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1 
2 /* @(#)e_asin.c 1.3 95/01/18 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  */
13 
14 /* __ieee754_asin(x)
15  * Method :
16  *	Since  ieee_asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
17  *	we approximate ieee_asin(x) on [0,0.5] by
18  *		asin(x) = x + x*x^2*R(x^2)
19  *	where
20  *		R(x^2) is a rational approximation of (ieee_asin(x)-x)/x^3
21  *	and its remez error is bounded by
22  *		|(ieee_asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
23  *
24  *	For x in [0.5,1]
25  *		asin(x) = pi/2-2*ieee_asin(ieee_sqrt((1-x)/2))
26  *	Let y = (1-x), z = y/2, s := ieee_sqrt(z), and pio2_hi+pio2_lo=pi/2;
27  *	then for x>0.98
28  *		asin(x) = pi/2 - 2*(s+s*z*R(z))
29  *			= pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
30  *	For x<=0.98, let pio4_hi = pio2_hi/2, then
31  *		f = hi part of s;
32  *		c = ieee_sqrt(z) - f = (z-f*f)/(s+f) 	...f+c=ieee_sqrt(z)
33  *	and
34  *		asin(x) = pi/2 - 2*(s+s*z*R(z))
35  *			= pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
36  *			= pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
37  *
38  * Special cases:
39  *	if x is NaN, return x itself;
40  *	if |x|>1, return NaN with invalid signal.
41  *
42  */
43 
44 
45 #include "fdlibm.h"
46 
47 #ifdef __STDC__
48 static const double
49 #else
50 static double
51 #endif
52 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
53 huge =  1.000e+300,
54 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
55 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
56 pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
57 	/* coefficient for R(x^2) */
58 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
59 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
60 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
61 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
62 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
63 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
64 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
65 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
66 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
67 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
68 
69 #ifdef __STDC__
__ieee754_asin(double x)70 	double __ieee754_asin(double x)
71 #else
72 	double __ieee754_asin(x)
73 	double x;
74 #endif
75 {
76 	double t,w,p,q,c,r,s;
77 	int hx,ix;
78 	hx = __HI(x);
79 	ix = hx&0x7fffffff;
80 	if(ix>= 0x3ff00000) {		/* |x|>= 1 */
81 	    if(((ix-0x3ff00000)|__LO(x))==0)
82 		    /* ieee_asin(1)=+-pi/2 with inexact */
83 		return x*pio2_hi+x*pio2_lo;
84 	    return (x-x)/(x-x);		/* ieee_asin(|x|>1) is NaN */
85 	} else if (ix<0x3fe00000) {	/* |x|<0.5 */
86 	    if(ix<0x3e400000) {		/* if |x| < 2**-27 */
87 		if(huge+x>one) return x;/* return x with inexact if x!=0*/
88 	    } else
89 		t = x*x;
90 		p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
91 		q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
92 		w = p/q;
93 		return x+x*w;
94 	}
95 	/* 1> |x|>= 0.5 */
96 	w = one-ieee_fabs(x);
97 	t = w*0.5;
98 	p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
99 	q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
100 	s = ieee_sqrt(t);
101 	if(ix>=0x3FEF3333) { 	/* if |x| > 0.975 */
102 	    w = p/q;
103 	    t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
104 	} else {
105 	    w  = s;
106 	    __LO(w) = 0;
107 	    c  = (t-w*w)/(s+w);
108 	    r  = p/q;
109 	    p  = 2.0*s*r-(pio2_lo-2.0*c);
110 	    q  = pio4_hi-2.0*w;
111 	    t  = pio4_hi-(p-q);
112 	}
113 	if(hx>0) return t; else return -t;
114 }
115