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
119
120
121
122
123
124
125
126
127
128
129
130
|
/* Quad-precision floating point cosine on <-pi/4,pi/4>.
Copyright (C) 1999-2021 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 <math.h>
#include <math_private.h>
static const _Float128 c[] = {
#define ONE c[0]
L(1.00000000000000000000000000000000000E+00), /* 3fff0000000000000000000000000000 */
/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 )
x in <0,1/256> */
#define SCOS1 c[1]
#define SCOS2 c[2]
#define SCOS3 c[3]
#define SCOS4 c[4]
#define SCOS5 c[5]
L(-5.00000000000000000000000000000000000E-01), /* bffe0000000000000000000000000000 */
L(4.16666666666666666666666666556146073E-02), /* 3ffa5555555555555555555555395023 */
L(-1.38888888888888888888309442601939728E-03), /* bff56c16c16c16c16c16a566e42c0375 */
L(2.48015873015862382987049502531095061E-05), /* 3fefa01a01a019ee02dcf7da2d6d5444 */
L(-2.75573112601362126593516899592158083E-07), /* bfe927e4f5dce637cb0b54908754bde0 */
/* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 )
x in <0,0.1484375> */
#define COS1 c[6]
#define COS2 c[7]
#define COS3 c[8]
#define COS4 c[9]
#define COS5 c[10]
#define COS6 c[11]
#define COS7 c[12]
#define COS8 c[13]
L(-4.99999999999999999999999999999999759E-01), /* bffdfffffffffffffffffffffffffffb */
L(4.16666666666666666666666666651287795E-02), /* 3ffa5555555555555555555555516f30 */
L(-1.38888888888888888888888742314300284E-03), /* bff56c16c16c16c16c16c16a463dfd0d */
L(2.48015873015873015867694002851118210E-05), /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
L(-2.75573192239858811636614709689300351E-07), /* bfe927e4fb7789f5aa8142a22044b51f */
L(2.08767569877762248667431926878073669E-09), /* 3fe21eed8eff881d1e9262d7adff4373 */
L(-1.14707451049343817400420280514614892E-11), /* bfda9397496922a9601ed3d4ca48944b */
L(4.77810092804389587579843296923533297E-14), /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */
/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 )
x in <0,1/256> */
#define SSIN1 c[14]
#define SSIN2 c[15]
#define SSIN3 c[16]
#define SSIN4 c[17]
#define SSIN5 c[18]
L(-1.66666666666666666666666666666666659E-01), /* bffc5555555555555555555555555555 */
L(8.33333333333333333333333333146298442E-03), /* 3ff81111111111111111111110fe195d */
L(-1.98412698412698412697726277416810661E-04), /* bff2a01a01a01a01a019e7121e080d88 */
L(2.75573192239848624174178393552189149E-06), /* 3fec71de3a556c640c6aaa51aa02ab41 */
L(-2.50521016467996193495359189395805639E-08), /* bfe5ae644ee90c47dc71839de75b2787 */
};
#define SINCOSL_COS_HI 0
#define SINCOSL_COS_LO 1
#define SINCOSL_SIN_HI 2
#define SINCOSL_SIN_LO 3
extern const _Float128 __sincosl_table[];
_Float128
__kernel_cosl(_Float128 x, _Float128 y)
{
_Float128 h, l, z, sin_l, cos_l_m1;
int64_t ix;
uint32_t tix, hix, index;
GET_LDOUBLE_MSW64 (ix, x);
tix = ((uint64_t)ix) >> 32;
tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
if (tix < 0x3ffc3000) /* |x| < 0.1484375 */
{
/* Argument is small enough to approximate it by a Chebyshev
polynomial of degree 16. */
if (tix < 0x3fc60000) /* |x| < 2^-57 */
if (!((int)x)) return ONE; /* generate inexact */
z = x * x;
return ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
}
else
{
/* So that we don't have to use too large polynomial, we find
l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83
possible values for h. We look up cosl(h) and sinl(h) in
pre-computed tables, compute cosl(l) and sinl(l) using a
Chebyshev polynomial of degree 10(11) and compute
cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */
index = 0x3ffe - (tix >> 16);
hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
if (signbit (x))
{
x = -x;
y = -y;
}
switch (index)
{
case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break;
case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break;
default:
case 2: index = (hix - 0x3ffc3000) >> 10; break;
}
SET_LDOUBLE_WORDS64(h, ((uint64_t)hix) << 32, 0);
l = y - (h - x);
z = l * l;
sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
return __sincosl_table [index + SINCOSL_COS_HI]
+ (__sincosl_table [index + SINCOSL_COS_LO]
- (__sincosl_table [index + SINCOSL_SIN_HI] * sin_l
- __sincosl_table [index + SINCOSL_COS_HI] * cos_l_m1));
}
}
|
