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/* Double-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
{
double third;
double tenth, two_over_five, two_over_fifteen;
double two_over_nine, two_over_fortyfive;
double max, shift;
} data = {
.third = 0x1.5555555555556p-2, /* used to compute 2/3 and 1/6 too. */
.two_over_fifteen = 0x1.1111111111111p-3,
.tenth = -0x1.999999999999ap-4,
.two_over_five = -0x1.999999999999ap-2,
.two_over_nine = -0x1.c71c71c71c71cp-3,
.two_over_fortyfive = 0x1.6c16c16c16c17p-5,
.max = 5.9921875, /* 6 - 1/128. */
.shift = 0x1p45,
};
#define SignMask (0x8000000000000000)
/* Double-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 (2 r^2 - 1) d^2
- 1/6 (r (2 r^2 - 3)) d^3
+ 1/30 (4 r^4 - 12 r^2 + 3) d^4
- 1/90 (4 r^4 - 20 r^2 + 15) d^5
]
Maximum measure error: 2.29 ULP
_ZGVsMxv_erf(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8
want -0x1.20dd59132ebafp-8. */
svfloat64_t SV_NAME_D1 (erf) (svfloat64_t x, const svbool_t pg)
{
const struct data *dat = ptr_barrier (&data);
/* |x| >= 6.0 - 1/128. Opposite conditions except none of them catch NaNs so
they can be used in lookup and BSLs to yield the expected results. */
svbool_t a_ge_max = svacge (pg, x, dat->max);
svbool_t a_lt_max = svaclt (pg, x, dat->max);
/* Set r to multiple of 1/128 nearest to |x|. */
svfloat64_t a = svabs_x (pg, x);
svfloat64_t shift = sv_f64 (dat->shift);
svfloat64_t z = svadd_x (pg, a, shift);
svuint64_t i
= svsub_x (pg, svreinterpret_u64 (z), svreinterpret_u64 (shift));
/* Lookup without shortcut for small values but with predicate to avoid
segfault for large values and NaNs. */
svfloat64_t r = svsub_x (pg, z, shift);
svfloat64_t erfr = svld1_gather_index (a_lt_max, __sv_erf_data.erf, i);
svfloat64_t scale = svld1_gather_index (a_lt_max, __sv_erf_data.scale, i);
/* erf(x) ~ erf(r) + scale * d * poly (r, d). */
svfloat64_t d = svsub_x (pg, a, r);
svfloat64_t d2 = svmul_x (pg, d, d);
svfloat64_t r2 = svmul_x (pg, r, r);
/* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */
svfloat64_t p1 = r;
svfloat64_t third = sv_f64 (dat->third);
svfloat64_t twothird = svmul_x (pg, third, 2.0);
svfloat64_t sixth = svmul_x (pg, third, 0.5);
svfloat64_t p2 = svmls_x (pg, third, r2, twothird);
svfloat64_t p3 = svmad_x (pg, r2, third, -0.5);
p3 = svmul_x (pg, r, p3);
svfloat64_t p4
= svmla_x (pg, sv_f64 (dat->two_over_five), r2, dat->two_over_fifteen);
p4 = svmls_x (pg, sv_f64 (dat->tenth), r2, p4);
svfloat64_t p5
= svmla_x (pg, sv_f64 (dat->two_over_nine), r2, dat->two_over_fortyfive);
p5 = svmla_x (pg, sixth, r2, p5);
p5 = svmul_x (pg, r, p5);
svfloat64_t p34 = svmla_x (pg, p3, d, p4);
svfloat64_t p12 = svmla_x (pg, p1, d, p2);
svfloat64_t y = svmla_x (pg, p34, d2, p5);
y = svmla_x (pg, p12, d2, y);
y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y));
/* Solves the |x| = inf and NaN cases. */
y = svsel (a_ge_max, sv_f64 (1.0), y);
/* Copy sign. */
svuint64_t ix = svreinterpret_u64 (x);
svuint64_t iy = svreinterpret_u64 (y);
svuint64_t sign = svand_x (pg, ix, SignMask);
return svreinterpret_f64 (svorr_x (pg, sign, iy));
}
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