD80C(0xdb26f9bbe31a2794, 3, -1.36970155085888424425e+1L), +rc3u = LD80C(0xb5f69a38f5747ac8, 6, -9.09816453742625888546e+1L), +rc4u = LD80C(0xd79676d970d0a21a, 7, -2.15587750997584074147e+2L), +rc5u = LD80C(0xfe528153c45ec97c, 6, -1.27161142938347796666e+2L), +sc1u = LD80C(0xc5e8cd46d5604a96, 4, 2.47386727842204312937e+1L), +sc2u = LD80C(0xc5f0f5a5484520eb, 7, 1.97941248254913378865e+2L), +sc3u = LD80C(0x964e3c7b34db9170, 9, 6.01222441484087787522e+2L), +sc4u = LD80C(0x99be1b89faa0596a, 9, 6.14970430845978077827e+2L), +sc5u = LD80C(0xf80dfcbf37ffc5ea, 6, 1.24027318931184605891e+2L); +#define rc0 (rc0u.e) +#define rc1 (rc1u.e) +#define rc2 (rc2u.e) +#define rc3 (rc3u.e) +#define rc4 (rc4u.e) +#define rc5 (rc5u.e) +#define sc1 (sc1u.e) +#define sc2 (sc2u.e) +#define sc3 (sc3u.e) +#define sc4 (sc4u.e) +#define sc5 (sc5u.e) + +long double +erfl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx; + int32_t i; + uint16_t hx; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfl(nan)=nan */ + i = (hx>>15)<<1; + return (1-i)+one/x; /* erfl(+-inf)=+-1 */ + } + + ENTERI(); + + ax = fabsl(x); + if(ax < 0.84375) { + if(ax < 0x1p-34L) { + if(ax < 0x1p-16373L) + RETURNI((8*x+efx8*x)/8); /* avoid spurious underflow */ + RETURNI(x + efx*x); + } + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); + y = r/s; + RETURNI(x + x*y); + } + if(ax < 1.25) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); + if(x>=0) RETURNI(erx + P/Q); else RETURNI(-erx - P/Q); + } + if(ax >= 7) { /* inf>|x|>= 7 */ + if(x>=0) RETURNI(one-tiny); else RETURNI(tiny-one); + } + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*ra9)))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*sa9)))))))); + } else { /* |x| >= 2.85715 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); + } + z=(float)ax; + r=expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>=0) RETURNI(one-r/ax); else RETURNI(r/ax-one); +} + +long double +erfcl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx; + uint16_t hx; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfcl(nan)=nan */ + /* erfcl(+-inf)=0,2 */ + return ((hx>>15)<<1)+one/x; + } + + ENTERI(); + + ax = fabsl(x); + if(ax < 0.84375L) { + if(ax < 0x1p-34L) + RETURNI(one-x); + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); + y = r/s; + if(ax < 0.25L) { /* x<1/4 */ + RETURNI(one-(x+x*y)); + } else { + r = x*y; + r += (x-half); + RETURNI(half - r); + } + } + if(ax < 1.25L) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); + if(x>=0) { + z = one-erx; RETURNI(z - P/Q); + } else { + z = (erx+P/Q); RETURNI(one+z); + } + } + + if(ax < 108) { /* |x| < 108 */ + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*ra9)))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*sa9)))))))); + } else if(ax < 7) { /* | |x| < 7 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); + } else { + if(x < -7) RETURNI(two-tiny);/* x < -7 */ + R=rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*rc5)))); + S=one+s*(sc1+s*(sc2+s*(sc3+s*(sc4+s*sc5)))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>0) RETURNI(r/ax); else RETURNI(two-r/ax); + } else { + if(x>0) RETURNI(tiny*tiny); else RETURNI(two-tiny); + } +} diff --git a/newlib/libm/ld80/s_exp2l.c b/newlib/libm/ld80/s_exp2l.c new file mode 100644 index 000000000..421d6e2e0 --- /dev/null +++ b/newlib/libm/ld80/s_exp2l.c @@ -0,0 +1,290 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2005-2008 David Schultz + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND + * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE + * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE + * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS + * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT + * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY + * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF + * SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +#include +#include + +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +#define TBLBITS 7 +#define TBLSIZE (1 << TBLBITS) + +#define BIAS (LDBL_MAX_EXP - 1) + +static volatile long double + huge = 0x1p10000L, + twom10000 = 0x1p-10000L; + +static const union IEEEl2bits +P1 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309429e-1L); + +static const double +redux = 0x1.8p63 / TBLSIZE, +/* + * Domain [-0.00390625, 0.00390625], range ~[-1.7079e-23, 1.7079e-23] + * |exp(x) - p(x)| < 2**-75.6 + */ +P2 = 2.4022650695910072e-1, /* 0x1ebfbdff82c58f.0p-55 */ +P3 = 5.5504108664816879e-2, /* 0x1c6b08d7049e1a.0p-57 */ +P4 = 9.6181291055695180e-3, /* 0x13b2ab6fa8321a.0p-59 */ +P5 = 1.3333563089183052e-3, /* 0x15d8806f67f251.0p-62 */ +P6 = 1.5413361552277414e-4; /* 0x1433ddacff3441.0p-65 */ + +static const double tbl[TBLSIZE * 2] = { + 0x1.6a09e667f3bcdp-1, -0x1.bdd3413b2648p-55, + 0x1.6c012750bdabfp-1, -0x1.2895667ff0cp-57, + 0x1.6dfb23c651a2fp-1, -0x1.bbe3a683c88p-58, + 0x1.6ff7df9519484p-1, -0x1.83c0f25860fp-56, + 0x1.71f75e8ec5f74p-1, -0x1.16e4786887bp-56, + 0x1.73f9a48a58174p-1, -0x1.0a8d96c65d5p-55, + 0x1.75feb564267c9p-1, -0x1.0245957316ep-55, + 0x1.780694fde5d3fp-1, 0x1.866b80a0216p-55, + 0x1.7a11473eb0187p-1, -0x1.41577ee0499p-56, + 0x1.7c1ed0130c132p-1, 0x1.f124cd1164ep-55, + 0x1.7e2f336cf4e62p-1, 0x1.05d02ba157ap-57, + 0x1.80427543e1a12p-1, -0x1.27c86626d97p-55, + 0x1.82589994cce13p-1, -0x1.d4c1dd41533p-55, + 0x1.8471a4623c7adp-1, -0x1.8d684a341cep-56, + 0x1.868d99b4492edp-1, -0x1.fc6f89bd4f68p-55, + 0x1.88ac7d98a6699p-1, 0x1.994c2f37cb5p-55, + 0x1.8ace5422aa0dbp-1, 0x1.6e9f156864bp-55, + 0x1.8cf3216b5448cp-1, -0x1.0d55e32e9e4p-57, + 0x1.8f1ae99157736p-1, 0x1.5cc13a2e397p-56, + 0x1.9145b0b91ffc6p-1, -0x1.dd6792e5825p-55, + 0x1.93737b0cdc5e5p-1, -0x1.75fc781b58p-58, + 0x1.95a44cbc8520fp-1, -0x1.64b7c96a5fp-57, + 0x1.97d829fde4e5p-1, -0x1.d185b7c1b86p-55, + 0x1.9a0f170ca07bap-1, -0x1.173bd91cee6p-55, + 0x1.9c49182a3f09p-1, 0x1.c7c46b071f2p-57, + 0x1.9e86319e32323p-1, 0x1.824ca78e64cp-57, + 0x1.a0c667b5de565p-1, -0x1.359495d1cd5p-55, + 0x1.a309bec4a2d33p-1, 0x1.6305c7ddc368p-55, + 0x1.a5503b23e255dp-1, -0x1.d2f6edb8d42p-55, + 