* [PATCH 1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f.
@ 2021-02-09 5:28 Paul Zimmermann
2021-02-09 5:28 ` [PATCH 2/4] Fix the inaccuracy of j0f and of y0f [BZ #14469 and #14471] Paul Zimmermann
` (2 more replies)
0 siblings, 3 replies; 4+ messages in thread
From: Paul Zimmermann @ 2021-02-09 5:28 UTC (permalink / raw)
To: libc-alpha
---
math/Makefile | 3 +-
sysdeps/ieee754/flt-32/e_reduce_pi.c | 186 +++++++++++++++++++++++++++
sysdeps/ieee754/flt-32/e_reduce_pi.h | 19 +++
3 files changed, 207 insertions(+), 1 deletion(-)
create mode 100644 sysdeps/ieee754/flt-32/e_reduce_pi.c
create mode 100644 sysdeps/ieee754/flt-32/e_reduce_pi.h
diff --git a/math/Makefile b/math/Makefile
index 687aa5d510..6f5dfcfd42 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -134,7 +134,8 @@ type-double-routines := branred doasin dosincos mpa mpatan2 \
# float support
type-float-suffix := f
type-float-routines := math_errf e_exp2f_data e_logf_data \
- e_log2f_data e_powf_log2_data s_sincosf_data
+ e_log2f_data e_powf_log2_data s_sincosf_data \
+ e_reduce_pi
# _Float128 support
type-float128-suffix := f128
diff --git a/sysdeps/ieee754/flt-32/e_reduce_pi.c b/sysdeps/ieee754/flt-32/e_reduce_pi.c
new file mode 100644
index 0000000000..2028d44817
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/e_reduce_pi.c
@@ -0,0 +1,186 @@
+/* Argument reduction modulo 2pi.
+ Copyright (C) 2021 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, see
+ <https://www.gnu.org/licenses/>. */
+
+#include "math_config.h"
+
+/* Argument reduction mod 2pi: T[i] ~ 2^i mod (2*pi). */
+static const double T[128] = {
+ 0x1p+0,
+ 0x2p+0,
+ -0x2.487ed5110b462p+0,
+ 0x1.b7812aeef4b9fp+0,
+ -0x2.d97c7f3321d24p+0,
+ 0x9.585d6aac7a1a8p-4,
+ 0x1.2b0bad558f435p+0,
+ 0x2.56175aab1e86ap+0,
+ -0x1.9c501fbace38dp+0,
+ 0x3.0fde959b6ed46p+0,
+ -0x2.8c1a9da2d9d3cp-4,
+ -0x5.18353b45b3a78p-4,
+ -0xa.306a768b674fp-4,
+ -0x1.460d4ed16ce9ep+0,
+ -0x2.8c1a9da2d9d3cp+0,
+ 0x1.304999cb579e8p+0,
+ 0x2.60933396af3dp+0,
+ -0x1.87586de3accc3p+0,
+ -0x3.0eb0dbc759986p+0,
+ 0x2.b1d1d8258155p-4,
+ 0x5.63a3b04b02aap-4,
+ 0xa.c74760960554p-4,
+ 0x1.58e8ec12c0aa8p+0,
+ 0x2.b1d1d8258155p+0,
+ -0xe.4db24c6089bf8p-4,
+ -0x1.c9b6498c1137fp+0,
+ 0x2.b51241f8e8d64p+0,
+ -0xd.e5a511f3999bp-4,
+ -0x1.bcb4a23e73336p+0,
+ 0x2.cf15909424df4p+0,
+ -0xa.a53b3e8c1877p-4,
+ -0x1.54a767d1830eep+0,
+ -0x2.a94ecfa3061dcp+0,
+ 0xf.5e135caff0a78p-4,
+ 0x1.ebc26b95fe14fp+0,
+ -0x2.70f9fde50f1c4p+0,
+ 0x1.668ad946ed0dap+0,
+ 0x2.cd15b28dda1b4p+0,
+ -0xa.e536ff5570fbp-4,
+ -0x1.5ca6dfeaae1f6p+0,
+ -0x2.b94dbfd55c3ecp+0,
+ 0xd.5e3556652c89p-4,
+ 0x1.abc6aacca5912p+0,
+ -0x2.f0f17f77c023ep+0,
+ 0x6.69bd6218afe64p-4,
+ 0xc.d37ac4315fcc8p-4,
+ 0x1.9a6f58862bf99p+0,
+ -0x3.13a02404b352ep+0,
+ 0x2.13e8d07a4a046p-4,
+ 0x4.27d1a0f49408cp-4,
+ 0x8.4fa341e928118p-4,
+ 0x1.09f4683d25023p+0,
+ 0x2.13e8d07a4a046p+0,
+ -0x2.20ad341c773d4p+0,
+ 0x2.07246cd81ccb8p+0,
+ -0x2.3a35fb60d1af2p+0,
+ 0x1.d412de4f67e7ep+0,
+ -0x2.a05918723b764p+0,
+ 0x1.07cca42c94599p+0,
+ 0x2.0f99485928b32p+0,
+ -0x2.294c445eb9dfep+0,
+ 0x1.f5e64c5397865p+0,
+ -0x2.5cb23c69dc396p+0,
+ 0x1.8f1a5c3d52d34p+0,
+ 0x3.1e34b87aa5a68p+0,
+ -0xc.15641bbff90bp-8,
+ -0x1.82ac8377ff216p-4,
+ -0x3.055906effe42cp-4,
+ -0x6.0ab20ddffc858p-4,
+ -0xc.15641bbff90bp-4,
+ -0x1.82ac8377ff216p+0,
+ -0x3.055906effe42cp+0,
+ 0x3.dccc7310ec09ap-4,
+ 0x7.b998e621d8134p-4,
+ 0xf.7331cc43b0268p-4,
+ 0x1.ee6639887604dp+0,
+ -0x2.6bb262001f3c6p+0,
+ 0x1.711a1110cccd4p+0,
+ 0x2.e2342221999a8p+0,
+ -0x8.41690cdd81128p-4,
+ -0x1.082d219bb0225p+0,
+ -0x2.105a43376044ap+0,
+ 0x2.27ca4ea24abcep+0,
+ -0x1.f8ea37cc75cc6p+0,
+ 0x2.56aa65781fad6p+0,
+ -0x1.9b2a0a20cbeb6p+0,
+ 0x3.122ac0cf736f6p+0,
+ -0x2.4295372246764p-4,
+ -0x4.852a6e448cec8p-4,
+ -0x9.0a54dc8919d9p-4,
+ -0x1.214a9b91233b2p+0,
+ -0x2.4295372246764p+0,
+ 0x1.c35466cc7e59ap+0,
+ -0x2.c1d607780e92ep+0,
+ 0xc.4d2c620ee206p-4,
+ 0x1.89a58c41dc40cp+0,
+ 0x3.134b1883b8818p+0,
+ -0x2.1e8a4099a4324p-4,
+ -0x4.3d14813348648p-4,
+ -0x8.7a29026690c9p-4,
+ -0x1.0f45204cd2192p+0,
+ -0x2.1e8a4099a4324p+0,
+ 0x2.0b6a53ddc2e18p+0,
+ -0x2.31aa2d558583p+0,
+ 0x1.e52a7a6600402p+0,
+ -0x2.7e29e0450ac5cp+0,
+ 0x1.4c2b1486f5ba8p+0,
+ 0x2.9856290deb75p+0,
+ -0x1.17d282f5345bfp+0,
+ -0x2.2fa505ea68b7ep+0,
+ 0x1.e934c93c39d64p+0,
+ -0x2.761542989799ap+0,
+ 0x1.5c544fdfdc12fp+0,
+ 0x2.b8a89fbfb825ep+0,
+ -0xd.72d95919afa58p-4,
+ -0x1.ae5b2b2335f4bp+0,
+ 0x2.ebc87eca9f5cap+0,
+ -0x7.0edd77bcc8ccp-4,
+ -0xe.1dbaef799198p-4,
+ -0x1.c3b75def3233p+0,
+ 0x2.c1101932a6e02p+0,
+ -0xc.65ea2abbd85fp-4,
+ -0x1.8cbd45577b0bep+0,
+ -0x3.197a8aaef617cp+0,
+ 0x1.589bfb31f1687p-4,
+ 0x2.b137f663e2d0ep-4,
+ 0x5.626fecc7c5a1cp-4,
+ 0xa.c4dfd98f8b438p-4
+};
+
+/* Return h such that x - pi/4 mod (2*pi) ~ h,
+ with maximal error about 2^-45. */
+attribute_hidden double
+__math_reduce_mod_twopi (double x)
+{
+ double sign = 1, h;
+ if (x < 0)
+ {
+ x = -x;
+ sign = -1;
+ }
+ h = 0; /* Invariant is x+h mod (2*pi). */
+ while (x >= 4)
+ {
+ int i = ilogb (x);
+ x -= ldexp (1.0f, i);
+ h += T[i];
+ }
+ /* Add the remainder x. */
+ h += x;
+ /* Subtract pi/4. */
+ double piover4 = 0xc.90fdaa22168cp-4;
+ h -= piover4;
+ /* Reduce mod 2*pi. */
+ double twopi = 0x6.487ed5110b46p+0;
+ while (fabs (h) > twopi * 0.5)
+ {
+ if (h > 0)
+ h -= twopi;
+ else
+ h += twopi;
+ }
+ return sign * h;
+}
diff --git a/sysdeps/ieee754/flt-32/e_reduce_pi.h b/sysdeps/ieee754/flt-32/e_reduce_pi.h
new file mode 100644
index 0000000000..93d6ce8764
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/e_reduce_pi.h
@@ -0,0 +1,19 @@
+/* Argument reduction modulo 2pi.
+ Copyright (C) 2021 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, see
+ <https://www.gnu.org/licenses/>. */
+
+attribute_hidden double __math_reduce_mod_twopi (double);
--
2.30.0
^ permalink raw reply [flat|nested] 4+ messages in thread
* [PATCH 2/4] Fix the inaccuracy of j0f and of y0f [BZ #14469 and #14471]
2021-02-09 5:28 [PATCH 1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f Paul Zimmermann
@ 2021-02-09 5:28 ` Paul Zimmermann
2021-02-09 5:28 ` [PATCH 3/4] Fix the inaccuracy of j1f and y1f [BZ #14470 and #14472] Paul Zimmermann
2021-02-09 5:28 ` [PATCH 4/4] Added new entries for j0/j1/y0/y1 and regenerate ulps Paul Zimmermann
2 siblings, 0 replies; 4+ messages in thread
From: Paul Zimmermann @ 2021-02-09 5:28 UTC (permalink / raw)
To: libc-alpha
For both j0f and y0f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/y0f computation, or for very large inputs, thus should not give any
visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/y0, approximation polynomials of degree 3
are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Tested on x86_64 with --disable-multi-arch.
---
sysdeps/ieee754/flt-32/e_j0f.c | 408 +++++++++++++++++++++++++++++----
1 file changed, 368 insertions(+), 40 deletions(-)
diff --git a/sysdeps/ieee754/flt-32/e_j0f.c b/sysdeps/ieee754/flt-32/e_j0f.c
index 5d29611eb7..83e2301fef 100644
--- a/sysdeps/ieee754/flt-32/e_j0f.c
+++ b/sysdeps/ieee754/flt-32/e_j0f.c
@@ -16,7 +16,9 @@
#include <math.h>
#include <math-barriers.h>
#include <math_private.h>
+#include <fenv_private.h>
#include <libm-alias-finite.h>
+#include <e_reduce_pi.h>
static float pzerof(float), qzerof(float);
@@ -37,6 +39,166 @@ S04 = 1.1661400734e-09; /* 0x30a045e8 */
static const float zero = 0.0;
+#define FIRST_ZERO_J0 2.404825f
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of j0 and degree-3 polynomial
+ approximations of j0 around these zeros: Pj[0] for the first zero (2.40482),
+ Pj[1] for the second one (5.520078), and so on. Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number
+ closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
+ polynomial. Each polynomial was generated using Sollya on the interval
+ [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+ for the alternate code. Degree 3 is enough to get an error <= 9 ulps.
