Line data    Source code

       1             : 
       2             : /* @(#)e_asin.c 1.3 95/01/18 */
       3             : /*
       4             :  * ====================================================
       5             :  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
       6             :  *
       7             :  * Developed at SunSoft, a Sun Microsystems, Inc. business.
       8             :  * Permission to use, copy, modify, and distribute this
       9             :  * software is freely granted, provided that this notice 
      10             :  * is preserved.
      11             :  * ====================================================
      12             :  */
      13             : 
      14             : //#include <sys/cdefs.h>
      15             : //__FBSDID("$FreeBSD$");
      16             : 
      17             : /* __ieee754_asin(x)
      18             :  * Method :                  
      19             :  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
      20             :  *      we approximate asin(x) on [0,0.5] by
      21             :  *              asin(x) = x + x*x^2*R(x^2)
      22             :  *      where
      23             :  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3 
      24             :  *      and its remez error is bounded by
      25             :  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
      26             :  *
      27             :  *      For x in [0.5,1]
      28             :  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
      29             :  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
      30             :  *      then for x>0.98
      31             :  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
      32             :  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
      33             :  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
      34             :  *              f = hi part of s;
      35             :  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
      36             :  *      and
      37             :  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
      38             :  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
      39             :  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
      40             :  *
      41             :  * Special cases:
      42             :  *      if x is NaN, return x itself;
      43             :  *      if |x|>1, return NaN with invalid signal.
      44             :  *
      45             :  */
      46             : 
      47             : #include <float.h>
      48             : 
      49             : #include "math_private.h"
      50             : 
      51             : static const double
      52             : one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
      53             : huge =  1.000e+300,
      54             : pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
      55             : pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
      56             : pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
      57             :         /* coefficient for R(x^2) */
      58             : pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
      59             : pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
      60             : pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
      61             : pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
      62             : pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
      63             : pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
      64             : qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
      65             : qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
      66             : qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
      67             : qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
      68             : 
      69             : double
      70           0 : __ieee754_asin(double x)
      71             : {
      72           0 :         double t=0.0,w,p,q,c,r,s;
      73             :         int32_t hx,ix;
      74           0 :         GET_HIGH_WORD(hx,x);
      75           0 :         ix = hx&0x7fffffff;
      76           0 :         if(ix>= 0x3ff00000) {                /* |x|>= 1 */
      77             :             u_int32_t lx;
      78           0 :             GET_LOW_WORD(lx,x);
      79           0 :             if(((ix-0x3ff00000)|lx)==0)
      80             :                     /* asin(1)=+-pi/2 with inexact */
      81           0 :                 return x*pio2_hi+x*pio2_lo;     
      82           0 :             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */   
      83           0 :         } else if (ix<0x3fe00000) {  /* |x|<0.5 */
      84           0 :             if(ix<0x3e500000) {              /* if |x| < 2**-26 */
      85           0 :                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
      86             :             }
      87           0 :             t = x*x;
      88           0 :             p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
      89           0 :             q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
      90           0 :             w = p/q;
      91           0 :             return x+x*w;
      92             :         }
      93             :         /* 1> |x|>= 0.5 */
      94           0 :         w = one-fabs(x);
      95           0 :         t = w*0.5;
      96           0 :         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
      97           0 :         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
      98           0 :         s = sqrt(t);
      99           0 :         if(ix>=0x3FEF3333) {         /* if |x| > 0.975 */
     100           0 :             w = p/q;
     101           0 :             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
     102             :         } else {
     103           0 :             w  = s;
     104           0 :             SET_LOW_WORD(w,0);
     105           0 :             c  = (t-w*w)/(s+w);
     106           0 :             r  = p/q;
     107           0 :             p  = 2.0*s*r-(pio2_lo-2.0*c);
     108           0 :             q  = pio4_hi-2.0*w;
     109           0 :             t  = pio4_hi-(p-q);
     110             :         }    
     111           0 :         if(hx>0) return t; else return -t;    
     112             : }