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