Atlas - k_tan.c
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1#include "SDL_internal.h"
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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/* __kernel_tan( x, y, k )
14 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
15 * Input x is assumed to be bounded by ~pi/4 in magnitude.
16 * Input y is the tail of x.
17 * Input k indicates whether tan (if k=1) or
18 * -1/tan (if k= -1) is returned.
19 *
20 * Algorithm
21 *	1. Since tan(-x) = -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. 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 * 	        |tan(x)         2     4            26   |     -59.2
30 * 	        |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
31 * 	        |  x 					|
32 *
33 *	   Note: tan(x+y) = tan(x) + tan'(x)*y
34 *		          ~ tan(x) + (1+x*x)*y
35 *	   Therefore, for better accuracy in computing 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) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
44 *		       = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
45 */
46
47#include "math_libm.h"
48#include "math_private.h"
49
50static const double
51one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
52pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
53pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
54T[] =  {
55  3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
56  1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
57  5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
58  2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
59  8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
60  3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
61  1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
62  5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
63  2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
64  7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
65  7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
66 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
67  2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
68};
69
70double attribute_hidden __kernel_tan(double x, double y, int iy)
71{
72	double z,r,v,w,s;
73	int32_t ix,hx;
74	GET_HIGH_WORD(hx,x);
75	ix = hx&0x7fffffff;	/* high word of |x| */
76	if(ix<0x3e300000)			/* x < 2**-28 */
77	    {if((int)x==0) {			/* generate inexact */
78	        u_int32_t low;
79		GET_LOW_WORD(low,x);
80		if(((ix|low)|(iy+1))==0) return one/fabs(x);
81		else return (iy==1)? x: -one/x;
82	    }
83	    }
84	if(ix>=0x3FE59428) { 			/* |x|>=0.6744 */
85	    if(hx<0) {x = -x; y = -y;}
86	    z = pio4-x;
87	    w = pio4lo-y;
88	    x = z+w; y = 0.0;
89	}
90	z	=  x*x;
91	w 	=  z*z;
92    /* Break x^5*(T[1]+x^2*T[2]+...) into
93     *	  x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
94     *	  x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
95     */
96	r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
97	v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
98	s = z*x;
99	r = y + z*(s*(r+v)+y);
100	r += T[0]*s;
101	w = x+r;
102	if(ix>=0x3FE59428) {
103	    v = (double)iy;
104	    return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
105	}
106	if(iy==1) return w;
107	else {		/* if allow error up to 2 ulp,
108			   simply return -1.0/(x+r) here */
109     /*  compute -1.0/(x+r) accurately */
110	    double a,t;
111	    z  = w;
112	    SET_LOW_WORD(z,0);
113	    v  = r-(z - x); 	/* z+v = r+x */
114	    t = a  = -1.0/w;	/* a = -1.0/w */
115	    SET_LOW_WORD(t,0);
116	    s  = 1.0+t*z;
117	    return t+a*(s+t*v);
118	}
119}
120
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