+*/
+static const float Pj[SMALL_SIZE][7] = {
+ { 0x1.31e5c6p+1, 0x1.33d152p+1, 0x1.3b58dep+1, 0xf.26247p-28, -0x8.4e6d7p-4, 0x1.ba2ac2p-4, 0xe.4b9a2p-8 }, /* 0 */
+ { 0x1.60eafap+2, 0x1.6148f6p+2, 0x1.62955cp+2, 0x6.9205fp-28, 0x5.71b98p-4, -0x7.e3e798p-8, -0xd.87d1p-8 }, /* 1 */
+ { 0x1.14cde2p+3, 0x1.14eb56p+3, 0x1.1525c6p+3, 0x1.bcc1cap-24, -0x4.57de6p-4, 0x4.03e7cp-8, 0xb.39a37p-8 }, /* 2 */
+ { 0x1.7931d8p+3, 0x1.79544p+3, 0x1.7998d6p+3, -0xf.2976fp-32, 0x3.b827ccp-4, -0x2.8603ep-8, -0x9.bf49bp-8 }, /* 3 */
+ { 0x1.ddb6d4p+3, 0x1.ddca14p+3, 0x1.ddf0c8p+3, -0x1.bd67d8p-28, -0x3.4e03ap-4, 0x1.c562a2p-8, 0x8.90ec2p-8 }, /* 4 */
+ { 0x1.2118e4p+4, 0x1.212314p+4, 0x1.21375p+4, 0x1.62209cp-28, 0x3.00efecp-4, -0x1.5458dap-8, -0x8.10063p-8 }, /* 5 */
+ { 0x1.535d28p+4, 0x1.5362dep+4, 0x1.536ceap+4, -0x2.853f74p-24, -0x2.c5b274p-4, 0x1.0b9db4p-8, 0x7.8c3578p-8 }, /* 6 */
+ { 0x1.859df6p+4, 0x1.85a3bap+4, 0x1.85aff4p+4, 0x2.19ed1cp-24, 0x2.96545cp-4, -0xd.997e6p-12, -0x6.d9af28p-8 }, /* 7 */
+ { 0x1.b7decap+4, 0x1.b7e54ap+4, 0x1.b7f038p+4, 0xe.959aep-28, -0x2.6f5594p-4, 0xb.538dp-12, 0x7.003ea8p-8 }, /* 8 */
+ { 0x1.ea21c6p+4, 0x1.ea275ap+4, 0x1.ea337ap+4, 0x2.0c3964p-24, 0x2.4e80fcp-4, -0x9.a2216p-12, -0x6.61e0a8p-8 }, /* 9 */
+ { 0x1.0e3316p+5, 0x1.0e34e2p+5, 0x1.0e379ap+5, -0x3.642554p-24, -0x2.325e48p-4, 0x8.4f49cp-12, 0x7.d37c3p-8 }, /* 10 */
+ { 0x1.275456p+5, 0x1.275638p+5, 0x1.2759e2p+5, 0x1.6c015ap-24, 0x2.19e7d8p-4, -0x7.4c1bf8p-12, -0x4.af7ef8p-8 }, /* 11 */
+ { 0x1.4075ecp+5, 0x1.4077a8p+5, 0x1.407b96p+5, -0x4.b18c9p-28, -0x2.046174p-4, 0x6.705618p-12, 0x5.f2d28p-8 }, /* 12 */
+ { 0x1.59973p+5, 0x1.59992cp+5, 0x1.599b2ap+5, -0x1.8b8792p-24, 0x1.f13fbp-4, -0x5.c14938p-12, -0x5.73e0cp-8 }, /* 13 */
+ { 0x1.72b9bp+5, 0x1.72bacp+5, 0x1.72bc5ap+5, 0x3.a26e0cp-24, -0x1.e018dap-4, 0x5.30e8dp-12, 0x2.81099p-8 }, /* 14 */
+ { 0x1.8bdb4ap+5, 0x1.8bdc62p+5, 0x1.8bde7ep+5, -0x2.18fabcp-24, 0x1.d09b22p-4, -0x4.b0b688p-12, -0x5.5fd308p-8 }, /* 15 */
+ { 0x1.a4fcecp+5, 0x1.a4fe0ep+5, 0x1.a50042p+5, 0x3.2370e8p-24, -0x1.c28614p-4, 0x4.4647e8p-12, 0x5.68a28p-8 }, /* 16 */
+ { 0x1.be1ebcp+5, 0x1.be1fc4p+5, 0x1.be21fp+5, -0x5.9eae3p-28, 0x1.b5a622p-4, -0x3.eb9054p-12, -0x5.12d8cp-8 }, /* 17 */
+ { 0x1.d7405p+5, 0x1.d7418p+5, 0x1.d74294p+5, 0x2.9fa1e8p-24, -0x1.a9d184p-4, 0x3.9d1e7p-12, 0x4.33d058p-8 }, /* 18 */
+ { 0x1.f0624p+5, 0x1.f06344p+5, 0x1.f06452p+5, 0x9.9ac67p-28, 0x1.9ee5eep-4, -0x3.5816e8p-12, -0x2.6e5004p-8 }, /* 19 */
+ { 0x1.04c23ep+6, 0x1.04c286p+6, 0x1.04c316p+6, 0xd.6ab94p-28, -0x1.94c6f6p-4, 0x3.174efcp-12, 0x7.9a092p-8 }, /* 20 */
+ { 0x1.1153p+6, 0x1.11536cp+6, 0x1.11541p+6, -0x4.4cb2d8p-24, 0x1.8b5cccp-4, -0x2.e3c238p-12, -0x4.e5437p-8 }, /* 21 */
+ { 0x1.1de3d8p+6, 0x1.1de456p+6, 0x1.1de4dap+6, -0x4.4aa8c8p-24, -0x1.829356p-4, 0x2.b45124p-12, 0x5.baf638p-8 }, /* 22 */
+ { 0x1.2a74f8p+6, 0x1.2a754p+6, 0x1.2a75bp+6, 0x2.077c38p-24, 0x1.7a597ep-4, -0x2.8a0414p-12, -0x2.838d3p-8 }, /* 23 */
+ { 0x1.3705d4p+6, 0x1.37062cp+6, 0x1.370682p+6, -0x2.6a6cd8p-24, -0x1.72a09ap-4, 0x2.623a3cp-12, 0x5.5256a8p-8 }, /* 24 */
+ { 0x1.4396dp+6, 0x1.439718p+6, 0x1.43976ep+6, -0x5.08287p-24, 0x1.6b5c06p-4, -0x2.3da154p-12, -0x7.a2254p-8 }, /* 25 */
+ { 0x1.5027acp+6, 0x1.502808p+6, 0x1.50288cp+6, -0x3.4598dcp-24, -0x1.6480c4p-4, 0x2.1cb944p-12, 0x7.27c77p-8 }, /* 26 */
+ { 0x1.5cb89ap+6, 0x1.5cb8f8p+6, 0x1.5cb97ep+6, 0x5.4e74bp-24, 0x1.5e0544p-4, -0x2.00b158p-12, -0x5.9bc4a8p-8 }, /* 27 */
+ { 0x1.69498cp+6, 0x1.6949e8p+6, 0x1.694a42p+6, -0x2.05751cp-24, -0x1.57e12p-4, 0x1.e78edcp-12, 0x9.9667dp-8 }, /* 28 */
+ { 0x1.75da7ep+6, 0x1.75dadap+6, 0x1.75db3p+6, 0x4.c5e278p-24, 0x1.520ceep-4, -0x1.d0127cp-12, -0xd.62681p-8 }, /* 29 */
+ { 0x1.826b7ep+6, 0x1.826bccp+6, 0x1.826c2cp+6, -0x3.50e62cp-24, -0x1.4c822p-4, 0x1.ba5832p-12, -0x1.eb2ee2p-8 }, /* 30 */
+ { 0x1.8efc84p+6, 0x1.8efcbep+6, 0x1.8efd16p+6, -0x1.c39f38p-24, 0x1.473ae6p-4, -0x1.a616c8p-12, 0xf.f352ap-12 }, /* 31 */
+ { 0x1.9b8d88p+6, 0x1.9b8db2p+6, 0x1.9b8e7p+6, -0x1.9245b6p-28, -0x1.42320ap-4, 0x1.932a04p-12, 0x2.dc113cp-8 }, /* 32 */
+ { 0x1.a81e74p+6, 0x1.a81ea6p+6, 0x1.a81edep+6, -0x1.0acf8p-24, 0x1.3d62e6p-4, -0x1.7c4b14p-12, -0x1.cfc5c2p-4 }, /* 33 */
+ { 0x1.b4af6ap+6, 0x1.b4af9ap+6, 0x1.b4afeep+6, 0x4.cd92d8p-24, -0x1.38c94ap-4, 0x1.643154p-12, 0x1.4c2a06p-4 }, /* 34 */
+ { 0x1.c1406p+6, 0x1.c1409p+6, 0x1.c140cp+6, -0x1.37bf8ap-24, 0x1.34617p-4, -0x1.5f504ap-12, -0x1.e2d324p-4 }, /* 35 */
+ { 0x1.cdd154p+6, 0x1.cdd186p+6, 0x1.cdd1e8p+6, -0x1.8f62dep-28, -0x1.3027fp-4, 0x1.534a02p-12, 0x2.c7f144p-12 }, /* 36 */
+ { 0x1.da6248p+6, 0x1.da627cp+6, 0x1.da62e6p+6, -0x9.81e79p-28, 0x1.2c19b4p-4, -0x1.4b8288p-12, 0x7.2d8bap-8 }, /* 37 */
+ { 0x1.e6f33ep+6, 0x1.e6f372p+6, 0x1.e6f3a8p+6, 0x3.103b3p-24, -0x1.2833eep-4, 0x1.36f4d2p-12, 0x9.29f91p-8 }, /* 38 */
+ { 0x1.f38434p+6, 0x1.f3846ap+6, 0x1.f384d8p+6, 0x2.07b058p-24, 0x1.24740ap-4, -0x1.2ee58ap-12, 0xd.f1393p-12 }, /* 39 */
+ { 0x1.000a98p+7, 0x1.000abp+7, 0x1.000ac8p+7, 0x3.87576cp-24, -0x1.20d7b6p-4, 0x1.2083e2p-12, 0x3.9a7aap-8 }, /* 40 */
+ { 0x1.06531p+7, 0x1.06532cp+7, 0x1.065348p+7, -0x1.691ecp-24, 0x1.1d5ccap-4, -0x1.166726p-12, -0x1.e4af48p-8 }, /* 41 */
+ { 0x1.0c9b9ap+7, 0x1.0c9ba8p+7, 0x1.0c9bb2p+7, 0x9.b406dp-28, -0x1.1a015p-4, 0x1.038f9cp-12, -0x4.021058p-4 }, /* 42 */
+ { 0x1.12e412p+7, 0x1.12e424p+7, 0x1.12e436p+7, -0xf.bfd8fp-28, 0x1.16c37ap-4, -0x1.039edep-12, 0x1.f0033p-4 }, /* 43 */
+ { 0x1.192c92p+7, 0x1.192cap+7, 0x1.192cb6p+7, 0x2.6d50c8p-24, -0x1.13a19ep-4, 0xf.9df8ap-16, 0x4.ecd978p-8 }, /* 44 */
+ { 0x1.1f7512p+7, 0x1.1f751cp+7, 0x1.1f753ap+7, -0x4.d475c8p-24, 0x1.109a32p-4, -0x1.04fb3ap-12, -0xd.c271p-12 }, /* 45 */
+ { 0x1.25bd8ep+7, 0x1.25bd98p+7, 0x1.25bdap+7, 0x8.1982p-24, -0x1.0dabc8p-4, 0xe.88eabp-16, -0x4.ed75dp-4 }, /* 46 */
+ { 0x1.2c060cp+7, 0x1.2c0616p+7, 0x1.2c063p+7, 0x4.864518p-24, 0x1.0ad51p-4, -0xe.27196p-16, 0xb.97a3ep-8 }, /* 47 */
+ { 0x1.324e86p+7, 0x1.324e92p+7, 0x1.324e9cp+7, 0x6.8917a8p-28, -0x1.0814d4p-4, 0xd.4fe7ep-16, -0x6.8d8d6p-4 }, /* 48 */
+ { 0x1.389702p+7, 0x1.38970ep+7, 0x1.38971cp+7, -0x5.fa18fp-24, 0x1.0569fp-4, -0xd.5b0d4p-16, 0x1.50353ap-4 }, /* 49 */
+ { 0x1.3edf84p+7, 0x1.3edf8cp+7, 0x1.3edf9cp+7, -0x4.0e5c98p-24, -0x1.02d354p-4, 0xb.7b255p-16, 0x7.8a916p-4 }, /* 50 */
+ { 0x1.4527fp+7, 0x1.452808p+7, 0x1.45281p+7, -0x2.c3ddbp-24, 0x1.005004p-4, -0xd.7729cp-16, -0x3.bcc354p-8 }, /* 51 */
+ { 0x1.4b7076p+7, 0x1.4b7086p+7, 0x1.4b70a4p+7, -0x5.d052p-24, -0xf.ddf16p-8, 0xc.318c1p-16, 0x5.7947p-8 }, /* 52 */
+ { 0x1.51b8f4p+7, 0x1.51b902p+7, 0x1.51b90ep+7, -0x2.0b97dcp-24, 0xf.b7fafp-8, -0xc.1429dp-16, -0x3.43c36p-4 }, /* 53 */
+ { 0x1.580168p+7, 0x1.58018p+7, 0x1.580188p+7, -0x5.4aab5p-24, -0xf.930fep-8, 0xa.ecc24p-16, 0x9.c62cdp-12 }, /* 54 */
+ { 0x1.5e49eap+7, 0x1.5e49fcp+7, 0x1.5e4a12p+7, -0x3.6dadd8p-24, 0xf.6f245p-8, -0xb.6816cp-16, 0xa.d731ap-8 }, /* 55 */
+ { 0x1.649272p+7, 0x1.64927ap+7, 0x1.64929p+7, -0x2.d7e038p-24, -0xf.4c2cep-8, 0xb.118bep-16, 0xb.69a4ep-8 }, /* 56 */
+ { 0x1.6adae6p+7, 0x1.6adaf6p+7, 0x1.6adb04p+7, -0x6.977a1p-24, 0xf.2a1fp-8, -0xa.a8911p-16, -0x4.bf6d2p-8 }, /* 57 */
+ { 0x1.712366p+7, 0x1.712374p+7, 0x1.71238ep+7, 0x1.3cc95ep-24, -0xf.08f0ap-8, 0x9.f0858p-16, 0x1.77f7f4p-4 }, /* 58 */
+ { 0x1.776beap+7, 0x1.776bf2p+7, 0x1.776bfap+7, 0x3.a4921p-24, 0xe.e8986p-8, -0xa.39dfp-16, -0x6.7ba3dp-4 }, /* 59 */
+ { 0x1.7db46ep+7, 0x1.7db46ep+7, 0x1.7db476p+7, 0x6.b45a7p-24, -0xe.c90d8p-8, 0xa.e586fp-16, -0x1.d66becp-4 }, /* 60 */
+ { 0x1.83fce2p+7, 0x1.83fcecp+7, 0x1.83fd0ep+7, -0x2.8f34a4p-24, 0xe.aa478p-8, -0x9.810bp-16, -0x3.a5f3fcp-8 }, /* 61 */
+ { 0x1.8a455cp+7, 0x1.8a456ap+7, 0x1.8a4588p+7, -0x1.325968p-24, -0xe.8c3eap-8, 0x9.0a765p-16, 0x1.29a54ap-4 }, /* 62 */
+ { 0x1.908dd8p+7, 0x1.908de8p+7, 0x1.908df4p+7, 0x4.96b808p-24, 0xe.6eeb5p-8, -0x9.0251bp-16, 0x1.41a488p-4 }, /* 63 */
+};
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ j0f(x) ~ sqrt(2/(pi*x))*beta0(x)*cos(x-pi/4-alpha0(x))
+ where beta0(x) = 1 - 1/(16*x^2) + 53/(512*x^4)
+ and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
+static float
+j0f_asympt (float x)
+{
+ /* The following code fails to give an error <= 9 ulps in only two cases,
+ for which we tabulate the result. */
+ if (x == 0x1.4665d2p+24f)
+ return 0xa.50206p-52f;
+ if (x == 0x1.a9afdep+7f)
+ return 0xf.47039p-28f;
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
+ double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
+ double h;
+ h = __math_reduce_mod_twopi ((double) x);
+ /* Subtract alpha0. */
+ h -= alpha0;
+ /* Now reduce mod pi/2. */
+ double piover2 = 0x1.921fb54442d18p+0;
+ int n = 0;
+ while (fabs (h) > piover2 / 2)
+ {
+ if (h > 0)
+ {
+ h -= piover2;
+ n ++;
+ }
+ else
+ {
+ h += piover2;
+ n --;
+ }
+ }
+ /* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ float t = cst / sqrtf (x) * (float) beta0;
+ if (n == 0)
+ return t * cosf (xr);
+ else if (n == 2) /* cos(x+pi) = -cos(x) */
+ return -t * cosf (xr);
+ else if (n == 1) /* cos(x+pi/2) = -sin(x) */
+ return -t * sinf (xr);
+ else /* cos(x+3pi/2) = sin(x) */
+ return t * sinf (xr);
+}
+
+/* Special code for x near a root of j0.
+ z is the value computed by the generic code.
+ For small x, we use a polynomial approximating j0 around its root.
+ For large x, we use an asymptotic formula (j0f_asympt). */
+static float
+j0f_near_root (float x, float z)
+{
+ float index_f;
+ int index;
+
+ index_f = roundf ((x - FIRST_ZERO_J0) / (float) M_PI);
+ /* j0f_asympt fails to give an error <= 9 ulps for x=0x1.324e92p+7 (index 48)
+ thus we can't reduce SMALL_SIZE below 49. */
+ if (index_f >= SMALL_SIZE)
+ return j0f_asympt (x);
+ index = (int) index_f;
+ const float *p = Pj[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If not in the interval [x0,x1] around xmid, we return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid;
+ return p[3] + y * (p[4] + y * (p[5] + y * p[6]));
+}
+
float
__ieee754_j0f(float x)
{
@@ -48,39 +210,33 @@ __ieee754_j0f(float x)
if(ix>=0x7f800000) return one/(x*x);
x = fabsf(x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
+ SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (x, &s, &c);
ss = s-c;
cc = s+c;
- if(ix<0x7f000000) { /* make sure x+x not overflow */
- z = -__cosf(x+x);
- if ((s*c)<zero) cc = z/ss;
- else ss = z/cc;
- } else {
- /* We subtract (exactly) a value x0 such that
- cos(x0)+sin(x0) is very near to 0, and use the identity
- sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
- sin(x) + cos(x) with extra accuracy. */
- float x0 = 0xe.d4108p+124f;
- float y = x - x0; /* exact */
- /* sin(y) = sin(x)*cos(x0)-cos(x)*sin(x0) */
- z = __sinf (y);
- float eps = 0x1.5f263ep-24f;
- /* cos(x0) ~ -sin(x0) + eps */
- z += eps * __cosf (x);
- /* now z ~ (sin(x)-cos(x))*cos(x0) */
- float cosx0 = -0xb.504f3p-4f;
- cc = z / cosx0;
- }
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return j0f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = -__cosf(x+x);
+ if ((s*c)<zero) cc = z/ss;
+ else ss = z/cc;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
- if(ix>0x5c000000) z = (invsqrtpi*cc)/sqrtf(x);
- else {
- u = pzerof(x); v = qzerof(x);
- z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
- }
- return z;
+ if (ix <= 0x5c000000)
+ {
+ u = pzerof(x); v = qzerof(x);
+ cc = u*cc-v*ss;
+ }
+ z = (invsqrtpi * cc) / sqrtf(x);
+ /* The following threshold is attained for x=0x1.3b58dep+1,
+ where z is 10 ulps wrong for rounding toward +Inf. */
+ float threshold = 0x1.579b26p-4;
+ if (fabsf (cc) > threshold)
+ return z;
+ else /* |cc| <= 2^-4 */
+ return j0f_near_root (x, z);
}
if(ix<0x39000000) { /* |x| < 2**-13 */
math_force_eval(huge+x); /* raise inexact if x != 0 */
@@ -112,6 +268,170 @@ v02 = 7.6006865129e-05, /* 0x389f65e0 */
v03 = 2.5915085189e-07, /* 0x348b216c */
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
+#define FIRST_ZERO_Y0 0.893576f
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of y0 and degree-3 polynomial
+ approximations of y0 around these zeros: Py[0] for the first zero (0.89358),
+ Py[1] for the second one (3.957678), and so on. Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number
+ closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
+ polynomial. Each polynomial was generated using Sollya on the interval
+ [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+ for the alternate code. Degree 3 is enough, except for index 0 where we use
+ degree 5, and the coefficients of degree 4 and 5 are hard-coded in
+ y0f_near_root.
