glibc/sysdeps/ieee754/dbl-64/e_pow.c

/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2018 Free Software Foundation, Inc.
 *
 * This program 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 program 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 program; if not, see <http://www.gnu.org/licenses/>.
 */
/***************************************************************************/
/*  MODULE_NAME: upow.c                                                    */
/*                                                                         */
/*  FUNCTIONS: upow                                                        */
/*             log1                                                        */
/*             checkint                                                    */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h                             */
/*               root.tbl uexp.tbl upow.tbl                                */
/* An ultimate power routine. Given two IEEE double machine numbers y,x    */
/* it computes the correctly rounded (to nearest) value of x^y.            */
/* Assumption: Machine arithmetic operations are performed in              */
/* round to nearest mode of IEEE 754 standard.                             */
/*                                                                         */
/***************************************************************************/
#include <math.h>
#include "endian.h"
#include "upow.h"
#include <dla.h>
#include "mydefs.h"
#include "MathLib.h"
#include "upow.tbl"
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <fenv.h>

#ifndef SECTION
# define SECTION
#endif

static const double huge = 1.0e300, tiny = 1.0e-300;

double __exp1 (double x, double xx);
static double log1 (double x, double *delta);
static int checkint (double x);

/* An ultimate power routine. Given two IEEE double machine numbers y, x it
   computes the correctly rounded (to nearest) value of X^y.  */
double
SECTION
__ieee754_pow (double x, double y)
{
  double z, a, aa, t, a1, a2, y1, y2;
  mynumber u, v;
  int k;
  int4 qx, qy;
  v.x = y;
  u.x = x;
  if (v.i[LOW_HALF] == 0)
    {				/* of y */
      qx = u.i[HIGH_HALF] & 0x7fffffff;
      /* Is x a NaN?  */
      if ((((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
	  && (y != 0 || issignaling (x)))
	return x + x;
      if (y == 1.0)
	return x;
      if (y == 2.0)
	return x * x;
      if (y == -1.0)
	return 1.0 / x;
      if (y == 0)
	return 1.0;
    }
  /* else */
  if (((u.i[HIGH_HALF] > 0 && u.i[HIGH_HALF] < 0x7ff00000) ||	/* x>0 and not x->0 */
       (u.i[HIGH_HALF] == 0 && u.i[LOW_HALF] != 0)) &&
      /*   2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
      (v.i[HIGH_HALF] & 0x7fffffff) < 0x4ff00000)
    {				/* if y<-1 or y>1   */
      double retval;

      {
	SET_RESTORE_ROUND (FE_TONEAREST);

	/* Avoid internal underflow for tiny y.  The exact value of y does
	   not matter if |y| <= 2**-64.  */
	if (fabs (y) < 0x1p-64)
	  y = y < 0 ? -0x1p-64 : 0x1p-64;
	z = log1 (x, &aa);	/* x^y  =e^(y log (X)) */
	t = y * CN;
	y1 = t - (t - y);
	y2 = y - y1;
	t = z * CN;
	a1 = t - (t - z);
	a2 = (z - a1) + aa;
	a = y1 * a1;
	aa = y2 * a1 + y * a2;
	a1 = a + aa;
	a2 = (a - a1) + aa;

	/* Maximum relative error RElog of log1 is 1.0e-21 (69.7 bits).
	   Maximum relative error REexp of __exp1 is 8.8e-22 (69.9 bits).
	   We actually compute exp ((1 + RElog) * log (x) * y) * (1 + REexp).
	   Since RElog/REexp are tiny and log (x) * y is at most log (DBL_MAX),
	   this is equivalent to pow (x, y) * (1 + 710 * RElog + REexp).
	   So the relative error is 710 * 1.0e-21 + 8.8e-22 = 7.1e-19
	   (60.2 bits).  The worst-case ULP error is 0.5064.  */

