1df930be7Sderaadt /* @(#)k_tan.c 5.1 93/09/24 */
2df930be7Sderaadt /*
3df930be7Sderaadt * ====================================================
4df930be7Sderaadt * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5df930be7Sderaadt *
6df930be7Sderaadt * Developed at SunPro, a Sun Microsystems, Inc. business.
7df930be7Sderaadt * Permission to use, copy, modify, and distribute this
8df930be7Sderaadt * software is freely granted, provided that this notice
9df930be7Sderaadt * is preserved.
10df930be7Sderaadt * ====================================================
11df930be7Sderaadt */
12df930be7Sderaadt
13df930be7Sderaadt /* __kernel_tan( x, y, k )
14df930be7Sderaadt * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
15df930be7Sderaadt * Input x is assumed to be bounded by ~pi/4 in magnitude.
16df930be7Sderaadt * Input y is the tail of x.
17df930be7Sderaadt * Input k indicates whether tan (if k=1) or
18df930be7Sderaadt * -1/tan (if k= -1) is returned.
19df930be7Sderaadt *
20df930be7Sderaadt * Algorithm
21df930be7Sderaadt * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
22df930be7Sderaadt * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
23df930be7Sderaadt * 3. tan(x) is approximated by a odd polynomial of degree 27 on
24df930be7Sderaadt * [0,0.67434]
25df930be7Sderaadt * 3 27
26df930be7Sderaadt * tan(x) ~ x + T1*x + ... + T13*x
27df930be7Sderaadt * where
28df930be7Sderaadt *
29df930be7Sderaadt * |tan(x) 2 4 26 | -59.2
30df930be7Sderaadt * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
31df930be7Sderaadt * | x |
32df930be7Sderaadt *
33df930be7Sderaadt * Note: tan(x+y) = tan(x) + tan'(x)*y
34df930be7Sderaadt * ~ tan(x) + (1+x*x)*y
35df930be7Sderaadt * Therefore, for better accuracy in computing tan(x+y), let
36df930be7Sderaadt * 3 2 2 2 2
37df930be7Sderaadt * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
38df930be7Sderaadt * then
39df930be7Sderaadt * 3 2
40df930be7Sderaadt * tan(x+y) = x + (T1*x + (x *(r+y)+y))
41df930be7Sderaadt *
42df930be7Sderaadt * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
43df930be7Sderaadt * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
44df930be7Sderaadt * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
45df930be7Sderaadt */
46df930be7Sderaadt
47df930be7Sderaadt #include "math.h"
48df930be7Sderaadt #include "math_private.h"
49*ee391e4fSmartynas
50*ee391e4fSmartynas static const double xxx[] = {
51*ee391e4fSmartynas 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
52*ee391e4fSmartynas 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
53*ee391e4fSmartynas 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
54*ee391e4fSmartynas 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
55*ee391e4fSmartynas 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
56*ee391e4fSmartynas 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
57*ee391e4fSmartynas 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
58*ee391e4fSmartynas 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
59*ee391e4fSmartynas 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
60*ee391e4fSmartynas 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
61*ee391e4fSmartynas 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
62*ee391e4fSmartynas -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
63*ee391e4fSmartynas 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
64*ee391e4fSmartynas /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
65*ee391e4fSmartynas /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
