1/* Double-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_f64.h"
22
23static const struct data
24{
25 float64x2_t poly[12];
26 float64x2_t pi, pi_over_2;
27 uint64x2_t abs_mask;
28} data = {
29 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
30 on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
31 .poly = { V2 (0x1.555555555554ep-3), V2 (0x1.3333333337233p-4),
32 V2 (0x1.6db6db67f6d9fp-5), V2 (0x1.f1c71fbd29fbbp-6),
33 V2 (0x1.6e8b264d467d6p-6), V2 (0x1.1c5997c357e9dp-6),
34 V2 (0x1.c86a22cd9389dp-7), V2 (0x1.856073c22ebbep-7),
35 V2 (0x1.fd1151acb6bedp-8), V2 (0x1.087182f799c1dp-6),
36 V2 (-0x1.6602748120927p-7), V2 (0x1.cfa0dd1f9478p-6), },
37 .pi = V2 (0x1.921fb54442d18p+1),
38 .pi_over_2 = V2 (0x1.921fb54442d18p+0),
39 .abs_mask = V2 (0x7fffffffffffffff),
40};
41
42#define AllMask v_u64 (0xffffffffffffffff)
43#define Oneu 0x3ff0000000000000
44#define Small 0x3e50000000000000 /* 2^-53. */
45
46#if WANT_SIMD_EXCEPT
47static float64x2_t VPCS_ATTR NOINLINE
48special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
49{
50 return v_call_f64 (acos, x, y, special);
51}
52#endif
53
54/* Double-precision implementation of vector acos(x).
55
56 For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct
57 rounding.
58 If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
59 approximation.
60
61 For |x| in [Small, 0.5], use an order 11 polynomial P such that the final
62 approximation of asin is an odd polynomial:
63
64 acos(x) ~ pi/2 - (x + x^3 P(x^2)).
65
66 The largest observed error in this region is 1.18 ulps,
67 _ZGVnN2v_acos (0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0
68 want 0x1.0d54d1985c069p+0.
69
70 For |x| in [0.5, 1.0], use same approximation with a change of variable
71
72 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
73
74 The largest observed error in this region is 1.52 ulps,
75 _ZGVnN2v_acos (0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1
76 want 0x1.edbbedf8a7d6cp-1. */
77float64x2_t VPCS_ATTR V_NAME_D1 (acos) (float64x2_t x)
78{
79 const struct data *d = ptr_barrier (&data);
80
81 float64x2_t ax = vabsq_f64 (x);
82
83#if WANT_SIMD_EXCEPT
84 /* A single comparison for One, Small and QNaN. */
85 uint64x2_t special
86 = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)),
87 v_u64 (Oneu - Small));
88 if (__glibc_unlikely (v_any_u64 (special)))
89 return special_case (x, x, AllMask);
90#endif
91
92 uint64x2_t a_le_half = vcleq_f64 (ax, v_f64 (0.5));
93
94 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
95 z2 = x ^ 2 and z = |x| , if |x| < 0.5
96 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
97 float64x2_t z2 = vbslq_f64 (a_le_half, vmulq_f64 (x, x),
98 vfmaq_f64 (v_f64 (0.5), v_f64 (-0.5), ax));
99 float64x2_t z = vbslq_f64 (a_le_half, ax, vsqrtq_f64 (z2));
100
101 /* Use a single polynomial approximation P for both intervals. */
102 float64x2_t z4 = vmulq_f64 (z2, z2);
103 float64x2_t z8 = vmulq_f64 (z4, z4);
104 float64x2_t z16 = vmulq_f64 (z8, z8);
105 float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly);
106
107 /* Finalize polynomial: z + z * z2 * P(z2). */
108 p = vfmaq_f64 (z, vmulq_f64 (z, z2), p);
109
110 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
111 = 2 Q(|x|) , for 0.5 < x < 1.0
112 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
113 float64x2_t y = vbslq_f64 (d->abs_mask, p, x);
114
115 uint64x2_t is_neg = vcltzq_f64 (x);
116 float64x2_t off = vreinterpretq_f64_u64 (
117 vandq_u64 (is_neg, vreinterpretq_u64_f64 (d->pi)));
118 float64x2_t mul = vbslq_f64 (a_le_half, v_f64 (-1.0), v_f64 (2.0));
119 float64x2_t add = vbslq_f64 (a_le_half, d->pi_over_2, off);
120
121 return vfmaq_f64 (add, mul, y);
122}
123

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