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 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  46 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  47 /* coefficients for R(x^2) */
  48 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  49 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  50 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  51 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  52 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  53 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  54 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  55 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  56 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  57 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  58
  59 static double R(double z)
  60 {
  61         double_t p, q;
  62         p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
  63         q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
  64         return p/q;
  65 }
  66
  67 double asin(double x)
  68 {
  69         double z,r,s;
  70         uint32_t hx,ix;
  71
  72         GET_HIGH_WORD(hx, x);
  73         ix = hx & 0x7fffffff;
  74         /* |x| >= 1 or nan */
  75         if (ix >= 0x3ff00000) {
  76                 uint32_t lx;
  77                 GET_LOW_WORD(lx, x);
  78                 if ((ix-0x3ff00000 | lx) == 0)
  79                         /* asin(1) = +-pi/2 with inexact */
  80                         return x*pio2_hi + 0x1p-120f;
  81                 return 0/(x-x);
  82         }
  83         /* |x| < 0.5 */
  84         if (ix < 0x3fe00000) {
  85                 /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
  86                 if (ix < 0x3e500000 && ix >= 0x00100000)
  87                         return x;
  88                 return x + x*R(x*x);
  89         }
  90         /* 1 > |x| >= 0.5 */
  91         z = (1 - fabs(x))*0.5;
  92         s = sqrt(z);
  93         r = R(z);
  94         if (ix >= 0x3fef3333) {  /* if |x| > 0.975 */
  95                 x = pio2_hi-(2*(s+s*r)-pio2_lo);
  96         } else {
  97                 double f,c;
  98                 /* f+c = sqrt(z) */
  99                 f = s;
 100                 SET_LOW_WORD(f,0);
 101                 c = (z-f*f)/(s+f);
 102                 x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f));
 103         }
 104         if (hx >> 31)
 105                 return -x;
 106         return x;
 107 }