+*/
+static const float Py[SMALL_SIZE][7] = {
+ { 0x1.a681dap-1, 0x1.c982ecp-1, 0x1.ef6bcap-1, 0x3.274468p-28, 0xe.121b8p-4, -0x7.df8b3p-4, 0x3.877be4p-4/*, -0x3.a46c9p-4, 0x3.735478p-4*/ }, /* 0 */
+ { 0x1.f957c6p+1, 0x1.fa9534p+1, 0x1.fd11b2p+1, 0xa.f1f83p-28, -0x6.70d098p-4, 0xd.04d48p-8, 0xe.f61a9p-8 }, /* 1 */
+ { 0x1.c51832p+2, 0x1.c581dcp+2, 0x1.c65164p+2, -0x5.e2a51p-28, 0x4.cd3328p-4, -0x5.6bbe08p-8, -0xc.46d8p-8 }, /* 2 */
+ { 0x1.46fd84p+3, 0x1.471d74p+3, 0x1.475bfcp+3, -0x1.4b0aeep-24, -0x3.fec6b8p-4, 0x3.2068a4p-8, 0xa.76e2dp-8 }, /* 3 */
+ { 0x1.ab7afap+3, 0x1.ab8e1cp+3, 0x1.abb294p+3, -0x8.179d7p-28, 0x3.7e6544p-4, -0x2.1799fp-8, -0x9.0e1c4p-8 }, /* 4 */
+ { 0x1.07f9aap+4, 0x1.0803c8p+4, 0x1.08170cp+4, -0x2.5b8078p-24, -0x3.24b868p-4, 0x1.8631ecp-8, 0x8.3cb46p-8 }, /* 5 */
+ { 0x1.3a38eap+4, 0x1.3a42cep+4, 0x1.3a4d8ap+4, 0x1.cd304ap-28, 0x2.e189ecp-4, -0x1.2c6954p-8, -0x7.8178ep-8 }, /* 6 */
+ { 0x1.6c7d42p+4, 0x1.6c833p+4, 0x1.6c99fp+4, -0x6.c63f1p-28, -0x2.acc9a8p-4, 0xf.09e31p-12, 0x7.0b5ab8p-8 }, /* 7 */
+ { 0x1.9ebec4p+4, 0x1.9ec47p+4, 0x1.9ed016p+4, 0x1.e53838p-24, 0x2.81f2p-4, -0xc.5ff51p-12, -0x7.05ep-8 }, /* 8 */
+ { 0x1.d1008ep+4, 0x1.d10644p+4, 0x1.d11262p+4, 0x1.636feep-24, -0x2.5e40dcp-4, 0xa.6f81dp-12, 0x5.ff6p-8 }, /* 9 */
+ { 0x1.01a27cp+5, 0x1.01a442p+5, 0x1.01a924p+5, -0x4.04e1bp-28, 0x2.3febd8p-4, -0x8.f11e2p-12, -0x6.0111ap-8 }, /* 10 */
+ { 0x1.1ac3bcp+5, 0x1.1ac588p+5, 0x1.1ac912p+5, 0x3.6063d8p-24, -0x2.25baacp-4, 0x7.c93cdp-12, 0x4.e7577p-8 }, /* 11 */
+ { 0x1.33e508p+5, 0x1.33e6ecp+5, 0x1.33ea1ap+5, -0x3.f39ebcp-24, 0x2.0ed04cp-4, -0x6.d9434p-12, -0x4.fc0b7p-8 }, /* 12 */
+ { 0x1.4d0672p+5, 0x1.4d0868p+5, 0x1.4d0c14p+5, -0x4.ea23p-28, -0x1.fa8b4p-4, 0x6.1470e8p-12, 0x5.97f71p-8 }, /* 13 */
+ { 0x1.6628dep+5, 0x1.6629f4p+5, 0x1.662e0ep+5, -0x3.5df0c8p-24, 0x1.e8727ep-4, -0x5.76a038p-12, -0x4.ee37c8p-8 }, /* 14 */
+ { 0x1.7f4a7cp+5, 0x1.7f4b9p+5, 0x1.7f4daap+5, 0x1.1ef09ep-24, -0x1.d8293ap-4, 0x4.ed8a28p-12, 0x4.d43708p-8 }, /* 15 */
+ { 0x1.986c5cp+5, 0x1.986d38p+5, 0x1.986f5ep+5, 0x1.b70cecp-24, 0x1.c967p-4, -0x4.7a70b8p-12, -0x5.6840e8p-8 }, /* 16 */
+ { 0x1.b18dcap+5, 0x1.b18ee8p+5, 0x1.b19122p+5, 0x1.abaadcp-24, -0x1.bbf246p-4, 0x4.1a35bp-12, 0x3.c2d46p-8 }, /* 17 */
+ { 0x1.caaf86p+5, 0x1.cab0a2p+5, 0x1.cab2fep+5, 0x1.63989ep-24, 0x1.af9cb4p-4, -0x3.c2f2dcp-12, -0x4.cf8108p-8 }, /* 18 */
+ { 0x1.e3d158p+5, 0x1.e3d262p+5, 0x1.e3d492p+5, -0x1.68a8ecp-24, -0x1.a4407ep-4, 0x3.7733ecp-12, 0x5.97916p-8 }, /* 19 */
+ { 0x1.fcf316p+5, 0x1.fcf428p+5, 0x1.fcf59ap+5, 0x1.e1bb5p-24, 0x1.99be74p-4, -0x3.37210cp-12, -0x5.d19bf8p-8 }, /* 20 */
+ { 0x1.0b0a7cp+6, 0x1.0b0afap+6, 0x1.0b0b9cp+6, -0x5.5bbcfp-24, -0x1.8ffc9ap-4, 0x2.ffe638p-12, 0x2.ed28e8p-8 }, /* 21 */
+ { 0x1.179b66p+6, 0x1.179bep+6, 0x1.179d0ap+6, -0x4.9e34a8p-24, 0x1.86e51cp-4, -0x2.cc7a68p-12, -0x3.3642c4p-8 }, /* 22 */
+ { 0x1.242c5cp+6, 0x1.242ccap+6, 0x1.242d68p+6, 0x1.966706p-24, -0x1.7e657p-4, 0x2.9aed4cp-12, 0x7.b87a58p-8 }, /* 23 */
+ { 0x1.30bd62p+6, 0x1.30bdb6p+6, 0x1.30beb2p+6, 0x3.4b3b68p-24, 0x1.766dc2p-4, -0x2.72651cp-12, -0x3.e347f8p-8 }, /* 24 */
+ { 0x1.3d4e56p+6, 0x1.3d4ea2p+6, 0x1.3d4f2ep+6, 0x6.a99008p-28, -0x1.6ef07ep-4, 0x2.53aec4p-12, 0x2.9e3d88p-12 }, /* 25 */
+ { 0x1.49df38p+6, 0x1.49df9p+6, 0x1.49e042p+6, -0x7.a9fa6p-32, 0x1.67e1dap-4, -0x2.324d7p-12, -0xc.0e669p-12 }, /* 26 */
+ { 0x1.56703ap+6, 0x1.56708p+6, 0x1.5670e6p+6, -0x5.026808p-24, -0x1.613798p-4, 0x2.114594p-12, 0x1.a22402p-8 }, /* 27 */
+ { 0x1.630126p+6, 0x1.63017p+6, 0x1.630226p+6, 0x4.46aa2p-24, 0x1.5ae8c2p-4, -0x1.f4aaa4p-12, -0x3.58593cp-8 }, /* 28 */
+ { 0x1.6f9234p+6, 0x1.6f926p+6, 0x1.6f92b2p+6, 0x1.5cfccp-24, -0x1.54ed76p-4, 0x1.dd540ap-12, -0xb.e9429p-12 }, /* 29 */
+ { 0x1.7c2302p+6, 0x1.7c2352p+6, 0x1.7c23c2p+6, -0xb.4dc4cp-28, 0x1.4f3ebcp-4, -0x1.c463fp-12, -0x1.e94c54p-8 }, /* 30 */
+ { 0x1.88b412p+6, 0x1.88b444p+6, 0x1.88b50ap+6, 0x3.f5343p-24, -0x1.49d668p-4, 0x1.a53f24p-12, 0x5.128198p-8 }, /* 31 */
+ { 0x1.9544dcp+6, 0x1.954538p+6, 0x1.95459p+6, -0x6.e6f32p-28, 0x1.44aefap-4, -0x1.9a6ef8p-12, -0x6.c639cp-8 }, /* 32 */
+ { 0x1.a1d5fap+6, 0x1.a1d62cp+6, 0x1.a1d672p+6, 0x1.f359c2p-28, -0x1.3fc386p-4, 0x1.887ebep-12, 0x1.6c813cp-8 }, /* 33 */
+ { 0x1.ae66cp+6, 0x1.ae672p+6, 0x1.ae6788p+6, -0x2.9de748p-24, 0x1.3b0fa4p-4, -0x1.777f26p-12, 0x1.c128ccp-8 }, /* 34 */
+ { 0x1.baf7eep+6, 0x1.baf816p+6, 0x1.baf86cp+6, -0x2.24ccc8p-24, -0x1.368f5cp-4, 0x1.62bd9ep-12, 0xa.df002p-8 }, /* 35 */
+ { 0x1.c788dap+6, 0x1.c7890cp+6, 0x1.c7896cp+6, 0x4.7dcea8p-24, 0x1.323f16p-4, -0x1.61abf4p-12, 0xa.ad73ep-8 }, /* 36 */
+ { 0x1.d419ccp+6, 0x1.d41a02p+6, 0x1.d41a68p+6, -0x4.b538p-24, -0x1.2e1b98p-4, 0x1.4a4d64p-12, 0x3.a47674p-8 }, /* 37 */
+ { 0x1.e0aacep+6, 0x1.e0aaf8p+6, 0x1.e0ab5ep+6, 0x3.09dc4cp-24, 0x1.2a21ecp-4, -0x1.3fa20cp-12, 0x2.216e8cp-8 }, /* 38 */
+ { 0x1.ed3bb8p+6, 0x1.ed3beep+6, 0x1.ed3c56p+6, 0x4.d5a58p-28, -0x1.264f66p-4, 0x1.32c4cep-12, 0x1.53cbb4p-8 }, /* 39 */
+ { 0x1.f9ccaep+6, 0x1.f9cce6p+6, 0x1.f9cd52p+6, 0x3.f4c44cp-24, 0x1.22a192p-4, -0x1.1f8514p-12, -0xc.0de32p-8 }, /* 40 */
+ { 0x1.032ed6p+7, 0x1.032eeep+7, 0x1.032f0cp+7, 0x2.4beae8p-24, -0x1.1f1634p-4, 0x1.171664p-12, 0x1.72a654p-4 }, /* 41 */
+ { 0x1.09776p+7, 0x1.09776ap+7, 0x1.09779cp+7, -0xd.a581ep-28, 0x1.1bab3cp-4, -0xf.9f507p-16, -0xc.ba2d4p-8 }, /* 42 */
+ { 0x1.0fbfdp+7, 0x1.0fbfe6p+7, 0x1.0fbff6p+7, 0xa.7c0bdp-28, -0x1.185eccp-4, 0x1.01d7dep-12, -0x1.a2186ep-4 }, /* 43 */
+ { 0x1.160856p+7, 0x1.160862p+7, 0x1.16087ap+7, -0x1.9452ecp-24, 0x1.152f26p-4, -0x1.07b4aap-12, 0x1.6bbd7ep-4 }, /* 44 */
+ { 0x1.1c50dp+7, 0x1.1c50dep+7, 0x1.1c5118p+7, 0x3.83b7b8p-24, -0x1.121ab2p-4, 0x1.0e938cp-12, -0x5.1a6dfp-8 }, /* 45 */
+ { 0x1.22995p+7, 0x1.22995ap+7, 0x1.229976p+7, -0x6.5ca3a8p-24, 0x1.0f1ff2p-4, -0xe.f198p-16, -0x3.8e98b8p-8 }, /* 46 */
+ { 0x1.28e1ccp+7, 0x1.28e1d8p+7, 0x1.28e1ep+7, -0x6.bb61ap-24, -0x1.0c3d8ap-4, 0xf.a14b9p-16, 0x9.81e82p-4 }, /* 47 */
+ { 0x1.2f2a48p+7, 0x1.2f2a54p+7, 0x1.2f2a72p+7, 0x2.2438p-24, 0x1.097236p-4, -0xd.fed5ep-16, -0x3.19eb5cp-8 }, /* 48 */
+ { 0x1.3572b8p+7, 0x1.3572dp+7, 0x1.3572ecp+7, 0x3.1e0054p-24, -0x1.06bcc8p-4, 0xd.d2596p-16, -0x1.67e00ap-4 }, /* 49 */
+ { 0x1.3bbb3ep+7, 0x1.3bbb4ep+7, 0x1.3bbb6ap+7, 0x7.46c908p-24, 0x1.041c28p-4, -0xd.04045p-16, -0x8.fb297p-8 }, /* 50 */
+ { 0x1.4203aep+7, 0x1.4203cap+7, 0x1.4203e6p+7, -0xb.4f158p-28, -0x1.018f52p-4, 0xc.ccf6fp-16, 0x1.4d5dp-4 }, /* 51 */
+ { 0x1.484c38p+7, 0x1.484c46p+7, 0x1.484c56p+7, -0x6.5a89c8p-24, 0xf.f155p-8, -0xc.5d21dp-16, -0xd.aca34p-8 }, /* 52 */
+ { 0x1.4e94b8p+7, 0x1.4e94c4p+7, 0x1.4e94d4p+7, -0x1.ef16c8p-24, -0xf.cad3fp-8, 0xb.d75f8p-16, 0x1.f36732p-4 }, /* 53 */
+ { 0x1.54dd36p+7, 0x1.54dd4p+7, 0x1.54dd52p+7, -0x6.1e7b68p-24, 0xf.a564cp-8, -0xb.ec1cfp-16, 0xe.e7421p-8 }, /* 54 */
+ { 0x1.5b25aep+7, 0x1.5b25bep+7, 0x1.5b25d4p+7, -0xf.8c858p-28, -0xf.80faep-8, 0xb.8b6c5p-16, -0x5.835ed8p-8 }, /* 55 */
+ { 0x1.616e34p+7, 0x1.616e3cp+7, 0x1.616e4ep+7, 0x7.75d858p-24, 0xf.5d8abp-8, -0xb.b3779p-16, 0x2.40b948p-4 }, /* 56 */
+ { 0x1.67b6bp+7, 0x1.67b6b8p+7, 0x1.67b6dp+7, 0x1.d78632p-24, -0xf.3b096p-8, 0xa.daf89p-16, 0x1.aa8548p-8 }, /* 57 */
+ { 0x1.6dff2cp+7, 0x1.6dff36p+7, 0x1.6dff54p+7, 0x3.b24794p-24, 0xf.196c7p-8, -0xb.1afe1p-16, -0x1.77538cp-8 }, /* 58 */
+ { 0x1.7447a2p+7, 0x1.7447b2p+7, 0x1.7447cap+7, 0x6.39cbc8p-24, -0xe.f8aa5p-8, 0xa.50daap-16, 0x1.9592c2p-8 }, /* 59 */
+ { 0x1.7a902p+7, 0x1.7a903p+7, 0x1.7a9038p+7, -0x1.820e3ap-24, 0xe.d8b9dp-8, -0xa.998cp-16, -0x2.c35d78p-4 }, /* 60 */
+ { 0x1.80d89ep+7, 0x1.80d8aep+7, 0x1.80d8bep+7, -0x2.c7e038p-24, -0xe.b9925p-8, 0x9.ce06p-16, -0x2.2b3054p-4 }, /* 61 */
+ { 0x1.87211cp+7, 0x1.87212cp+7, 0x1.872144p+7, 0x6.ab31c8p-24, 0xe.9b2bep-8, -0x9.4de7p-16, -0x1.32cb5ep-4 }, /* 62 */
+ { 0x1.8d699ap+7, 0x1.8d69a8p+7, 0x1.8d69bp+7, 0x4.4ef25p-24, -0xe.7d7ecp-8, 0x9.a0f1ep-16, 0x1.6ac076p-4 }, /* 63 */
+};
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ y0(x) ~ sqrt(2/(pi*x))*beta0(x)*sin(x-pi/4-alpha0(x))
+ where beta0(x) = 1 - 1/(16*x^2) + 53/(512*x^4)
+ and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
+static float
+y0f_asympt (float x)
+{
+ /* The following code fails to give an error <= 9 ulps in only two cases,
+ for which we tabulate the correctly-rounded result. */
+ if (x == 0x1.bfad96p+7f)
+ return -0x7.f32bdp-32f;
+ if (x == 0x1.2e2a42p+17f)
+ return 0x1.a48974p-40f;
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
+ double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
+ double h;
+ h = __math_reduce_mod_twopi ((double) x);
+ /* Subtract alpha0. */
+ h -= alpha0;
+ /* Now reduce mod pi/2. */
+ double piover2 = 0x1.921fb54442d18p+0;
+ int n = 0;
+ while (fabs (h) > piover2 / 2)
+ {
+ if (h > 0)
+ {
+ h -= piover2;
+ n ++;
+ }
+ else
+ {
+ h += piover2;
+ n --;
+ }
+ }
+ /* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ float t = cst / sqrtf (x) * (float) beta0;
+ if (n == 0)
+ return t * sinf (xr);
+ else if (n == 2) /* sin(x+pi) = -sin(x) */
+ return -t * sinf (xr);
+ else if (n == 1) /* sin(x+pi/2) = cos(x) */
+ return t * cosf (xr);
+ else /* sin(x+3pi/2) = -cos(x) */
+ return -t * cosf (xr);
+}
+
+/* Special code for x near a root of y0.
