glibc/sysdeps/ieee754/ldbl-128/e_acosl.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>
   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, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */

/* __ieee754_acosl(x)
 * Method :
 *      acos(x)  = pi/2 - asin(x)
 *      acos(-x) = pi/2 + asin(x)
 * For |x| <= 0.375
 *      acos(x) = pi/2 - asin(x)
 * Between .375 and .5 the approximation is
 *      acos(0.4375 + x) = acos(0.4375) + x P(x) / Q(x)
 * Between .5 and .625 the approximation is
 *      acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
 * For x > 0.625,
 *      acos(x) = 2 asin(sqrt((1-x)/2))
 *      computed with an extended precision square root in the leading term.
 * For x < -0.625
 *      acos(x) = pi - 2 asin(sqrt((1-|x|)/2))
 *
 * Special cases:
 *      if x is NaN, return x itself;
 *      if |x|>1, return NaN with invalid signal.
 *
 * Functions needed: __ieee754_sqrtl.
 */

#include "math.h"
#include "math_private.h"

#ifdef __STDC__
static const long double
#else
static long double
#endif
  one = 1.0L,
  pio2_hi = 1.5707963267948966192313216916397514420986L,
  pio2_lo = 4.3359050650618905123985220130216759843812E-35L,

  /* acos(0.5625 + x) = acos(0.5625) + x rS(x) / sS(x)
     -0.0625 <= x <= 0.0625
     peak relative error 3.3e-35  */

  rS0 =  5.619049346208901520945464704848780243887E0L,
  rS1 = -4.460504162777731472539175700169871920352E1L,
  rS2 =  1.317669505315409261479577040530751477488E2L,
  rS3 = -1.626532582423661989632442410808596009227E2L,
  rS4 =  3.144806644195158614904369445440583873264E1L,
  rS5 =  9.806674443470740708765165604769099559553E1L,
  rS6 = -5.708468492052010816555762842394927806920E1L,
  rS7 = -1.396540499232262112248553357962639431922E1L,
  rS8 =  1.126243289311910363001762058295832610344E1L,
  rS9 =  4.956179821329901954211277873774472383512E-1L,
  rS10 = -3.313227657082367169241333738391762525780E-1L,

  sS0 = -4.645814742084009935700221277307007679325E0L,
  sS1 =  3.879074822457694323970438316317961918430E1L,
  sS2 = -1.221986588013474694623973554726201001066E2L,
  sS3 =  1.658821150347718105012079876756201905822E2L,
  sS4 = -4.804379630977558197953176474426239748977E1L,
  sS5 = -1.004296417397316948114344573811562952793E2L,
  sS6 =  7.530281592861320234941101403870010111138E1L,
  sS7 =  1.270735595411673647119592092304357226607E1L,
  sS8 = -1.815144839646376500705105967064792930282E1L,
  sS9 = -7.821597334910963922204235247786840828217E-2L,
  /* 1.000000000000000000000000000000000000000E0 */

  acosr5625 = 9.7338991014954640492751132535550279812151E-1L,
  pimacosr5625 = 2.1682027434402468335351320579240000860757E0L,

  /* acos(0.4375 + x) = acos(0.4375) + x rS(x) / sS(x)
     -0.0625 <= x <= 0.0625
     peak relative error 2.1e-35  */

  P0 =  2.177690192235413635229046633751390484892E0L,
  P1 = -2.848698225706605746657192566166142909573E1L,
  P2 =  1.040076477655245590871244795403659880304E2L,
  P3 = -1.400087608918906358323551402881238180553E2L,
  P4 =  2.221047917671449176051896400503615543757E1L,
  P5 =  9.643714856395587663736110523917499638702E1L,
  P6 = -5.158406639829833829027457284942389079196E1L,
  P7 = -1.578651828337585944715290382181219741813E1L,
  P8 =  1.093632715903802870546857764647931045906E1L,
  P9 =  5.448925479898460003048760932274085300103E-1L,
  P10 = -3.315886001095605268470690485170092986337E-1L,
  Q0 = -1.958219113487162405143608843774587557016E0L,
  Q1 =  2.614577866876185080678907676023269360520E1L,
  Q2 = -9.990858606464150981009763389881793660938E1L,
  Q3 =  1.443958741356995763628660823395334281596E2L,
  Q4 = -3.206441012484232867657763518369723873129E1L,
  Q5 = -1.048560885341833443564920145642588991492E2L,
  Q6 =  6.745883931909770880159915641984874746358E1L,
  Q7 =  1.806809656342804436118449982647641392951E1L,
  Q8 = -1.770150690652438294290020775359580915464E1L,
  Q9 = -5.659156469628629327045433069052560211164E-1L,
  /* 1.000000000000000000000000000000000000000E0 */

  acosr4375 = 1.1179797320499710475919903296900511518755E0L,
  pimacosr4375 = 2.0236129215398221908706530535894517323217E0L,

  /* asin(x) = x + x^3 pS(x^2) / qS(x^2)
     0 <= x <= 0.5
     peak relative error 1.9e-35  */
  pS0 = -8.358099012470680544198472400254596543711E2L,
  pS1 =  3.674973957689619490312782828051860366493E3L,
  pS2 = -6.730729094812979665807581609853656623219E3L,
  pS3 =  6.643843795209060298375552684423454077633E3L,
  pS4 = -3.817341990928606692235481812252049415993E3L,
  pS5 =  1.284635388402653715636722822195716476156E3L,
  pS6 = -2.410736125231549204856567737329112037867E2L,
  pS7 =  2.219191969382402856557594215833622156220E1L,
  pS8 = -7.249056260830627156600112195061001036533E-1L,
  pS9 =  1.055923570937755300061509030361395604448E-3L,

  qS0 = -5.014859407482408326519083440151745519205E3L,
  qS1 =  2.430653047950480068881028451580393430537E4L,
  qS2 = -4.997904737193653607449250593976069726962E4L,
  qS3 =  5.675712336110456923807959930107347511086E4L,
  qS4 = -3.881523118339661268482937768522572588022E4L,
  qS5 =  1.634202194895541569749717032234510811216E4L,
  qS6 = -4.151452662440709301601820849901296953752E3L,
  qS7 =  5.956050864057192019085175976175695342168E2L,
  qS8 = -4.175375777334867025769346564600396877176E1L;
  /* 1.000000000000000000000000000000000000000E0 */

