nsz Git - libm/blob - src/math/asin.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, 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 /* 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 "libm.h"
  43
  44 static const double
  45 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  46 huge = 1.000e+300,
  47 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  48 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  49 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  50 /* coefficients for R(x^2) */
  51 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  52 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  53 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  54 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  55 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  56 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  57 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  58 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  59 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  60 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  61
  62 double asin(double x)
  63 {
  64         double t=0.0,w,p,q,c,r,s;
  65         int32_t hx,ix;
  66
  67         GET_HIGH_WORD(hx, x);
  68         ix = hx & 0x7fffffff;
  69         if (ix >= 0x3ff00000) {           /* |x|>= 1 */
  70                 uint32_t lx;
  71
  72                 GET_LOW_WORD(lx, x);
  73                 if ((ix-0x3ff00000 | lx) == 0)
  74                         /* asin(1) = +-pi/2 with inexact */
  75                         return x*pio2_hi + x*pio2_lo;
  76                 return (x-x)/(x-x);  /* asin(|x|>1) is NaN */
  77         } else if (ix < 0x3fe00000) {  /* |x|<0.5 */
  78                 if (ix < 0x3e500000) {  /* if |x| < 2**-26 */
  79                         if (huge+x > one)
  80                                 return x; /* return x with inexact if x!=0*/
  81                 }
  82                 t = x*x;
  83                 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  84                 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  85                 w = p/q;
  86                 return x + x*w;
  87         }
  88         /* 1 > |x| >= 0.5 */
  89         w = one - fabs(x);
  90         t = w*0.5;
  91         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  92         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  93         s = sqrt(t);
  94         if (ix >= 0x3FEF3333) {  /* if |x| > 0.975 */
  95                 w = p/q;
  96                 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
  97         } else {
  98                 w = s;
  99                 SET_LOW_WORD(w,0);
 100                 c = (t-w*w)/(s+w);
 101                 r = p/q;
 102                 p = 2.0*s*r-(pio2_lo-2.0*c);
 103                 q = pio4_hi - 2.0*w;
 104                 t = pio4_hi - (p-q);
 105         }
 106         if (hx > 0)
 107                 return t;
 108         return -t;
 109 }