+ z is the value computed by the generic code.
+ For small x, use a polynomial approximating y0 around its root.
+ For large x, use an asymptotic formula (y0f_asympt). */
+static float
+y0f_near_root (float x, float z)
+{
+ float index_f;
+ int index;
+
+ index_f = roundf ((x - FIRST_ZERO_Y0) / (float) M_PI);
+ if (index_f >= SMALL_SIZE)
+ return y0f_asympt (x);
+ index = (int) index_f;
+ const float *p = Py[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If not in the interval [x0,x1] around xmid, return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid;
+ /* For degree 0 use a degree-5 polynomial, where the coefficients of
+ degree 4 and 5 are hard-coded. */
+ float p6 = (index > 0) ? p[6] : p[6] + y * (-0x3.a46c9p-4 + y * 0x3.735478p-4);
+ float res = p[3] + y * (p[4] + y * (p[5] + y * p6));
+ return res;
+}
+
float
__ieee754_y0f(float x)
{
@@ -124,7 +444,8 @@ __ieee754_y0f(float x)
if(ix>=0x7f800000) return one/(x+x*x);
if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
if(hx<0) return zero/(zero*x);
- if(ix >= 0x40000000) { /* |x| >= 2.0 */
+ if(ix >= 0x40000000 || (0x3f5340ed <= ix && ix <= 0x3f77b5e5)) {
+ /* |x| >= 2.0 or 0x1.a681dap-1 <= |x| <= 0x1.ef6bcap-1 (around 1st zero) */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
@@ -136,6 +457,7 @@ __ieee754_y0f(float x)
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
+ SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (x, &s, &c);
ss = s-c;
cc = s+c;
@@ -143,17 +465,23 @@ __ieee754_y0f(float x)
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
- if(ix<0x7f000000) { /* make sure x+x not overflow */
- z = -__cosf(x+x);
- if ((s*c)<zero) cc = z/ss;
- else ss = z/cc;
- }
- if(ix>0x5c000000) z = (invsqrtpi*ss)/sqrtf(x);
- else {
- u = pzerof(x); v = qzerof(x);
- z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
- }
- return z;
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return y0f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = -__cosf(x+x);
+ if ((s*c)<zero) cc = z/ss;
+ else ss = z/cc;
+ if (ix <= 0x5c000000)
+ {
+ u = pzerof(x); v = qzerof(x);
+ ss = u*ss+v*cc;
+ }
+ z = (invsqrtpi*ss)/sqrtf(x);
+ float threshold = 0x1.be5858p-4;
+ if (fabsf (ss) > threshold)
+ return z;
+ else /* |ss| <= 2^-4 */
+ return y0f_near_root (x, z);
}
if(ix<=0x39800000) { /* x < 2**-13 */
return(u00 + tpi*__ieee754_logf(x));
@@ -165,7 +493,7 @@ __ieee754_y0f(float x)
}
libm_alias_finite (__ieee754_y0f, __y0f)
-/* The asymptotic expansions of pzero is
+/* The asymptotic expansion of pzero is
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
* For x >= 2, We approximate pzero by
* pzero(x) = 1 + (R/S)
@@ -257,7 +585,7 @@ pzerof(float x)
}
-/* For x >= 8, the asymptotic expansions of qzero is
+/* For x >= 8, the asymptotic expansion of qzero is
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
* We approximate pzero by
* qzero(x) = s*(-1.25 + (R/S))
--
2.30.0
^ permalink raw reply [flat|nested] 4+ messages in thread
* [PATCH 3/4] Fix the inaccuracy of j1f and y1f [BZ #14470 and #14472]
2021-02-09 5:28 [PATCH 1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f Paul Zimmermann
2021-02-09 5:28 ` [PATCH 2/4] Fix the inaccuracy of j0f and of y0f [BZ #14469 and #14471] Paul Zimmermann
@ 2021-02-09 5:28 ` Paul Zimmermann
2021-02-09 5:28 ` [PATCH 4/4] Added new entries for j0/j1/y0/y1 and regenerate ulps Paul Zimmermann
2 siblings, 0 replies; 4+ messages in thread
From: Paul Zimmermann @ 2021-02-09 5:28 UTC (permalink / raw)
To: libc-alpha
For both j1f and y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j1f/y1f computation, or for very large inputs, thus should not give any
visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j1/y1, approximation polynomials of degree 3
are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Tested on x86_64 with --disable-multi-arch.
---
sysdeps/ieee754/flt-32/e_j1f.c | 403 ++++++++++++++++++++++++++++++---
1 file changed, 373 insertions(+), 30 deletions(-)
diff --git a/sysdeps/ieee754/flt-32/e_j1f.c b/sysdeps/ieee754/flt-32/e_j1f.c
index ac5bb76828..92ebbd3a9a 100644
--- a/sysdeps/ieee754/flt-32/e_j1f.c
+++ b/sysdeps/ieee754/flt-32/e_j1f.c
@@ -21,6 +21,7 @@
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-finite.h>
+#include <e_reduce_pi.h>
static float ponef(float), qonef(float);
@@ -42,6 +43,184 @@ s05 = 1.2354227016e-11; /* 0x2d59567e */
static const float zero = 0.0;
+#define FIRST_ZERO_J1 3.831705f /* First positive zero of j1. */
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of j1 and degree-3 polynomial
+ approximations of j1 around these zeros: Pj[0] for the first positive zero
+ (3.831705), Pj[1] for the second one (7.015586), and so on.
+ Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number
+ closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
+ polynomial. Each polynomial was generated using Sollya on the interval
+ [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+ for the alternate code. Degree 3 is enough to get an error at most 9 ulps,
+ except around the first zero.
+*/
+static const float Pj[SMALL_SIZE][7] = {
+ /* For index 0, we use a degree-4 polynomial generated by Sollya, with the
+ coefficient of degree 4 hard-coded in j1f_near_root(). */
+ { 0x1.e09e5ep+1, 0x1.ea7558p+1, 0x1.ef7352p+1, -0x8.4f069p-28, -0x6.71b3d8p-4, 0xd.744a2p-8, 0xd.acd9p-8/*, -0x1.3e51aap-8*/ }, /* 0 */
+ { 0x1.bdb4c2p+2, 0x1.c0ff6p+2, 0x1.c27a8cp+2, 0xe.c2858p-28, 0x4.cd464p-4, -0x5.79b71p-8, -0xc.11124p-8 }, /* 1 */
+ { 0x1.43b214p+3, 0x1.458d0ep+3, 0x1.460ccep+3, -0x1.e7acecp-24, -0x3.feca9p-4, 0x3.2470f8p-8, 0xa.625b7p-8 }, /* 2 */
+ { 0x1.a9c98p+3, 0x1.aa5bbp+3, 0x1.aaa492p+3, 0x1.698158p-24, 0x3.7e666cp-4, -0x2.1900ap-8, -0x9.2755p-8 }, /* 3 */
+ { 0x1.073be4p+4, 0x1.0787b4p+4, 0x1.07aed8p+4, -0x1.f5f658p-24, -0x3.24b8ep-4, 0x1.86e35cp-8, 0x8.4e4bbp-8 }, /* 4 */
+ { 0x1.39ae2ap+4, 0x1.39da8ep+4, 0x1.39f3dap+4, -0x1.4e744p-24, 0x2.e18a24p-4, -0x1.2ccd16p-8, -0x7.a27ep-8 }, /* 5 */
+ { 0x1.6bfa46p+4, 0x1.6c294ep+4, 0x1.6c412p+4, 0xa.3fb7fp-28, -0x2.acc9c4p-4, 0xf.0b783p-12, 0x7.1c0d3p-8 }, /* 6 */
+ { 0x1.9e44a2p+4, 0x1.9e757p+4, 0x1.9e876ep+4, -0x2.29f6f4p-24, 0x2.81f21p-4, -0xc.641bp-12, -0x6.a7ea58p-8 }, /* 7 */
+ { 0x1.d08a3ep+4, 0x1.d0bfdp+4, 0x1.d0cd3cp+4, -0x1.b5d196p-24, -0x2.5e40e4p-4, 0xa.7059fp-12, 0x6.4d6bfp-8 }, /* 8 */
+ { 0x1.017794p+5, 0x1.018476p+5, 0x1.018b8cp+5, -0x4.0e001p-24, 0x2.3febep-4, -0x8.f23aap-12, -0x6.0102cp-8 }, /* 9 */
+ { 0x1.1a9e78p+5, 0x1.1aa89p+5, 0x1.1aaf88p+5, 0x3.b26f2p-24, -0x2.25babp-4, 0x7.c6d948p-12, 0x5.a1d988p-8 }, /* 10 */
+ { 0x1.33bddep+5, 0x1.33cc52p+5, 0x1.33d2e4p+5, -0xf.c8cdap-28, 0x2.0ed05p-4, -0x6.d97dbp-12, -0x5.8da498p-8 }, /* 11 */
+ { 0x1.4ce7cp+5, 0x1.4cefdp+5, 0x1.4cf7d4p+5, -0x3.9940e4p-24, -0x1.fa8b4p-4, 0x6.16108p-12, 0x5.4355e8p-8 }, /* 12 */
+ { 0x1.6603e8p+5, 0x1.661316p+5, 0x1.66173p+5, 0x9.da15dp-28, 0x1.e8727ep-4, -0x5.742468p-12, -0x5.117c28p-8 }, /* 13 */
+ { 0x1.7f2ebcp+5, 0x1.7f3632p+5, 0x1.7f3a7ep+5, -0x3.39b218p-24, -0x1.d8293ap-4, 0x4.ee3348p-12, 0x4.f9bep-8 }, /* 14 */
+ { 0x1.9850e6p+5, 0x1.985928p+5, 0x1.985d9ep+5, -0x3.7b5108p-24, 0x1.c96702p-4, -0x4.7b0d08p-12, -0x4.c784a8p-8 }, /* 15 */
+ { 0x1.b172e8p+5, 0x1.b17c04p+5, 0x1.b1805cp+5, -0x1.91e43ep-24, -0x1.bbf246p-4, 0x4.18ad78p-12, 0x4.9bfae8p-8 }, /* 16 */
+ { 0x1.ca955ap+5, 0x1.ca9ec6p+5, 0x1.caa2a4p+5, 0x1.28453cp-24, 0x1.af9cb4p-4, -0x3.c3a494p-12, -0x4.78b69p-8 }, /* 17 */
+ { 0x1.e3bc94p+5, 0x1.e3c174p+5, 0x1.e3c64p+5, -0x2.e7fef4p-24, -0x1.a4407ep-4, 0x3.79b228p-12, 0x4.874f7p-8 }, /* 18 */
+ { 0x1.fcdf16p+5, 0x1.fce40ep+5, 0x1.fce71p+5, -0x3.23b2fcp-24, 0x1.99be76p-4, -0x3.39ad7cp-12, -0x4.92a55p-8 }, /* 19 */
+ { 0x1.0afe34p+6, 0x1.0b034ep+6, 0x1.0b0532p+6, -0xd.19e93p-28, -0x1.8ffc9cp-4, 0x2.fee7f8p-12, 0x4.2d33b8p-8 }, /* 20 */
+ { 0x1.179344p+6, 0x1.17948ep+6, 0x1.1795a2p+6, 0x1.c2ac48p-24, 0x1.86e51cp-4, -0x2.cc5abp-12, -0x4.866d08p-8 }, /* 21 */
+ { 0x1.24231ep+6, 0x1.2425c8p+6, 0x1.2426e2p+6, -0xd.31027p-28, -0x1.7e656ep-4, 0x2.9db23cp-12, 0x3.cc63c8p-8 }, /* 22 */
+ { 0x1.30b5a8p+6, 0x1.30b6fep+6, 0x1.30b84ep+6, 0x5.b5e53p-24, 0x1.766dc2p-4, -0x2.754cfcp-12, -0x3.c39bb4p-8 }, /* 23 */
+ { 0x1.3d46ccp+6, 0x1.3d482ep+6, 0x1.3d495ep+6, -0x1.340a8ap-24, -0x1.6ef07ep-4, 0x2.4ff9d4p-12, 0x3.9b36e4p-8 }, /* 24 */
+ { 0x1.49d688p+6, 0x1.49d95ap+6, 0x1.49dabep+6, -0x3.ba66p-24, 0x1.67e1dcp-4, -0x2.2f32b8p-12, -0x3.e2aaf4p-8 }, /* 25 */
+ { 0x1.566916p+6, 0x1.566a84p+6, 0x1.566bcp+6, 0x6.47ca5p-28, -0x1.61379ap-4, 0x2.1096acp-12, 0x4.2d0968p-8 }, /* 26 */
+ { 0x1.62f8dap+6, 0x1.62fbaap+6, 0x1.62fc9cp+6, -0x2.12affp-24, 0x1.5ae8c4p-4, -0x1.f32444p-12, -0x3.9e592p-8 }, /* 27 */
