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/* Single-precision vector (SVE) erf function
Copyright (C) 2024 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 "sv_math.h"
static const struct data
{
float min, max, scale, shift, third;
} data = {
.min = 0x1.cp-7f, /* 1/64 - 1/512. */
.max = 3.9375, /* 4 - 8/128. */
.scale = 0x1.20dd76p+0f, /* 2/sqrt(pi). */
.shift = 0x1p16f,
.third = 0x1.555556p-2f, /* 1/3. */
};
#define SignMask (0x80000000)
/* Single-precision implementation of vector erf(x).
Approximation based on series expansion near x rounded to
nearest multiple of 1/128.
Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
erf(x) ~ erf(r) + scale * d * [1 - r * d - 1/3 * d^2]
Values of erf(r) and scale are read from lookup tables.
For |x| < 0x1.cp-7, the algorithm sets r = 0, erf(r) = 0, and scale = 2 /
sqrt(pi), so it simply boils down to a Taylor series expansion near 0. For
|x| > 3.9375, erf(|x|) rounds to 1.0f.
Maximum error on each interval:
- [0, 0x1.cp-7]: 1.93 ULP
_ZGVsMxv_erff(0x1.c373e6p-9) got 0x1.fd686cp-9 want 0x1.fd6868p-9
- [0x1.cp-7, 4.0]: 1.26 ULP
_ZGVsMxv_erff(0x1.1d002ep+0) got 0x1.c4eb9ap-1 want 0x1.c4eb98p-1. */
svfloat32_t SV_NAME_F1 (erf) (svfloat32_t x, const svbool_t pg)
{
const struct data *dat = ptr_barrier (&data);
/* |x| > 1/64 - 1/512. */
svbool_t a_gt_min = svacgt (pg, x, dat->min);
/* |x| >= 4.0 - 8/128. */
svbool_t a_ge_max = svacge (pg, x, dat->max);
svfloat32_t a = svabs_x (pg, x);
svfloat32_t shift = sv_f32 (dat->shift);
svfloat32_t z = svadd_x (pg, a, shift);
svuint32_t i
= svsub_x (pg, svreinterpret_u32 (z), svreinterpret_u32 (shift));
/* Saturate lookup index. */
i = svsel (a_ge_max, sv_u32 (512), i);
/* r and erf(r) set to 0 for |x| below min. */
svfloat32_t r = svsub_z (a_gt_min, z, shift);
svfloat32_t erfr = svld1_gather_index (a_gt_min, __sv_erff_data.erf, i);
/* scale set to 2/sqrt(pi) for |x| below min. */
svfloat32_t scale = svld1_gather_index (a_gt_min, __sv_erff_data.scale, i);
scale = svsel (a_gt_min, scale, sv_f32 (dat->scale));
/* erf(x) ~ erf(r) + scale * d * (1 - r * d + 1/3 * d^2). */
svfloat32_t d = svsub_x (pg, a, r);
svfloat32_t d2 = svmul_x (pg, d, d);
svfloat32_t y = svmla_x (pg, r, d, dat->third);
y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y));
/* Solves the |x| = inf case. */
y = svsel (a_ge_max, sv_f32 (1.0f), y);
/* Copy sign. */
svuint32_t ix = svreinterpret_u32 (x);
svuint32_t iy = svreinterpret_u32 (y);
svuint32_t sign = svand_x (pg, ix, SignMask);
return svreinterpret_f32 (svorr_x (pg, sign, iy));
}
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