1/* Single-precision SVE inverse cos
2
3 Copyright (C) 2023-2024 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#include "poly_sve_f32.h"
22
23static const struct data
24{
25 float32_t poly[5];
26 float32_t pi, pi_over_2;
27} data = {
28 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on
29 [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */
30 .poly = { 0x1.55555ep-3, 0x1.33261ap-4, 0x1.70d7dcp-5, 0x1.b059dp-6,
31 0x1.3af7d8p-5, },
32 .pi = 0x1.921fb6p+1f,
33 .pi_over_2 = 0x1.921fb6p+0f,
34};
35
36/* Single-precision SVE implementation of vector acos(x).
37
38 For |x| in [0, 0.5], use order 4 polynomial P such that the final
39 approximation of asin is an odd polynomial:
40
41 acos(x) ~ pi/2 - (x + x^3 P(x^2)).
42
43 The largest observed error in this region is 1.16 ulps,
44 _ZGVsMxv_acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0
45 want 0x1.0c27f6p+0.
46
47 For |x| in [0.5, 1.0], use same approximation with a change of variable
48
49 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
50
51 The largest observed error in this region is 1.32 ulps,
52 _ZGVsMxv_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
53 want 0x1.feb32ep-1. */
54svfloat32_t SV_NAME_F1 (acos) (svfloat32_t x, const svbool_t pg)
55{
56 const struct data *d = ptr_barrier (&data);
57
58 svuint32_t sign = svand_x (pg, svreinterpret_u32 (x), 0x80000000);
59 svfloat32_t ax = svabs_x (pg, x);
60 svbool_t a_gt_half = svacgt (pg, x, 0.5);
61
62 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
63 z2 = x ^ 2 and z = |x| , if |x| < 0.5
64 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
65 svfloat32_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f32 (x: 0.5), ax, 0.5),
66 svmul_x (pg, x, x));
67 svfloat32_t z = svsqrt_m (ax, a_gt_half, z2);
68
69 /* Use a single polynomial approximation P for both intervals. */
70 svfloat32_t p = sv_horner_4_f32_x (pg, x: z2, poly: d->poly);
71 /* Finalize polynomial: z + z * z2 * P(z2). */
72 p = svmla_x (pg, z, svmul_x (pg, z, z2), p);
73
74 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
75 = 2 Q(|x|) , for 0.5 < x < 1.0
76 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
77 svfloat32_t y
78 = svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (p), sign));
79
80 svbool_t is_neg = svcmplt (pg, x, 0.0);
81 svfloat32_t off = svdup_f32_z (is_neg, d->pi);
82 svfloat32_t mul = svsel (a_gt_half, sv_f32 (x: 2.0), sv_f32 (x: -1.0));
83 svfloat32_t add = svsel (a_gt_half, off, sv_f32 (x: d->pi_over_2));
84
85 return svmla_x (pg, add, mul, y);
86}
87

source code of glibc/sysdeps/aarch64/fpu/acosf_sve.c