#ifdef __STDC__
long double
__ieee754_acosl (long double x)
#else
long double
__ieee754_acosl (x)
     long double x;
#endif
{
  long double z, r, w, p, q, s, t, f2;
  int32_t ix, sign;
  ieee854_long_double_shape_type u;

  u.value = x;
  sign = u.parts32.w0;
  ix = sign & 0x7fffffff;
  u.parts32.w0 = ix;		/* |x| */
  if (ix >= 0x3fff0000)		/* |x| >= 1 */
    {
      if (ix == 0x3fff0000
	  && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
	{			/* |x| == 1 */
	  if ((sign & 0x80000000) == 0)
	    return 0.0;		/* acos(1) = 0  */
	  else
	    return (2.0 * pio2_hi) + (2.0 * pio2_lo);	/* acos(-1)= pi */
	}
      return (x - x) / (x - x);	/* acos(|x| > 1) is NaN */
    }
  else if (ix < 0x3ffe0000)	/* |x| < 0.5 */
    {
      if (ix < 0x3fc60000)	/* |x| < 2**-57 */
	return pio2_hi + pio2_lo;
      if (ix < 0x3ffde000)	/* |x| < .4375 */
	{
	  /* Arcsine of x.  */
	  z = x * x;
	  p = (((((((((pS9 * z
		       + pS8) * z
		      + pS7) * z
		     + pS6) * z
		    + pS5) * z
		   + pS4) * z
		  + pS3) * z
		 + pS2) * z
		+ pS1) * z
	       + pS0) * z;
	  q = (((((((( z
		       + qS8) * z
		     + qS7) * z
		    + qS6) * z
		   + qS5) * z
		  + qS4) * z
		 + qS3) * z
		+ qS2) * z
	       + qS1) * z
	    + qS0;
	  r = x + x * p / q;
	  z = pio2_hi - (r - pio2_lo);
	  return z;
	}
      /* .4375 <= |x| < .5 */
      t = u.value - 0.4375L;
      p = ((((((((((P10 * t
		    + P9) * t
		   + P8) * t
		  + P7) * t
		 + P6) * t
		+ P5) * t
	       + P4) * t
	      + P3) * t
	     + P2) * t
	    + P1) * t
	   + P0) * t;

      q = (((((((((t
		   + Q9) * t
		  + Q8) * t
		 + Q7) * t
		+ Q6) * t
	       + Q5) * t
	      + Q4) * t
	     + Q3) * t
	    + Q2) * t
	   + Q1) * t
	+ Q0;
      r = p / q;
      if (sign & 0x80000000)
	r = pimacosr4375 - r;
      else
	r = acosr4375 + r;
      return r;
    }
  else if (ix < 0x3ffe4000)	/* |x| < 0.625 */
    {
      t = u.value - 0.5625L;
      p = ((((((((((rS10 * t
		    + rS9) * t
		   + rS8) * t
		  + rS7) * t
		 + rS6) * t
		+ rS5) * t
	       + rS4) * t
	      + rS3) * t
	     + rS2) * t
	    + rS1) * t
	   + rS0) * t;

      q = (((((((((t
		   + sS9) * t
		  + sS8) * t
		 + sS7) * t
		+ sS6) * t
	       + sS5) * t
	      + sS4) * t
	     + sS3) * t
	    + sS2) * t
	   + sS1) * t
	+ sS0;
      if (sign & 0x80000000)
	r = pimacosr5625 - p / q;
      else
	r = acosr5625 + p / q;
      return r;
    }
  else
    {				/* |x| >= .625 */
      z = (one - u.value) * 0.5;
      s = __ieee754_sqrtl (z);
      /* Compute an extended precision square root from
	 the Newton iteration  s -> 0.5 * (s + z / s).
         The change w from s to the improved value is
	    w = 0.5 * (s + z / s) - s  = (s^2 + z)/2s - s = (z - s^2)/2s.
          Express s = f1 + f2 where f1 * f1 is exactly representable.
	  w = (z - s^2)/2s = (z - f1^2 - 2 f1 f2 - f2^2)/2s .
          s + w has extended precision.  */
      u.value = s;
      u.parts32.w2 = 0;
      u.parts32.w3 = 0;
      f2 = s - u.value;
      w = z - u.value * u.value;
      w = w - 2.0 * u.value * f2;
      w = w - f2 * f2;
      w = w / (2.0 * s);
      /* Arcsine of s.  */
      p = (((((((((pS9 * z
		   + pS8) * z
		  + pS7) * z
		 + pS6) * z
		+ pS5) * z
	       + pS4) * z
	      + pS3) * z
	     + pS2) * z
	    + pS1) * z
	   + pS0) * z;
      q = (((((((( z
		   + qS8) * z
		 + qS7) * z
		+ qS6) * z
	       + qS5) * z
	      + qS4) * z
	     + qS3) * z
	    + qS2) * z
	   + qS1) * z
	+ qS0;
      r = s + (w + s * p / q);

      if (sign & 0x80000000)
	w = pio2_hi + (pio2_lo - r);
      else
	w = r;
      return 2.0 * w;
    }
}