e_asin.c revision 17141 
187866Ssheldonh/* @(#)e_asin.c 5.1 93/09/24 */
287866Ssheldonh/*
387866Ssheldonh * ====================================================
487866Ssheldonh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
587866Ssheldonh *
687866Ssheldonh * Developed at SunPro, a Sun Microsystems, Inc. business.
787866Ssheldonh * Permission to use, copy, modify, and distribute this
887866Ssheldonh * software is freely granted, provided that this notice
987866Ssheldonh * is preserved.
1087866Ssheldonh * ====================================================
1187866Ssheldonh */
1287866Ssheldonh
1387866Ssheldonh#ifndef lint
1487866Ssheldonhstatic char rcsid[] = "$Id: e_asin.c,v 1.2 1995/05/30 05:47:55 rgrimes Exp $";
1587866Ssheldonh#endif
1687866Ssheldonh
1787866Ssheldonh/* __ieee754_asin(x)
1887866Ssheldonh * Method :
1987866Ssheldonh *	Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
2087866Ssheldonh *	we approximate asin(x) on [0,0.5] by
21 *		asin(x) = x + x*x^2*R(x^2)
22 *	where
23 *		R(x^2) is a rational approximation of (asin(x)-x)/x^3
24 *	and its remez error is bounded by
25 *		|(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
26 *
27 *	For x in [0.5,1]
28 *		asin(x) = pi/2-2*asin(sqrt((1-x)/2))
29 *	Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
30 *	then for x>0.98
31 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
32 *			= pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
33 *	For x<=0.98, let pio4_hi = pio2_hi/2, then
34 *		f = hi part of s;
35 *		c = sqrt(z) - f = (z-f*f)/(s+f) 	...f+c=sqrt(z)
36 *	and
37 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
38 *			= pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
39 *			= pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
40 *
41 * Special cases:
42 *	if x is NaN, return x itself;
43 *	if |x|>1, return NaN with invalid signal.
44 *
45 */
46
47
48#include "math.h"
49#include "math_private.h"
50
51#ifdef __STDC__
52static const double
53#else
54static double
55#endif
56one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
57huge =  1.000e+300,
58pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
59pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
60pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
61	/* coefficient for R(x^2) */
62pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
63pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
64pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
65pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
66pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
67pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
68qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
69qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
70qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
71qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
72
73#ifdef __STDC__
74	double __ieee754_asin(double x)
75#else
76	double __ieee754_asin(x)
77	double x;
78#endif
79{
80	double t=0.0,w,p,q,c,r,s;
81	int32_t hx,ix;
82	GET_HIGH_WORD(hx,x);
83	ix = hx&0x7fffffff;
84	if(ix>= 0x3ff00000) {		/* |x|>= 1 */
85	    u_int32_t lx;
86	    GET_LOW_WORD(lx,x);
87	    if(((ix-0x3ff00000)|lx)==0)
88		    /* asin(1)=+-pi/2 with inexact */
89		return x*pio2_hi+x*pio2_lo;
90	    return (x-x)/(x-x);		/* asin(|x|>1) is NaN */
91	} else if (ix<0x3fe00000) {	/* |x|<0.5 */
92	    if(ix<0x3e400000) {		/* if |x| < 2**-27 */
93		if(huge+x>one) return x;/* return x with inexact if x!=0*/
94	    } else
95		t = x*x;
96		p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
97		q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
98		w = p/q;
99		return x+x*w;
100	}
101	/* 1> |x|>= 0.5 */
102	w = one-fabs(x);
103	t = w*0.5;
104	p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
105	q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
106	s = sqrt(t);
107	if(ix>=0x3FEF3333) { 	/* if |x| > 0.975 */
108	    w = p/q;
109	    t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
110	} else {
111	    w  = s;
112	    SET_LOW_WORD(w,0);
113	    c  = (t-w*w)/(s+w);
114	    r  = p/q;
115	    p  = 2.0*s*r-(pio2_lo-2.0*c);
116	    q  = pio4_hi-2.0*w;
117	    t  = pio4_hi-(p-q);
118	}
119	if(hx>0) return t; else return -t;
120}
121