sysdeps/aarch64/fpu/erff_sve.c

   1 /* Single-precision vector (SVE) erf function
   2
   3    Copyright (C) 2024 Free Software Foundation, Inc.
   4    This file is part of the GNU C Library.
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library is distributed in the hope that it will be useful,
  12    but WITHOUT ANY WARRANTY; without even the implied warranty of
  13    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, see
  18    <https://www.gnu.org/licenses/>.  */
  19
  20 #include "sv_math.h"
  21
  22 static const struct data
  23 {
  24   float min, max, scale, shift, third;
  25 } data = {
  26   .min = 0x1.cp-7f,        /* 1/64 - 1/512.  */
  27   .max = 3.9375,           /* 4 - 8/128.  */
  28   .scale = 0x1.20dd76p+0f, /* 2/sqrt(pi).  */
  29   .shift = 0x1p16f,
  30   .third = 0x1.555556p-2f, /* 1/3.  */
  31 };
  32
  33 #define SignMask (0x80000000)
  34
  35 /* Single-precision implementation of vector erf(x).
  36    Approximation based on series expansion near x rounded to
  37    nearest multiple of 1/128.
  38    Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
  39
  40    erf(x) ~ erf(r) + scale * d * [1 - r * d - 1/3 * d^2]
  41
  42    Values of erf(r) and scale are read from lookup tables.
  43    For |x| < 0x1.cp-7, the algorithm sets r = 0, erf(r) = 0, and scale = 2 /
  44    sqrt(pi), so it simply boils down to a Taylor series expansion near 0. For
  45    |x| > 3.9375, erf(|x|) rounds to 1.0f.
  46
  47    Maximum error on each interval:
  48    - [0, 0x1.cp-7]: 1.93 ULP
  49      _ZGVsMxv_erff(0x1.c373e6p-9) got 0x1.fd686cp-9 want 0x1.fd6868p-9
  50    - [0x1.cp-7, 4.0]: 1.26 ULP
  51      _ZGVsMxv_erff(0x1.1d002ep+0) got 0x1.c4eb9ap-1 want 0x1.c4eb98p-1.  */
  52 svfloat32_t SV_NAME_F1 (erf) (svfloat32_t x, const svbool_t pg)
  53 {
  54   const struct data *dat = ptr_barrier (&data);
  55
  56   /* |x| > 1/64 - 1/512.  */
  57   svbool_t a_gt_min = svacgt (pg, x, dat->min);
  58
  59   /* |x| >= 4.0 - 8/128.  */
  60   svbool_t a_ge_max = svacge (pg, x, dat->max);
  61   svfloat32_t a = svabs_x (pg, x);
  62
  63   svfloat32_t shift = sv_f32 (dat->shift);
  64   svfloat32_t z = svadd_x (pg, a, shift);
  65   svuint32_t i
  66       = svsub_x (pg, svreinterpret_u32 (z), svreinterpret_u32 (shift));
  67
  68   /* Saturate lookup index.  */
  69   i = svsel (a_ge_max, sv_u32 (512), i);
  70
  71   /* r and erf(r) set to 0 for |x| below min.  */
  72   svfloat32_t r = svsub_z (a_gt_min, z, shift);
  73   svfloat32_t erfr = svld1_gather_index (a_gt_min, __sv_erff_data.erf, i);
  74
  75   /* scale set to 2/sqrt(pi) for |x| below min.  */
  76   svfloat32_t scale = svld1_gather_index (a_gt_min, __sv_erff_data.scale, i);
  77   scale = svsel (a_gt_min, scale, sv_f32 (dat->scale));
  78
  79   /* erf(x) ~ erf(r) + scale * d * (1 - r * d + 1/3 * d^2).  */
  80   svfloat32_t d = svsub_x (pg, a, r);
  81   svfloat32_t d2 = svmul_x (pg, d, d);
  82   svfloat32_t y = svmla_x (pg, r, d, dat->third);
  83   y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y));
  84
  85   /* Solves the |x| = inf case.  */
  86   y = svsel (a_ge_max, sv_f32 (1.0f), y);
  87
  88   /* Copy sign.  */
  89   svuint32_t ix = svreinterpret_u32 (x);
  90   svuint32_t iy = svreinterpret_u32 (y);
  91   svuint32_t sign = svand_x (pg, ix, SignMask);
  92   return svreinterpret_f32 (svorr_x (pg, sign, iy));
  93 }