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/* Double-precision SVE inverse cos
Copyright (C) 2023-2025 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
{
float64_t c1, c3, c5, c7, c9, c11;
float64_t c0, c2, c4, c6, c8, c10;
float64_t pi_over_2;
} data = {
/* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
.c0 = 0x1.555555555554ep-3, .c1 = 0x1.3333333337233p-4,
.c2 = 0x1.6db6db67f6d9fp-5, .c3 = 0x1.f1c71fbd29fbbp-6,
.c4 = 0x1.6e8b264d467d6p-6, .c5 = 0x1.1c5997c357e9dp-6,
.c6 = 0x1.c86a22cd9389dp-7, .c7 = 0x1.856073c22ebbep-7,
.c8 = 0x1.fd1151acb6bedp-8, .c9 = 0x1.087182f799c1dp-6,
.c10 = -0x1.6602748120927p-7, .c11 = 0x1.cfa0dd1f9478p-6,
.pi_over_2 = 0x1.921fb54442d18p+0,
};
/* Double-precision SVE implementation of vector acos(x).
For |x| in [0, 0.5], use an order 11 polynomial P such that the final
approximation of asin is an odd polynomial:
acos(x) ~ pi/2 - (x + x^3 P(x^2)).
The largest observed error in this region is 1.18 ulp:
_ZGVsMxv_acos (0x1.fbb7c9079b429p-2) got 0x1.0d51266607582p+0
want 0x1.0d51266607583p+0.
For |x| in [0.5, 1.0], use same approximation with a change of variable
acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
The largest observed error in this region is 1.50 ulp:
_ZGVsMxv_acos (0x1.252a2cf3fb9acp-1) got 0x1.ec1a46aa82901p-1
want 0x1.ec1a46aa829p-1. */
svfloat64_t SV_NAME_D1 (acos) (svfloat64_t x, const svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
svbool_t ptrue = svptrue_b64 ();
svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000);
svfloat64_t ax = svabs_x (pg, x);
svbool_t a_gt_half = svacgt (pg, x, 0.5);
/* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
z2 = x ^ 2 and z = |x| , if |x| < 0.5
z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
svfloat64_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f64 (0.5), ax, 0.5),
svmul_x (pg, x, x));
svfloat64_t z = svsqrt_m (ax, a_gt_half, z2);
/* Use a single polynomial approximation P for both intervals. */
svfloat64_t z3 = svmul_x (ptrue, z2, z);
svfloat64_t z4 = svmul_x (ptrue, z2, z2);
svfloat64_t z8 = svmul_x (ptrue, z4, z4);
svfloat64_t c13 = svld1rq (ptrue, &d->c1);
svfloat64_t c57 = svld1rq (ptrue, &d->c5);
svfloat64_t c911 = svld1rq (ptrue, &d->c9);
svfloat64_t p01 = svmla_lane (sv_f64 (d->c0), z2, c13, 0);
svfloat64_t p23 = svmla_lane (sv_f64 (d->c2), z2, c13, 1);
svfloat64_t p03 = svmla_x (pg, p01, z4, p23);
svfloat64_t p45 = svmla_lane (sv_f64 (d->c4), z2, c57, 0);
svfloat64_t p67 = svmla_lane (sv_f64 (d->c6), z2, c57, 1);
svfloat64_t p47 = svmla_x (pg, p45, z4, p67);
svfloat64_t p89 = svmla_lane (sv_f64 (d->c8), z2, c911, 0);
svfloat64_t p1011 = svmla_lane (sv_f64 (d->c10), z2, c911, 1);
svfloat64_t p811 = svmla_x (pg, p89, z4, p1011);
svfloat64_t p411 = svmla_x (pg, p47, z8, p811);
svfloat64_t p = svmad_x (pg, p411, z8, p03);
/* Finalize polynomial: z + z * z2 * P(z2). */
p = svmad_x (pg, p, z3, z);
/* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
= 2 Q(|x|) , for 0.5 < x < 1.0
= pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
svfloat64_t mul = svreinterpret_f64 (
svlsl_m (a_gt_half, svreinterpret_u64 (sv_f64 (1.0)), 10));
mul = svreinterpret_f64 (sveor_x (ptrue, svreinterpret_u64 (mul), sign));
svfloat64_t add = svreinterpret_f64 (
svorr_x (ptrue, sign, svreinterpret_u64 (sv_f64 (d->pi_over_2))));
add = svsub_m (a_gt_half, sv_f64 (d->pi_over_2), add);
return svmsb_x (pg, p, mul, add);
}
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