k_tan.c revision 141296 
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#ifndef lint
15static char rcsid[] = "$FreeBSD: head/lib/msun/src/k_tan.c 141296 2005-02-04 18:26:06Z das $";
16#endif
17
18/* __kernel_tan( x, y, k )
19 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
20 * Input x is assumed to be bounded by ~pi/4 in magnitude.
21 * Input y is the tail of x.
22 * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
23 *
24 * Algorithm
25 *	1. Since tan(-x) = -tan(x), we need only to consider positive x.
26 *	2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
27 *	3. tan(x) is approximated by a odd polynomial of degree 27 on
28 *	   [0,0.67434]
29 *		  	         3             27
30 *	   	tan(x) ~ x + T1*x + ... + T13*x
31 *	   where
32 *
33 * 	        |tan(x)         2     4            26   |     -59.2
34 * 	        |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
35 * 	        |  x 					|
36 *
37 *	   Note: tan(x+y) = tan(x) + tan'(x)*y
38 *		          ~ tan(x) + (1+x*x)*y
39 *	   Therefore, for better accuracy in computing tan(x+y), let
40 *		     3      2      2       2       2
41 *		r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
42 *	   then
43 *		 		    3    2
44 *		tan(x+y) = x + (T1*x + (x *(r+y)+y))
45 *
46 *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
47 *		tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
48 *		       = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
49 */
50
51#include "math.h"
52#include "math_private.h"
53static 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 */
70};
71#define	one	xxx[13]
72#define	pio4	xxx[14]
73#define	pio4lo	xxx[15]
74#define	T	xxx
75/* INDENT ON */
76
77double
78__kernel_tan(double x, double y, int iy) {
79	double z, r, v, w, s;
80	int32_t ix, hx;
81
82	GET_HIGH_WORD(hx,x);
83	ix = hx & 0x7fffffff;			/* high word of |x| */
84	if (ix < 0x3e300000) {			/* x < 2**-28 */
85		if ((int) x == 0) {		/* generate inexact */
86			u_int32_t low;
87			GET_LOW_WORD(low,x);
88			if (((ix | low) | (iy + 1)) == 0)
89				return one / fabs(x);
90			else {
91				if (iy == 1)
92					return x;
93				else {	/* compute -1 / (x+y) carefully */
94					double a, t;
95
96					z = w = x + y;
97					SET_LOW_WORD(z, 0);
98					v = y - (z - x);
99					t = a = -one / w;
100					SET_LOW_WORD(t, 0);
101					s = one + t * z;
102					return t + a * (s + t * v);
103				}
104			}
105		}
106	}
107	if (ix >= 0x3FE59428) {	/* |x| >= 0.6744 */
108		if (hx < 0) {
109			x = -x;
110			y = -y;
111		}
112		z = pio4 - x;
113		w = pio4lo - y;
114		x = z + w;
115		y = 0.0;
116	}
117	z = x * x;
118	w = z * z;
119	/*
120	 * Break x^5*(T[1]+x^2*T[2]+...) into
121	 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
122	 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
123	 */
124	r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
125		w * T[11]))));
126	v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
127		w * T[12])))));
128	s = z * x;
129	r = y + z * (s * (r + v) + y);
130	r += T[0] * s;
131	w = x + r;
132	if (ix >= 0x3FE59428) {
133		v = (double) iy;
134		return (double) (1 - ((hx >> 30) & 2)) *
135			(v - 2.0 * (x - (w * w / (w + v) - r)));
136	}
137	if (iy == 1)
138		return w;
139	else {
140		/*
141		 * if allow error up to 2 ulp, simply return
142		 * -1.0 / (x+r) here
143		 */
144		/* compute -1.0 / (x+r) accurately */
145		double a, t;
146		z = w;
147		SET_LOW_WORD(z,0);
148		v = r - (z - x);	/* z+v = r+x */
149		t = a = -1.0 / w;	/* a = -1.0/w */
150		SET_LOW_WORD(t,0);
151		s = 1.0 + t * z;
152		return t + a * (s + t * v);
153	}
154}
155