StdLib/LibC/Math/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 #include  <LibConfig.h>
  13 #include  <sys/EfiCdefs.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 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  19 // C4723: potential divide by zero.
  20 #pragma warning ( disable : 4723 )
  21 #endif
  22
  23 /* __ieee754_asin(x)
  24  * Method :
  25  *  Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  26  *  we approximate asin(x) on [0,0.5] by
  27  *    asin(x) = x + x*x^2*R(x^2)
  28  *  where
  29  *    R(x^2) is a rational approximation of (asin(x)-x)/x^3
  30  *  and its remez error is bounded by
  31  *    |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  32  *
  33  *  For x in [0.5,1]
  34  *    asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  35  *  Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  36  *  then for x>0.98
  37  *    asin(x) = pi/2 - 2*(s+s*z*R(z))
  38  *      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  39  *  For x<=0.98, let pio4_hi = pio2_hi/2, then
  40  *    f = hi part of s;
  41  *    c = sqrt(z) - f = (z-f*f)/(s+f)   ...f+c=sqrt(z)
  42  *  and
  43  *    asin(x) = pi/2 - 2*(s+s*z*R(z))
  44  *      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  45  *      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  46  *
  47  * Special cases:
  48  *  if x is NaN, return x itself;
  49  *  if |x|>1, return NaN with invalid signal.
  50  *
  51  */
  52
  53
  54 #include "math.h"
  55 #include "math_private.h"
  56
  57 static const double
  58 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  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 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  65 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  66 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  67 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  68 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  69 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  70 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  71 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  72 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  73 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  74
  75 double
  76 __ieee754_asin(double x)
  77 {
  78   double t,w,p,q,c,r,s;
  79   int32_t hx,ix;
  80
  81   t = 0;
  82   GET_HIGH_WORD(hx,x);
  83   ix = hx&0x7fffffff;
  84   if(ix>= 0x3ff00000) {   /* |x|>= 1 */
  85       u_int32_t lx;
  86       GET_LOW_WORD(lx,x);
  87       if(((ix-0x3ff00000)|lx)==0)
  88         /* asin(1)=+-pi/2 with inexact */
  89     return x*pio2_hi+x*pio2_lo;
  90       return (x-x)/(x-x);   /* asin(|x|>1) is NaN */
  91   } else if (ix<0x3fe00000) { /* |x|<0.5 */
  92       if(ix<0x3e400000) {   /* if |x| < 2**-27 */
  93     if(huge+x>one) return x;/* return x with inexact if x!=0*/
  94       } else
  95     t = x*x;
  96     p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  97     q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  98     w = p/q;
  99     return x+x*w;
 100   }
 101   /* 1> |x|>= 0.5 */
 102   w = one-fabs(x);
 103   t = w*0.5;
 104   p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
 105   q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
 106   s = __ieee754_sqrt(t);
 107   if(ix>=0x3FEF3333) {  /* if |x| > 0.975 */
 108       w = p/q;
 109       t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
 110   } else {
 111       w  = s;
 112       SET_LOW_WORD(w,0);
 113       c  = (t-w*w)/(s+w);
 114       r  = p/q;
 115       p  = 2.0*s*r-(pio2_lo-2.0*c);
 116       q  = pio4_hi-2.0*w;
 117       t  = pio4_hi-(p-q);
 118   }
 119   if(hx>0) return t; else return -t;
 120 }