0x1.a799e1330b358p-1, 0x1.bcb7ecac564p-55, + 0x1.a9e6b5579fdbfp-1, 0x1.0fac90ef7fdp-55, + 0x1.ac36bbfd3f37ap-1, -0x1.f9234cae76dp-56, + 0x1.ae89f995ad3adp-1, 0x1.7a1cd345dcc8p-55, + 0x1.b0e07298db666p-1, -0x1.bdef54c80e4p-55, + 0x1.b33a2b84f15fbp-1, -0x1.2805e3084d8p-58, + 0x1.b59728de5593ap-1, -0x1.c71dfbbba6ep-55, + 0x1.b7f76f2fb5e47p-1, -0x1.5584f7e54acp-57, + 0x1.ba5b030a1064ap-1, -0x1.efcd30e5429p-55, + 0x1.bcc1e904bc1d2p-1, 0x1.23dd07a2d9fp-56, + 0x1.bf2c25bd71e09p-1, -0x1.efdca3f6b9c8p-55, + 0x1.c199bdd85529cp-1, 0x1.11065895049p-56, + 0x1.c40ab5fffd07ap-1, 0x1.b4537e083c6p-55, + 0x1.c67f12e57d14bp-1, 0x1.2884dff483c8p-55, + 0x1.c8f6d9406e7b5p-1, 0x1.1acbc48805cp-57, + 0x1.cb720dcef9069p-1, 0x1.503cbd1e94ap-57, + 0x1.cdf0b555dc3fap-1, -0x1.dd83b53829dp-56, + 0x1.d072d4a07897cp-1, -0x1.cbc3743797a8p-55, + 0x1.d2f87080d89f2p-1, -0x1.d487b719d858p-55, + 0x1.d5818dcfba487p-1, 0x1.2ed02d75b37p-56, + 0x1.d80e316c98398p-1, -0x1.11ec18bedep-55, + 0x1.da9e603db3285p-1, 0x1.c2300696db5p-55, + 0x1.dd321f301b46p-1, 0x1.2da5778f019p-55, + 0x1.dfc97337b9b5fp-1, -0x1.1a5cd4f184b8p-55, + 0x1.e264614f5a129p-1, -0x1.7b627817a148p-55, + 0x1.e502ee78b3ff6p-1, 0x1.39e8980a9cdp-56, + 0x1.e7a51fbc74c83p-1, 0x1.2d522ca0c8ep-55, + 0x1.ea4afa2a490dap-1, -0x1.e9c23179c288p-55, + 0x1.ecf482d8e67f1p-1, -0x1.c93f3b411ad8p-55, + 0x1.efa1bee615a27p-1, 0x1.dc7f486a4b68p-55, + 0x1.f252b376bba97p-1, 0x1.3a1a5bf0d8e8p-55, + 0x1.f50765b6e454p-1, 0x1.9d3e12dd8a18p-55, + 0x1.f7bfdad9cbe14p-1, -0x1.dbb12d00635p-55, + 0x1.fa7c1819e90d8p-1, 0x1.74853f3a593p-56, + 0x1.fd3c22b8f71f1p-1, 0x1.2eb74966578p-58, + 0x1p+0, 0x0p+0, + 0x1.0163da9fb3335p+0, 0x1.b61299ab8cd8p-54, + 0x1.02c9a3e778061p+0, -0x1.19083535b08p-56, + 0x1.04315e86e7f85p+0, -0x1.0a31c1977c98p-54, + 0x1.059b0d3158574p+0, 0x1.d73e2a475b4p-55, + 0x1.0706b29ddf6dep+0, -0x1.c91dfe2b13cp-55, + 0x1.0874518759bc8p+0, 0x1.186be4bb284p-57, + 0x1.09e3ecac6f383p+0, 0x1.14878183161p-54, + 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc61p-54, + 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f8p-54, + 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c58p-59, + 0x1.0fb66affed31bp+0, -0x1.b9bedc44ebcp-57, + 0x1.11301d0125b51p+0, -0x1.6c51039449bp-54, + 0x1.12abdc06c31ccp+0, -0x1.1b514b36ca8p-58, + 0x1.1429aaea92dep+0, -0x1.32fbf9af1368p-54, + 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeabp-55, + 0x1.172b83c7d517bp+0, -0x1.19041b9d78ap-55, + 0x1.18af9388c8deap+0, -0x1.11023d1970f8p-54, + 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4969p-55, + 0x1.1bbe084045cd4p+0, -0x1.95386352ef6p-54, + 0x1.1d4873168b9aap+0, 0x1.e016e00a264p-54, + 0x1.1ed5022fcd91dp+0, -0x1.1df98027bb78p-54, + 0x1.2063b88628cd6p+0, 0x1.dc775814a85p-55, + 0x1.21f49917ddc96p+0, 0x1.2a97e9494a6p-55, + 0x1.2387a6e756238p+0, 0x1.9b07eb6c7058p-54, + 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f5p-55, + 0x1.26b4565e27cddp+0, 0x1.2bd339940eap-55, + 0x1.284dfe1f56381p+0, -0x1.a4c3a8c3f0d8p-54, + 0x1.29e9df51fdee1p+0, 0x1.612e8afad12p-55, + 0x1.2b87fd0dad99p+0, -0x1.10adcd6382p-59, + 0x1.2d285a6e4030bp+0, 0x1.0024754db42p-54, + 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d524p-56, + 0x1.306fe0a31b715p+0, 0x1.6f46ad23183p-55, + 0x1.32170fc4cd831p+0, 0x1.a9ce78e1804p-55, + 0x1.33c08b26416ffp+0, 0x1.327218436598p-54, + 0x1.356c55f929ff1p+0, -0x1.b5cee5c4e46p-55, + 0x1.371a7373aa9cbp+0, -0x1.63aeabf42ebp-54, + 0x1.38cae6d05d866p+0, -0x1.e958d3c99048p-54, + 0x1.3a7db34e59ff7p+0, -0x1.5e436d661f6p-56, + 0x1.3c32dc313a8e5p+0, -0x1.efff8375d2ap-54, + 0x1.3dea64c123422p+0, 0x1.ada0911f09fp-55, + 0x1.3fa4504ac801cp+0, -0x1.7d023f956fap-54, + 0x1.4160a21f72e2ap+0, -0x1.ef3691c309p-58, + 0x1.431f5d950a897p+0, -0x1.1c7dde35f7ap-55, + 0x1.44e086061892dp+0, 0x1.89b7a04ef8p-59, + 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a8p-54, + 0x1.486a2b5c13cdp+0, 0x1.3c1a3b69062p-56, + 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be8p-54, + 0x1.4bfdad5362a27p+0, 0x1.d4397afec42p-56, + 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a78p-54, + 0x1.4f9b2769d2ca7p+0, -0x1.4b309d25958p-54, + 0x1.516daa2cf6642p+0, -0x1.f768569bd94p-55, + 0x1.5342b569d4f82p+0, -0x1.07abe1db13dp-55, + 0x1.551a4ca5d920fp+0, -0x1.d689cefede6p-55, + 0x1.56f4736b527dap+0, 0x1.9bb2c011d938p-54, + 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1ep-55, + 0x1.5ab07dd485429p+0, 0x1.6324c0546478p-54, + 0x1.5c9268a5946b7p+0, 0x1.c4b1b81698p-60, + 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e68p-54, + 0x1.605e1b976dc09p+0, -0x1.3e2429b56de8p-54, + 0x1.6247eb03a5585p+0, -0x1.383c17e40b48p-54, + 0x1.6434634ccc32p+0, -0x1.c483c759d89p-55, + 0x1.6623882552225p+0, -0x1.bb60987591cp-54, + 0x1.68155d44ca973p+0, 0x1.038ae44f74p-57, +}; + +/** + * Compute the base 2 exponential of x for Intel 80-bit format. + * + * Accuracy: Peak error < 0.511 ulp. + * + * Method: (equally-spaced tables) + * + * Reduce x: + * x = 2**k + y, for integer k and |y| <= 1/2. + * Thus we have exp2l(x) = 2**k * exp2(y). + * + * Reduce y: + * y = i/TBLSIZE + z for integer i near y * TBLSIZE. + * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), + * with |z| <= 2**-(TBLBITS+1). + * + * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a + * degree-6 minimax polynomial with maximum error under 2**-75.6. + * The table entries each have 104 bits of accuracy, encoded as + * a pair of double precision values. + */ +long double +exp2l(long double x) +{ + union IEEEl2bits u, v; + long double r, twopk, twopkp10000, z; + uint32_t hx, ix, i0; + int k; + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 14) { /* |x| >= 16384 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000 && u.xbits.man == 1ULL << 63) + return (0.0L); /* x is -Inf */ + return (x + x); /* x is +Inf, NaN or unsupported */ + } + if (x >= 16384) + return (huge * huge); /* overflow */ + if (x <= -16446) + return (twom10000 * twom10000); /* underflow */ + } else if (ix <= BIAS - 66) { /* |x| < 0x1p-65 (includes pseudos) */ + return (1.0L + x); /* 1 with inexact */ + } + + ENTERI(); + + /* + * Reduce x, computing z, i0, and