1 /* @(#)k_tan.c 1.5 04/04/22 SMI */
4 * ====================================================
5 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
14 //#include <sys/cdefs.h>
15 //__FBSDID("$FreeBSD$");
17 /* __kernel_tan( x, y, k )
18 * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
19 * Input x is assumed to be bounded by ~pi/4 in magnitude.
20 * Input y is the tail of x.
21 * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
24 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
25 * 2. Callers must return tan(-0) = -0 without calling here since our
26 * odd polynomial is not evaluated in a way that preserves -0.
27 * Callers may do the optimization tan(x) ~ x for tiny x.
28 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
31 * tan(x) ~ x + T1*x + ... + T13*x
34 * |tan(x) 2 4 26 | -59.2
35 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
38 * Note: tan(x+y) = tan(x) + tan'(x)*y
39 * ~ tan(x) + (1+x*x)*y
40 * Therefore, for better accuracy in computing tan(x+y), let
42 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
45 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
47 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
48 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
49 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
52 #include "math_private.h"
53 static const double xxx
[] = {
54 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
55 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
56 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
57 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
58 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
59 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
60 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
61 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
62 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
63 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
64 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
65 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
66 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
67 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
68 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
69 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
73 #define pio4lo xxx[15]
78 __kernel_tan(double x
, double y
, int iy
) {
83 ix
= hx
& 0x7fffffff; /* high word of |x| */
84 if (ix
>= 0x3FE59428) { /* |x| >= 0.6744 */
97 * Break x^5*(T[1]+x^2*T[2]+...) into
98 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
99 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
101 r
= T
[1] + w
* (T
[3] + w
* (T
[5] + w
* (T
[7] + w
* (T
[9] +
103 v
= z
* (T
[2] + w
* (T
[4] + w
* (T
[6] + w
* (T
[8] + w
* (T
[10] +
106 r
= y
+ z
* (s
* (r
+ v
) + y
);
109 if (ix
>= 0x3FE59428) {
111 return (double) (1 - ((hx
>> 30) & 2)) *
112 (v
- 2.0 * (x
- (w
* w
/ (w
+ v
) - r
)));
118 * if allow error up to 2 ulp, simply return
121 /* compute -1.0 / (x+r) accurately */
125 v
= r
- (z
- x
); /* z+v = r+x */
126 t
= a
= -1.0 / w
; /* a = -1.0/w */
129 return t
+ a
* (s
+ t
* v
);