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