66*ee391e4fSmartynas /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
67df930be7Sderaadt };
68*ee391e4fSmartynas #define one xxx[13]
69*ee391e4fSmartynas #define pio4 xxx[14]
70*ee391e4fSmartynas #define pio4lo xxx[15]
71*ee391e4fSmartynas #define T xxx
72df930be7Sderaadt
73e7beb4a7Smillert double
__kernel_tan(double x,double y,int iy)74e7beb4a7Smillert __kernel_tan(double x, double y, int iy)
75df930be7Sderaadt {
76df930be7Sderaadt double z, r, v, w, s;
77df930be7Sderaadt int32_t ix, hx;
78*ee391e4fSmartynas
79*ee391e4fSmartynas GET_HIGH_WORD(hx, x); /* high word of x */
80df930be7Sderaadt ix = hx & 0x7fffffff; /* high word of |x| */
81*ee391e4fSmartynas if (ix < 0x3e300000) { /* x < 2**-28 */
82*ee391e4fSmartynas if ((int) x == 0) { /* generate inexact */
83df930be7Sderaadt u_int32_t low;
84df930be7Sderaadt GET_LOW_WORD(low, x);
85*ee391e4fSmartynas if(((ix | low) | (iy + 1)) == 0)
86*ee391e4fSmartynas return one / fabs(x);
87*ee391e4fSmartynas else {
88*ee391e4fSmartynas if (iy == 1)
89*ee391e4fSmartynas return x;
90*ee391e4fSmartynas else { /* compute -1 / (x+y) carefully */
91*ee391e4fSmartynas double a, t;
92*ee391e4fSmartynas
93*ee391e4fSmartynas z = w = x + y;
94*ee391e4fSmartynas SET_LOW_WORD(z, 0);
95*ee391e4fSmartynas v = y - (z - x);
96*ee391e4fSmartynas t = a = -one / w;
97*ee391e4fSmartynas SET_LOW_WORD(t, 0);
98*ee391e4fSmartynas s = one + t * z;
99*ee391e4fSmartynas return t + a * (s + t * v);
100*ee391e4fSmartynas }
101*ee391e4fSmartynas }
102df930be7Sderaadt }
103df930be7Sderaadt }
104df930be7Sderaadt if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
105*ee391e4fSmartynas if (hx < 0) {
106*ee391e4fSmartynas x = -x;
107*ee391e4fSmartynas y = -y;
108*ee391e4fSmartynas }
109df930be7Sderaadt z = pio4 - x;
110df930be7Sderaadt w = pio4lo - y;
111*ee391e4fSmartynas x = z + w;
112*ee391e4fSmartynas y = 0.0;
113df930be7Sderaadt }
114df930be7Sderaadt z = x * x;
115df930be7Sderaadt w = z * z;
116*ee391e4fSmartynas /*
117*ee391e4fSmartynas * Break x^5*(T[1]+x^2*T[2]+...) into
118df930be7Sderaadt * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
119df930be7Sderaadt * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
120df930be7Sderaadt */
121*ee391e4fSmartynas r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
122*ee391e4fSmartynas w * T[11]))));
123*ee391e4fSmartynas v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
124*ee391e4fSmartynas w * T[12])))));
125df930be7Sderaadt s = z * x;
126df930be7Sderaadt r = y + z * (s * (r + v) + y);
127df930be7Sderaadt r += T[0] * s;
128df930be7Sderaadt w = x + r;
129df930be7Sderaadt if (ix >= 0x3FE59428) {
130df930be7Sderaadt v = (double) iy;
131*ee391e4fSmartynas return (double) (1 - ((hx >> 30) & 2)) *
132*ee391e4fSmartynas (v - 2.0 * (x - (w * w / (w + v) - r)));
133df930be7Sderaadt }
134*ee391e4fSmartynas if (iy == 1)
135*ee391e4fSmartynas return w;
136*ee391e4fSmartynas else {
137*ee391e4fSmartynas /*
138*ee391e4fSmartynas * if allow error up to 2 ulp, simply return
139*ee391e4fSmartynas * -1.0 / (x+r) here
140*ee391e4fSmartynas */
141df930be7Sderaadt /* compute -1.0 / (x+r) accurately */
142df930be7Sderaadt double a, t;
143df930be7Sderaadt z = w;
144df930be7Sderaadt SET_LOW_WORD(z, 0);
145df930be7Sderaadt v = r - (z - x); /* z+v = r+x */
146df930be7Sderaadt t = a = -1.0 / w; /* a = -1.0/w */
147df930be7Sderaadt SET_LOW_WORD(t, 0);
148df930be7Sderaadt s = 1.0 + t * z;
149df930be7Sderaadt return t + a * (s + t * v);
150df930be7Sderaadt }
151df930be7Sderaadt }
152