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  1.  
  2. /* @(#)e_asin.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, 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. /* __ieee754_asin(x)
  15.  * Method :                  
  16.  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  17.  *      we approximate asin(x) on [0,0.5] by
  18.  *              asin(x) = x + x*x^2*R(x^2)
  19.  *      where
  20.  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  21.  *      and its remez error is bounded by
  22.  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  23.  *
  24.  *      For x in [0.5,1]
  25.  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  26.  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  27.  *      then for x>0.98
  28.  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  29.  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  30.  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  31.  *              f = hi part of s;
  32.  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  33.  *      and
  34.  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  35.  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  36.  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  37.  *
  38.  * Special cases:
  39.  *      if x is NaN, return x itself;
  40.  *      if |x|>1, return NaN with invalid signal.
  41.  *
  42.  */
  43.  
  44.  
  45. #include "fdlibm.h"
  46.  
  47. #ifndef _DOUBLE_IS_32BITS
  48.  
  49. #ifdef __STDC__
  50. static const double
  51. #else
  52. static double
  53. #endif
  54. one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  55. huge =  1.000e+300,
  56. pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  57. pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  58. pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  59.         /* coefficient for R(x^2) */
  60. pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  61. pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  62. pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  63. pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  64. pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  65. pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  66. qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  67. qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  68. qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  69. qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  70.  
  71. #ifdef __STDC__
  72.         double __ieee754_asin(double x)
  73. #else
  74.         double __ieee754_asin(x)
  75.         double x;
  76. #endif
  77. {
  78.         double t,w,p,q,c,r,s;
  79.         __int32_t hx,ix;
  80.         GET_HIGH_WORD(hx,x);
  81.         ix = hx&0x7fffffff;
  82.         if(ix>= 0x3ff00000) {           /* |x|>= 1 */
  83.             __uint32_t lx;
  84.             GET_LOW_WORD(lx,x);
  85.             if(((ix-0x3ff00000)|lx)==0)
  86.                     /* asin(1)=+-pi/2 with inexact */
  87.                 return x*pio2_hi+x*pio2_lo;    
  88.             return (x-x)/(x-x);         /* asin(|x|>1) is NaN */  
  89.         } else if (ix<0x3fe00000) {     /* |x|<0.5 */
  90.             if(ix<0x3e400000) {         /* if |x| < 2**-27 */
  91.                 if(huge+x>one) return x;/* return x with inexact if x!=0*/
  92.           } else {
  93.                 t = x*x;
  94.                 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  95.                 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  96.                 w = p/q;
  97.                 return x+x*w;
  98.           }
  99.         }
  100.         /* 1> |x|>= 0.5 */
  101.         w = one-fabs(x);
  102.         t = w*0.5;
  103.         p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
  104.         q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
  105.         s = __ieee754_sqrt(t);
  106.         if(ix>=0x3FEF3333) {    /* if |x| > 0.975 */
  107.             w = p/q;
  108.             t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
  109.         } else {
  110.             w  = s;
  111.             SET_LOW_WORD(w,0);
  112.             c  = (t-w*w)/(s+w);
  113.             r  = p/q;
  114.             p  = 2.0*s*r-(pio2_lo-2.0*c);
  115.             q  = pio4_hi-2.0*w;
  116.             t  = pio4_hi-(p-q);
  117.         }    
  118.         if(hx>0) return t; else return -t;    
  119. }
  120.  
  121. #endif /* defined(_DOUBLE_IS_32BITS) */
  122.