1 /* @(#)k_tan.c 1.5 04/04/22 SMI */
2
3 /*
4 * ====================================================
5 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
6 *
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 /* INDENT OFF */
14 #include <sys/cdefs.h>
15 __FBSDID("$FreeBSD$");
16
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.
22 *
23 * Algorithm
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
29 * [0,0.67434]
30 * 3 27
31 * tan(x) ~ x + T1*x + ... + T13*x
32 * where
33 *
34 * |tan(x) 2 4 26 | -59.2
35 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
36 * | x |
37 *
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
41 * 3 2 2 2 2
42 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
43 * then
44 * 3 2
45 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
46 *
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)))
50 */
51
52 #include "math.h"
53 #include "math_private.h"
54 static const double xxx[] = {
55 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
56 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
57 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
58 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
59 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
60 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
61 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
62 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
63 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
64 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
65 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
66 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
67 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
68 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
69 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
70 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
71 };
72 #define one xxx[13]
73 #define pio4 xxx[14]
74 #define pio4lo xxx[15]
75 #define T xxx
76 /* INDENT ON */
77
78 double
__kernel_tan(double x,double y,int iy)79 __kernel_tan(double x, double y, int iy) {
80 double z, r, v, w, s;
81 int32_t ix, hx;
82
83 GET_HIGH_WORD(hx,x);
84 ix = hx & 0x7fffffff; /* high word of |x| */
85 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
86 if (hx < 0) {
87 x = -x;
88 y = -y;
89 }
90 z = pio4 - x;
91 w = pio4lo - y;
92 x = z + w;
93 y = 0.0;
94 }
95 z = x * x;
96 w = z * z;
97 /*
98 * Break x^5*(T[1]+x^2*T[2]+...) into
99 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
100 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
101 */
102 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
103 w * T[11]))));
104 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
105 w * T[12])))));
106 s = z * x;
107 r = y + z * (s * (r + v) + y);
108 r += T[0] * s;
109 w = x + r;
110 if (ix >= 0x3FE59428) {
111 v = (double) iy;
112 return (double) (1 - ((hx >> 30) & 2)) *
113 (v - 2.0 * (x - (w * w / (w + v) - r)));
114 }
115 if (iy == 1)
116 return w;
117 else {
118 /*
119 * if allow error up to 2 ulp, simply return
120 * -1.0 / (x+r) here
121 */
122 /* compute -1.0 / (x+r) accurately */
123 double a, t;
124 z = w;
125 SET_LOW_WORD(z,0);
126 v = r - (z - x); /* z+v = r+x */
127 t = a = -1.0 / w; /* a = -1.0/w */
128 SET_LOW_WORD(t,0);
129 s = 1.0 + t * z;
130 return t + a * (s + t * v);
131 }
132 }
133