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
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
|
/* Quad-precision floating point cosine on <-pi/4,pi/4>.
Copyright (C) 1999-2020 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Jakub Jelinek <jj@ultra.linux.cz>
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 long double c[] = {
#define ONE c[0]
1.00000000000000000000000000000000000E+00L, /* 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]
-5.00000000000000000000000000000000000E-01L, /* bffe0000000000000000000000000000 */
4.16666666666666666666666666556146073E-02L, /* 3ffa5555555555555555555555395023 */
-1.38888888888888888888309442601939728E-03L, /* bff56c16c16c16c16c16a566e42c0375 */
2.48015873015862382987049502531095061E-05L, /* 3fefa01a01a019ee02dcf7da2d6d5444 */
-2.75573112601362126593516899592158083E-07L, /* 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]
-4.99999999999999999999999999999999759E-01L, /* bffdfffffffffffffffffffffffffffb */
4.16666666666666666666666666651287795E-02L, /* 3ffa5555555555555555555555516f30 */
-1.38888888888888888888888742314300284E-03L, /* bff56c16c16c16c16c16c16a463dfd0d */
2.48015873015873015867694002851118210E-05L, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
-2.75573192239858811636614709689300351E-07L, /* bfe927e4fb7789f5aa8142a22044b51f */
2.08767569877762248667431926878073669E-09L, /* 3fe21eed8eff881d1e9262d7adff4373 */
-1.14707451049343817400420280514614892E-11L, /* bfda9397496922a9601ed3d4ca48944b */
4.77810092804389587579843296923533297E-14L, /* 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]
-1.66666666666666666666666666666666659E-01L, /* bffc5555555555555555555555555555 */
8.33333333333333333333333333146298442E-03L, /* 3ff81111111111111111111110fe195d */
-1.98412698412698412697726277416810661E-04L, /* bff2a01a01a01a01a019e7121e080d88 */
2.75573192239848624174178393552189149E-06L, /* 3fec71de3a556c640c6aaa51aa02ab41 */
-2.50521016467996193495359189395805639E-08L, /* bfe5ae644ee90c47dc71839de75b2787 */
};
#define SINCOSL_COS_HI 0
#define SINCOSL_COS_LO 1
#define SINCOSL_SIN_HI 2
#define SINCOSL_SIN_LO 3
extern const long double __sincosl_table[];
long double
__kernel_cosl(long double x, long double y)
{
long double h, l, z, sin_l, cos_l_m1;
int64_t ix;
uint32_t tix, hix, index;
double xhi, hhi;
xhi = ldbl_high (x);
EXTRACT_WORDS64 (ix, xhi);
tix = ((uint64_t)ix) >> 32;
tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
if (tix < 0x3fc30000) /* |x| < 0.1484375 */
{
/* Argument is small enough to approximate it by a Chebyshev
polynomial of degree 16. */
if (tix < 0x3c600000) /* |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). */
int six = tix;
tix = ((six - 0x3ff00000) >> 4) + 0x3fff0000;
index = 0x3ffe - (tix >> 16);
hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
x = fabsl (x);
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;
}
hix = (hix << 4) & 0x3fffffff;
/*
The following should work for double but generates the wrong index.
For now the code above converts double to ieee extended to compute
the index back to double for the h value.
index = 0x3fe - (tix >> 20);
hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
if (signbit (x))
{
x = -x;
y = -y;
}
switch (index)
{
case 0: index = ((45 << 14) + hix - 0x3fe00000) >> 12; break;
case 1: index = ((13 << 15) + hix - 0x3fd00000) >> 13; break;
default:
case 2: index = (hix - 0x3fc30000) >> 14; break;
}
*/
INSERT_WORDS64 (hhi, ((uint64_t)hix) << 32);
h = hhi;
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));
}
}
|
