Re: [PATCH] aarch64: Improve SVE sin polynomial

[Paul Zimmermann <Paul.Zimmermann@inria.fr>]

       Hi Joe,

you can still improve the polynomial with the following one generated by
Sollya (https://www.sollya.org/) with relative error < 2^-52.454 on
[-pi/2, pi/2], instead of 2^-51.765 for the one you propose:

$ cat sin.sollya
// polynomial proposed by Joe Ramsay
p3=-0x1.555555555547bp-3;
p5=0x1.1111111108a4dp-7;
p7=-0x1.a01a019936f27p-13;
p9=0x1.71de37a97d93ep-19;
p11=-0x1.ae633919987c6p-26;
p13=0x1.60e277ae07cecp-33;
p15=-0x1.9e9540300a1p-41;

p=x+p3*x^3+p5*x^5+p7*x^7+p9*x^9+p11*x^11+p13*x^13+p15*x^15;

pretty = proc(u) {
  return ~(floor(u*1000)/1000);
};

d = [0,pi/2];
f = 1;
w = 1/sin(x);
err_p = -log2(dirtyinfnorm(p*w-f, d));
display = hexadecimal;
print (p);
display = decimal;
print (pretty(err_p));

// polynomial generated by Sollya
n = [|1,3,5,7,9,11,13,15|];
p =  remez(f, n, d, w);
pf = fpminimax(sin(x), n, [|1,53...|], d, relative, floating, 0, p);
err_p = -log2(dirtyinfnorm(pf*w-f, d));
display = hexadecimal;
print (pf);
display = decimal;
print (pretty(err_p));

$ sollya sin.sollya
Display mode is hexadecimal numbers.
x + -0x1.555555555547bp-3 * x^0x1.8p1 + 0x1.1111111108a4dp-7 * x^0x1.4p2 + -0x1.a01a019936f27p-13 * x^0x1.cp2 + 0x1.71de37a97d93ep-19 * x^0x1.2p3 + -0x1.ae633919987c6p-26 * x^0x1.6p3 + 0x1.60e277ae07cecp-33 * x^0x1.ap3 + -0x1.9e9540300a1p-41 * x^0x1.ep3
Display mode is decimal numbers.
51.765
Display mode is hexadecimal numbers.
-0x1.9f0ef1c39804ap-41 * x^0x1.ep3 + 0x1.60e6871043ae8p-33 * x^0x1.ap3 + -0x1.ae635418ed56dp-26 * x^0x1.6p3 + 0x1.71de3801036c7p-19 * x^0x1.2p3 + -0x1.a01a019a50542p-13 * x^0x1.cp2 + 0x1.111111110a30bp-7 * x^0x1.4p2 + -0x1.55555555554a2p-3 * x^0x1.8p1 + x
Display mode is decimal numbers.
52.454