k. The low bits of x + redux + * contain the 16-bit integer part of the exponent (k) followed by + * TBLBITS fractional bits (i0). We use bit tricks to extract these + * as integers, then set z to the remainder. + * + * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8. + * Then the low-order word of x + redux is 0x000abc12, + * We split this into k = 0xabc and i0 = 0x12 (adjusted to + * index into the table), then we compute z = 0x0.003456p0. + * + * XXX If the exponent is negative, the computation of k depends on + * '>>' doing sign extension. + */ + u.e = x + redux; + i0 = u.bits.manl + TBLSIZE / 2; + k = (int)i0 >> TBLBITS; + i0 = (i0 & (TBLSIZE - 1)) << 1; + u.e -= redux; + z = x - u.e; + v.xbits.man = 1ULL << 63; + if (k >= LDBL_MIN_EXP) { + v.xbits.expsign = BIAS + k; + twopk = v.e; + } else { + v.xbits.expsign = BIAS + k + 10000; + twopkp10000 = v.e; + } + + /* Compute r = exp2l(y) = exp2lt[i0] * p(z). */ + long double t_hi = tbl[i0]; + long double t_lo = tbl[i0 + 1]; + r = t_lo + (t_hi + t_lo) * z * (P1.e + z * (P2 + z * (P3 + z * (P4 + + z * (P5 + z * P6))))) + t_hi; + + /* Scale by 2**k. */ + if (k >= LDBL_MIN_EXP) { + if (k == LDBL_MAX_EXP) + RETURNI(r * 2.0 * 0x1p16383L); + RETURNI(r * twopk); + } else { + RETURNI(r * twopkp10000 * twom10000); + } +} diff --git a/newlib/libm/ld80/s_expl.c b/newlib/libm/ld80/s_expl.c new file mode 100644 index 000000000..e46e73f0c --- /dev/null +++ b/newlib/libm/ld80/s_expl.c @@ -0,0 +1,279 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include +__FBSDID("$FreeBSD$"); + +/** + * Compute the exponential of x for Intel 80-bit format. This is based on: + * + * PTP Tang, "Table-driven implementation of the exponential function + * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, + * 144-157 (1989). + * + * where the 32 table entries have been expanded to INTERVALS (see below). + */ + +#include + +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double +huge = 0x1p10000L, +tiny = 0x1p-10000L; + +static const long double +twom10000 = 0x1p-10000L; + +static const union IEEEl2bits +/* log(2**16384 - 0.5) rounded towards zero: */ +/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ +o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), +#define o_threshold (o_thresholdu.e) +/* log(2**(-16381-64-1)) rounded towards zero: */ +u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); +#define u_threshold (u_thresholdu.e) + +long double +expl(long double x) +{ + union IEEEl2bits u; + long double hi, lo, t, twopk; + int k; + uint16_t hx, ix; + + DOPRINT_START(&x); + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ + RETURNP(-1 / x); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ + } + if (x > o_threshold) + RETURNP(huge * huge); + if (x < u_threshold) + RETURNP(tiny * tiny); + } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ + RETURN2P(1, x); /* 1 with inexact iff x != 0 */ + } + + ENTERI(); + + twopk = 1; + __k_expl(x, &hi, &lo, &k); + t = SUM2P(hi, lo); + + /* Scale by 2**k. */ + if (k >= LDBL_MIN_EXP) { + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L); + SET_LDBL_EXPSIGN(twopk, BIAS + k); + RETURNI(t * twopk); + } else { + SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); + RETURNI(t * twopk * twom10000); + } +} + +/** + * Compute expm1l(x) for Intel 80-bit format. This is based on: + * + * PTP Tang, "Table-driven implementation of the Expm1 function + * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, + * 211-222 (1992). + */ + +/* + * Our T1 and T2 are chosen to be approximately the points where method + * A and method B have the same accuracy. Tang's T1 and T2 are the + * points where method A's accuracy changes by a full bit. For Tang, + * this drop in accuracy makes method A immediately less accurate than + * method B, but our larger INTERVALS makes method A 2 bits more + * accurate so it remains the most accurate method significantly + * closer to the origin despite losing the full bit in our extended + * range for it. + */ +static const double +T1 = -0.1659, /* ~-30.625/128 * log(2) */ +T2 = 0.1659; /* ~30.625/128 * log(2) */ + +/* + * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 + * + * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, + * but unlike for ld128 we can't drop any terms. + */ +static const union IEEEl2bits +B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), +B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); + +static const double +B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ +B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ +B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ +B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ +B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ +B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ +B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ +B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ + +long double +expm1l(long double x) +{ + union IEEEl2bits u, v; + long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; + long double x_lo, x2, z; + long double x4; + int k, n, n2; + uint16_t hx, ix; + + DOPRINT_START(&x); + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ + RETURNP(-1 / x - 1); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ + } + if (x > o_threshold) + RETURNP(huge * huge); + /* + * expm1l() never underflows, but it must avoid + * unrepresentable large negative exponents. We used a + * much smaller threshold for large |x| above than in + * expl() so as to handle not so large negative exponents + * in the same way as large ones here. + */ + if (hx & 0x8000) /* x <= -64 */ + RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ + } + + ENTERI(); + + if (T1 < x && x < T2) { + if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ + /* x (rounded) with inexact if x != 0: */ + RETURNPI(x == 0 ? x : + (0x1p100 * x + fabsl(x)) * 0x1p-100); + } + + x2 = x * x; + x4 = x2 * x2; + q = x4 * (x2 * (x4 * + /* + * XXX the number of terms is no longer good for + * pairwise grouping of all except B3, and the + * grouping