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
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
|
/* Double-Precision vector (Advanced SIMD) inverse tanpi function
Copyright (C) 2025 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 "v_math.h"
static const struct data
{
double c2, c4, c6, c8, c10, c12, c14, c16, c18, c20;
float64x2_t c0, c1, c3, c5, c7, c9, c11, c13, c15, c17, c19;
} data = {
/* Coefficients of polynomial P such that atanpi(x)~x*P(x^2) on
[2^-1022, 1.0]. */
.c0 = V2 (0x1.45f306dc9c883p-2), .c1 = V2 (-0x1.b2995e7b7ba4ap-4),
.c2 = 0x1.04c26be3d2c1p-4, .c3 = V2 (-0x1.7483759c17ea1p-5),
.c4 = 0x1.21bb95c315d57p-5, .c5 = V2 (-0x1.da1bdc3d453f3p-6),
.c6 = 0x1.912d20459b4bfp-6, .c7 = V2 (-0x1.5bbd4545cad1fp-6),
.c8 = 0x1.331b83bec30a1p-6, .c9 = V2 (-0x1.13d6457f44de3p-6),
.c10 = 0x1.f8e802974db94p-7, .c11 = V2 (-0x1.d7e173ab04a1ap-7),
.c12 = 0x1.bdfa47d6a4f28p-7, .c13 = V2 (-0x1.9ba78f3232ceep-7),
.c14 = 0x1.5e6044590ab4fp-7, .c15 = V2 (-0x1.01ccfdeb9f77fp-7),
.c16 = 0x1.345cf0d4eb1c1p-8, .c17 = V2 (-0x1.19e5f00f67e3ap-9),
.c18 = 0x1.6d3035ac7625bp-11, .c19 = V2 (-0x1.286bb9ae4ed79p-13),
.c20 = 0x1.c37ec36da0e1ap-17,
};
#define SignMask v_u64 (0x8000000000000000)
/* Fast implementation of vector atanpi.
atanpi(x) ~ shift + z * P(z^2) with reduction to [0,1] using
z=1/x and shift = +-1/2. Maximum observed error is 2.76 ulps:
_ZGVnN2v_atanpi(0x1.fa2d6912cd64fp-1) got 0x1.fc45a51bd497fp-3
want 0x1.fc45a51bd497cp-3. */
float64x2_t VPCS_ATTR V_NAME_D1 (atanpi) (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
uint64x2_t ix = vreinterpretq_u64_f64 (x);
uint64x2_t sign = vandq_u64 (ix, SignMask);
/* Argument Reduction:
y := arctanpi(x) for |x| < 1
y := arctanpi(-1/x) + 1/2 for x > 1
y := arctanpi(-1/x) - 1/2 for x < -1
Hence, use z=-1/a if |x|>=|-1|, otherwise z=a. */
uint64x2_t red = vcagtq_f64 (x, v_f64 (-1.0));
float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (-1.0), x), x);
/* Shift is calculated as +1/2 or 0, depending on the argument case. */
float64x2_t shift = vreinterpretq_f64_u64 (
vandq_u64 (red, vreinterpretq_u64_f64 (v_f64 (0.5))));
/* Reinsert sign bit from argument into the shift value. */
shift = vreinterpretq_f64_u64 (
veorq_u64 (vreinterpretq_u64_f64 (shift), sign));
/* Calculate polynomial approximation P(z^2) with deg(P)=19. */
float64x2_t z2 = vmulq_f64 (z, z);
float64x2_t z4 = vmulq_f64 (z2, z2);
float64x2_t z8 = vmulq_f64 (z4, z4);
float64x2_t z16 = vmulq_f64 (z8, z8);
float64x2_t c24 = vld1q_f64 (&d->c2);
float64x2_t c68 = vld1q_f64 (&d->c6);
/* Order-7 Estrin. */
float64x2_t p12 = vfmaq_laneq_f64 (d->c1, z2, c24, 0);
float64x2_t p34 = vfmaq_laneq_f64 (d->c3, z2, c24, 1);
float64x2_t p56 = vfmaq_laneq_f64 (d->c5, z2, c68, 0);
float64x2_t p78 = vfmaq_laneq_f64 (d->c7, z2, c68, 1);
float64x2_t p14 = vfmaq_f64 (p12, z4, p34);
float64x2_t p58 = vfmaq_f64 (p56, z4, p78);
float64x2_t p18 = vfmaq_f64 (p14, z8, p58);
/* Order-11 Estrin. */
float64x2_t c1012 = vld1q_f64 (&d->c10);
float64x2_t c1416 = vld1q_f64 (&d->c14);
float64x2_t c1820 = vld1q_f64 (&d->c18);
float64x2_t p910 = vfmaq_laneq_f64 (d->c9, z2, c1012, 0);
float64x2_t p1112 = vfmaq_laneq_f64 (d->c11, z2, c1012, 1);
float64x2_t p912 = vfmaq_f64 (p910, z4, p1112);
float64x2_t p1314 = vfmaq_laneq_f64 (d->c13, z2, c1416, 0);
float64x2_t p1516 = vfmaq_laneq_f64 (d->c15, z2, c1416, 1);
float64x2_t p1316 = vfmaq_f64 (p1314, z4, p1516);
float64x2_t p1718 = vfmaq_laneq_f64 (d->c17, z2, c1820, 0);
float64x2_t p1920 = vfmaq_laneq_f64 (d->c19, z2, c1820, 1);
float64x2_t p1720 = vfmaq_f64 (p1718, z4, p1920);
float64x2_t p916 = vfmaq_f64 (p912, z8, p1316);
float64x2_t p920 = vfmaq_f64 (p916, z16, p1720);
float64x2_t y = vfmaq_f64 (p18, p920, z16);
y = vfmaq_f64 (d->c0, z2, y);
/* y = shift + z * p(z^2). */
return vfmaq_f64 (shift, z, y);
}
|