+ { 0x1.6f89e6p+6, 0x1.6f8ccep+6, 0x1.6f8e34p+6, -0x7.4853ap-28, -0x1.54ed76p-4, 0x1.db004ap-12, 0x3.907034p-8 }, /* 28 */
+ { 0x1.7c1c6ap+6, 0x1.7c1deep+6, 0x1.7c1f1ap+6, -0x4.f0a998p-24, 0x1.4f3ebcp-4, -0x1.c26808p-12, -0x2.da8df8p-8 }, /* 29 */
+ { 0x1.88adaep+6, 0x1.88af0ep+6, 0x1.88afc4p+6, -0x1.80c246p-24, -0x1.49d668p-4, 0x1.aebc26p-12, 0x3.af7b5cp-8 }, /* 30 */
+ { 0x1.953d7p+6, 0x1.95402ap+6, 0x1.9540a6p+6, -0x2.22aff8p-24, 0x1.44aefap-4, -0x1.99f25p-12, -0x3.5e9198p-8 }, /* 31 */
+ { 0x1.a1d01ep+6, 0x1.a1d146p+6, 0x1.a1d202p+6, -0x3.aad6d4p-24, -0x1.3fc386p-4, 0x1.892858p-12, 0x3.fe0184p-8 }, /* 32 */
+ { 0x1.ae60ecp+6, 0x1.ae625ep+6, 0x1.ae6322p+6, -0x2.010be4p-24, 0x1.3b0fa4p-4, -0x1.7539ap-12, -0x2.b2c9bp-8 }, /* 33 */
+ { 0x1.baf234p+6, 0x1.baf376p+6, 0x1.baf442p+6, -0xd.4fd17p-32, -0x1.368f5cp-4, 0x1.6734e4p-12, 0x3.59f514p-8 }, /* 34 */
+ { 0x1.c782e6p+6, 0x1.c7848cp+6, 0x1.c78516p+6, -0xa.d662dp-28, 0x1.323f18p-4, -0x1.571c02p-12, -0x3.2be5bp-8 }, /* 35 */
+ { 0x1.d4144ep+6, 0x1.d415ap+6, 0x1.d41622p+6, 0x4.9f217p-24, -0x1.2e1b9ap-4, 0x1.4a2edap-12, 0x3.a4e96p-8 }, /* 36 */
+ { 0x1.e0a60cp+6, 0x1.e0a6b4p+6, 0x1.e0a788p+6, -0x2.d167p-24, 0x1.2a21eep-4, -0x1.3c4b46p-12, -0x4.9e0978p-8 }, /* 37 */
+ { 0x1.ed36eep+6, 0x1.ed37c8p+6, 0x1.ed3892p+6, -0x4.15a83p-24, -0x1.264f66p-4, 0x1.31dea4p-12, 0x3.d125ecp-8 }, /* 38 */
+ { 0x1.f9c77p+6, 0x1.f9c8d8p+6, 0x1.f9c9acp+6, -0x2.a5bbbp-24, 0x1.22a192p-4, -0x1.25e59ep-12, -0x2.ef6934p-8 }, /* 39 */
+ { 0x1.032c54p+7, 0x1.032cf4p+7, 0x1.032d66p+7, 0x4.e828bp-24, -0x1.1f1634p-4, 0x1.1c2394p-12, 0x3.6d744cp-8 }, /* 40 */
+ { 0x1.09750cp+7, 0x1.09757cp+7, 0x1.0975b2p+7, -0x3.28a3bcp-24, 0x1.1bab3ep-4, -0x1.1569cep-12, -0x5.84da7p-8 }, /* 41 */
+ { 0x1.0fbd9ap+7, 0x1.0fbe04p+7, 0x1.0fbe5ep+7, -0x2.2f667p-24, -0x1.185eccp-4, 0x1.07f42cp-12, 0x2.87896cp-8 }, /* 42 */
+ { 0x1.160628p+7, 0x1.16068ap+7, 0x1.1606cep+7, -0x6.9097dp-24, 0x1.152f28p-4, -0x1.0227fep-12, -0x5.da6e6p-8 }, /* 43 */
+ { 0x1.1c4e9ap+7, 0x1.1c4f12p+7, 0x1.1c4f7cp+7, -0x5.1b408p-24, -0x1.121abp-4, 0xf.6efcp-16, 0x2.c5e954p-8 }, /* 44 */
+ { 0x1.2296aap+7, 0x1.229798p+7, 0x1.2297cap+7, 0x2.70d0dp-24, 0x1.0f1ffp-4, -0xf.523f5p-16, -0x3.5c0568p-8 }, /* 45 */
+ { 0x1.28dfa4p+7, 0x1.28e01ep+7, 0x1.28e054p+7, -0x2.7c176p-24, -0x1.0c3d8ap-4, 0xe.8329ap-16, 0x3.5eb34p-8 }, /* 46 */
+ { 0x1.2f282ap+7, 0x1.2f28a4p+7, 0x1.2f28dep+7, 0x4.fd6368p-24, 0x1.097236p-4, -0xe.17299p-16, -0x3.73a2e4p-8 }, /* 47 */
+ { 0x1.3570bp+7, 0x1.357128p+7, 0x1.35715p+7, 0x6.b05f68p-24, -0x1.06bccap-4, 0xd.527b8p-16, 0x2.b8bf9cp-8 }, /* 48 */
+ { 0x1.3bb932p+7, 0x1.3bb9aep+7, 0x1.3bb9eap+7, 0x4.0d622p-28, 0x1.041c28p-4, -0xd.0ac11p-16, -0x1.65f2ccp-8 }, /* 49 */
+ { 0x1.4201b6p+7, 0x1.420232p+7, 0x1.42027p+7, 0x7.0d98cp-24, -0x1.018f52p-4, 0xc.c4d8ep-16, 0x2.8f250cp-8 }, /* 50 */
+ { 0x1.484a78p+7, 0x1.484ab8p+7, 0x1.484ad8p+7, 0x3.93d10cp-24, 0xf.f154fp-8, -0xc.7b7fep-16, -0x3.6b6e4cp-8 }, /* 51 */
+ { 0x1.4e92c8p+7, 0x1.4e933cp+7, 0x1.4e9368p+7, 0xd.88185p-32, -0xf.cad3fp-8, 0xc.1462p-16, 0x2.bd66p-8 }, /* 52 */
+ { 0x1.54db84p+7, 0x1.54dbcp+7, 0x1.54dbf4p+7, -0x1.fe6b92p-24, 0xf.a564cp-8, -0xb.c4e6cp-16, -0x3.d51decp-8 }, /* 53 */
+ { 0x1.5b23c4p+7, 0x1.5b2444p+7, 0x1.5b2486p+7, 0x2.6137f4p-24, -0xf.80faep-8, 0xb.5199ep-16, 0x1.9ca85ap-8 }, /* 54 */
+ { 0x1.616c88p+7, 0x1.616cc8p+7, 0x1.616d0ap+7, -0x1.55468p-24, 0xf.5d8acp-8, -0xb.21d16p-16, -0x1.b8809ap-8 }, /* 55 */
+ { 0x1.67b4fp+7, 0x1.67b54cp+7, 0x1.67b588p+7, -0x1.08c6bep-24, -0xf.3b096p-8, 0xa.e65efp-16, 0x3.642304p-8 }, /* 56 */
+ { 0x1.6dfd8ep+7, 0x1.6dfddp+7, 0x1.6dfdfap+7, 0x4.9ebfa8p-24, 0xf.196c7p-8, -0xa.ba8c8p-16, -0x5.ad6a2p-8 }, /* 57 */
+ { 0x1.74461p+7, 0x1.744652p+7, 0x1.744692p+7, 0x5.a4017p-24, -0xe.f8aa5p-8, 0xa.49748p-16, 0x2.a86498p-8 }, /* 58 */
+ { 0x1.7a8e5ep+7, 0x1.7a8ed6p+7, 0x1.7a8ef8p+7, 0x3.bcb2a8p-28, 0xe.d8b9dp-8, -0x9.c43bep-16, -0x1.e7124ap-8 }, /* 59 */
+ { 0x1.80d714p+7, 0x1.80d75ap+7, 0x1.80d78ap+7, -0x7.1091fp-24, -0xe.b9925p-8, 0x9.c43dap-16, 0x1.aba86p-8 }, /* 60 */
+ { 0x1.871f58p+7, 0x1.871fdcp+7, 0x1.87201ap+7, 0x2.ca1cf4p-28, 0xe.9b2bep-8, -0x9.843b3p-16, -0x2.093e68p-8 }, /* 61 */
+ { 0x1.8d6836p+7, 0x1.8d685ep+7, 0x1.8d6882p+7, 0x5.aa8908p-24, -0xe.7d7ecp-8, 0x9.501a8p-16, 0x2.54a754p-8 }, /* 62 */
+ { 0x1.93b09cp+7, 0x1.93b0e2p+7, 0x1.93b10ep+7, 0x3.d9cd9cp-24, 0xe.6083ap-8, -0x9.45dadp-16, -0x5.112908p-8 }, /* 63 */
+};
+
+/* Reduce h mod (pi/2), and add k to n if k*pi/2 was subtracted from h. */
+static double
+reduce_mod_piover2 (double h, int *n)
+{
+ double piover2 = 0x1.921fb54442d18p+0;
+ while (fabs (h) > piover2 / 2)
+ {
+ if (h > 0)
+ {
+ h -= piover2;
+ (*n) ++;
+ }
+ else
+ {
+ h += piover2;
+ (*n) --;
+ }
+ }
+ return h;
+}
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ j1f(x) ~ sqrt(2/(pi*x))*beta1(x)*cos(x-3pi/4-alpha1(x))
+ where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
+ and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
+static float
+j1f_asympt (float x)
+{
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ if (x < 0)
+ {
+ x = -x;
+ cst = -cst;
+ }
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
+ double alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
+ double h;
+ h = __math_reduce_mod_twopi ((double) x);
+ int n = -1; /* Accounts for -pi/2 (pi/4 was subtracted in
+ __math_reduce_mod_twopi). */
+ /* Now reduce mod pi/2. */
+ h = reduce_mod_piover2 (h, &n);
+ /* Subtract alpha1. */
+ h -= alpha1;
+ /* Reduce again mod pi/2 if needed. */
+ h = reduce_mod_piover2 (h, &n);
+ /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float t = cst / sqrtf (x) * (float) beta1;
+ if (n == 0)
+ return t * cosf (xr);
+ else if (n == 2) /* cos(x+pi) = -cos(x) */
+ return -t * cosf (xr);
+ else if (n == 1) /* cos(x+pi/2) = -sin(x) */
+ return -t * sinf (xr);
+ else /* cos(x+3pi/2) = sin(x) */
+ return t * sinf (xr);
+}
+
+/* Special code for x near a root of j1.
+ z is the value computed by the generic code.
+ For small x, we use a polynomial approximating j1 around its root.
+ For large x, we use an asymptotic formula (j1f_asympt). */
+static float
+j1f_near_root (float x, float z)
+{
+ float index_f, sign = 1.0f;
+ int index;
+
+ if (x < 0)
+ {
+ x = -x;
+ sign = -1.0f;
+ }
+ index_f = roundf ((x - FIRST_ZERO_J1) / (float) M_PI);
+ if (index_f >= SMALL_SIZE)
+ return sign * j1f_asympt (x);
+ index = (int) index_f;
+ const float *p = Pj[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If not in the interval [x0,x1] around xmid, return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid;
+ float p6 = (index > 0) ? p[6] : p[6] + y * -0x1.3e51aap-8f;
+ return sign * (p[3] + y * (p[4] + y * (p[5] + y * p6)));
+}
+
float
__ieee754_j1f(float x)
{
@@ -53,25 +232,32 @@ __ieee754_j1f(float x)
if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
y = fabsf(x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
+ SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (y, &s, &c);
ss = -s-c;
cc = s-c;
- if(ix<0x7f000000) { /* make sure y+y not overflow */
- z = __cosf(y+y);
- if ((s*c)>zero) cc = z/ss;
- else ss = z/cc;
- }
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return j1f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = __cosf(y+y);
+ if ((s*c)>zero) cc = z/ss;
+ else ss = z/cc;
/*
* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
*/
- if(ix>0x5c000000) z = (invsqrtpi*cc)/sqrtf(y);
- else {
+ if (ix <= 0x5c000000)
+ {
u = ponef(y); v = qonef(y);
- z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
- }
- if(hx<0) return -z;
- else return z;
+ cc = u*cc-v*ss;
+ }
+ z = (invsqrtpi * cc) / sqrtf(y);
+ /* Adjust sign of z. */
+ z = (hx < 0) ? -z : z;
+ if (fabsf (cc) > 0x1p-3) /* Most likely. */
+ return z;
+ else /* |cc| <= 2^-3 */
+ return j1f_near_root (x, z);
}
if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */
if(huge+x>one) { /* inexact if x!=0 necessary */
@@ -105,6 +291,157 @@ static const float V0[5] = {
1.6655924903e-11, /* 0x2d9281cf */
};
+#define FIRST_ZERO_Y1 2.197141f /* First zero of y1. */
+
+/* The following table contains successive zeros of y1 and degree-3 polynomial
+ approximations of y1 around these zeros: Py[0] for the first positive zero
+ (2.197141), Py[1] for the second one (5.429681), and so on.
+ Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number
+ closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
+ polynomial. Each polynomial was generated using Sollya on the interval
+ [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+ for the alternate code. Degree 3 is enough (except around the first roots).
