1
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, 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/* asin(x)
14 * Method :
15 *	Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
16 *	we approximate asin(x) on [0,0.5] by
17 *		asin(x) = x + x*x^2*R(x^2)
18 *	where
19 *		R(x^2) is a rational approximation of (asin(x)-x)/x^3
20 *	and its remez error is bounded by
21 *		|(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
22 *
23 *	For x in [0.5,1]
24 *		asin(x) = pi/2-2*asin(sqrt((1-x)/2))
25 *	Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
26 *	then for x>0.98
27 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
28 *			= pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
29 *	For x<=0.98, let pio4_hi = pio2_hi/2, then
30 *		f = hi part of s;
31 *		c = sqrt(z) - f = (z-f*f)/(s+f) 	...f+c=sqrt(z)
32 *	and
33 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
34 *			= pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
35 *			= pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
36 *
37 * Special cases:
38 *	if x is NaN, return x itself;
39 *	if |x|>1, return NaN with invalid signal.
40 *
41 */
42
43#include <float.h>
44
45#include "math.h"
46#include "math_private.h"
47
48static const double
49one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
50huge =  1.000e+300,
51pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
52pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
53pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
54	/* coefficient for R(x^2) */
55pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
56pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
57pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
58pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
59pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
60pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
61qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
62qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
63qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
64qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
65
66double
67asin(double x)
68{
69	double t=0.0,w,p,q,c,r,s;
70	int32_t hx,ix;
71	GET_HIGH_WORD(hx,x);
72	ix = hx&0x7fffffff;
73	if(ix>= 0x3ff00000) {		/* |x|>= 1 */
74	    u_int32_t lx;
75	    GET_LOW_WORD(lx,x);
76	    if(((ix-0x3ff00000)|lx)==0)
77		    /* asin(1)=+-pi/2 with inexact */
78		return x*pio2_hi+x*pio2_lo;
79	    return (x-x)/(x-x);		/* asin(|x|>1) is NaN */
80	} else if (ix<0x3fe00000) {	/* |x|<0.5 */
81	    if(ix<0x3e500000) {		/* if |x| < 2**-26 */
82		if(huge+x>one) return x;/* return x with inexact if x!=0*/
83	    }
84	    t = x*x;
85	    p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
86	    q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
87	    w = p/q;
88	    return x+x*w;
89	}
90	/* 1> |x|>= 0.5 */
91	w = one-fabs(x);
92	t = w*0.5;
93	p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
94	q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
95	s = sqrt(t);
96	if(ix>=0x3FEF3333) { 	/* if |x| > 0.975 */
97	    w = p/q;
98	    t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
99	} else {
100	    w  = s;
101	    SET_LOW_WORD(w,0);
102	    c  = (t-w*w)/(s+w);
103	    r  = p/q;
104	    p  = 2.0*s*r-(pio2_lo-2.0*c);
105	    q  = pio4_hi-2.0*w;
106	    t  = pio4_hi-(p-q);
107	}
108	if(hx>0) return t; else return -t;
109}
110
111#if LDBL_MANT_DIG == 53
112__weak_reference(asin, asinl);
113#endif
114