glibc/sysdeps/ieee754/ldbl-96/e_asinl.c

/*
 * ====================================================
 * 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.
 * ====================================================
 */

/*
  Long double expansions contributed by
  Stephen L. Moshier <moshier@na-net.ornl.gov>
*/

/* __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)
 *
 *	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 "math.h"
#include "math_private.h"

#ifdef __STDC__
static const long double
#else
static long double
#endif
  one = 1.0L,
  huge = 1.0e+4932L,
 pio2_hi = 1.5707963267948966192021943710788178805159986950457096099853515625L,
  pio2_lo = 2.9127320560933561582586004641843300502121E-20L,
  pio4_hi = 7.8539816339744830960109718553940894025800E-1L,

	/* coefficient for R(x^2) */

  /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
     0 <= x <= 0.5
     peak relative error 1.9e-21  */
  pS0 =  -1.008714657938491626019651170502036851607E1L,
  pS1 =   2.331460313214179572063441834101394865259E1L,
  pS2 =  -1.863169762159016144159202387315381830227E1L,
  pS3 =   5.930399351579141771077475766877674661747E0L,
  pS4 =  -6.121291917696920296944056882932695185001E-1L,
  pS5 =   3.776934006243367487161248678019350338383E-3L,

  qS0 =  -6.052287947630949712886794360635592886517E1L,
  qS1 =   1.671229145571899593737596543114258558503E2L,
  qS2 =  -1.707840117062586426144397688315411324388E2L,
  qS3 =   7.870295154902110425886636075950077640623E1L,
  qS4 =  -1.568433562487314651121702982333303458814E1L;
    /* 1.000000000000000000000000000000000000000E0 */

#ifdef __STDC__
long double
__ieee754_asinl (long double x)
#else
double
__ieee754_asinl (x)
     long double x;
#endif
{
  long double t, w, p, q, c, r, s;
  int32_t ix;
  u_int32_t se, i0, i1, k;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  ix = (ix << 16) | (i0 >> 16);
  if (ix >= 0x3fff8000)
    {				/* |x|>= 1 */
      if (((i0 - 0x80000000) | i1) == 0)
	/* asin(1)=+-pi/2 with inexact */
	return x * pio2_hi + x * pio2_lo;
      return (x - x) / (x - x);	/* asin(|x|>1) is NaN */
    }
  else if (ix < 0x3ffe8000)
    {				/* |x|<0.5 */
      if (ix < 0x3fde8000)
	{			/* if |x| < 2**-33 */
	  if (huge + 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 = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
	  w = p / q;
	  return x + x * w;
	}
    }
  /* 1> |x|>= 0.5 */
  w = one - fabsl (x);
  t = w * 0.5;
  p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
  q = qS0 + t * (qS1 + t * (qS2 + t * (qS3 + t * (qS4 + t))));
  s = __ieee754_sqrtl (t);
  if (ix >= 0x3ffef999)
    {				/* if |x| > 0.975 */
      w = p / q;
      t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
    }
  else
    {
      GET_LDOUBLE_WORDS (k, i0, i1, s);
      i1 = 0;
      SET_LDOUBLE_WORDS (w,k,i0,i1);
      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 ((se & 0x8000) == 0)
    return t;
  else
    return -t;
}