+*/
+static const float Py[SMALL_SIZE][7] = {
+ /* For index 0, we use a degree-5 polynomial generated by Sollya, with the
+ coefficients of degree 4 and 5 hard-coded in y1f_near_root(). */
+ { 0x1.f7f16ap+0, 0x1.193beep+1, 0x1.2105dcp+1, 0xb.96749p-28, 0x8.55241p-4, -0x1.e570bp-4, -0x8.68b61p-8/*, -0x1.28043p-8, 0x2.50e83p-8*/ }, /* 0 */
+ /* For index 1, we use a degree-4 polynomial generated by Sollya, with the
+ coefficient of degree 4 hard-coded in y1f_near_root(). */
+ { 0x1.55c6d2p+2, 0x1.5b7fe4p+2, 0x1.5cf8cap+2, 0x1.3c7822p-24, -0x5.71f158p-4, 0x8.05cb4p-8, 0xd.0b15p-8/*, -0xf.ff6b8p-12*/ }, /* 1 */
+ { 0x1.113c6p+3, 0x1.13127ap+3, 0x1.138718p+3, -0x1.f3ad8ep-24, 0x4.57e66p-4, -0x4.0afb58p-8, -0xb.29207p-8 }, /* 2 */
+ { 0x1.76e7dep+3, 0x1.77f914p+3, 0x1.786a6ap+3, -0xd.5608fp-28, -0x3.b829d4p-4, 0x2.8852cp-8, 0x9.b70e3p-8 }, /* 3 */
+ { 0x1.dc2794p+3, 0x1.dcb7d8p+3, 0x1.dd032p+3, -0xe.a7c04p-28, 0x3.4e0458p-4, -0x1.c64b18p-8, -0x8.b0e7fp-8 }, /* 4 */
+ { 0x1.20874p+4, 0x1.20b1c6p+4, 0x1.20c71p+4, 0x1.c2626p-24, -0x3.00f03cp-4, 0x1.54f806p-8, 0x7.f9cf9p-8 }, /* 5 */
+ { 0x1.52d89ap+4, 0x1.530254p+4, 0x1.531962p+4, -0x1.9503ecp-24, 0x2.c5b29cp-4, -0x1.0bf28p-8, -0x7.562e58p-8 }, /* 6 */
+ { 0x1.851e64p+4, 0x1.854fa4p+4, 0x1.85679p+4, -0x2.8d40fcp-24, -0x2.96547p-4, 0xd.9c38bp-12, 0x6.dcbf8p-8 }, /* 7 */
+ { 0x1.b7808ep+4, 0x1.b79acep+4, 0x1.b7b2a8p+4, -0x2.36df5cp-24, 0x2.6f55ap-4, -0xb.57f9fp-12, -0x6.82569p-8 }, /* 8 */
+ { 0x1.e9c916p+4, 0x1.e9e48p+4, 0x1.e9f24p+4, 0xd.e2eb7p-28, -0x2.4e8104p-4, 0x9.a4be2p-12, 0x6.2541fp-8 }, /* 9 */
+ { 0x1.0e0808p+5, 0x1.0e169p+5, 0x1.0e1d92p+5, -0x2.3070f4p-24, 0x2.325e4cp-4, -0x8.53604p-12, -0x5.ca03a8p-8 }, /* 10 */
+ { 0x1.273318p+5, 0x1.273a7cp+5, 0x1.2741fcp+5, -0x3.525508p-24, -0x2.19e7dcp-4, 0x7.49d1dp-12, 0x5.9cb02p-8 }, /* 11 */
+ { 0x1.405122p+5, 0x1.405e18p+5, 0x1.4065cep+5, -0xe.6e158p-28, 0x2.046174p-4, -0x6.71b3dp-12, -0x5.4c3c8p-8 }, /* 12 */
+ { 0x1.5971dcp+5, 0x1.598178p+5, 0x1.598592p+5, 0x1.e72698p-24, -0x1.f13fb2p-4, 0x5.c0f938p-12, 0x5.28ca78p-8 }, /* 13 */
+ { 0x1.729c4ep+5, 0x1.72a4a8p+5, 0x1.72a8eap+5, -0x1.5bed9cp-24, 0x1.e018dcp-4, -0x5.2f11e8p-12, -0x5.16ce48p-8 }, /* 14 */
+ { 0x1.8bbf4ep+5, 0x1.8bc7b2p+5, 0x1.8bcc1p+5, -0x3.6b654cp-24, -0x1.d09b2p-4, 0x4.b1747p-12, 0x4.bd22fp-8 }, /* 15 */
+ { 0x1.a4e272p+5, 0x1.a4ea9ap+5, 0x1.a4eef4p+5, 0x1.6f11bp-24, 0x1.c28612p-4, -0x4.47462p-12, -0x4.947c5p-8 }, /* 16 */
+ { 0x1.be08bep+5, 0x1.be0d68p+5, 0x1.be1088p+5, -0x2.0bc074p-24, -0x1.b5a622p-4, 0x3.ed52d4p-12, 0x4.b76fc8p-8 }, /* 17 */
+ { 0x1.d7272ap+5, 0x1.d7301ep+5, 0x1.d734aep+5, -0x2.87dd4p-24, 0x1.a9d184p-4, -0x3.9cf494p-12, -0x4.6303ep-8 }, /* 18 */
+ { 0x1.f0499ap+5, 0x1.f052c4p+5, 0x1.f05758p+5, -0x2.fb964p-24, -0x1.9ee5eep-4, 0x3.5800dp-12, 0x4.4e9f9p-8 }, /* 19 */
+ { 0x1.04b63ap+6, 0x1.04baacp+6, 0x1.04bc92p+6, 0x2.cf5adp-24, 0x1.94c6f4p-4, -0x3.1a83e4p-12, -0x4.2311fp-8 }, /* 20 */
+ { 0x1.1146dp+6, 0x1.114beep+6, 0x1.114e12p+6, 0x3.6766fp-24, -0x1.8b5cccp-4, 0x2.e4a4e4p-12, 0x4.20bf9p-8 }, /* 21 */
+ { 0x1.1dda8cp+6, 0x1.1ddd2cp+6, 0x1.1dde7ap+6, 0x3.501424p-24, 0x1.829356p-4, -0x2.b47524p-12, -0x4.04bf18p-8 }, /* 22 */
+ { 0x1.2a6bcp+6, 0x1.2a6e64p+6, 0x1.2a6faap+6, -0x5.c05808p-24, -0x1.7a597ep-4, 0x2.8a0498p-12, 0x4.187258p-8 }, /* 23 */
+ { 0x1.36fcd6p+6, 0x1.36ff96p+6, 0x1.3700f6p+6, 0x7.1e1478p-28, 0x1.72a09ap-4, -0x2.61a7fp-12, -0x3.c0b54p-8 }, /* 24 */
+ { 0x1.438f46p+6, 0x1.4390c4p+6, 0x1.4391e2p+6, 0x3.e36e6cp-24, -0x1.6b5c06p-4, 0x2.3f612p-12, 0x4.18f868p-8 }, /* 25 */
+ { 0x1.501f4cp+6, 0x1.5021fp+6, 0x1.50235p+6, 0x1.3f9e5ap-24, 0x1.6480c4p-4, -0x2.1f28fcp-12, -0x3.bb4e3cp-8 }, /* 26 */
+ { 0x1.5cb07cp+6, 0x1.5cb318p+6, 0x1.5cb464p+6, -0x2.39e41cp-24, -0x1.5e0544p-4, 0x2.0189f4p-12, 0x3.8b55acp-8 }, /* 27 */
+ { 0x1.694166p+6, 0x1.69443cp+6, 0x1.694594p+6, -0x2.912f84p-24, 0x1.57e12p-4, -0x1.e6fabep-12, -0x3.850174p-8 }, /* 28 */
+ { 0x1.75d27cp+6, 0x1.75d55ep+6, 0x1.75d642p+6, 0x3.d5b00cp-24, -0x1.520ceep-4, 0x1.d0286ep-12, 0x3.8e7d1p-8 }, /* 29 */
+ { 0x1.82653ep+6, 0x1.82667ep+6, 0x1.82674p+6, -0x3.1726ecp-24, 0x1.4c8222p-4, -0x1.b98206p-12, -0x3.f34978p-8 }, /* 30 */
+ { 0x1.8ef4c8p+6, 0x1.8ef79cp+6, 0x1.8ef888p+6, 0x1.949e22p-24, -0x1.473ae6p-4, 0x1.a47388p-12, 0x3.69eefcp-8 }, /* 31 */
+ { 0x1.9b8728p+6, 0x1.9b88b8p+6, 0x1.9b896cp+6, -0x5.5553bp-28, 0x1.42320ap-4, -0x1.90f0b8p-12, -0x3.6565p-8 }, /* 32 */
+ { 0x1.a8183cp+6, 0x1.a819d2p+6, 0x1.a81aecp+6, 0x3.2df7ecp-28, -0x1.3d62e4p-4, 0x1.7dae28p-12, 0x2.9eb128p-8 }, /* 33 */
+ { 0x1.b4aa1cp+6, 0x1.b4aaeap+6, 0x1.b4abb8p+6, -0x1.e13fcep-24, 0x1.38c948p-4, -0x1.6eb0ecp-12, -0x1.f9ddf8p-8 }, /* 34 */
+ { 0x1.c13a94p+6, 0x1.c13c02p+6, 0x1.c13cbp+6, -0x3.ad9974p-24, -0x1.34616ep-4, 0x1.5e36ecp-12, 0x2.a9fc5p-8 }, /* 35 */
+ { 0x1.cdcb76p+6, 0x1.cdcd16p+6, 0x1.cdcde4p+6, -0x3.6977e8p-24, 0x1.3027fp-4, -0x1.4f703p-12, -0x2.9817d4p-8 }, /* 36 */
+ { 0x1.da5cf2p+6, 0x1.da5e2ap+6, 0x1.da5efp+6, 0x4.654cbp-24, -0x1.2c19b6p-4, 0x1.455982p-12, 0x3.f1c564p-8 }, /* 37 */
+ { 0x1.e6edccp+6, 0x1.e6ef3ep+6, 0x1.e6f00ap+6, 0x8.825c8p-32, 0x1.2833eep-4, -0x1.39097p-12, -0x3.b2646p-8 }, /* 38 */
+ { 0x1.f37f86p+6, 0x1.f3805p+6, 0x1.f3812ap+6, -0x2.0d11d8p-28, -0x1.24740ap-4, 0x1.2c16p-12, 0x1.fc3804p-8 }, /* 39 */
+ { 0x1.000842p+7, 0x1.0008bp+7, 0x1.000908p+7, -0x4.4e495p-24, 0x1.20d7b6p-4, -0x1.20816p-12, -0x2.d1ebe8p-8 }, /* 40 */
+ { 0x1.06505cp+7, 0x1.065138p+7, 0x1.06518p+7, 0x4.81c1c8p-24, -0x1.1d5ccap-4, 0x1.17ad5ap-12, 0x2.fda33p-8 }, /* 41 */
+ { 0x1.0c98dap+7, 0x1.0c99cp+7, 0x1.0c9a28p+7, -0xe.99386p-28, 0x1.1a015p-4, -0x1.0bd50ap-12, -0x2.9dfb68p-8 }, /* 42 */
+ { 0x1.12e212p+7, 0x1.12e248p+7, 0x1.12e29p+7, -0x6.16f1c8p-24, -0x1.16c37ap-4, 0x1.0303dcp-12, 0x4.34316p-8 }, /* 43 */
+ { 0x1.192a6ep+7, 0x1.192acep+7, 0x1.192b02p+7, -0x1.129336p-24, 0x1.13a19ep-4, -0xf.bd247p-16, -0x3.851d18p-8 }, /* 44 */
+ { 0x1.1f727p+7, 0x1.1f7354p+7, 0x1.1f73ap+7, 0x5.19c09p-24, -0x1.109a32p-4, 0xf.09644p-16, 0x2.d78194p-8 }, /* 45 */
+ { 0x1.25bb8p+7, 0x1.25bbdap+7, 0x1.25bc12p+7, -0x6.497dp-24, 0x1.0dabc8p-4, -0xe.a1d25p-16, -0x2.3378bp-8 }, /* 46 */
+ { 0x1.2c04p+7, 0x1.2c046p+7, 0x1.2c04ap+7, 0x4.e4f338p-24, -0x1.0ad512p-4, 0xe.52d84p-16, 0x4.3bfa08p-8 }, /* 47 */
+ { 0x1.324cbp+7, 0x1.324ce6p+7, 0x1.324d4p+7, -0x1.287c58p-24, 0x1.0814d4p-4, -0xe.03a95p-16, 0x3.9930ap-12 }, /* 48 */
+ { 0x1.3894f6p+7, 0x1.38956cp+7, 0x1.38959ap+7, -0x4.b594ep-24, -0x1.0569fp-4, 0xd.6787ep-16, 0x4.0a5148p-8 }, /* 49 */
+ { 0x1.3edd98p+7, 0x1.3eddfp+7, 0x1.3ede2ap+7, -0x3.a8f164p-24, 0x1.02d354p-4, -0xd.0309dp-16, -0x3.2ebfb4p-8 }, /* 50 */
+ { 0x1.452638p+7, 0x1.452676p+7, 0x1.4526b4p+7, -0x6.12505p-24, -0x1.005004p-4, 0xc.a0045p-16, 0x4.87c67p-8 }, /* 51 */
+ { 0x1.4b6e8p+7, 0x1.4b6efap+7, 0x1.4b6f34p+7, 0x1.8acf4ep-24, 0xf.ddf16p-8, -0xc.2d207p-16, -0x1.da6c36p-8 }, /* 52 */
+ { 0x1.51b742p+7, 0x1.51b77ep+7, 0x1.51b7b2p+7, 0x1.39cf86p-24, -0xf.b7faep-8, 0xb.db598p-16, -0x8.945b1p-12 }, /* 53 */
+ { 0x1.57ffc4p+7, 0x1.580002p+7, 0x1.58003cp+7, -0x2.5f8de8p-24, 0xf.930fep-8, -0xb.91889p-16, -0xa.30df9p-12 }, /* 54 */
+ { 0x1.5e483p+7, 0x1.5e4886p+7, 0x1.5e48c8p+7, 0x2.073d64p-24, -0xf.6f245p-8, 0xb.4085fp-16, 0x2.128188p-8 }, /* 55 */
+ { 0x1.64908cp+7, 0x1.64910ap+7, 0x1.64912ap+7, -0x4.ed26ep-28, 0xf.4c2cep-8, -0xa.fe719p-16, -0x2.9374b8p-8 }, /* 56 */
+ { 0x1.6ad91ep+7, 0x1.6ad98ep+7, 0x1.6ad9cep+7, -0x2.ae5204p-24, -0xf.2a1efp-8, 0xa.aa585p-16, 0x2.1c0834p-8 }, /* 57 */
+ { 0x1.7121cep+7, 0x1.712212p+7, 0x1.712238p+7, 0x6.d72168p-24, 0xf.08f09p-8, -0xa.7da49p-16, -0x3.4f5f1cp-8 }, /* 58 */
+ { 0x1.776a0cp+7, 0x1.776a94p+7, 0x1.776accp+7, 0x2.d3f294p-24, -0xe.e8986p-8, 0xa.23ccdp-16, 0x2.2a6678p-8 }, /* 59 */
+ { 0x1.7db2e8p+7, 0x1.7db318p+7, 0x1.7db35ap+7, 0x3.88c0fp-24, 0xe.c90d7p-8, -0x9.eaeap-16, -0x2.86438cp-8 }, /* 60 */
+ { 0x1.83fb56p+7, 0x1.83fb9ap+7, 0x1.83fbep+7, 0x3.d94d34p-24, -0xe.aa478p-8, 0x9.abac7p-16, 0x1.ac2d84p-8 }, /* 61 */
+ { 0x1.8a43eep+7, 0x1.8a441ep+7, 0x1.8a446p+7, 0x4.66b7ep-24, 0xe.8c3e9p-8, -0x9.87682p-16, -0x7.9ab4a8p-12 }, /* 62 */
+ { 0x1.908c6p+7, 0x1.908cap+7, 0x1.908cd6p+7, 0xf.f7ac9p-28, -0xe.6eeb6p-8, 0x9.4423p-16, 0x4.54c4d8p-8 }, /* 63 */
+};
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ y1f(x) ~ sqrt(2/(pi*x))*beta1(x)*sin(x-3pi/4-alpha1(x))
+ where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
+ and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
+static float
+y1f_asympt (float x)
+{
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
+ double alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
+ double h;
+ h = __math_reduce_mod_twopi ((double) x);
+ int n = -1; /* Accounts for -pi/2 (pi/4 was subtracted in
+ __math_reduce_mod_twopi). */
+ /* Now reduce mod pi/2. */
+ h = reduce_mod_piover2 (h, &n);
+ /* Subtract alpha1. */
+ h -= alpha1;
+ /* Reduce again mod pi/2 if needed. */
+ h = reduce_mod_piover2 (h, &n);
+ /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float t = cst / sqrtf (x) * (float) beta1;
+ if (n == 0)
+ return t * sinf (xr);
+ else if (n == 2) /* sin(x+pi) = -sin(x) */
+ return -t * sinf (xr);
+ else if (n == 1) /* sin(x+pi/2) = cos(x) */
+ return t * cosf (xr);
+ else /* sin(x+3pi/2) = -cos(x) */
+ return -t * cosf (xr);
+}
+
+/* Special code for x near a root of y1.
+ z is the value computed by the generic code.
+ For small x, we use a polynomial approximating y1 around its root.
