1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
|
/* Double-precision SVE inverse sin
Copyright (C) 2023 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include "sv_math.h"
#include "poly_sve_f64.h"
static const struct data
{
float64_t poly[12];
float64_t pi_over_2f;
} data = {
/* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
.poly = { 0x1.555555555554ep-3, 0x1.3333333337233p-4,
0x1.6db6db67f6d9fp-5, 0x1.f1c71fbd29fbbp-6,
0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6,
0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7,
0x1.fd1151acb6bedp-8, 0x1.087182f799c1dp-6,
-0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, },
.pi_over_2f = 0x1.921fb54442d18p+0,
};
#define P(i) sv_f64 (d->poly[i])
/* Double-precision SVE implementation of vector asin(x).
For |x| in [0, 0.5], use an order 11 polynomial P such that the final
approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
The largest observed error in this region is 0.52 ulps,
_ZGVsMxv_asin(0x1.d95ae04998b6cp-2) got 0x1.ec13757305f27p-2
want 0x1.ec13757305f26p-2.
For |x| in [0.5, 1.0], use same approximation with a change of variable
asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
The largest observed error in this region is 2.69 ulps,
_ZGVsMxv_asin(0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1
want 0x1.110d7e85fdd53p-1. */
svfloat64_t SV_NAME_D1 (asin) (svfloat64_t x, const svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000);
svfloat64_t ax = svabs_x (pg, x);
svbool_t a_ge_half = svacge (pg, x, 0.5);
/* Evaluate polynomial Q(x) = y + y * z * P(z) with
z = x ^ 2 and y = |x| , if |x| < 0.5
z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
svfloat64_t z2 = svsel (a_ge_half, svmls_x (pg, sv_f64 (0.5), ax, 0.5),
svmul_x (pg, x, x));
svfloat64_t z = svsqrt_m (ax, a_ge_half, z2);
/* Use a single polynomial approximation P for both intervals. */
svfloat64_t z4 = svmul_x (pg, z2, z2);
svfloat64_t z8 = svmul_x (pg, z4, z4);
svfloat64_t z16 = svmul_x (pg, z8, z8);
svfloat64_t p = sv_estrin_11_f64_x (pg, z2, z4, z8, z16, d->poly);
/* Finalize polynomial: z + z * z2 * P(z2). */
p = svmla_x (pg, z, svmul_x (pg, z, z2), p);
/* asin(|x|) = Q(|x|) , for |x| < 0.5
= pi/2 - 2 Q(|x|), for |x| >= 0.5. */
svfloat64_t y = svmad_m (a_ge_half, p, sv_f64 (-2.0), d->pi_over_2f);
/* Copy sign. */
return svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign));
}
|
