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
118
|
/* Single-precision vector (SVE) tan function
Copyright (C) 2023-2024 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"
static const struct data
{
float pio2_1, pio2_2, pio2_3, invpio2;
float c1, c3, c5;
float c0, c2, c4, range_val, shift;
} data = {
/* Coefficients generated using:
poly = fpminimax((tan(sqrt(x))-sqrt(x))/x^(3/2),
deg,
[|single ...|],
[a*a;b*b]);
optimize relative error
final prec : 23 bits
deg : 5
a : 0x1p-126 ^ 2
b : ((pi) / 0x1p2) ^ 2
dirty rel error: 0x1.f7c2e4p-25
dirty abs error: 0x1.f7c2ecp-25. */
.c0 = 0x1.55555p-2, .c1 = 0x1.11166p-3,
.c2 = 0x1.b88a78p-5, .c3 = 0x1.7b5756p-6,
.c4 = 0x1.4ef4cep-8, .c5 = 0x1.0e1e74p-7,
.pio2_1 = 0x1.921fb6p+0f, .pio2_2 = -0x1.777a5cp-25f,
.pio2_3 = -0x1.ee59dap-50f, .invpio2 = 0x1.45f306p-1f,
.range_val = 0x1p15f, .shift = 0x1.8p+23f
};
static svfloat32_t NOINLINE
special_case (svfloat32_t x, svfloat32_t y, svbool_t cmp)
{
return sv_call_f32 (tanf, x, y, cmp);
}
/* Fast implementation of SVE tanf.
Maximum error is 3.45 ULP:
SV_NAME_F1 (tan)(-0x1.e5f0cap+13) got 0x1.ff9856p-1
want 0x1.ff9850p-1. */
svfloat32_t SV_NAME_F1 (tan) (svfloat32_t x, const svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
/* Determine whether input is too large to perform fast regression. */
svbool_t cmp = svacge (pg, x, d->range_val);
svfloat32_t odd_coeffs = svld1rq (svptrue_b32 (), &d->c1);
svfloat32_t pi_vals = svld1rq (svptrue_b32 (), &d->pio2_1);
/* n = rint(x/(pi/2)). */
svfloat32_t q = svmla_lane (sv_f32 (d->shift), x, pi_vals, 3);
svfloat32_t n = svsub_x (pg, q, d->shift);
/* n is already a signed integer, simply convert it. */
svint32_t in = svcvt_s32_x (pg, n);
/* Determine if x lives in an interval, where |tan(x)| grows to infinity. */
svint32_t alt = svand_x (pg, in, 1);
svbool_t pred_alt = svcmpne (pg, alt, 0);
/* r = x - n * (pi/2) (range reduction into 0 .. pi/4). */
svfloat32_t r;
r = svmls_lane (x, n, pi_vals, 0);
r = svmls_lane (r, n, pi_vals, 1);
r = svmls_lane (r, n, pi_vals, 2);
/* If x lives in an interval, where |tan(x)|
- is finite, then use a polynomial approximation of the form
tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2).
- grows to infinity then use symmetries of tangent and the identity
tan(r) = cotan(pi/2 - r) to express tan(x) as 1/tan(-r). Finally, use
the same polynomial approximation of tan as above. */
/* Perform additional reduction if required. */
svfloat32_t z = svneg_m (r, pred_alt, r);
/* Evaluate polynomial approximation of tangent on [-pi/4, pi/4],
using Estrin on z^2. */
svfloat32_t z2 = svmul_x (pg, z, z);
svfloat32_t p01 = svmla_lane (sv_f32 (d->c0), z2, odd_coeffs, 0);
svfloat32_t p23 = svmla_lane (sv_f32 (d->c2), z2, odd_coeffs, 1);
svfloat32_t p45 = svmla_lane (sv_f32 (d->c4), z2, odd_coeffs, 2);
svfloat32_t z4 = svmul_x (pg, z2, z2);
svfloat32_t p = svmla_x (pg, p01, z4, p23);
svfloat32_t z8 = svmul_x (pg, z4, z4);
p = svmla_x (pg, p, z8, p45);
svfloat32_t y = svmla_x (pg, z, p, svmul_x (pg, z, z2));
/* Transform result back, if necessary. */
svfloat32_t inv_y = svdivr_x (pg, y, 1.0f);
/* No need to pass pg to specialcase here since cmp is a strict subset,
guaranteed by the cmpge above. */
if (__glibc_unlikely (svptest_any (pg, cmp)))
return special_case (x, svsel (pred_alt, inv_y, y), cmp);
return svsel (pred_alt, inv_y, y);
}
|