+ For large x, we use an asymptotic formula (y1f_asympt). */
+static float
+y1f_near_root (float x, float z)
+{
+ float index_f;
+ int index;
+
+ index_f = roundf ((x - FIRST_ZERO_Y1) / (float) M_PI);
+ if (index_f >= SMALL_SIZE)
+ return y1f_asympt (x);
+ index = (int) index_f;
+ const float *p = Py[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If not in the interval [x0,x1] around xmid, return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid, p6;
+ if (index == 0)
+ p6 = p[6] + y * (-0x1.28043p-8 + y * 0x2.50e83p-8);
+ else if (index == 1)
+ p6 = p[6] + y * -0xf.ff6b8p-12;
+ else
+ p6 = p[6];
+ return p[3] + y * (p[4] + y * (p[5] + y * p6));
+}
+
float
__ieee754_y1f(float x)
{
@@ -118,16 +455,17 @@ __ieee754_y1f(float x)
if(__builtin_expect(ix==0, 0))
return -1/zero; /* -inf and divide by zero exception. */
if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
- if(ix >= 0x40000000) { /* |x| >= 2.0 */
+ if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */
SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (x, &s, &c);
ss = -s-c;
cc = s-c;
- if(ix<0x7f000000) { /* make sure x+x not overflow */
- z = __cosf(x+x);
- if ((s*c)>zero) cc = z/ss;
- else ss = z/cc;
- }
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return y1f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = __cosf(x+x);
+ if ((s*c)>zero) cc = z/ss;
+ else ss = z/cc;
/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
* where x0 = x-3pi/4
* Better formula:
@@ -139,12 +477,17 @@ __ieee754_y1f(float x)
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
- if(ix>0x5c000000) z = (invsqrtpi*ss)/sqrtf(x);
- else {
- u = ponef(x); v = qonef(x);
- z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
- }
- return z;
+ if (ix <= 0x5c000000)
+ {
+ u = ponef(x); v = qonef(x);
+ ss = u*ss+v*cc;
+ }
+ z = (invsqrtpi * ss) / sqrtf(x);
+ /* The following value is attained for x=0x1.f7f16ap+0. */
+ if (fabsf (ss) > 0x1.3e014cp-2)
+ return z;
+ else /* |cc| <= 2^-2 */
+ return y1f_near_root (x, z);
}
if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */
z = -tpi / x;
@@ -152,6 +495,7 @@ __ieee754_y1f(float x)
__set_errno (ERANGE);
return z;
}
+ /* Now 2**-25 <= x < 0x1.c1bf78p+0. */
z = x*x;
u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
@@ -159,7 +503,7 @@ __ieee754_y1f(float x)
}
libm_alias_finite (__ieee754_y1f, __y1f)
-/* For x >= 8, the asymptotic expansions of pone is
+/* For x >= 8, the asymptotic expansion of pone is
* 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
* We approximate pone by
* pone(x) = 1 + (R/S)
@@ -252,8 +596,7 @@ ponef(float x)
return one+ r/s;
}
-
-/* For x >= 8, the asymptotic expansions of qone is
+/* For x >= 8, the asymptotic expansion of qone is
* 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
* We approximate pone by
* qone(x) = s*(0.375 + (R/S))
@@ -340,10 +683,10 @@ qonef(float x)
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff;
/* ix >= 0x40000000 for all calls to this function. */
- if(ix>=0x40200000) {p = qr8; q= qs8;}
- else if(ix>=0x40f71c58){p = qr5; q= qs5;}
- else if(ix>=0x4036db68){p = qr3; q= qs3;}
- else {p = qr2; q= qs2;}
+ if(ix>=0x41000000) {p = qr8; q= qs8;} /* x >= 8 */
+ else if(ix>=0x40f71c58){p = qr5; q= qs5;} /* x >= 7.722209930e+00 */
+ else if(ix>=0x4036db68){p = qr3; q= qs3;} /* x >= 2.857141495e+00 */
+ else {p = qr2; q= qs2;} /* x >= 2 */
z = one/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
--
2.30.0
^ permalink raw reply [flat|nested] 4+ messages in thread
* [PATCH 4/4] Added new entries for j0/j1/y0/y1 and regenerate ulps.
2021-02-09 5:28 [PATCH 1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f Paul Zimmermann
2021-02-09 5:28 ` [PATCH 2/4] Fix the inaccuracy of j0f and of y0f [BZ #14469 and #14471] Paul Zimmermann
2021-02-09 5:28 ` [PATCH 3/4] Fix the inaccuracy of j1f and y1f [BZ #14470 and #14472] Paul Zimmermann
@ 2021-02-09 5:28 ` Paul Zimmermann
2 siblings, 0 replies; 4+ messages in thread
From: Paul Zimmermann @ 2021-02-09 5:28 UTC (permalink / raw)
To: libc-alpha
---
math/auto-libm-test-in | 15 ++++++
math/auto-libm-test-out-j0 | 50 ++++++++++++++++++++
math/auto-libm-test-out-j1 | 50 ++++++++++++++++++++
math/auto-libm-test-out-y0 | 50 ++++++++++++++++++++
math/auto-libm-test-out-y1 | 75 ++++++++++++++++++++++++++++++
sysdeps/x86_64/fpu/libm-test-ulps | 76 +++++++++++++++----------------
6 files changed, 278 insertions(+), 38 deletions(-)
diff --git a/math/auto-libm-test-in b/math/auto-libm-test-in
index 73840b8bef..3e1611c3d6 100644
--- a/math/auto-libm-test-in
+++ b/math/auto-libm-test-in
@@ -5758,6 +5758,9 @@ j0 0x1p16382
j0 0x1p16383
# the next value generates larger error bounds on x86_64 (binary32)
j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
+# the next values yield the largest error (9 ulps) for binary32 (x86_64)
+j0 0x1.04c39cp+6
+j0 0x1.4b7066p+7
# the next value exercises the flt-32 code path for x >= 2^127
j0 0x8.2f4ecp+124
@@ -5793,6 +5796,9 @@ j1 0x1p-600
j1 0x1p-10000
# the next value generates larger error bounds on x86_64 (binary32)
j1 0x3.ae4b2p+0 xfail-rounding:ibm128-libgcc
+# the next two values yield the largest error in the binary32 format
+j1 0x1.2f28eap+7 xfail-rounding:binary64 xfail-rounding:intel96
+j1 0x1.a1d20ap+6 xfail-rounding:binary128 xfail-rounding:intel96
j1 min
j1 -min
j1 min_subnorm
@@ -8227,6 +8233,11 @@ y0 0x1p-600
y0 0x1p-10000
# the next value generates larger error bounds on x86_64 (binary32)
y0 0xd.3432bp-4
+# the next two values yield the largest error (9 ulps) for binary32 (x86_64)
+# the first one fails for binary64 and ldbl-96 because y0 is known to have
+# large errors (#16492)
+y0 0x1.33eaacp+5 xfail:binary64 xfail:intel96
+y0 0x1.a681cep-1
y0 min
y0 min_subnorm
@@ -8255,6 +8266,10 @@ y1 0x1p-100
y1 0x1p-110
y1 0x1p-600
y1 0x1p-10000
+# the next three values yield the largest error in the binary32 format
+y1 0x1.065194p+7 xfail-rounding:binary64 xfail-rounding:intel96
+y1 0x1.c1badep+0
+y1 0x1.c1bc2ep+0
y1 min
y1 min_subnorm
diff --git a/math/auto-libm-test-out-j0 b/math/auto-libm-test-out-j0
index cc1fea074e..85c4148de3 100644
--- a/math/auto-libm-test-out-j0
+++ b/math/auto-libm-test-out-j0
@@ -1359,6 +1359,56 @@ j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
= j0 tonearest ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : inexact-ok
= j0 towardzero ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : xfail:ibm128-libgcc inexact-ok
= j0 upward ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab8p-8 : xfail:ibm128-libgcc inexact-ok
+j0 0x1.04c39cp+6
+= j0 downward binary32 0x4.130e7p+4 : -0x6.dd65d8p-16 : inexact-ok
+= j0 tonearest binary32 0x4.130e7p+4 : -0x6.dd65dp-16 : inexact-ok
+= j0 towardzero binary32 0x4.130e7p+4 : -0x6.dd65dp-16 : inexact-ok
+= j0 upward binary32 0x4.130e7p+4 : -0x6.dd65dp-16 : inexact-ok
+= j0 downward binary64 0x4.130e7p+4 : -0x6.dd65d0005a19p-16 : inexact-ok
+= j0 tonearest binary64 0x4.130e7p+4 : -0x6.dd65d0005a18cp-16 : inexact-ok
+= j0 towardzero binary64 0x4.130e7p+4 : -0x6.dd65d0005a18cp-16 : inexact-ok
+= j0 upward binary64 0x4.130e7p+4 : -0x6.dd65d0005a18cp-16 : inexact-ok
+= j0 downward intel96 0x4.130e7p+4 : -0x6.dd65d0005a18c04p-16 : inexact-ok
+= j0 tonearest intel96 0x4.130e7p+4 : -0x6.dd65d0005a18c04p-16 : inexact-ok
+= j0 towardzero intel96 0x4.130e7p+4 : -0x6.dd65d0005a18c038p-16 : inexact-ok
+= j0 upward intel96 0x4.130e7p+4 : -0x6.dd65d0005a18c038p-16 : inexact-ok
+= j0 downward m68k96 0x4.130e7p+4 : -0x6.dd65d0005a18c04p-16 : inexact-ok
+= j0 tonearest m68k96 0x4.130e7p+4 : -0x6.dd65d0005a18c04p-16 : inexact-ok
+= j0 towardzero m68k96 0x4.130e7p+4 : -0x6.dd65d0005a18c038p-16 : inexact-ok
+= j0 upward m68k96 0x4.130e7p+4 : -0x6.dd65d0005a18c038p-16 : inexact-ok
+= j0 downward binary128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bb64p-16 : inexact-ok
+= j0 tonearest binary128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bb6p-16 : inexact-ok
+= j0 towardzero binary128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bb6p-16 : inexact-ok
+= j0 upward binary128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bb6p-16 : inexact-ok
+= j0 downward ibm128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bcp-16 : inexact-ok
+= j0 tonearest ibm128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bcp-16 : inexact-ok
+= j0 towardzero ibm128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bap-16 : inexact-ok
+= j0 upward ibm128 0x4.130e7p+4 : -0x6.dd65d0005a18c03fc1c61141bap-16 : inexact-ok
+j0 0x1.4b7066p+7
+= j0 downward binary32 0xa.5b833p+4 : 0xf.80edp-20 : inexact-ok
+= j0 tonearest binary32 0xa.5b833p+4 : 0xf.80edp-20 : inexact-ok
+= j0 towardzero binary32 0xa.5b833p+4 : 0xf.80edp-20 : inexact-ok
+= j0 upward binary32 0xa.5b833p+4 : 0xf.80ed1p-20 : inexact-ok
+= j0 downward binary64 0xa.5b833p+4 : 0xf.80ed0055a8e58p-20 : inexact-ok
+= j0 tonearest binary64 0xa.5b833p+4 : 0xf.80ed0055a8e6p-20 : inexact-ok
+= j0 towardzero binary64 0xa.5b833p+4 : 0xf.80ed0055a8e58p-20 : inexact-ok
+= j0 upward binary64 0xa.5b833p+4 : 0xf.80ed0055a8e6p-20 : inexact-ok
+= j0 downward intel96 0xa.5b833p+4 : 0xf.80ed0055a8e5c88p-20 : inexact-ok
+= j0 tonearest intel96 0xa.5b833p+4 : 0xf.80ed0055a8e5c89p-20 : inexact-ok
+= j0 towardzero intel96 0xa.5b833p+4 : 0xf.80ed0055a8e5c88p-20 : inexact-ok
+= j0 upward intel96 0xa.5b833p+4 : 0xf.80ed0055a8e5c89p-20 : inexact-ok
+= j0 downward m68k96 0xa.5b833p+4 : 0xf.80ed0055a8e5c88p-20 : inexact-ok
+= j0 tonearest m68k96 0xa.5b833p+4 : 0xf.80ed0055a8e5c89p-20 : inexact-ok
+= j0 towardzero m68k96 0xa.5b833p+4 : 0xf.80ed0055a8e5c88p-20 : inexact-ok
+= j0 upward m68k96 0xa.5b833p+4 : 0xf.80ed0055a8e5c89p-20 : inexact-ok
+= j0 downward binary128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe12ap-20 : inexact-ok
+= j0 tonearest binary128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe12a8p-20 : inexact-ok
+= j0 towardzero binary128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe12ap-20 : inexact-ok
+= j0 upward binary128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe12a8p-20 : inexact-ok
+= j0 downward ibm128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe1p-20 : inexact-ok
+= j0 tonearest ibm128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe14p-20 : inexact-ok
+= j0 towardzero ibm128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe1p-20 : inexact-ok
+= j0 upward ibm128 0xa.5b833p+4 : 0xf.80ed0055a8e5c88af9c0edfe14p-20 : inexact-ok
j0 0x8.2f4ecp+124
= j0 downward binary32 0x8.2f4ecp+124 : 0xd.331efp-68 : inexact-ok
= j0 tonearest binary32 0x8.2f4ecp+124 : 0xd.331efp-68 : inexact-ok
diff --git a/math/auto-libm-test-out-j1 b/math/auto-libm-test-out-j1
index 6bc3bbe5a4..4311320189 100644
--- a/math/auto-libm-test-out-j1
+++ b/math/auto-libm-test-out-j1
@@ -993,6 +993,56 @@ j1 0x3.ae4b2p+0 xfail-rounding:ibm128-libgcc
= j1 tonearest ibm128 0x3.ae4b2p+0 : 0xf.d085c66e86f30267f22d6f787cp-8 : inexact-ok
= j1 towardzero ibm128 0x3.ae4b2p+0 : 0xf.d085c66e86f30267f22d6f787cp-8 : xfail:ibm128-libgcc inexact-ok
= j1 upward ibm128 0x3.ae4b2p+0 : 0xf.d085c66e86f30267f22d6f788p-8 : xfail:ibm128-libgcc inexact-ok
+j1 0x1.2f28eap+7 xfail-rounding:binary64 xfail-rounding:intel96
+= j1 downward binary32 0x9.79475p+4 : 0x2.49a6fp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest binary32 0x9.79475p+4 : 0x2.49a6f4p-16 : inexact-ok
+= j1 towardzero binary32 0x9.79475p+4 : 0x2.49a6fp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward binary32 0x9.79475p+4 : 0x2.49a6f4p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 downward binary64 0x9.79475p+4 : 0x2.49a6f3fb00346p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest binary64 0x9.79475p+4 : 0x2.49a6f3fb00346p-16 : inexact-ok
+= j1 towardzero binary64 0x9.79475p+4 : 0x2.49a6f3fb00346p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward binary64 0x9.79475p+4 : 0x2.49a6f3fb00348p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 downward intel96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest intel96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : inexact-ok
+= j1 towardzero intel96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward intel96 0x9.79475p+4 : 0x2.49a6f3fb003462a4p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 downward m68k96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest m68k96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : inexact-ok
+= j1 towardzero m68k96 0x9.79475p+4 : 0x2.49a6f3fb003462ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward m68k96 0x9.79475p+4 : 0x2.49a6f3fb003462a4p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 downward binary128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cec05ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest binary128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cec05cp-16 : inexact-ok
+= j1 towardzero binary128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cec05ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward binary128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cec05cp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 downward ibm128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cecp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 tonearest ibm128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cecp-16 : inexact-ok
+= j1 towardzero ibm128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cecp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= j1 upward ibm128 0x9.79475p+4 : 0x2.49a6f3fb003462a1f15135cec1p-16 : xfail:binary64 xfail:intel96 inexact-ok
+j1 0x1.a1d20ap+6 xfail-rounding:binary128 xfail-rounding:intel96
+= j1 downward binary32 0x6.874828p+4 : -0x3.d6f0b8p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest binary32 0x6.874828p+4 : -0x3.d6f0b4p-16 : inexact-ok
+= j1 towardzero binary32 0x6.874828p+4 : -0x3.d6f0b4p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward binary32 0x6.874828p+4 : -0x3.d6f0b4p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 downward binary64 0x6.874828p+4 : -0x3.d6f0b408112dp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest binary64 0x6.874828p+4 : -0x3.d6f0b408112dp-16 : inexact-ok
+= j1 towardzero binary64 0x6.874828p+4 : -0x3.d6f0b408112cep-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward binary64 0x6.874828p+4 : -0x3.d6f0b408112cep-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 downward intel96 0x6.874828p+4 : -0x3.d6f0b408112cf3dp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest intel96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : inexact-ok
+= j1 towardzero intel96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward intel96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 downward m68k96 0x6.874828p+4 : -0x3.d6f0b408112cf3dp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest m68k96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : inexact-ok
+= j1 towardzero m68k96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward m68k96 0x6.874828p+4 : -0x3.d6f0b408112cf3ccp-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 downward binary128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7088p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest binary128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7086p-16 : inexact-ok
