1/* Single-precision AdvSIMD inverse sin
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 "v_math.h"
21#include "poly_advsimd_f32.h"
22
23static const struct data
24{
25 float32x4_t poly[5];
26 float32x4_t pi_over_2f;
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 = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5),
31 V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) },
32 .pi_over_2f = V4 (0x1.921fb6p+0f),
33};
34
35#define AbsMask 0x7fffffff
36#define Half 0x3f000000
37#define One 0x3f800000
38#define Small 0x39800000 /* 2^-12. */
39
40#if WANT_SIMD_EXCEPT
41static float32x4_t VPCS_ATTR NOINLINE
42special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
43{
44 return v_call_f32 (asinf, x, y, special);
45}
46#endif
47
48/* Single-precision implementation of vector asin(x).
49
50 For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct
51 rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the
52 following approximation.
53
54 For |x| in [Small, 0.5], use order 4 polynomial P such that the final
55 approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
56
57 The largest observed error in this region is 0.83 ulps,
58 _ZGVnN4v_asinf (0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2.
59
60 For |x| in [0.5, 1.0], use same approximation with a change of variable
61
62 asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
63
64 The largest observed error in this region is 2.41 ulps,
65 _ZGVnN4v_asinf (0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */
66float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (asin) (float32x4_t x)
67{
68 const struct data *d = ptr_barrier (&data);
69
70 uint32x4_t ix = vreinterpretq_u32_f32 (x);
71 uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
72
73#if WANT_SIMD_EXCEPT
74 /* Special values need to be computed with scalar fallbacks so
75 that appropriate fp exceptions are raised. */
76 uint32x4_t special
77 = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
78 if (__glibc_unlikely (v_any_u32 (special)))
79 return special_case (x, x, v_u32 (0xffffffff));
80#endif
81
82 float32x4_t ax = vreinterpretq_f32_u32 (ia);
83 uint32x4_t a_lt_half = vcltq_u32 (ia, v_u32 (Half));
84
85 /* Evaluate polynomial Q(x) = y + y * z * P(z) with
86 z = x ^ 2 and y = |x| , if |x| < 0.5
87 z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
88 float32x4_t z2 = vbslq_f32 (a_lt_half, vmulq_f32 (x, x),
89 vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
90 float32x4_t z = vbslq_f32 (a_lt_half, ax, vsqrtq_f32 (z2));
91
92 /* Use a single polynomial approximation P for both intervals. */
93 float32x4_t p = v_horner_4_f32 (z2, d->poly);
94 /* Finalize polynomial: z + z * z2 * P(z2). */
95 p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
96
97 /* asin(|x|) = Q(|x|) , for |x| < 0.5
98 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
99 float32x4_t y
100 = vbslq_f32 (a_lt_half, p, vfmsq_n_f32 (d->pi_over_2f, p, 2.0));
101
102 /* Copy sign. */
103 return vbslq_f32 (v_u32 (AbsMask), y, x);
104}
105libmvec_hidden_def (V_NAME_F1 (asin))
106HALF_WIDTH_ALIAS_F1 (asin)
107

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