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
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
|
/* s_tanl.c -- long double version of s_tan.c.
* Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
*/
/* @(#)s_tan.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include <config.h>
/* Specification. */
#include <math.h>
/* tanl(x)
* Return tangent function of x.
*
* kernel function:
* __kernel_tanl ... tangent function on [-pi/4,pi/4]
* __ieee754_rem_pio2l ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
#include "trigl.h"
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
Long double expansions contributed by
Stephen L. Moshier <moshier@na-net.ornl.gov>
*/
/* __kernel_tanl( x, y, k )
* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input k indicates whether tan (if k=1) or
* -1/tan (if k= -1) is returned.
*
* Algorithm
* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
* 2. if x < 2^-57, return x with inexact if x!=0.
* 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
* on [0,0.67433].
*
* Note: tan(x+y) = tan(x) + tan'(x)*y
* ~ tan(x) + (1+x*x)*y
* Therefore, for better accuracy in computing tan(x+y), let
* r = x^3 * R(x^2)
* then
* tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
*
* 4. For x in [0.67433,pi/4], let y = pi/4 - x, then
* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
static const long double
pio4hi = 7.8539816339744830961566084581987569936977E-1L,
pio4lo = 2.1679525325309452561992610065108379921906E-35L,
/* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
0 <= x <= 0.6743316650390625
Peak relative error 8.0e-36 */
TH = 3.333333333333333333333333333333333333333E-1L,
T0 = -1.813014711743583437742363284336855889393E7L,
T1 = 1.320767960008972224312740075083259247618E6L,
T2 = -2.626775478255838182468651821863299023956E4L,
T3 = 1.764573356488504935415411383687150199315E2L,
T4 = -3.333267763822178690794678978979803526092E-1L,
U0 = -1.359761033807687578306772463253710042010E8L,
U1 = 6.494370630656893175666729313065113194784E7L,
U2 = -4.180787672237927475505536849168729386782E6L,
U3 = 8.031643765106170040139966622980914621521E4L,
U4 = -5.323131271912475695157127875560667378597E2L;
/* 1.000000000000000000000000000000000000000E0 */
static long double
kernel_tanl (long double x, long double y, int iy)
{
long double z, r, v, w, s, u, u1;
int invert = 0, sign;
sign = 1;
if (x < 0)
{
x = -x;
y = -y;
sign = -1;
}
if (x < 0.000000000000000006938893903907228377647697925567626953125L) /* x < 2**-57 */
{
if ((int) x == 0)
{ /* generate inexact */
if (iy == -1 && x == 0.0)
return 1.0L / fabs (x);
else
return (iy == 1) ? x : -1.0L / x;
}
}
if (x >= 0.6743316650390625) /* |x| >= 0.6743316650390625 */
{
invert = 1;
z = pio4hi - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x * x;
r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
r = r / v;
s = z * x;
r = y + z * (s * r + y);
r += TH * s;
w = x + r;
if (invert)
{
v = (long double) iy;
w = (v - 2.0 * (x - (w * w / (w + v) - r)));
if (sign < 0)
w = -w;
return w;
}
if (iy == 1)
return w;
else
{ /* if allow error up to 2 ulp,
simply return -1.0/(x+r) here */
/* compute -1.0/(x+r) accurately */
u1 = (double) w;
v = r - (u1 - x);
z = -1.0 / w;
u = (double) z;
s = 1.0 + u * u1;
return u + z * (s + u * v);
}
}
long double
tanl (long double x)
{
long double y[2], z = 0.0L;
int n;
/* tanl(NaN) is NaN */
if (isnanl (x))
return x;
/* |x| ~< pi/4 */
if (x >= -0.7853981633974483096156608458198757210492 &&
x <= 0.7853981633974483096156608458198757210492)
return kernel_tanl (x, z, 1);
/* tanl(Inf) is NaN, tanl(0) is 0 */
else if (x + x == x)
return x - x; /* NaN */
/* argument reduction needed */
else
{
n = ieee754_rem_pio2l (x, y);
/* 1 -- n even, -1 -- n odd */
return kernel_tanl (y[0], y[1], 1 - ((n & 1) << 1));
}
}
#if 0
int
main (void)
{
printf ("%.16Lg\n", tanl (0.7853981633974483096156608458198757210492));
printf ("%.16Lg\n", tanl (-0.7853981633974483096156608458198757210492));
printf ("%.16Lg\n", tanl (0.7853981633974483096156608458198757210492 *3));
printf ("%.16Lg\n", tanl (-0.7853981633974483096156608458198757210492 *31));
printf ("%.16Lg\n", tanl (0.7853981633974483096156608458198757210492 / 2));
printf ("%.16Lg\n", tanl (0.7853981633974483096156608458198757210492 * 3/2));
printf ("%.16Lg\n", tanl (0.7853981633974483096156608458198757210492 * 5/2));
}
#endif
|
