1*2fe8fb19SBen Gras /* @(#)k_tan.c 5.1 93/09/24 */
2*2fe8fb19SBen Gras /*
3*2fe8fb19SBen Gras * ====================================================
4*2fe8fb19SBen Gras * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*2fe8fb19SBen Gras *
6*2fe8fb19SBen Gras * Developed at SunPro, a Sun Microsystems, Inc. business.
7*2fe8fb19SBen Gras * Permission to use, copy, modify, and distribute this
8*2fe8fb19SBen Gras * software is freely granted, provided that this notice
9*2fe8fb19SBen Gras * is preserved.
10*2fe8fb19SBen Gras * ====================================================
11*2fe8fb19SBen Gras */
12*2fe8fb19SBen Gras
13*2fe8fb19SBen Gras #include <sys/cdefs.h>
14*2fe8fb19SBen Gras #if defined(LIBM_SCCS) && !defined(lint)
15*2fe8fb19SBen Gras __RCSID("$NetBSD: k_tan.c,v 1.12 2004/07/22 18:24:09 drochner Exp $");
16*2fe8fb19SBen Gras #endif
17*2fe8fb19SBen Gras
18*2fe8fb19SBen Gras /* __kernel_tan( x, y, k )
19*2fe8fb19SBen Gras * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
20*2fe8fb19SBen Gras * Input x is assumed to be bounded by ~pi/4 in magnitude.
21*2fe8fb19SBen Gras * Input y is the tail of x.
22*2fe8fb19SBen Gras * Input k indicates whether tan (if k=1) or
23*2fe8fb19SBen Gras * -1/tan (if k= -1) is returned.
24*2fe8fb19SBen Gras *
25*2fe8fb19SBen Gras * Algorithm
26*2fe8fb19SBen Gras * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
27*2fe8fb19SBen Gras * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
28*2fe8fb19SBen Gras * 3. tan(x) is approximated by a odd polynomial of degree 27 on
29*2fe8fb19SBen Gras * [0,0.67434]
30*2fe8fb19SBen Gras * 3 27
31*2fe8fb19SBen Gras * tan(x) ~ x + T1*x + ... + T13*x
32*2fe8fb19SBen Gras * where
33*2fe8fb19SBen Gras *
34*2fe8fb19SBen Gras * |tan(x) 2 4 26 | -59.2
35*2fe8fb19SBen Gras * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
36*2fe8fb19SBen Gras * | x |
37*2fe8fb19SBen Gras *
38*2fe8fb19SBen Gras * Note: tan(x+y) = tan(x) + tan'(x)*y
39*2fe8fb19SBen Gras * ~ tan(x) + (1+x*x)*y
40*2fe8fb19SBen Gras * Therefore, for better accuracy in computing tan(x+y), let
41*2fe8fb19SBen Gras * 3 2 2 2 2
42*2fe8fb19SBen Gras * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
43*2fe8fb19SBen Gras * then
44*2fe8fb19SBen Gras * 3 2
45*2fe8fb19SBen Gras * tan(x+y) = x + (T1*x + (x *(r+y)+y))
46*2fe8fb19SBen Gras *
47*2fe8fb19SBen Gras * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
48*2fe8fb19SBen Gras * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
49*2fe8fb19SBen Gras * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
50*2fe8fb19SBen Gras */
51*2fe8fb19SBen Gras
52*2fe8fb19SBen Gras #include "math.h"
53*2fe8fb19SBen Gras #include "math_private.h"
54*2fe8fb19SBen Gras
55*2fe8fb19SBen Gras static const double xxx[] = {
56*2fe8fb19SBen Gras 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
57*2fe8fb19SBen Gras 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
58*2fe8fb19SBen Gras 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
59*2fe8fb19SBen Gras 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
60*2fe8fb19SBen Gras 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
61*2fe8fb19SBen Gras 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
62*2fe8fb19SBen Gras 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
63*2fe8fb19SBen Gras 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
64*2fe8fb19SBen Gras 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
65*2fe8fb19SBen Gras 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
66*2fe8fb19SBen Gras 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
67*2fe8fb19SBen Gras -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
68*2fe8fb19SBen Gras 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