Paul

> CC: Joe Ramsay <Joe.Ramsay@arm.com>
> Date: Thu, 3 Aug 2023 12:54:53 +0100
> NoDisclaimer: true
> From: Joe Ramsay via Libc-alpha <libc-alpha@sourceware.org>
> 
> The polynomial used in the AdvSIMD variant was found to be more
> efficient than FTRIG intrinsics, with acceptable loss of precision.
> ---
> Thanks,
> Joe
>  sysdeps/aarch64/fpu/sin_sve.c | 99 ++++++++++++++++++-----------------
>  1 file changed, 50 insertions(+), 49 deletions(-)
> 
> diff --git a/sysdeps/aarch64/fpu/sin_sve.c b/sysdeps/aarch64/fpu/sin_sve.c
> index c3f450d0ea..4d93e761af 100644
> --- a/sysdeps/aarch64/fpu/sin_sve.c
> +++ b/sysdeps/aarch64/fpu/sin_sve.c
> @@ -21,20 +21,23 @@
>  
>  static const struct data
>  {
> -  double inv_pi, half_pi, inv_pi_over_2, pi_over_2_1, pi_over_2_2, pi_over_2_3,
> -      shift;
> +  double inv_pi, pi_1, pi_2, pi_3, shift, range_val;
> +  double poly[7];
>  } data = {
> -  /* Polynomial coefficients are hard-wired in the FTMAD instruction.  */
> +  /* Worst-case error is 2.8+0.5 ulp in [-pi/2, pi/2].  */
> +  .poly = { -0x1.555555555547bp-3, 0x1.1111111108a4dp-7, -0x1.a01a019936f27p-13,
> +            0x1.71de37a97d93ep-19, -0x1.ae633919987c6p-26,
> +            0x1.60e277ae07cecp-33, -0x1.9e9540300a1p-41, },
> +
>    .inv_pi = 0x1.45f306dc9c883p-2,
> -  .half_pi = 0x1.921fb54442d18p+0,
> -  .inv_pi_over_2 = 0x1.45f306dc9c882p-1,
> -  .pi_over_2_1 = 0x1.921fb50000000p+0,
> -  .pi_over_2_2 = 0x1.110b460000000p-26,
> -  .pi_over_2_3 = 0x1.1a62633145c07p-54,
> -  .shift = 0x1.8p52
> +  .pi_1 = 0x1.921fb54442d18p+1,
> +  .pi_2 = 0x1.1a62633145c06p-53,
> +  .pi_3 = 0x1.c1cd129024e09p-106,
> +  .shift = 0x1.8p52,
> +  .range_val = 0x1p23,
>  };
>  
> -#define RangeVal 0x4160000000000000 /* asuint64 (0x1p23).  */
> +#define C(i) sv_f64 (d->poly[i])
>  
>  static svfloat64_t NOINLINE
>  special_case (svfloat64_t x, svfloat64_t y, svbool_t cmp)
> @@ -42,55 +45,53 @@ special_case (svfloat64_t x, svfloat64_t y, svbool_t cmp)
>    return sv_call_f64 (sin, x, y, cmp);
>  }
>  
> -/* A fast SVE implementation of sin based on trigonometric
> -   instructions (FTMAD, FTSSEL, FTSMUL).
> -   Maximum observed error in 2.52 ULP:
> -   SV_NAME_D1 (sin)(0x1.2d2b00df69661p+19) got 0x1.10ace8f3e786bp-40
> -					  want 0x1.10ace8f3e7868p-40.  */
> +/* A fast SVE implementation of sin.
> +   Maximum observed error in 3.22 ULP:
> +   _ZGVsMxv_sin (0x1.d70eef40f39b1p+12) got -0x1.ffe9537d5dbb7p-3
> +				       want -0x1.ffe9537d5dbb4p-3.  */
>  svfloat64_t SV_NAME_D1 (sin) (svfloat64_t x, const svbool_t pg)
>  {
>    const struct data *d = ptr_barrier (&data);
>  
> -  svfloat64_t r = svabs_f64_x (pg, x);
> -  svuint64_t sign
> -      = sveor_u64_x (pg, svreinterpret_u64_f64 (x), svreinterpret_u64_f64 (r));
> -  svbool_t cmp = svcmpge_n_u64 (pg, svreinterpret_u64_f64 (r), RangeVal);
> -
