prime/src/Runtime/e_asin.c

/* @(#)e_asin.c 1.2 95/01/04 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* __ieee754_asin(x)
 * Method :
 *	Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
 *	we approximate asin(x) on [0,0.5] by
 *		asin(x) = x + x*x^2*R(x^2)
 *	where
 *		R(x^2) is a rational approximation of (asin(x)-x)/x^3
 *	and its remez error is bounded by
 *		|(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
 *
 *	For x in [0.5,1]
 *		asin(x) = pi/2-2*asin(sqrt((1-x)/2))
 *	Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
 *	then for x>0.98
 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
 *			= pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
 *	For x<=0.98, let pio4_hi = pio2_hi/2, then
 *		f = hi part of s;
 *		c = sqrt(z) - f = (z-f*f)/(s+f) 	...f+c=sqrt(z)
 *	and
 *		asin(x) = pi/2 - 2*(s+s*z*R(z))
 *			= pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
 *			= pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
 *
 * Special cases:
 *	if x is NaN, return x itself;
 *	if |x|>1, return NaN with invalid signal.
 *
 */

#include "fdlibm.h"

#ifdef __STDC__
static const double
#else
static double
#endif
    one = 1.00000000000000000000e+00,                       /* 0x3FF00000, 0x00000000 */
    big = 1.000e+300, pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
    pio2_lo = 6.12323399573676603587e-17,                   /* 0x3C91A626, 0x33145C07 */
    pio4_hi = 7.85398163397448278999e-01,                   /* 0x3FE921FB, 0x54442D18 */
                                                            /* coefficient for R(x^2) */
    pS0 = 1.66666666666666657415e-01,                       /* 0x3FC55555, 0x55555555 */
    pS1 = -3.25565818622400915405e-01,                      /* 0xBFD4D612, 0x03EB6F7D */
    pS2 = 2.01212532134862925881e-01,                       /* 0x3FC9C155, 0x0E884455 */
    pS3 = -4.00555345006794114027e-02,                      /* 0xBFA48228, 0xB5688F3B */
    pS4 = 7.91534994289814532176e-04,                       /* 0x3F49EFE0, 0x7501B288 */
    pS5 = 3.47933107596021167570e-05,                       /* 0x3F023DE1, 0x0DFDF709 */
    qS1 = -2.40339491173441421878e+00,                      /* 0xC0033A27, 0x1C8A2D4B */
    qS2 = 2.02094576023350569471e+00,                       /* 0x40002AE5, 0x9C598AC8 */
    qS3 = -6.88283971605453293030e-01,                      /* 0xBFE6066C, 0x1B8D0159 */
    qS4 = 7.70381505559019352791e-02;                       /* 0x3FB3B8C5, 0xB12E9282 */

#ifdef __STDC__
double __ieee754_asin(double x)
#else
double __ieee754_asin(x)
double x;
#endif
{
  double t, w, p, q, c, r, s;
  _INT32 hx, ix;
  hx = __HI(x);
  ix = hx & 0x7fffffff;
  if (ix >= 0x3ff00000) { /* |x|>= 1 */
    if (((ix - 0x3ff00000) | __LO(x)) == 0)
      /* asin(1)=+-pi/2 with inexact */
      return x * pio2_hi + x * pio2_lo;
    return NAN;                 /* asin(|x|>1) is NaN */
  } else if (ix < 0x3fe00000) { /* |x|<0.5 */
    if (ix < 0x3e400000) {      /* if |x| < 2**-27 */
      if (big + x > one)
        return x; /* return x with inexact if x!=0*/
    } else
      t = x * x;
    p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
    q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
    w = p / q;
    return x + x * w;
  }
  /* 1> |x|>= 0.5 */
  w = one - fabs(x);
  t = w * 0.5;
  p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
  q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
  s = sqrt(t);
  if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
    w = p / q;
    t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
  } else {
    w = s;
    __LO(w) = 0;
    c = (t - w * w) / (s + w);
    r = p / q;
    p = 2.0 * s * r - (pio2_lo - 2.0 * c);
    q = pio4_hi - 2.0 * w;
    t = pio4_hi - (p - q);
  }
  if (hx > 0)
    return t;
  else
    return -t;
}