	retval = __exp1 (a1, a2);
      }

      if (isinf (retval))
	retval = huge * huge;
      else if (retval == 0)
	retval = tiny * tiny;
      else
	math_check_force_underflow_nonneg (retval);
      return retval;
    }

  if (x == 0)
    {
      if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
	  || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)	/* NaN */
	return y + y;
      if (fabs (y) > 1.0e20)
	return (y > 0) ? 0 : 1.0 / 0.0;
      k = checkint (y);
      if (k == -1)
	return y < 0 ? 1.0 / x : x;
      else
	return y < 0 ? 1.0 / 0.0 : 0.0;	/* return 0 */
    }

  qx = u.i[HIGH_HALF] & 0x7fffffff;	/*   no sign   */
  qy = v.i[HIGH_HALF] & 0x7fffffff;	/*   no sign   */

  if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0))	/* NaN */
    return x + y;
  if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))	/* NaN */
    return x == 1.0 && !issignaling (y) ? 1.0 : y + y;

  /* if x<0 */
  if (u.i[HIGH_HALF] < 0)
    {
      k = checkint (y);
      if (k == 0)
	{
	  if (qy == 0x7ff00000)
	    {
	      if (x == -1.0)
		return 1.0;
	      else if (x > -1.0)
		return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
	      else
		return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
	    }
	  else if (qx == 0x7ff00000)
	    return y < 0 ? 0.0 : INF.x;
	  return (x - x) / (x - x);	/* y not integer and x<0 */
	}
      else if (qx == 0x7ff00000)
	{
	  if (k < 0)
	    return y < 0 ? nZERO.x : nINF.x;
	  else
	    return y < 0 ? 0.0 : INF.x;
	}
      /* if y even or odd */
      if (k == 1)
	return __ieee754_pow (-x, y);
      else
	{
	  double retval;
	  {
	    SET_RESTORE_ROUND (FE_TONEAREST);
	    retval = -__ieee754_pow (-x, y);
	  }
	  if (isinf (retval))
	    retval = -huge * huge;
	  else if (retval == 0)
	    retval = -tiny * tiny;
	  return retval;
	}
    }
  /* x>0 */

  if (qx == 0x7ff00000)		/* x= 2^-0x3ff */
    return y > 0 ? x : 0;

  if (qy > 0x45f00000 && qy < 0x7ff00000)
    {
      if (x == 1.0)
	return 1.0;
      if (y > 0)
	return (x > 1.0) ? huge * huge : tiny * tiny;
      if (y < 0)
	return (x < 1.0) ? huge * huge : tiny * tiny;
    }

  if (x == 1.0)
    return 1.0;
  if (y > 0)
    return (x > 1.0) ? INF.x : 0;
  if (y < 0)
    return (x < 1.0) ? INF.x : 0;
  return 0;			/* unreachable, to make the compiler happy */
}

#ifndef __ieee754_pow
strong_alias (__ieee754_pow, __pow_finite)
#endif

/* Compute log(x) (x is left argument). The result is the returned double + the
   parameter DELTA.  */
static double
SECTION
log1 (double x, double *delta)
{
  unsigned int i, j;
  int m;
  double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
  mynumber u, v;
#ifdef BIG_ENDI
  mynumber /**/ two52 = {{0x43300000, 0x00000000}};	/* 2**52  */
#else
# ifdef LITTLE_ENDI
  mynumber /**/ two52 = {{0x00000000, 0x43300000}};	/* 2**52  */
# endif
#endif

  u.x = x;
  m = u.i[HIGH_HALF];
  if (m < 0x00100000)		/* Handle denormal x.  */
    {
      x = x * t52.x;
      add = -52.0;
      u.x = x;
      m = u.i[HIGH_HALF];
    }

  if ((m & 0x000fffff) < 0x0006a09e)
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
      two52.i[LOW_HALF] = (m >> 20);
    }
  else
    {
      u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
      two52.i[LOW_HALF] = (m >> 20) + 1;
    }