> -  /* Load first two pio2-related constants to one vector.  */
> -  svfloat64_t invpio2_and_pio2_1
> -      = svld1rq_f64 (svptrue_b64 (), &d->inv_pi_over_2);
> -
> -  /* n = rint(|x|/(pi/2)).  */
> -  svfloat64_t q = svmla_lane_f64 (sv_f64 (d->shift), r, invpio2_and_pio2_1, 0);
> -  svfloat64_t n = svsub_n_f64_x (pg, q, d->shift);
> +  /* Load some values in quad-word chunks to minimise memory access.  */
> +  const svbool_t ptrue = svptrue_b64 ();
> +  svfloat64_t shift = sv_f64 (d->shift);
> +  svfloat64_t inv_pi_and_pi1 = svld1rq (ptrue, &d->inv_pi);
> +  svfloat64_t pi2_and_pi3 = svld1rq (ptrue, &d->pi_2);
> +
> +  /* n = rint(|x|/pi).  */
> +  svfloat64_t n = svmla_lane (shift, x, inv_pi_and_pi1, 0);
> +  svuint64_t odd = svlsl_x (pg, svreinterpret_u64 (n), 63);
> +  odd = sveor_m (
> +      svcmpeq (pg, svreinterpret_u64 (x), svreinterpret_u64 (sv_f64 (-0.0))),
> +      odd, 0x8000000000000000);
> +  n = svsub_x (pg, n, shift);
> +
> +  /* r = |x| - n*(pi/2)  (range reduction into -pi/2 .. pi/2).  */
> +  svfloat64_t r = x;
> +  r = svmls_lane (r, n, inv_pi_and_pi1, 1);
> +  r = svmls_lane (r, n, pi2_and_pi3, 0);
> +  r = svmls_lane (r, n, pi2_and_pi3, 1);
>  
> -  /* r = |x| - n*(pi/2)  (range reduction into -pi/4 .. pi/4).  */
> -  r = svmls_lane_f64 (r, n, invpio2_and_pio2_1, 1);
> -  r = svmls_n_f64_x (pg, r, n, d->pi_over_2_2);
> -  r = svmls_n_f64_x (pg, r, n, d->pi_over_2_3);
> +  /* sin(r) poly approx.  */
> +  svfloat64_t r2 = svmul_x (pg, r, r);
> +  svfloat64_t r3 = svmul_x (pg, r2, r);
> +  svfloat64_t r4 = svmul_x (pg, r2, r2);
>  
> -  /* Final multiplicative factor: 1.0 or x depending on bit #0 of q.  */
> -  svfloat64_t f = svtssel_f64 (r, svreinterpret_u64_f64 (q));
> +  svfloat64_t t1 = svmla_x (pg, C (4), C (5), r2);
> +  svfloat64_t t2 = svmla_x (pg, C (2), C (3), r2);
> +  svfloat64_t t3 = svmla_x (pg, C (0), C (1), r2);
>  
> -  /* sin(r) poly approx.  */
> -  svfloat64_t r2 = svtsmul_f64 (r, svreinterpret_u64_f64 (q));
> -  svfloat64_t y = sv_f64 (0.0);
> -  y = svtmad_f64 (y, r2, 7);
> -  y = svtmad_f64 (y, r2, 6);
> -  y = svtmad_f64 (y, r2, 5);
> -  y = svtmad_f64 (y, r2, 4);
> -  y = svtmad_f64 (y, r2, 3);
> -  y = svtmad_f64 (y, r2, 2);
> -  y = svtmad_f64 (y, r2, 1);
> -  y = svtmad_f64 (y, r2, 0);
> -
> -  /* Apply factor.  */
> -  y = svmul_f64_x (pg, f, y);
> +  svfloat64_t y = svmla_x (pg, t1, C (6), r4);
> +  y = svmla_x (pg, t2, y, r4);
> +  y = svmla_x (pg, t3, y, r4);
> +  y = svmla_x (pg, r, y, r3);
>  
>    /* sign = y^sign.  */
> -  y = svreinterpret_f64_u64 (
> -      sveor_u64_x (pg, svreinterpret_u64_f64 (y), sign));
> +  y = svreinterpret_f64 (sveor_z (pg, svreinterpret_u64 (y), odd));
>  
> +  svbool_t cmp = svacle (pg, x, d->range_val);
> +  cmp = svnot_z (pg, cmp);
>    if (__glibc_unlikely (svptest_any (pg, cmp)))
>      return special_case (x, y, cmp);
>    return y;
> -- 
> 2.27.0