69*2fe8fb19SBen Gras /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
70*2fe8fb19SBen Gras /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
71*2fe8fb19SBen Gras /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
72*2fe8fb19SBen Gras };
73*2fe8fb19SBen Gras #define one xxx[13]
74*2fe8fb19SBen Gras #define pio4 xxx[14]
75*2fe8fb19SBen Gras #define pio4lo xxx[15]
76*2fe8fb19SBen Gras #define T xxx
77*2fe8fb19SBen Gras
78*2fe8fb19SBen Gras double
__kernel_tan(double x,double y,int iy)79*2fe8fb19SBen Gras __kernel_tan(double x, double y, int iy)
80*2fe8fb19SBen Gras {
81*2fe8fb19SBen Gras double z, r, v, w, s;
82*2fe8fb19SBen Gras int32_t ix, hx;
83*2fe8fb19SBen Gras
84*2fe8fb19SBen Gras GET_HIGH_WORD(hx, x); /* high word of x */
85*2fe8fb19SBen Gras ix = hx & 0x7fffffff; /* high word of |x| */
86*2fe8fb19SBen Gras if (ix < 0x3e300000) { /* x < 2**-28 */
87*2fe8fb19SBen Gras if ((int) x == 0) { /* generate inexact */
88*2fe8fb19SBen Gras u_int32_t low;
89*2fe8fb19SBen Gras GET_LOW_WORD(low, x);
90*2fe8fb19SBen Gras if(((ix | low) | (iy + 1)) == 0)
91*2fe8fb19SBen Gras return one / fabs(x);
92*2fe8fb19SBen Gras else {
93*2fe8fb19SBen Gras if (iy == 1)
94*2fe8fb19SBen Gras return x;
95*2fe8fb19SBen Gras else { /* compute -1 / (x+y) carefully */
96*2fe8fb19SBen Gras double a, t;
97*2fe8fb19SBen Gras
98*2fe8fb19SBen Gras z = w = x + y;
99*2fe8fb19SBen Gras SET_LOW_WORD(z, 0);
100*2fe8fb19SBen Gras v = y - (z - x);
101*2fe8fb19SBen Gras t = a = -one / w;
102*2fe8fb19SBen Gras SET_LOW_WORD(t, 0);
103*2fe8fb19SBen Gras s = one + t * z;
104*2fe8fb19SBen Gras return t + a * (s + t * v);
105*2fe8fb19SBen Gras }
106*2fe8fb19SBen Gras }
107*2fe8fb19SBen Gras }
108*2fe8fb19SBen Gras }
109*2fe8fb19SBen Gras if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
110*2fe8fb19SBen Gras if (hx < 0) {
111*2fe8fb19SBen Gras x = -x;
112*2fe8fb19SBen Gras y = -y;
113*2fe8fb19SBen Gras }
114*2fe8fb19SBen Gras z = pio4 - x;
115*2fe8fb19SBen Gras w = pio4lo - y;
116*2fe8fb19SBen Gras x = z + w;
117*2fe8fb19SBen Gras y = 0.0;
118*2fe8fb19SBen Gras }
119*2fe8fb19SBen Gras z = x * x;
120*2fe8fb19SBen Gras w = z * z;
121*2fe8fb19SBen Gras /*
122*2fe8fb19SBen Gras * Break x^5*(T[1]+x^2*T[2]+...) into
123*2fe8fb19SBen Gras * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
124*2fe8fb19SBen Gras * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
125*2fe8fb19SBen Gras */
126*2fe8fb19SBen Gras r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
127*2fe8fb19SBen Gras w * T[11]))));
128*2fe8fb19SBen Gras v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
129*2fe8fb19SBen Gras w * T[12])))));
130*2fe8fb19SBen Gras s = z * x;
131*2fe8fb19SBen Gras r = y + z * (s * (r + v) + y);
132*2fe8fb19SBen Gras r += T[0] * s;
133*2fe8fb19SBen Gras w = x + r;
134*2fe8fb19SBen Gras if (ix >= 0x3FE59428) {
135*2fe8fb19SBen Gras v = (double) iy;
136*2fe8fb19SBen Gras return (double) (1 - ((hx >> 30) & 2)) *
137*2fe8fb19SBen Gras (v - 2.0 * (x - (w * w / (w + v) - r)));
138*2fe8fb19SBen Gras }
139*2fe8fb19SBen Gras if (iy == 1)
140*2fe8fb19SBen Gras return w;
141*2fe8fb19SBen Gras else {
142*2fe8fb19SBen Gras /*
143*2fe8fb19SBen Gras * if allow error up to 2 ulp, simply return
144*2fe8fb19SBen Gras * -1.0 / (x+r) here
145*2fe8fb19SBen Gras */
146*2fe8fb19SBen Gras /* compute -1.0 / (x+r) accurately */
147*2fe8fb19SBen Gras double a, t;
148*2fe8fb19SBen Gras z = w;
149*2fe8fb19SBen Gras SET_LOW_WORD(z, 0);
150*2fe8fb19SBen Gras v = r - (z - x); /* z+v = r+x */
151*2fe8fb19SBen Gras t = a = -1.0 / w; /* a = -1.0/w */
152*2fe8fb19SBen Gras SET_LOW_WORD(t, 0);
153*2fe8fb19SBen Gras s = 1.0 + t * z;
154*2fe8fb19SBen Gras return t + a * (s + t * v);
155*2fe8fb19SBen Gras }
156*2fe8fb19SBen Gras }
157