is no longer from highest down. + */ + (x2 * B12 + (x * B11 + B10)) + + (x2 * (x * B9 + B8) + (x * B7 + B6))) + + (x * B5 + B4.e)) + x2 * x * B3.e; + + x_hi = (float)x; + x_lo = x - x_hi; + hx2_hi = x_hi * x_hi / 2; + hx2_lo = x_lo * (x + x_hi) / 2; + if (ix >= BIAS - 7) + RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); + else + RETURN2PI(x, hx2_lo + q + hx2_hi); + } + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + fn = rnintl(x * INV_L); + n = irint(fn); + n2 = (unsigned)n % INTERVALS; + k = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Prepare scale factor. */ + v.e = 1; + v.xbits.expsign = BIAS + k; + twopk = v.e; + + /* + * Evaluate lower terms of + * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). + */ + z = r * r; + q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; + + t = (long double)tbl[n2].lo + tbl[n2].hi; + + if (k == 0) { + t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); + RETURNI(t); + } + if (k == -1) { + t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); + RETURNI(t / 2); + } + if (k < -7) { + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); + RETURNI(t * twopk - 1); + } + if (k > 2 * LDBL_MANT_DIG - 1) { + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L - 1); + RETURNI(t * twopk - 1); + } + + v.xbits.expsign = BIAS - k; + twomk = v.e; + + if (k > LDBL_MANT_DIG - 1) + t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); + else + t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); + RETURNI(t * twopk); +} diff --git a/newlib/libm/ld80/s_logl.c b/newlib/libm/ld80/s_logl.c new file mode 100644 index 000000000..c74519caf --- /dev/null +++ b/newlib/libm/ld80/s_logl.c @@ -0,0 +1,722 @@ +/*- + * SPDX-License-Identifier: BSD-2-Clause-FreeBSD + * + * Copyright (c) 2007-2013 Bruce D. Evans + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +#include +__FBSDID("$FreeBSD$"); + +/** + * Implementation of the natural logarithm of x for Intel 80-bit format. + * + * First decompose x into its base 2 representation: + * + * log(x) = log(X * 2**k), where X is in [1, 2) + * = log(X) + k * log(2). + * + * Let X = X_i + e, where X_i is the center of one of the intervals + * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256) + * and X is in this interval. Then + * + * log(X) = log(X_i + e) + * = log(X_i * (1 + e / X_i)) + * = log(X_i) + log(1 + e / X_i). + * + * The values log(X_i) are tabulated below. Let d = e / X_i and use + * + * log(1 + d) = p(d) + * + * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of + * suitably high degree. + * + * To get sufficiently small roundoff errors, k * log(2), log(X_i), and + * sometimes (if |k| is not large) the first term in p(d) must be evaluated + * and added up in extra precision. Extra precision is not needed for the + * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final + * error is controlled mainly by the error in the second term in p(d). The + * error in this term itself is at most 0.5 ulps from the d*d operation in + * it. The error in this term relative to the first term is thus at most + * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of + * at most twice this at the point of the final rounding step. Thus the + * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive + * testing of a float variant of this function showed a maximum final error + * of 0.5008 ulps. Non-exhaustive testing of a double variant of this + * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256). + * + * We made the maximum of |d| (and thus the total relative error and the + * degree of p(d)) small by using a large number of intervals. Using + * centers of intervals instead of endpoints reduces this maximum by a + * factor of 2 for a given number of intervals. p(d) is special only + * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen + * naturally. The most accurate minimax polynomial of a given degree might + * be different, but then we wouldn't want it since we would have to do + * extra work to avoid roundoff error (especially for P0*d instead of d). + */ + +#ifdef DEBUG +#include +#include +#endif + +#ifdef __i386__ +#include +#endif + +#include "fpmath.h" +#include "math.h" +#define i386_SSE_GOOD +#ifndef NO_STRUCT_RETURN +#define STRUCT_RETURN +#endif +#include "math_private.h" + +#if !defined(NO_UTAB) && !defined(NO_UTABL) +#define USE_UTAB +#endif + +/* + * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]: + * |log(1 + d)/d - p(d)| < 2**-70.7 + */ +static const double +P2 = -0.5, +P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */ +P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */ +P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */ +P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */ +P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */ +P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */ + +static volatile const double zero = 0; + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define TSIZE (INTERVALS + 1) +#define G(i) (T[(i)].G) +#define F_hi(i) (T[(i)].F_hi) +#define F_lo(i) (T[(i)].F_lo) +#define ln2_hi F_hi(TSIZE - 1) +#define ln2_lo F_lo(TSIZE - 1) +#define E(i) (U[(i)].E) +#define H(i) (U[(i)].H) + +static const struct { + float G; /* 1/(1 + i/128) rounded to 8/9 bits */ + float F_hi; /* log(1 / G_i) rounded (see below) */ + double F_lo; /* next 53 bits for log(1 / G_i) */ +} T[TSIZE] = { + /* + * ln2_hi and each F_hi(i) are rounded to a number of bits that + * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. + * + * The last entry (for X just below 2) is used to define ln2_hi + * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly + * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. + * This is needed for accuracy when x is just below 1. (To avoid + * special cases, such x are "reduced" strangely to X just below + * 2 and dk = -1, and then the exact cancellation is needed + * because any the error from any non-exactness would be too + * large). + * + * We want to share this table between double precision and ld80, + * so the relevant range of dk is the larger