1 | /* Double-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_f64.h" |
22 | |
23 | static const struct data |
24 | { |
25 | float64_t poly[12]; |
26 | float64_t pi, pi_over_2; |
27 | } data = { |
28 | /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) |
29 | on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */ |
30 | .poly = { 0x1.555555555554ep-3, 0x1.3333333337233p-4, 0x1.6db6db67f6d9fp-5, |
31 | 0x1.f1c71fbd29fbbp-6, 0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6, |
32 | 0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7, 0x1.fd1151acb6bedp-8, |
33 | 0x1.087182f799c1dp-6, -0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, }, |
34 | .pi = 0x1.921fb54442d18p+1, |
35 | .pi_over_2 = 0x1.921fb54442d18p+0, |
36 | }; |
37 | |
38 | /* Double-precision SVE implementation of vector acos(x). |
39 | |
40 | For |x| in [0, 0.5], use an order 11 polynomial P such that the final |
41 | approximation of asin is an odd polynomial: |
42 | |
43 | acos(x) ~ pi/2 - (x + x^3 P(x^2)). |
44 | |
45 | The largest observed error in this region is 1.18 ulps, |
46 | _ZGVsMxv_acos (0x1.fbc5fe28ee9e3p-2) got 0x1.0d4d0f55667f6p+0 |
47 | want 0x1.0d4d0f55667f7p+0. |
48 | |
49 | For |x| in [0.5, 1.0], use same approximation with a change of variable |
50 | |
51 | acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). |
52 | |
53 | The largest observed error in this region is 1.52 ulps, |
54 | _ZGVsMxv_acos (0x1.24024271a500ap-1) got 0x1.ed82df4243f0dp-1 |
55 | want 0x1.ed82df4243f0bp-1. */ |
56 | svfloat64_t SV_NAME_D1 (acos) (svfloat64_t x, const svbool_t pg) |
57 | { |
58 | const struct data *d = ptr_barrier (&data); |
59 | |
60 | svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000); |
61 | svfloat64_t ax = svabs_x (pg, x); |
62 | |
63 | svbool_t a_gt_half = svacgt (pg, x, 0.5); |
64 | |
65 | /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with |
66 | z2 = x ^ 2 and z = |x| , if |x| < 0.5 |
67 | z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ |
68 | svfloat64_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f64 (x: 0.5), ax, 0.5), |
69 | svmul_x (pg, x, x)); |
70 | svfloat64_t z = svsqrt_m (ax, a_gt_half, z2); |
71 | |
72 | /* Use a single polynomial approximation P for both intervals. */ |
73 | svfloat64_t z4 = svmul_x (pg, z2, z2); |
74 | svfloat64_t z8 = svmul_x (pg, z4, z4); |
75 | svfloat64_t z16 = svmul_x (pg, z8, z8); |
76 | svfloat64_t p = sv_estrin_11_f64_x (pg, x: z2, x2: z4, x4: z8, x8: z16, poly: d->poly); |
77 | |
78 | /* Finalize polynomial: z + z * z2 * P(z2). */ |
79 | p = svmla_x (pg, z, svmul_x (pg, z, z2), p); |
80 | |
81 | /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 |
82 | = 2 Q(|x|) , for 0.5 < x < 1.0 |
83 | = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ |
84 | svfloat64_t y |
85 | = svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (p), sign)); |
86 | |
87 | svbool_t is_neg = svcmplt (pg, x, 0.0); |
88 | svfloat64_t off = svdup_f64_z (is_neg, d->pi); |
89 | svfloat64_t mul = svsel (a_gt_half, sv_f64 (x: 2.0), sv_f64 (x: -1.0)); |
90 | svfloat64_t add = svsel (a_gt_half, off, sv_f64 (x: d->pi_over_2)); |
91 | |
92 | return svmla_x (pg, add, mul, y); |
93 | } |
94 | |