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 are
  Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  and are incorporated herein by permission of the author.  The author
  reserves the right to distribute this material elsewhere under different
  copying permissions.  These modifications are distributed here under
  the following terms:

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, see
    <https://www.gnu.org/licenses/>.  */

/* __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 <float.h>
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <libm-alias-finite.h>

static const long double
  one = 1.0L,
  huge = 1.0e+4932L,
  pio2_hi = 0x1.921fb54442d1846ap+0L, /* pi/2 rounded to nearest to 64
					 bits.  */
  pio2_lo = -0x7.6733ae8fe47c65d8p-68L, /* pi/2 - pio2_hi rounded to
					   nearest to 64 bits.  */
  pio4_hi = 0xc.90fdaa22168c235p-4L, /* pi/4 rounded to nearest to 64
					bits.  */

	/* 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 */

long double
__ieee754_asinl (long double x)
{
  long double t, w, p, q, c, r, s;
  int32_t ix;
  uint32_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 (ix == 0x3fff8000 && ((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 */
	  math_check_force_underflow (x);
	  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 = 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;
}
libm_alias_finite (__ieee754_asinl, __asinl)