libm/e_asin.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* __ieee754_asin(x)
  13  * Method :
  14  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  15  *      we approximate asin(x) on [0,0.5] by
  16  *              asin(x) = x + x*x^2*R(x^2)
  17  *      where
  18  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  19  *      and its remez error is bounded by
  20  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  21  *
  22  *      For x in [0.5,1]
  23  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  24  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  25  *      then for x>0.98
  26  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  27  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  28  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  29  *              f = hi part of s;
  30  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  31  *      and
  32  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  33  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  34  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  35  *
  36  * Special cases:
  37  *      if x is NaN, return x itself;
  38  *      if |x|>1, return NaN with invalid signal.
  39  *
  40  */
  41
  42 #include "math.h"
  43 #include "math_private.h"
  44
  45 static const double
  46 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  47 huge =  1.000e+300,
  48 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  49 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  50 pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  51         /* coefficient for R(x^2) */
  52 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  53 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  54 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  55 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  56 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  57 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  58 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  59 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  60 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  61 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  62
  63 double __ieee754_asin(double x)
  64 {
  65         double t=0.0,w,p,q,c,r,s;
  66         int32_t hx,ix;
  67         GET_HIGH_WORD(hx,x);
  68         ix = hx&0x7fffffff;
  69         if(ix>= 0x3ff00000) {           /* |x|>= 1 */
  70             u_int32_t lx;
  71             GET_LOW_WORD(lx,x);
  72             if(((ix-0x3ff00000)|lx)==0)
  73                     /* asin(1)=+-pi/2 with inexact */
  74                 return x*pio2_hi+x*pio2_lo;
  75             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
  76         } else if (ix<0x3fe00000) {     /* |x|<0.5 */
  77             if(ix<0x3e400000) {         /* if |x| < 2**-27 */
  78                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
  79             } else {
  80                 t = x*x;
  81                 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  82                 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  83                 w = p/q;
  84                 return x+x*w;
  85             }
  86         }
  87         /* 1> |x|>= 0.5 */
  88         w = one-fabs(x);
  89         t = w*0.5;
  90         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  91         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  92         s = __ieee754_sqrt(t);
  93         if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
  94             w = p/q;
  95             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
  96         } else {
  97             w  = s;
  98             SET_LOW_WORD(w,0);
  99             c  = (t-w*w)/(s+w);
 100             r  = p/q;
 101             p  = 2.0*s*r-(pio2_lo-2.0*c);
 102             q  = pio4_hi-2.0*w;
 103             t  = pio4_hi-(p-q);
 104         }
 105         if(hx>0) return t; else return -t;
 106 }