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
|
/*
* ====================================================
* 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 are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
This 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.
This 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 this library; if not, see
<http://www.gnu.org/licenses/>. */
/* __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)))
*/
#include <math.h>
#include <math_private.h>
static const long double
one = 1.0L,
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 */
long double
__kernel_tanl (long double x, long double y, int iy)
{
long double z, r, v, w, s;
int32_t ix, sign, hx, lx;
double xhi;
xhi = ldbl_high (x);
EXTRACT_WORDS (hx, lx, xhi);
ix = hx & 0x7fffffff;
if (ix < 0x3c600000) /* x < 2**-57 */
{
if ((int) x == 0) /* generate inexact */
{
if ((ix | lx | (iy + 1)) == 0)
return one / fabs (x);
else
return (iy == 1) ? x : -one / x;
}
}
if (ix >= 0x3fe59420) /* |x| >= 0.6743316650390625 */
{
if ((hx & 0x80000000) != 0)
{
x = -x;
y = -y;
sign = -1;
}
else
sign = 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 (ix >= 0x3fe59420)
{
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 */
long double u1, z1;
u1 = ldbl_high (w);
v = r - (u1 - x); /* u1+v = r+x */
z = -1.0 / w;
z1 = ldbl_high (z);
s = 1.0 + z1 * u1;
return z1 + z * (s + z1 * v);
}
}
|
