1/*
2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26/* __ieee754_asin(x)
27 * Method :
28 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
29 * we approximate asin(x) on [0,0.5] by
30 * asin(x) = x + x*x^2*R(x^2)
31 * where
32 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
33 * and its remez error is bounded by
34 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
35 *
36 * For x in [0.5,1]
37 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
38 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
39 * then for x>0.98
40 * asin(x) = pi/2 - 2*(s+s*z*R(z))
41 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
42 * For x<=0.98, let pio4_hi = pio2_hi/2, then
43 * f = hi part of s;
44 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
45 * and
46 * asin(x) = pi/2 - 2*(s+s*z*R(z))
47 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
48 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
49 *
50 * Special cases:
51 * if x is NaN, return x itself;
52 * if |x|>1, return NaN with invalid signal.
53 *
54 */
55
56
57#include "fdlibm.h"
58
59#ifdef __STDC__
60static const double
61#else
62static double
63#endif
64one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
65huge = 1.000e+300,
66pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
67pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
68pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
69 /* coefficient for R(x^2) */
70pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
71pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
72pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
73pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
74pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
75pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
76qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
77qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
78qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
79qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
80
81#ifdef __STDC__
82 double __ieee754_asin(double x)
83#else
84 double __ieee754_asin(x)
85 double x;
86#endif
87{
88 double t=0,w,p,q,c,r,s;
89 int hx,ix;
90 hx = __HI(x);
91 ix = hx&0x7fffffff;
92 if(ix>= 0x3ff00000) { /* |x|>= 1 */
93 if(((ix-0x3ff00000)|__LO(x))==0)
94 /* asin(1)=+-pi/2 with inexact */
95 return x*pio2_hi+x*pio2_lo;
96 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
97 } else if (ix<0x3fe00000) { /* |x|<0.5 */
98 if(ix<0x3e400000) { /* if |x| < 2**-27 */
99 if(huge+x>one) return x;/* return x with inexact if x!=0*/
100 } else
101 t = x*x;
102 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
103 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
104 w = p/q;
105 return x+x*w;
106 }
107 /* 1> |x|>= 0.5 */
108 w = one-fabs(x);
109 t = w*0.5;
110 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
111 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
112 s = sqrt(t);
113 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
114 w = p/q;
115 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
116 } else {
117 w = s;
118 __LO(w) = 0;
119 c = (t-w*w)/(s+w);
120 r = p/q;
121 p = 2.0*s*r-(pio2_lo-2.0*c);
122 q = pio4_hi-2.0*w;
123 t = pio4_hi-(p-q);
124 }
125 if(hx>0) return t; else return -t;
126}
127