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
|
/* origin: FreeBSD /usr/src/lib/msun/ld80/k_tanl.c */
/* origin: FreeBSD /usr/src/lib/msun/ld128/k_tanl.c */
/*
* ====================================================
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include "libm.h"
#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
#if LDBL_MANT_DIG == 64
/*
* ld80 version of __tan.c. See __tan.c for most comments.
*/
/*
* Domain [-0.67434, 0.67434], range ~[-2.25e-22, 1.921e-22]
* |tan(x)/x - t(x)| < 2**-71.9
*
* See __cosl.c for more details about the polynomial.
*/
static const long double
T3 = 0.333333333333333333180L, /* 0xaaaaaaaaaaaaaaa5.0p-65 */
T5 = 0.133333333333333372290L, /* 0x88888888888893c3.0p-66 */
T7 = 0.0539682539682504975744L, /* 0xdd0dd0dd0dc13ba2.0p-68 */
pio4 = 0.785398163397448309628L, /* 0xc90fdaa22168c235.0p-64 */
pio4lo = -1.25413940316708300586e-20L; /* -0xece675d1fc8f8cbb.0p-130 */
static const double
T9 = 0.021869488536312216, /* 0x1664f4882cc1c2.0p-58 */
T11 = 0.0088632355256619590, /* 0x1226e355c17612.0p-59 */
T13 = 0.0035921281113786528, /* 0x1d6d3d185d7ff8.0p-61 */
T15 = 0.0014558334756312418, /* 0x17da354aa3f96b.0p-62 */
T17 = 0.00059003538700862256, /* 0x13559358685b83.0p-63 */
T19 = 0.00023907843576635544, /* 0x1f56242026b5be.0p-65 */
T21 = 0.000097154625656538905, /* 0x1977efc26806f4.0p-66 */
T23 = 0.000038440165747303162, /* 0x14275a09b3ceac.0p-67 */
T25 = 0.000018082171885432524, /* 0x12f5e563e5487e.0p-68 */
T27 = 0.0000024196006108814377, /* 0x144c0d80cc6896.0p-71 */
T29 = 0.0000078293456938132840, /* 0x106b59141a6cb3.0p-69 */
T31 = -0.0000032609076735050182, /* -0x1b5abef3ba4b59.0p-71 */
T33 = 0.0000023261313142559411; /* 0x13835436c0c87f.0p-71 */
#define RPOLY(w) (T5 + w * (T9 + w * (T13 + w * (T17 + w * (T21 + \
w * (T25 + w * (T29 + w * T33)))))))
#define VPOLY(w) (T7 + w * (T11 + w * (T15 + w * (T19 + w * (T23 + \
w * (T27 + w * T31))))))
#elif LDBL_MANT_DIG == 113
/*
* ld128 version of __tan.c. See __tan.c for most comments.
*/
/*
* Domain [-0.67434, 0.67434], range ~[-3.37e-36, 1.982e-37]
* |tan(x)/x - t(x)| < 2**-117.8 (XXX should be ~1e-37)
*
* See __cosl.c for more details about the polynomial.
*/
static const long double
T3 = 0x1.5555555555555555555555555553p-2L,
T5 = 0x1.1111111111111111111111111eb5p-3L,
T7 = 0x1.ba1ba1ba1ba1ba1ba1ba1b694cd6p-5L,
T9 = 0x1.664f4882c10f9f32d6bbe09d8bcdp-6L,
T11 = 0x1.226e355e6c23c8f5b4f5762322eep-7L,
T13 = 0x1.d6d3d0e157ddfb5fed8e84e27b37p-9L,
T15 = 0x1.7da36452b75e2b5fce9ee7c2c92ep-10L,
T17 = 0x1.355824803674477dfcf726649efep-11L,
T19 = 0x1.f57d7734d1656e0aceb716f614c2p-13L,
T21 = 0x1.967e18afcb180ed942dfdc518d6cp-14L,
T23 = 0x1.497d8eea21e95bc7e2aa79b9f2cdp-15L,
T25 = 0x1.0b132d39f055c81be49eff7afd50p-16L,
T27 = 0x1.b0f72d33eff7bfa2fbc1059d90b6p-18L,
T29 = 0x1.5ef2daf21d1113df38d0fbc00267p-19L,
T31 = 0x1.1c77d6eac0234988cdaa04c96626p-20L,
T33 = 0x1.cd2a5a292b180e0bdd701057dfe3p-22L,
T35 = 0x1.75c7357d0298c01a31d0a6f7d518p-23L,
T37 = 0x1.2f3190f4718a9a520f98f50081fcp-24L,
pio4 = 0x1.921fb54442d18469898cc51701b8p-1L,
pio4lo = 0x1.cd129024e088a67cc74020bbea60p-116L;
static const double
T39 = 0.000000028443389121318352, /* 0x1e8a7592977938.0p-78 */
T41 = 0.000000011981013102001973, /* 0x19baa1b1223219.0p-79 */
T43 = 0.0000000038303578044958070, /* 0x107385dfb24529.0p-80 */
T45 = 0.0000000034664378216909893, /* 0x1dc6c702a05262.0p-81 */
T47 = -0.0000000015090641701997785, /* -0x19ecef3569ebb6.0p-82 */
T49 = 0.0000000029449552300483952, /* 0x194c0668da786a.0p-81 */
T51 = -0.0000000022006995706097711, /* -0x12e763b8845268.0p-81 */
T53 = 0.0000000015468200913196612, /* 0x1a92fc98c29554.0p-82 */
T55 = -0.00000000061311613386849674, /* -0x151106cbc779a9.0p-83 */
T57 = 1.4912469681508012e-10; /* 0x147edbdba6f43a.0p-85 */
#define RPOLY(w) (T5 + w * (T9 + w * (T13 + w * (T17 + w * (T21 + \
w * (T25 + w * (T29 + w * (T33 + w * (T37 + w * (T41 + \
w * (T45 + w * (T49 + w * (T53 + w * T57)))))))))))))
#define VPOLY(w) (T7 + w * (T11 + w * (T15 + w * (T19 + w * (T23 + \
w * (T27 + w * (T31 + w * (T35 + w * (T39 + w * (T43 + \
w * (T47 + w * (T51 + w * T55))))))))))))
#endif
long double __tanl(long double x, long double y, int odd) {
long double z, r, v, w, s, a, t;
int big, sign;
big = fabsl(x) >= 0.67434;
if (big) {
sign = 0;
if (x < 0) {
sign = 1;
x = -x;
y = -y;
}
x = (pio4 - x) + (pio4lo - y);
y = 0.0;
}
z = x * x;
w = z * z;
r = RPOLY(w);
v = z * VPOLY(w);
s = z * x;
r = y + z * (s * (r + v) + y) + T3 * s;
w = x + r;
if (big) {
s = 1 - 2*odd;
v = s - 2.0 * (x + (r - w * w / (w + s)));
return sign ? -v : v;
}
if (!odd)
return w;
/*
* if allow error up to 2 ulp, simply return
* -1.0 / (x+r) here
*/
/* compute -1.0 / (x+r) accurately */
z = w;
z = z + 0x1p32 - 0x1p32;
v = r - (z - x); /* z+v = r+x */
t = a = -1.0 / w; /* a = -1.0/w */
t = t + 0x1p32 - 0x1p32;
s = 1.0 + t * z;
return t + a * (s + t * v);
}
#endif
|
