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 }
--- EOF ---