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
|
/* @(#)k_tan.c 1.5 04/04/22 SMI */
/*
* ====================================================
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* INDENT OFF */
#ifndef lint
static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_tan.c,v 1.12 2005/11/02 14:01:45 bde Exp $";
#endif
/* __kernel_tan( x, y, k )
* kernel tan function on ~[-pi/4, pi/4] (except on -0), 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. Callers must return tan(-0) = -0 without calling here since our
* odd polynomial is not evaluated in a way that preserves -0.
* Callers may do the optimization tan(x) ~ x for tiny x.
* 3. tan(x) is approximated by a odd polynomial of degree 27 on
* [0,0.67434]
* 3 27
* tan(x) ~ x + T1*x + ... + T13*x
* where
*
* |tan(x) 2 4 26 | -59.2
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
* | x |
*
* 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
* 3 2 2 2 2
* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
* then
* 3 2
* tan(x+y) = x + (T1*x + (x *(r+y)+y))
*
* 4. For x in [0.67434,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 double xxx[] = {
3.33333333333334091986e-01, /* 3FD55555, 55555563 */
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
};
#define one xxx[13]
#define pio4 xxx[14]
#define pio4lo xxx[15]
#define T xxx
/* INDENT ON */
double
__kernel_tan(double x, double y, int iy) {
double z, r, v, w, s;
int32_t ix, hx;
GET_HIGH_WORD(hx,x);
ix = hx & 0x7fffffff; /* high word of |x| */
if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
if (hx < 0) {
x = -x;
y = -y;
}
z = pio4 - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x * x;
w = z * z;
/*
* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
w * T[11]))));
v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
w * T[12])))));
s = z * x;
r = y + z * (s * (r + v) + y);
r += T[0] * s;
w = x + r;
if (ix >= 0x3FE59428) {
v = (double) iy;
return (double) (1 - ((hx >> 30) & 2)) *
(v - 2.0 * (x - (w * w / (w + v) - r)));
}
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 */
double a, t;
z = w;
SET_LOW_WORD(z,0);
v = r - (z - x); /* z+v = r+x */
t = a = -1.0 / w; /* a = -1.0/w */
SET_LOW_WORD(t,0);
s = 1.0 + t * z;
return t + a * (s + t * v);
}
}
|
