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
  13 #include <sys/cdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: e_asin.c,v 1.12 2002/05/26 22:01:48 wiz Exp $");
  16 #endif
  17
  18 /* __ieee754_asin(x)
  19  * Method :
  20  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  21  *      we approximate asin(x) on [0,0.5] by
  22  *              asin(x) = x + x*x^2*R(x^2)
  23  *      where
  24  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  25  *      and its remez error is bounded by
  26  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  27  *
  28  *      For x in [0.5,1]
  29  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  30  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  31  *      then for x>0.98
  32  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  33  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  34  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  35  *              f = hi part of s;
  36  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  37  *      and
  38  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  39  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  40  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  41  *
  42  * Special cases:
  43  *      if x is NaN, return x itself;
  44  *      if |x|>1, return NaN with invalid signal.
  45  *
  46  */
  47
  48
  49 #include "math.h"
  50 #include "math_private.h"
  51
  52 static const double
  53 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  54 huge =  1.000e+300,
  55 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  56 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  57 pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  58         /* coefficient for R(x^2) */
  59 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  60 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  61 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  62 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  63 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  64 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  65 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  66 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  67 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  68 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  69
  70 double
  71 __ieee754_asin(double x)
  72 {
  73         double t,w,p,q,c,r,s;
  74         int32_t hx,ix;
  75
  76         t = 0;
  77         GET_HIGH_WORD(hx,x);
  78         ix = hx&0x7fffffff;
  79         if(ix>= 0x3ff00000) {           /* |x|>= 1 */
  80             u_int32_t lx;
  81             GET_LOW_WORD(lx,x);
  82             if(((ix-0x3ff00000)|lx)==0)
  83                     /* asin(1)=+-pi/2 with inexact */
  84                 return x*pio2_hi+x*pio2_lo;
  85             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
  86         } else if (ix<0x3fe00000) {     /* |x|<0.5 */
  87             if(ix<0x3e400000) {         /* if |x| < 2**-27 */
  88                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
  89             } else
  90                 t = x*x;
  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                 w = p/q;
  94                 return x+x*w;
  95         }
  96         /* 1> |x|>= 0.5 */
  97         w = one-fabs(x);
  98         t = w*0.5;
  99         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
 100         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
 101         s = __ieee754_sqrt(t);
 102         if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
 103             w = p/q;
 104             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
 105         } else {
 106             w  = s;
 107             SET_LOW_WORD(w,0);
 108             c  = (t-w*w)/(s+w);
 109             r  = p/q;
 110             p  = 2.0*s*r-(pio2_lo-2.0*c);
 111             q  = pio4_hi-2.0*w;
 112             t  = pio4_hi-(p-q);
 113         }
 114         if(hx>0) return t; else return -t;
 115 }