+= j1 towardzero binary128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7086p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward binary128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7086p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 downward ibm128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca71p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 tonearest ibm128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca71p-16 : inexact-ok
+= j1 towardzero ibm128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7p-16 : xfail:binary128 xfail:intel96 inexact-ok
+= j1 upward ibm128 0x6.874828p+4 : -0x3.d6f0b408112cf3cdb53ca3ca7p-16 : xfail:binary128 xfail:intel96 inexact-ok
j1 min
= j1 downward binary32 0x4p-128 : 0x1.fffff8p-128 : inexact-ok underflow errno-erange-ok
= j1 tonearest binary32 0x4p-128 : 0x2p-128 : inexact-ok underflow errno-erange-ok
diff --git a/math/auto-libm-test-out-y0 b/math/auto-libm-test-out-y0
index 8ebb585170..a254bb16b9 100644
--- a/math/auto-libm-test-out-y0
+++ b/math/auto-libm-test-out-y0
@@ -820,6 +820,56 @@ y0 0xd.3432bp-4
= y0 tonearest ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
= y0 towardzero ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
= y0 upward ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
+y0 0x1.33eaacp+5 xfail:binary64 xfail:intel96
+= y0 downward binary32 0x2.67d558p+4 : 0xf.6b0ep-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest binary32 0x2.67d558p+4 : 0xf.6b0ep-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero binary32 0x2.67d558p+4 : 0xf.6b0ep-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward binary32 0x2.67d558p+4 : 0xf.6b0e1p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 downward binary64 0x2.67d558p+4 : 0xf.6b0e005e95ad8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest binary64 0x2.67d558p+4 : 0xf.6b0e005e95ad8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero binary64 0x2.67d558p+4 : 0xf.6b0e005e95ad8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward binary64 0x2.67d558p+4 : 0xf.6b0e005e95aep-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 downward intel96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest intel96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero intel96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward intel96 0x2.67d558p+4 : 0xf.6b0e005e95ad82bp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 downward m68k96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest m68k96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero m68k96 0x2.67d558p+4 : 0xf.6b0e005e95ad82ap-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward m68k96 0x2.67d558p+4 : 0xf.6b0e005e95ad82bp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 downward binary128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b86p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest binary128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b86p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero binary128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b86p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward binary128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b868p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 downward ibm128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 tonearest ibm128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 towardzero ibm128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6b8p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y0 upward ibm128 0x2.67d558p+4 : 0xf.6b0e005e95ad82a5093316a6bcp-16 : xfail:binary64 xfail:intel96 inexact-ok
+y0 0x1.a681cep-1
+= y0 downward binary32 0xd.340e7p-4 : -0xf.ffff8p-8 : inexact-ok
+= y0 tonearest binary32 0xd.340e7p-4 : -0xf.ffff8p-8 : inexact-ok
+= y0 towardzero binary32 0xd.340e7p-4 : -0xf.ffff7p-8 : inexact-ok
+= y0 upward binary32 0xd.340e7p-4 : -0xf.ffff7p-8 : inexact-ok
+= y0 downward binary64 0xd.340e7p-4 : -0xf.ffff7f7e5aba8p-8 : inexact-ok
+= y0 tonearest binary64 0xd.340e7p-4 : -0xf.ffff7f7e5aba8p-8 : inexact-ok
+= y0 towardzero binary64 0xd.340e7p-4 : -0xf.ffff7f7e5abap-8 : inexact-ok
+= y0 upward binary64 0xd.340e7p-4 : -0xf.ffff7f7e5abap-8 : inexact-ok
+= y0 downward intel96 0xd.340e7p-4 : -0xf.ffff7f7e5aba736p-8 : inexact-ok
+= y0 tonearest intel96 0xd.340e7p-4 : -0xf.ffff7f7e5aba736p-8 : inexact-ok
+= y0 towardzero intel96 0xd.340e7p-4 : -0xf.ffff7f7e5aba735p-8 : inexact-ok
+= y0 upward intel96 0xd.340e7p-4 : -0xf.ffff7f7e5aba735p-8 : inexact-ok
+= y0 downward m68k96 0xd.340e7p-4 : -0xf.ffff7f7e5aba736p-8 : inexact-ok
+= y0 tonearest m68k96 0xd.340e7p-4 : -0xf.ffff7f7e5aba736p-8 : inexact-ok
+= y0 towardzero m68k96 0xd.340e7p-4 : -0xf.ffff7f7e5aba735p-8 : inexact-ok
+= y0 upward m68k96 0xd.340e7p-4 : -0xf.ffff7f7e5aba735p-8 : inexact-ok
+= y0 downward binary128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25de8p-8 : inexact-ok
+= y0 tonearest binary128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25dep-8 : inexact-ok
+= y0 towardzero binary128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25dep-8 : inexact-ok
+= y0 upward binary128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25dep-8 : inexact-ok
+= y0 downward ibm128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b26p-8 : inexact-ok
+= y0 tonearest ibm128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25cp-8 : inexact-ok
+= y0 towardzero ibm128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25cp-8 : inexact-ok
+= y0 upward ibm128 0xd.340e7p-4 : -0xf.ffff7f7e5aba735ccb0b13b25cp-8 : inexact-ok
y0 min
= y0 downward binary32 0x4p-128 : -0x3.7ac89cp+4 : inexact-ok
= y0 tonearest binary32 0x4p-128 : -0x3.7ac89cp+4 : inexact-ok
diff --git a/math/auto-libm-test-out-y1 b/math/auto-libm-test-out-y1
index af68e6c05a..b1469213d2 100644
--- a/math/auto-libm-test-out-y1
+++ b/math/auto-libm-test-out-y1
@@ -795,6 +795,81 @@ y1 0x1p-10000
= y1 tonearest binary128 0x1p-10000 : -0xa.2f9836e4e441529fc2757d1f535p+9996 : inexact-ok
= y1 towardzero binary128 0x1p-10000 : -0xa.2f9836e4e441529fc2757d1f5348p+9996 : inexact-ok
= y1 upward binary128 0x1p-10000 : -0xa.2f9836e4e441529fc2757d1f5348p+9996 : inexact-ok
+y1 0x1.065194p+7 xfail-rounding:binary64 xfail-rounding:intel96
+= y1 downward binary32 0x8.328cap+4 : -0x3.2fe874p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest binary32 0x8.328cap+4 : -0x3.2fe87p-16 : inexact-ok
+= y1 towardzero binary32 0x8.328cap+4 : -0x3.2fe87p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward binary32 0x8.328cap+4 : -0x3.2fe87p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 downward binary64 0x8.328cap+4 : -0x3.2fe87001a1df4p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest binary64 0x8.328cap+4 : -0x3.2fe87001a1df4p-16 : inexact-ok
+= y1 towardzero binary64 0x8.328cap+4 : -0x3.2fe87001a1df2p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward binary64 0x8.328cap+4 : -0x3.2fe87001a1df2p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 downward intel96 0x8.328cap+4 : -0x3.2fe87001a1df3754p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest intel96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : inexact-ok
+= y1 towardzero intel96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward intel96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 downward m68k96 0x8.328cap+4 : -0x3.2fe87001a1df3754p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest m68k96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : inexact-ok
+= y1 towardzero m68k96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward m68k96 0x8.328cap+4 : -0x3.2fe87001a1df375p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 downward binary128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecb2p-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest binary128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecb2p-16 : inexact-ok
+= y1 towardzero binary128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecbp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward binary128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecbp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 downward ibm128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aedp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 tonearest ibm128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aedp-16 : inexact-ok
+= y1 towardzero ibm128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecp-16 : xfail:binary64 xfail:intel96 inexact-ok
+= y1 upward ibm128 0x8.328cap+4 : -0x3.2fe87001a1df375068d0356aecp-16 : xfail:binary64 xfail:intel96 inexact-ok
+y1 0x1.c1badep+0
+= y1 downward binary32 0x1.c1badep+0 : -0x3.ff6378p-4 : inexact-ok
+= y1 tonearest binary32 0x1.c1badep+0 : -0x3.ff6374p-4 : inexact-ok
+= y1 towardzero binary32 0x1.c1badep+0 : -0x3.ff6374p-4 : inexact-ok
+= y1 upward binary32 0x1.c1badep+0 : -0x3.ff6374p-4 : inexact-ok
+= y1 downward binary64 0x1.c1badep+0 : -0x3.ff6375c9f4404p-4 : inexact-ok
+= y1 tonearest binary64 0x1.c1badep+0 : -0x3.ff6375c9f4402p-4 : inexact-ok
+= y1 towardzero binary64 0x1.c1badep+0 : -0x3.ff6375c9f4402p-4 : inexact-ok
+= y1 upward binary64 0x1.c1badep+0 : -0x3.ff6375c9f4402p-4 : inexact-ok
+= y1 downward intel96 0x1.c1badep+0 : -0x3.ff6375c9f440230cp-4 : inexact-ok
+= y1 tonearest intel96 0x1.c1badep+0 : -0x3.ff6375c9f440230cp-4 : inexact-ok
+= y1 towardzero intel96 0x1.c1badep+0 : -0x3.ff6375c9f4402308p-4 : inexact-ok
+= y1 upward intel96 0x1.c1badep+0 : -0x3.ff6375c9f4402308p-4 : inexact-ok
+= y1 downward m68k96 0x1.c1badep+0 : -0x3.ff6375c9f440230cp-4 : inexact-ok
+= y1 tonearest m68k96 0x1.c1badep+0 : -0x3.ff6375c9f440230cp-4 : inexact-ok
+= y1 towardzero m68k96 0x1.c1badep+0 : -0x3.ff6375c9f4402308p-4 : inexact-ok
+= y1 upward m68k96 0x1.c1badep+0 : -0x3.ff6375c9f4402308p-4 : inexact-ok
+= y1 downward binary128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1ebp-4 : inexact-ok
+= y1 tonearest binary128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1ebp-4 : inexact-ok
+= y1 towardzero binary128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1eaep-4 : inexact-ok
+= y1 upward binary128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1eaep-4 : inexact-ok
+= y1 downward ibm128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1fp-4 : inexact-ok
+= y1 tonearest ibm128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1fp-4 : inexact-ok
+= y1 towardzero ibm128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1ep-4 : inexact-ok
+= y1 upward ibm128 0x1.c1badep+0 : -0x3.ff6375c9f440230a8962842c1ep-4 : inexact-ok
+y1 0x1.c1bc2ep+0
+= y1 downward binary32 0x1.c1bc2ep+0 : -0x3.ff56acp-4 : inexact-ok
+= y1 tonearest binary32 0x1.c1bc2ep+0 : -0x3.ff56a8p-4 : inexact-ok
+= y1 towardzero binary32 0x1.c1bc2ep+0 : -0x3.ff56a8p-4 : inexact-ok
+= y1 upward binary32 0x1.c1bc2ep+0 : -0x3.ff56a8p-4 : inexact-ok
+= y1 downward binary64 0x1.c1bc2ep+0 : -0x3.ff56a991ca78ap-4 : inexact-ok
+= y1 tonearest binary64 0x1.c1bc2ep+0 : -0x3.ff56a991ca788p-4 : inexact-ok
+= y1 towardzero binary64 0x1.c1bc2ep+0 : -0x3.ff56a991ca788p-4 : inexact-ok
+= y1 upward binary64 0x1.c1bc2ep+0 : -0x3.ff56a991ca788p-4 : inexact-ok
+= y1 downward intel96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd8p-4 : inexact-ok
+= y1 tonearest intel96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 towardzero intel96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 upward intel96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 downward m68k96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd8p-4 : inexact-ok
+= y1 tonearest m68k96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 towardzero m68k96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 upward m68k96 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd4p-4 : inexact-ok
+= y1 downward binary128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd5084138201442p-4 : inexact-ok
+= y1 tonearest binary128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd5084138201442p-4 : inexact-ok
+= y1 towardzero binary128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd508413820144p-4 : inexact-ok
+= y1 upward binary128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd508413820144p-4 : inexact-ok
+= y1 downward ibm128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd50841382015p-4 : inexact-ok
+= y1 tonearest ibm128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd50841382014p-4 : inexact-ok
+= y1 towardzero ibm128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd50841382014p-4 : inexact-ok
+= y1 upward ibm128 0x1.c1bc2ep+0 : -0x3.ff56a991ca788cd50841382014p-4 : inexact-ok
y1 min
= y1 downward binary32 0x4p-128 : -0x2.8be61p+124 : inexact-ok
= y1 tonearest binary32 0x4p-128 : -0x2.8be60cp+124 : inexact-ok
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index 633d2ab8e4..fcc3fad9ae 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -1313,50 +1313,50 @@ ldouble: 1
Function: "j0":
double: 2
-float: 8
+float: 9
float128: 2
-ldouble: 2
+ldouble: 8
Function: "j0_downward":
-double: 2
-float: 4
-float128: 4
+double: 5
+float: 9
+float128: 9
ldouble: 6
Function: "j0_towardzero":
-double: 5
-float: 6
-float128: 4
+double: 6
+float: 9
+float128: 9
ldouble: 6
Function: "j0_upward":
-double: 4
-float: 5
-float128: 5
+double: 9
+float: 9
+float128: 7
ldouble: 6
Function: "j1":
-double: 2
+double: 4
float: 9
float128: 4
-ldouble: 5
+ldouble: 9
Function: "j1_downward":
-double: 3
-float: 5
-float128: 4
-ldouble: 4
+double: 6
+float: 8
+float128: 6
+ldouble: 8
Function: "j1_towardzero":
-double: 3
-float: 2
-float128: 4
+double: 4
+float: 9
+float128: 9
ldouble: 4
Function: "j1_upward":
-double: 3
-float: 5
-float128: 3
+double: 9
+float: 9
+float128: 9
ldouble: 3
Function: "jn":
@@ -1749,27 +1749,27 @@ ldouble: 5
Function: "y0":
double: 3
-float: 8
+float: 9
float128: 3
-ldouble: 1
+ldouble: 2
Function: "y0_downward":
-double: 3
-float: 6
-float128: 4
-ldouble: 5
+double: 4
+float: 9
+float128: 7
+ldouble: 7
Function: "y0_towardzero":
-double: 3
-float: 3
+double: 4
+float: 9
float128: 3
-ldouble: 6
+ldouble: 8
Function: "y0_upward":
double: 3
-float: 6
-float128: 3
-ldouble: 5
+float: 9
+float128: 4
+ldouble: 7
Function: "y1":
double: 6
@@ -1778,14 +1778,14 @@ float128: 5
ldouble: 3
Function: "y1_downward":
-double: 3
-float: 2
+double: 6
+float: 9
float128: 5
ldouble: 7
Function: "y1_towardzero":
double: 4
-float: 5
+float: 9
float128: 6
ldouble: 5
--
2.30.0
^ permalink raw reply [flat|nested] 4+ messages in thread
end of thread, other threads:[~2021-02-09 5:28 UTC | newest]
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2021-02-09 5:28 [PATCH 1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f Paul Zimmermann
2021-02-09 5:28 ` [PATCH 2/4] Fix the inaccuracy of j0f and of y0f [BZ #14469 and #14471] Paul Zimmermann
2021-02-09 5:28 ` [PATCH 3/4] Fix the inaccuracy of j1f and y1f [BZ #14470 and #14472] Paul Zimmermann
2021-02-09 5:28 ` [PATCH 4/4] Added new entries for j0/j1/y0/y1 and regenerate ulps Paul Zimmermann
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