xref: /minix3/lib/libm/src/k_tan.c (revision 2fe8fb192fe7e8720e3e7a77f928da545e872a6a)
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