1/* Single-precision AdvSIMD 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 "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, pif;
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 .pif = V4 (0x1.921fb6p+1f),
34};
35
36#define AbsMask 0x7fffffff
37#define Half 0x3f000000
38#define One 0x3f800000
39#define Small 0x32800000 /* 2^-26. */
40
41#if WANT_SIMD_EXCEPT
42static float32x4_t VPCS_ATTR NOINLINE
43special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
44{
45 return v_call_f32 (acosf, x, y, special);
46}
47#endif
48
49/* Single-precision implementation of vector acos(x).
50
51 For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
52 rounding.
53 If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
54 approximation.
55
56 For |x| in [Small, 0.5], use order 4 polynomial P such that the final
57 approximation of asin is an odd polynomial:
58
59 acos(x) ~ pi/2 - (x + x^3 P(x^2)).
60
61 The largest observed error in this region is 1.26 ulps,
62 _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0.
63
64 For |x| in [0.5, 1.0], use same approximation with a change of variable
65
66 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
67
68 The largest observed error in this region is 1.32 ulps,
69 _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
70 want 0x1.feb32ep-1. */
71float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (acos) (float32x4_t x)
72{
73 const struct data *d = ptr_barrier (&data);
74
75 uint32x4_t ix = vreinterpretq_u32_f32 (x);
76 uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
77
78#if WANT_SIMD_EXCEPT
79 /* A single comparison for One, Small and QNaN. */
80 uint32x4_t special
81 = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
82 if (__glibc_unlikely (v_any_u32 (special)))
83 return special_case (x, x, v_u32 (0xffffffff));
84#endif
85
86 float32x4_t ax = vreinterpretq_f32_u32 (ia);
87 uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half));
88
89 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
90 z2 = x ^ 2 and z = |x| , if |x| < 0.5
91 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
92 float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x),
93 vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
94 float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
95
96 /* Use a single polynomial approximation P for both intervals. */
97 float32x4_t p = v_horner_4_f32 (z2, d->poly);
98 /* Finalize polynomial: z + z * z2 * P(z2). */
99 p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
100
101 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
102 = 2 Q(|x|) , for 0.5 < x < 1.0
103 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
104 float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x);
105
106 uint32x4_t is_neg = vcltzq_f32 (x);
107 float32x4_t off = vreinterpretq_f32_u32 (
108 vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg));
109 float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0));
110 float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off);
111
112 return vfmaq_f32 (add, mul, y);
113}
114libmvec_hidden_def (V_NAME_F1(acos))
115HALF_WIDTH_ALIAS_F1 (acos)
116

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