lib/libm/src/e_asin.c

   1 /* @(#)e_asin.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, 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  * $NetBSD: e_asin.c,v 1.12 2002/05/26 22:01:48 wiz Exp $
  13  * $DragonFly: src/lib/libm/src/e_asin.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  14  */
  15
  16 /* asin(x)
  17  * Method :
  18  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  19  *      we approximate asin(x) on [0,0.5] by
  20  *              asin(x) = x + x*x^2*R(x^2)
  21  *      where
  22  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  23  *      and its remez error is bounded by
  24  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  25  *
  26  *      For x in [0.5,1]
  27  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  28  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  29  *      then for x>0.98
  30  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  31  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  32  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  33  *              f = hi part of s;
  34  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  35  *      and
  36  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  37  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  38  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  39  *
  40  * Special cases:
  41  *      if x is NaN, return x itself;
  42  *      if |x|>1, return NaN with invalid signal.
  43  *
  44  */
  45
  46
  47 #include <math.h>
  48 #include "math_private.h"
  49
  50 static const double
  51 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  52 huge =  1.000e+300,
  53 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  54 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  55 pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  56         /* coefficient for R(x^2) */
  57 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  58 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  59 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  60 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  61 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  62 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  63 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  64 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  65 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  66 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  67
  68 double
  69 asin(double x)
  70 {
  71         double t,w,p,q,c,r,s;
  72         int32_t hx,ix;
  73
  74         t = 0;
  75         GET_HIGH_WORD(hx,x);
  76         ix = hx&0x7fffffff;
  77         if(ix>= 0x3ff00000) {           /* |x|>= 1 */
  78             u_int32_t lx;
  79             GET_LOW_WORD(lx,x);
  80             if(((ix-0x3ff00000)|lx)==0)
  81                     /* asin(1)=+-pi/2 with inexact */
  82                 return x*pio2_hi+x*pio2_lo;
  83             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
  84         } else if (ix<0x3fe00000) {     /* |x|<0.5 */
  85             if(ix<0x3e400000) {         /* if |x| < 2**-27 */
  86                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
  87             } else
  88                 t = x*x;
  89                 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  90                 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  91                 w = p/q;
  92                 return x+x*w;
  93         }
  94         /* 1> |x|>= 0.5 */
  95         w = one-fabs(x);
  96         t = w*0.5;
  97         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  98         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  99         s = sqrt(t);
 100         if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
 101             w = p/q;
 102             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
 103         } else {
 104             w  = s;
 105             SET_LOW_WORD(w,0);
 106             c  = (t-w*w)/(s+w);
 107             r  = p/q;
 108             p  = 2.0*s*r-(pio2_lo-2.0*c);
 109             q  = pio4_hi-2.0*w;
 110             t  = pio4_hi-(p-q);
 111         }
 112         if(hx>0) return t; else return -t;
 113 }