  v.x = u.x + bigu.x;
  uu = v.x - bigu.x;
  i = (v.i[LOW_HALF] & 0x000003ff) << 2;
  if (two52.i[LOW_HALF] == 1023)	/* Exponent of x is 0.  */
    {
      if (i > 1192 && i < 1208)	/* |x-1| < 1.5*2**-10  */
	{
	  t = x - 1.0;
	  t1 = (t + 5.0e6) - 5.0e6;
	  t2 = t - t1;
	  e1 = t - 0.5 * t1 * t1;
	  e2 = (t * t * t * (r3 + t * (r4 + t * (r5 + t * (r6 + t
							   * (r7 + t * r8)))))
		- 0.5 * t2 * (t + t1));
	  res = e1 + e2;
	  *delta = (e1 - res) + e2;
	  /* Max relative error is 1.464844e-24, so accurate to 79.1 bits.  */
	  return res;
	}			/* |x-1| < 1.5*2**-10  */
      else
	{
	  v.x = u.x * (ui.x[i] + ui.x[i + 1]) + bigv.x;
	  vv = v.x - bigv.x;
	  j = v.i[LOW_HALF] & 0x0007ffff;
	  j = j + j + j;
	  eps = u.x - uu * vv;
	  e1 = eps * ui.x[i];
	  e2 = eps * (ui.x[i + 1] + vj.x[j] * (ui.x[i] + ui.x[i + 1]));
	  e = e1 + e2;
	  e2 = ((e1 - e) + e2);
	  t = ui.x[i + 2] + vj.x[j + 1];
	  t1 = t + e;
	  t2 = ((((t - t1) + e) + (ui.x[i + 3] + vj.x[j + 2])) + e2 + e * e
		* (p2 + e * (p3 + e * p4)));
	  res = t1 + t2;
	  *delta = (t1 - res) + t2;
	  /* Max relative error is 1.0e-24, so accurate to 79.7 bits.  */
	  return res;
	}
    }
  else				/* Exponent of x != 0.  */
    {
      eps = u.x - uu;
      nx = (two52.x - two52e.x) + add;
      e1 = eps * ui.x[i];
      e2 = eps * ui.x[i + 1];
      e = e1 + e2;
      e2 = (e1 - e) + e2;
      t = nx * ln2a.x + ui.x[i + 2];
      t1 = t + e;
      t2 = ((((t - t1) + e) + nx * ln2b.x + ui.x[i + 3] + e2) + e * e
	    * (q2 + e * (q3 + e * (q4 + e * (q5 + e * q6)))));
      res = t1 + t2;
      *delta = (t1 - res) + t2;
      /* Max relative error is 1.0e-21, so accurate to 69.7 bits.  */
      return res;
    }
}


/* This function receives a double x and checks if it is an integer.  If not,
   it returns 0, else it returns 1 if even or -1 if odd.  */
static int
SECTION
checkint (double x)
{
  union
  {
    int4 i[2];
    double x;
  } u;
  int k;
  unsigned int m, n;
  u.x = x;
  m = u.i[HIGH_HALF] & 0x7fffffff;	/* no sign */
  if (m >= 0x7ff00000)
    return 0;			/*  x is +/-inf or NaN  */
  if (m >= 0x43400000)
    return 1;			/*  |x| >= 2**53   */
  if (m < 0x40000000)
    return 0;			/* |x| < 2,  can not be 0 or 1  */
  n = u.i[LOW_HALF];
  k = (m >> 20) - 1023;		/*  1 <= k <= 52   */
  if (k == 52)
    return (n & 1) ? -1 : 1;	/* odd or even */
  if (k > 20)
    {
      if (n << (k - 20) != 0)
	return 0;		/* if not integer */
      return (n << (k - 21) != 0) ? -1 : 1;
    }
  if (n)
    return 0;			/*if  not integer */
  if (k == 20)
    return (m & 1) ? -1 : 1;
  if (m << (k + 12) != 0)
    return 0;
  return (m << (k + 11) != 0) ? -1 : 1;
}