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/* Double-precision SVE expm1
Copyright (C) 2023 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"
#include "poly_sve_f64.h"
#define SpecialBound 0x1.62b7d369a5aa9p+9
#define ExponentBias 0x3ff0000000000000
static const struct data
{
double poly[11];
double shift, inv_ln2, special_bound;
/* To be loaded in one quad-word. */
double ln2_hi, ln2_lo;
} data = {
/* Generated using fpminimax. */
.poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5,
0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13,
0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22,
0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, },
.special_bound = SpecialBound,
.inv_ln2 = 0x1.71547652b82fep0,
.ln2_hi = 0x1.62e42fefa39efp-1,
.ln2_lo = 0x1.abc9e3b39803fp-56,
.shift = 0x1.8p52,
};
static svfloat64_t NOINLINE
special_case (svfloat64_t x, svfloat64_t y, svbool_t pg)
{
return sv_call_f64 (expm1, x, y, pg);
}
/* Double-precision vector exp(x) - 1 function.
The maximum error observed error is 2.18 ULP:
_ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
want 0x1.a8b9ea8d66e2p-2. */
svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
/* Large, Nan/Inf. */
svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound));
/* Reduce argument to smaller range:
Let i = round(x / ln2)
and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
where 2^i is exact because i is an integer. */
svfloat64_t shift = sv_f64 (d->shift);
svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift);
svint64_t i = svcvt_s64_x (pg, n);
svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
svfloat64_t f = svmls_lane (x, n, ln2, 0);
f = svmls_lane (f, n, ln2, 1);
/* Approximate expm1(f) using polynomial.
Taylor expansion for expm1(x) has the form:
x + ax^2 + bx^3 + cx^4 ....
So we calculate the polynomial P(f) = a + bf + cf^2 + ...
and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
svfloat64_t f2 = svmul_x (pg, f, f);
svfloat64_t f4 = svmul_x (pg, f2, f2);
svfloat64_t f8 = svmul_x (pg, f4, f4);
svfloat64_t p
= svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, f, f2, f4, f8, d->poly));
/* Assemble the result.
expm1(x) ~= 2^i * (p + 1) - 1
Let t = 2^i. */
svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias);
svfloat64_t t = svreinterpret_f64 (u);
/* expm1(x) ~= p * t + (t - 1). */
svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t);
if (__glibc_unlikely (svptest_any (pg, special)))
return special_case (x, y, special);
return y;
}
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