sysdeps/libm-ieee754/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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_asin.c,v 1.9 1995/05/12 04:57:22 jtc Exp $";
  18 #endif
  19
  20 /* __ieee754_asin(x)
  21  * Method :
  22  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  23  *      we approximate asin(x) on [0,0.5] by
  24  *              asin(x) = x + x*x^2*R(x^2)
  25  *      where
  26  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  27  *      and its remez error is bounded by
  28  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  29  *
  30  *      For x in [0.5,1]
  31  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  32  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  33  *      then for x>0.98
  34  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  35  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  36  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  37  *              f = hi part of s;
  38  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  39  *      and
  40  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  41  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  42  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  43  *
  44  * Special cases:
  45  *      if x is NaN, return x itself;
  46  *      if |x|>1, return NaN with invalid signal.
  47  *
  48  */
  49
  50
  51 #include "math.h"
  52 #include "math_private.h"
  53 #define one qS[0]
  54 #ifdef __STDC__
  55 static const double
  56 #else
  57 static double
  58 #endif
  59 huge =  1.000e+300,
  60 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  61 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  62 pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  63         /* coefficient for R(x^2) */
  64 pS[] =  {1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  65  -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  66   2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  67  -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  68   7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  69   3.47933107596021167570e-05}, /* 0x3F023DE1, 0x0DFDF709 */
  70 qS[] = {1.0, -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  71   2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  72  -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  73   7.70381505559019352791e-02}; /* 0x3FB3B8C5, 0xB12E9282 */
  74
  75 #ifdef __STDC__
  76         double __ieee754_asin(double x)
  77 #else
  78         double __ieee754_asin(x)
  79         double x;
  80 #endif
  81 {
  82         double t,w,p,q,c,r,s,p1,p2,p3,q1,q2,z2,z4,z6;
  83         int32_t hx,ix;
  84         GET_HIGH_WORD(hx,x);
  85         ix = hx&0x7fffffff;
  86         if(ix>= 0x3ff00000) {           /* |x|>= 1 */
  87             u_int32_t lx;
  88             GET_LOW_WORD(lx,x);
  89             if(((ix-0x3ff00000)|lx)==0)
  90                     /* asin(1)=+-pi/2 with inexact */
  91                 return x*pio2_hi+x*pio2_lo;
  92             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */
  93         } else if (ix<0x3fe00000) {     /* |x|<0.5 */
  94             if(ix<0x3e400000) {         /* if |x| < 2**-27 */
  95                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
  96             } else {
  97                 t = x*x;
  98 #ifdef DO_NOT_USE_THIS
  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 #else
 102                 p1 = t*pS[0]; z2=t*t;
 103                 p2 = pS[1]+t*pS[2]; z4=z2*z2;
 104                 p3 = pS[3]+t*pS[4]; z6=z4*z2;
 105                 q1 = one+t*qS[1];
 106                 q2 = qS[2]+t*qS[3];
 107                 p = p1 + z2*p2 + z4*p3 + z6*pS[5];
 108                 q = q1 + z2*q2 + z4*qS[4];
 109 #endif
 110                 w = p/q;
 111                 return x+x*w;
 112             }
 113         }
 114         /* 1> |x|>= 0.5 */
 115         w = one-fabs(x);
 116         t = w*0.5;
 117 #ifdef DO_NOT_USE_THIS
 118         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
 119         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
 120 #else
 121         p1 = t*pS[0]; z2=t*t;
 122         p2 = pS[1]+t*pS[2]; z4=z2*z2;
 123         p3 = pS[3]+t*pS[4]; z6=z4*z2;
 124         q1 = one+t*qS[1];
 125         q2 = qS[2]+t*qS[3];
 126         p = p1 + z2*p2 + z4*p3 + z6*pS[5];
 127         q = q1 + z2*q2 + z4*qS[4];
 128 #endif
 129         s = __ieee754_sqrt(t);
 130         if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
 131             w = p/q;
 132             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
 133         } else {
 134             w  = s;
 135             SET_LOW_WORD(w,0);
 136             c  = (t-w*w)/(s+w);
 137             r  = p/q;
 138             p  = 2.0*s*r-(pio2_lo-2.0*c);
 139             q  = pio4_hi-2.0*w;
 140             t  = pio4_hi-(p-q);
 141         }
 142         if(hx>0) return t; else return -t;
 143 }