one of ld80 + * ([-16445, 16383]) and the relevant exactness requirement is + * the stricter one of double precision. The maximum number of + * bits in F_hi(i) that works is very dependent on i but has + * a minimum of 33. We only need about 12 bits in F_hi(i) for + * it to provide enough extra precision in double precision (11 + * more than that are required for ld80). + * + * We round F_hi(i) to 24 bits so that it can have type float, + * mainly to minimize the size of the table. Using all 24 bits + * in a float for it automatically satisfies the above constraints. + */ + { 0x800000.0p-23, 0, 0 }, + { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84 }, + { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84 }, + { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83 }, + { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82 }, + { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82 }, + { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83 }, + { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82 }, + { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91 }, + { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81 }, + { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82 }, + { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83 }, + { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81 }, + { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83 }, + { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85 }, + { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84 }, + { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81 }, + { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82 }, + { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80 }, + { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83 }, + { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82 }, + { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80 }, + { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82 }, + { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81 }, + { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84 }, + { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80 }, + { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81 }, + { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80 }, + { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82 }, + { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81 }, + { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80 }, + { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81 }, + { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81 }, + { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85 }, + { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87 }, + { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81 }, + { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80 }, + { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79 }, + { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79 }, + { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81 }, + { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79 }, + { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81 }, + { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79 }, + { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80 }, + { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81 }, + { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79 }, + { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79 }, + { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81 }, + { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81 }, + { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82 }, + { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80 }, + { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79 }, + { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83 }, + { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79 }, + { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82 }, + { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80 }, + { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79 }, + { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79 }, + { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80 }, + { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79 }, + { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80 }, + { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80 }, + { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79 }, + { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81 }, + { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79 }, + { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81 }, + { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81 }, + { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79 }, + { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80 }, + { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80 }, + { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82 }, + { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80 }, + { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80 }, + { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80 }, + { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80 }, + { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79 }, + { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79 }, + { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80 }, + { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81 }, + { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79 }, + { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84 }, + { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79 }, + { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79 }, + { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79 }, + { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78 }, + { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81 }, + { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 }, + { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 }, + { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 }, + { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 }, + { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 }, + { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 }, + { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 }, + { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 }, + { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 }, + { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 }, + { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 }, + { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 }, + { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 }, + { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 }, + { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 }, + { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 }, + { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 }, + { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 }, + { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 }, + { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 }, + { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 }, + { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 }, + { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 }, + { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 }, + { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 }, + { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 }, + { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 }, + { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 }, + { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 }, + { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 }, + { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 }, + { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 }, + { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 }, + { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 }, + { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 }, + { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 }, + { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 }, + { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 }, + { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 }, + { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 }, + { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 }, + { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 }, + { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 }, +}; + +#ifdef USE_UTAB +static const struct { + float H; /* 1 + i/INTERVALS (exact) */ + float E; /* H(i) * G(i) - 1 (exact) */ +} U[TSIZE] = { + { 0x800000.0p-23, 0 }, + { 0x810000.0p-23, -0x800000.0p-37 }, + { 0x820000.0p-23, -0x800000.0p-35 }, + { 0x830000.0p-23, -0x900000.0p-34 }, + { 0x840000.0p-23, -0x800000.0p-33 }, + { 0x850000.0p-23, -0xc80000.0p-33 }, + { 0x860000.0p-23, -0xa00000.0p-36 }, + { 0x870000.0p-23, 0x940000.0p-33 }, + { 0x880000.0p-23, 0x800000.0p-35 }, + { 0x890000.0p-23, -0xc80000.0p-34 }, + { 0x8a0000.0p-23, 0xe00000.0p-36 }, + { 0x8b0000.0p-23, 0x900000.0p-33 }, + { 0x8c0000.0p-23, -0x800000.0p-35 }, + { 0x8d0000.0p-23, -0xe00000.0p-33 }, + { 0x8e0000.0p-23, 0x880000.0p-33 }, + { 0x8f0000.0p-23, -0xa80000.0p-34 }, + { 0x900000.0p-23, -0x800000.0p-35 }, + { 0x910000.0p-23, 0x800000.0p-37 }, + { 0x920000.0p-23, 0x900000.0p-35 }, + { 0x930000.0p-23, 0xd00000.0p-35 }, + { 0x940000.0p-23, 0xe00000.0p-35 }, + { 0x950000.0p-23, 0xc00000.0p-35 }, + { 0x960000.0p-23, 0xe00000.0p-36 }, + { 0x970000.0p-23, -0x800000.0p-38 }, + { 0x980000.0p-23, -0xc00000.0p-35 }, + { 0x990000.0p-23, -0xd00000.0p-34 }, + { 0x9a0000.0p-23, 0x880000.0p-33 }, + { 0x9b0000.0p-23, 0xe80000.0p-35 }, + { 0x9c0000.0p-23, -0x800000.0p-35 }, + { 0x9d0000.0p-23, 0xb40000.0p-33 }, + { 0x9e0000.0p-23, 0x880000.0p-34 }, + { 0x9f0000.0p-23, -0xe00000.0p-35 }, + { 0xa00000.0p-23, 0x800000.0p-33 }, + { 0xa10000.0p-23, -0x900000.0p-36 }, + { 0xa20000.0p-23, -0xb00000.0p-33 }, + { 0xa30000.0p-23, -0xa00000.0p-36 }, + { 0xa40000.0p-23, 0x800000.0p-33 }, + { 0xa50000.0p-23, -0xf80000.0p-35 }, + { 0xa60000.0p-23, 0x880000.0p-34 }, + { 0xa70000.0p-23, -0x900000.0p-33 }, + { 0xa80000.0p-23, -0x800000.0p-35 }, + { 0xa90000.0p-23, 0x900000.0p-34 }, + { 0xaa0000.0p-23, 0xa80000.0p-33 }, + { 0xab0000.0p-23, -0xac0000.0p-34 }, + { 0xac0000.0p-23, -0x800000.0p-37 }, + { 0xad0000.0p-23, 0xf80000.0p-35 }, + { 0xae0000.0p-23, 0xf80000.0p-34 }, + { 0xaf0000.0p-23, -0xac0000.0p-33 }, + { 0xb00000.0p-23, -0x800000.0p-33 }, + { 0xb10000.0p-23, -0xb80000.0p-34 }, + { 0xb20000.0p-23, -0x800000.0p-34 }, + { 0xb30000.0p-23, -0xb00000.0p-35 }, + { 0xb40000.0p-23, -0x800000.0p-35 }, + { 0xb50000.0p-23, -0xe00000.0p-36 }, + { 0xb60000.0p-23, -0x800000.0p-35 }, + { 0xb70000.0p-23, -0xb00000.0p-35 }, + { 0xb80000.0p-23, -0x800000.0p-34 }, + { 0xb90000.0p-23, -0xb80000.0p-34 }, + { 0xba0000.0p-23, -0x800000.0p-33 }, + { 0xbb0000.0p-23, -0xac0000.0p-33 }, + { 0xbc0000.0p-23, 0x980000.0p-33 }, + { 0xbd0000.0p-23, 0xbc0000.0p-34 }, + { 0xbe0000.0p-23, 0xe00000.0p-36 }, + { 0xbf0000.0p-23, -0xb80000.0p-35 }, + { 0xc00000.0p-23, -0x800000.0p-33 }, + { 0xc10000.0p-23, 0xa80000.0p-33 }, + { 0xc20000.0p-23, 0x900000.0p-34 }, + { 0xc30000.0p-23, -0x800000.0p-35 }, + { 0xc40000.0p-23, -0x900000.0p-33 }, + { 0xc50000.0p-23, 0x820000.0p-33 }, + { 0xc60000.0p-23, 0x800000.0p-38 }, + { 0xc70000.0p-23, -0x820000.0p-33 }, + { 0xc80000.0p-23, 0x800000.0p-33 }, + { 0xc90000.0p-23, -0xa00000.0p-36 }, + { 0xca0000.0p-23, -0xb00000.0p-33 }, + { 0xcb0000.0p-23, 0x840000.0p-34 }, + { 0xcc0000.0p-23, -0xd00000.0p-34 }, + { 0xcd0000.0p-23, 0x800000.0p-33 }, + { 0xce0000.0p-23, -0xe00000.0p-35 }, + { 0xcf0000.0p-23, 0xa60000.0p-33 }, + { 0xd00000.0p-23, -0x800000.0p-35 }, + { 0xd10000.0p-23, 0xb40000.0p-33 }, + { 0xd20000.0p-23, -0x800000.0p-35 }, + { 0xd30000.0p-23, 0xaa0000.0p-33 }, + { 0xd40000.0p-23, -0xe00000.0p-35 }, + { 0xd50000.0p-23, 0x880000.0p-33 }, + { 0xd60000.0p-23, -0xd00000.0p-34 }, + { 0xd70000.0p-23, 0x9c0000.0p-34 }, + { 0xd80000.0p-23, -0xb00000.0p-33 }, + { 0xd90000.0p-23, -0x800000.0p-38 }, + { 0xda0000.0p-23, 0xa40000.0p-33 }, + { 0xdb0000.0p-23, -0xdc0000.0p-34 }, + { 0xdc0000.0p-23, 0xc00000.0p-35 }, + { 0xdd0000.0p-23, 0xca0000.0p-33 }, + { 0xde0000.0p-23, -0xb80000.0p-34 }, + { 0xdf0000.0p-23, 0xd00000.0p-35 }, + { 0xe00000.0p-23, 0xc00000.0p-33 }, + { 0xe10000.0p-23, -0xf40000.0p-34 }, + { 0xe20000.0p-23, 0x800000.0p-37 }, + { 0xe30000.0p-23, 0x860000.0p-33 }, + { 0xe40000.0p-23, -0xc80000.0p-33 }, + { 0xe50000.0p-23, -0xa80000.0p-34 }, + { 0xe60000.0p-23, 0xe00000.0p-36 }, + { 0xe70000.0p-23, 0x880000.0p-33 }, + { 0xe80000.0p-23, -0xe00000.0p-33 }, + { 0xe90000.0p-23, -0xfc0000.0p-34 }, + { 0xea0000.0p-23, -0x800000.0p-35 }, + { 0xeb0000.0p-23, 0xe80000.0p-35 }, + { 0xec0000.0p-23, 0x900000.0p-33 }, + { 0xed0000.0p-23, 0xe20000.0p-33 }, + { 0xee0000.0p-23, -0xac0000.0p-33 }, + { 0xef0000.0p-23, -0xc80000.0p-34 }, + { 0xf00000.0p-23, -0x800000.0p-35 }, + { 0xf10000.0p-23, 0x800000.0p-35 }, + { 0xf20000.0p-23, 0xb80000.0p-34 }, + { 0xf30000.0p-23, 0x940000.0p-33 }, + { 0xf40000.0p-23, 0xc80000.0p-33 }, + { 0xf50000.0p-23, -0xf20000.0p-33 }, + { 0xf60000.0p-23, -0xc80000.0p-33 }, + { 0xf70000.0p-23, -0xa20000.0p-33 }, + { 0xf80000.0p-23, -0x800000.0p-33 }, + { 0xf90000.0p-23, -0xc40000.0p-34 }, + { 0xfa0000.0p-23, -0x900000.0p-34 }, + { 0xfb0000.0p-23, -0xc80000.0p-35 }, + { 0xfc0000.0p-23, -0x800000.0p-35 }, + { 0xfd0000.0p-23, -0x900000.0p-36 }, + { 0xfe0000.0p-23, -0x800000.0p-37 }, + { 0xff0000.0p-23, -0x800000.0p-39 }, + { 0x800000.0p-22, 0 }, +}; +#endif /* USE_UTAB */ + +#ifdef STRUCT_RETURN +#define RETURN1(rp, v) do { \ + (rp)->hi = (v); \ + (rp)->lo_set = 0; \ + return; \ +} while (0) + +#define RETURN2(rp, h, l) do { \ + (rp)->hi = (h); \ + (rp)->lo = (l); \ + (rp)->lo_set = 1; \ + return; \ +} while (0) + +struct ld { + long double hi; + long double lo; + int lo_set; +}; +#else +#define RETURN1(rp, v) RETURNF(v) +#define RETURN2(rp, h, l) RETURNI((h) + (l)) +#endif + +#ifdef STRUCT_RETURN +static inline __always_inline void +k_logl(long double x, struct ld *rp) +#else +long double +logl(long double x) +#endif +{ + long double d, dk, val_hi, val_lo, z; + uint64_t ix, lx; + int i, k; + uint16_t hx; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + k = -16383; +#if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ + if (x == 1) + RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ +#endif + if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ + if (((hx & 0x7fff) | lx) == 0) + RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ + if (hx != 0) + /* log(neg or [pseudo-]NaN) = qNaN: */ + RETURN1(rp, (x - x) / zero); + x *= 0x1.0p65; /* subnormal; scale up x */ + /* including pseudo-subnormals */ + EXTRACT_LDBL80_WORDS(hx, lx, x); + k = -16383 - 65; + } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0) + RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ + /* log(pseudo-Inf) = qNaN */ + /* log(pseudo-NaN) = qNaN */ + /* log(unnormal) = qNaN */ +#ifndef STRUCT_RETURN + ENTERI(); +#endif + k += hx; + ix = lx & 0x7fffffffffffffffULL; + dk = k; + + /* Scale x to be in [1, 2). */ + SET_LDBL_EXPSIGN(x, 0x3fff); + + /* 0 <= i <= INTERVALS: */ +#define L2I (64 - LOG2_INTERVALS) + i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); + + /* + * -0.005280 < d < 0.004838. In particular, the infinite- + * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits + * ensures that d is representable without extra precision for + * this bound on |d| (since when this calculation is expressed + * as x*G(i)-1, the multiplication needs as many extra bits as + * G(i) has and the subtraction cancels 8 bits). But for + * most i (107 cases out of 129), the infinite-precision |d| + * is <= 2**-8. G(i) is rounded to 9 bits for such i to give + * better accuracy (this works by improving the bound on |d|, + * which in turn allows rounding to 9 bits in more cases). + * This is only important when the original x is near 1 -- it + * lets us avoid using a special method to give the desired + * accuracy for such x. + */ + if (0) + d = x * G(i) - 1; + else { +#ifdef USE_UTAB + d = (x - H(i)) * G(i) + E(i); +#else + long double x_hi, x_lo; + float fx_hi; + + /* + * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. + * G(i) has at most 9 bits, so the splitting point is not + * critical. + */ + SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); + x_hi = fx_hi; + x_lo = x - x_hi; + d = x_hi * G(i) - 1 + x_lo * G(i); +#endif + } + + /* + * Our algorithm depends on exact cancellation of F_lo(i) and + * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is + * at the end of the table. This and other technical complications + * make it difficult to avoid the double scaling in (dk*ln2) * + * log(base) for base != e without losing more accuracy and/or + * efficiency than is gained. + */ + z = d * d; + val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + + (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2; + val_hi = d; +#ifdef DEBUG + if (fetestexcept(FE_UNDERFLOW)) + breakpoint(); +#endif + + _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); + RETURN2(rp, val_hi, val_lo); +} + +long double +log1pl(long double x) +{ + long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z; + long double f_hi, twopminusk; + uint64_t ix, lx; + int i, k; + int16_t ax, hx; + + DOPRINT_START(&x); + EXTRACT_LDBL80_WORDS(hx, lx, x); + if (hx < 0x3fff) { /* x < 1, or x neg NaN */ + ax = hx & 0x7fff; + if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ + if (ax == 0x3fff && lx == 0x8000000000000000ULL) + RETURNP(-1 / zero); /* log1p(-1) = -Inf */ + /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */ + RETURNP((x - x) / (x - x)); + } + if (ax <= 0x3fbe) { /* |x| < 2**-64 */ + if ((int)x == 0) + RETURNP(x); /* x with inexact if x != 0 */ + } + f_hi = 1; + f_lo = x; + } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ + RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ + /* log1p(pseudo-Inf) = qNaN */ + /* log1p(pseudo-NaN) = qNaN */ + /* log1p(unnormal) = qNaN */ + } else if (hx < 0x407f) { /* 1 <= x < 2**128 */ + f_hi = x; + f_lo = 1; + } else { /* 2**128 <= x < +Inf */ + f_hi = x; + f_lo = 0; /* avoid underflow of the P5 term */ + } + ENTERI(); + x = f_hi + f_lo; + f_lo = (f_hi - x) + f_lo; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + k = -16383; + + k += hx; + ix = lx & 0x7fffffffffffffffULL; + dk = k; + + SET_LDBL_EXPSIGN(x, 0x3fff); + twopminusk = 1; + SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); + f_lo *= twopminusk; + + i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); + + /* + * x*G(i)-1 (with a reduced x) can be represented exactly, as + * above, but now we need to evaluate the polynomial on d = + * (x+f_lo)*G(i)-1 and extra precision is needed for that. + * Since x+x_lo is a hi+lo decomposition and subtracting 1 + * doesn't lose too many bits, an inexact calculation for + * f_lo*G(i) is good enough. + */ + if (0) + d_hi = x * G(i) - 1; + else { +#ifdef USE_UTAB + d_hi = (x - H(i)) * G(i) + E(i); +#else + long double x_hi, x_lo; + float fx_hi; + + SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); + x_hi = fx_hi; + x_lo = x - x_hi; + d_hi = x_hi * G(i) - 1 + x_lo * G(i); +#endif + } + d_lo = f_lo * G(i); + + /* + * This is _2sumF(d_hi, d_lo) inlined. The condition + * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not + * always satisifed, so it is not clear that this works, but + * it works in practice. It works even if it gives a wrong + * normalized d_lo, since |d_lo| > |d_hi| implies that i is + * nonzero and d is tiny, so the F(i) term dominates d_lo. + * In float precision: + * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. + * And if d is only a little tinier than that, we would have + * another underflow problem for the P3 term; this is also ruled + * out by exhaustive testing.) + */ + d = d_hi + d_lo; + d_lo = d_hi - d + d_lo; + d_hi = d; + + z = d * d; + val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + + (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2; + val_hi = d_hi; +#ifdef DEBUG + if (fetestexcept(FE_UNDERFLOW)) + breakpoint(); +#endif + + _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); + RETURN2PI(val_hi, val_lo); +} + +#ifdef STRUCT_RETURN + +long double +logl(long double x) +{ + struct ld r; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + RETURNSPI(&r); +} + +/* Use macros since GCC < 8 rejects static const expressions in initializers. */ +#define invln10_hi 4.3429448190317999e-1 /* 0x1bcb7b1526e000.0p-54 */ +#define invln10_lo 7.1842412889749798e-14 /* 0x1438ca9aadd558.0p-96 */ +#define invln2_hi 1.4426950408887933e0 /* 0x171547652b8000.0p-52 */ +#define invln2_lo 1.7010652264631490e-13 /* 0x17f0bbbe87fed0.0p-95 */ +/* Let the compiler pre-calculate this sum to avoid FE_INEXACT at run time. */ +static const double invln10_lo_plus_hi = invln10_lo + invln10_hi; +static const double invln2_lo_plus_hi = invln2_lo + invln2_hi; + +long double +log10l(long double x) +{ + struct ld r; + long double hi, lo; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + if (!r.lo_set) + RETURNPI(r.hi); + _2sumF(r.hi, r.lo); + hi = (float)r.hi; + lo = r.lo + (r.hi - hi); + RETURN2PI(invln10_hi * hi, + invln10_lo_plus_hi * lo + invln10_lo * hi); +} + +long double +log2l(long double x) +{ + struct ld r; + long double hi, lo; + + ENTERI(); + DOPRINT_START(&x); + k_logl(x, &r); + if (!r.lo_set) + RETURNPI(r.hi); + _2sumF(r.hi, r.lo); + hi = (float)r.hi; + lo = r.lo + (r.hi - hi); + RETURN2PI(invln2_hi * hi, + invln2_lo_plus_hi * lo + invln2_lo * hi); +} + +#endif /* STRUCT_RETURN */ diff --git a/newlib/libm/ld80/s_sinpil.c b/newlib/libm/ld80/s_sinpil.c new file mode 100644 index 000000000..4cefa9235 --- /dev/null +++ b/newlib/libm/ld80/s_sinpil.c @@ -0,0 +1,140 @@ +/*- + * Copyright (c) 2017 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + */ + +/* + * See ../src/s_sinpi.c for implementation details. + */ + +#ifdef __i386__ +#include +#endif +#include + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +static const union IEEEl2bits +pi_hi_u = LD80C(0xc90fdaa200000000, 1, 3.14159265346825122833e+00L), +pi_lo_u = LD80C(0x85a308d313198a2e, -33, 1.21542010130123852029e-10L); +#define pi_hi (pi_hi_u.e) +#define pi_lo (pi_lo_u.e) + +#include "k_cospil.h" +#include "k_sinpil.h" + +volatile static const double vzero = 0; + +long double +sinpil(long double x) +{ + long double ax, hi, lo, s; + uint64_t lx, m; + uint32_t j0; + uint16_t hx, ix; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + ix = hx & 0x7fff; + INSERT_LDBL80_WORDS(ax, ix, lx); + + ENTERI(); + + if (ix < 0x3fff) { /* |x| < 1 */ + if (ix < 0x3ffd) { /* |x| < 0.25 */ + if (ix < 0x3fdd) { /* |x| < 0x1p-34 */ + if (x == 0) + RETURNI(x); + INSERT_LDBL80_WORDS(hi, hx, + lx & 0xffffffff00000000ull); + hi *= 0x1p63L; + lo = x * 0x1p63L - hi; + s = (pi_lo + pi_hi) * lo + pi_lo * hi + + pi_hi * hi; + RETURNI(s * 0x1p-63L); + } + s = __kernel_sinpil(ax); + RETURNI((hx & 0x8000) ? -s : s); + } + + if (ix < 0x3ffe) /* |x| < 0.5 */ + s = __kernel_cospil(0.5 - ax); + else if (lx < 0xc000000000000000ull) /* |x| < 0.75 */ + s = __kernel_cospil(ax - 0.5); + else + s = __kernel_sinpil(1 - ax); + RETURNI((hx & 0x8000) ? -s : s); + } + + if (ix < 0x403e) { /* 1 <= |x| < 0x1p63 */ + /* Determine integer part of ax. */ + j0 = ix - 0x3fff + 1; + if (j0 < 32) { + lx = (lx >> 32) << 32; + lx &= ~(((lx << 32)-1) >> j0); + } else { + m = (uint64_t)-1 >> (j0 + 1); + if (lx & m) lx &= ~m; + } + INSERT_LDBL80_WORDS(x, ix, lx); + + ax -= x; + EXTRACT_LDBL80_WORDS(ix, lx, ax); + + if (ix == 0) { + s = 0; + } else { + if (ix < 0x3ffe) { /* |x| < 0.5 */ + if (ix < 0x3ffd) /* |x| < 0.25 */ + s = __kernel_sinpil(ax); + else + s = __kernel_cospil(0.5 - ax); + } else { + /* |x| < 0.75 */ + if (lx < 0xc000000000000000ull) + s = __kernel_cospil(ax - 0.5); + else + s = __kernel_sinpil(1 - ax); + } + + if (j0 > 40) + x -= 0x1p40; + if (j0 > 30) + x -= 0x1p30; + j0 = (uint32_t)x; + if (j0 & 1) s = -s; + } + RETURNI((hx & 0x8000) ? -s : s); + } + + /* x = +-inf or nan. */ + if (ix >= 0x7fff) + RETURNI(vzero / vzero); + + /* + * |x| >= 0x1p63 is always an integer, so return +-0. + */ + RETURNI(copysignl(0, x)); +} -- 2.31.1