1 | /* Double-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_f64.h" |
22 | |
23 | static const struct data |
24 | { |
25 | float64x2_t poly[12]; |
26 | float64x2_t 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_over_2 = V2 (0x1.921fb54442d18p+0), |
38 | .abs_mask = V2 (0x7fffffffffffffff), |
39 | }; |
40 | |
41 | #define AllMask v_u64 (0xffffffffffffffff) |
42 | #define One 0x3ff0000000000000 |
43 | #define Small 0x3e50000000000000 /* 2^-12. */ |
44 | |
45 | #if WANT_SIMD_EXCEPT |
46 | static float64x2_t VPCS_ATTR NOINLINE |
47 | special_case (float64x2_t x, float64x2_t y, uint64x2_t special) |
48 | { |
49 | return v_call_f64 (asin, x, y, special); |
50 | } |
51 | #endif |
52 | |
53 | /* Double-precision implementation of vector asin(x). |
54 | |
55 | For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct |
56 | rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the |
57 | following approximation. |
58 | |
59 | For |x| in [Small, 0.5], use an order 11 polynomial P such that the final |
60 | approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). |
61 | |
62 | The largest observed error in this region is 1.01 ulps, |
63 | _ZGVnN2v_asin (0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2 |
64 | want 0x1.ed78525a927eep-2. |
65 | |
66 | For |x| in [0.5, 1.0], use same approximation with a change of variable |
67 | |
68 | asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z). |
69 | |
70 | The largest observed error in this region is 2.69 ulps, |
71 | _ZGVnN2v_asin (0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1 |
72 | want 0x1.110d7e85fdd53p-1. */ |
73 | float64x2_t VPCS_ATTR V_NAME_D1 (asin) (float64x2_t x) |
74 | { |
75 | const struct data *d = ptr_barrier (&data); |
76 | |
77 | float64x2_t ax = vabsq_f64 (x); |
78 | |
79 | #if WANT_SIMD_EXCEPT |
80 | /* Special values need to be computed with scalar fallbacks so |
81 | that appropriate exceptions are raised. */ |
82 | uint64x2_t special |
83 | = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)), |
84 | v_u64 (One - Small)); |
85 | if (__glibc_unlikely (v_any_u64 (special))) |
86 | return special_case (x, x, AllMask); |
87 | #endif |
88 | |
89 | uint64x2_t a_lt_half = vcltq_f64 (ax, v_f64 (0.5)); |
90 | |
91 | /* Evaluate polynomial Q(x) = y + y * z * P(z) with |
92 | z = x ^ 2 and y = |x| , if |x| < 0.5 |
93 | z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */ |
94 | float64x2_t z2 = vbslq_f64 (a_lt_half, vmulq_f64 (x, x), |
95 | vfmsq_n_f64 (v_f64 (0.5), ax, 0.5)); |
96 | float64x2_t z = vbslq_f64 (a_lt_half, ax, vsqrtq_f64 (z2)); |
97 | |
98 | /* Use a single polynomial approximation P for both intervals. */ |
99 | float64x2_t z4 = vmulq_f64 (z2, z2); |
100 | float64x2_t z8 = vmulq_f64 (z4, z4); |
101 | float64x2_t z16 = vmulq_f64 (z8, z8); |
102 | float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly); |
103 | |
104 | /* Finalize polynomial: z + z * z2 * P(z2). */ |
105 | p = vfmaq_f64 (z, vmulq_f64 (z, z2), p); |
106 | |
107 | /* asin(|x|) = Q(|x|) , for |x| < 0.5 |
108 | = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ |
109 | float64x2_t y = vbslq_f64 (a_lt_half, p, vfmsq_n_f64 (d->pi_over_2, p, 2.0)); |
110 | |
111 | /* Copy sign. */ |
112 | return vbslq_f64 (d->abs_mask, y, x); |
113 | } |
114 | |