e_pow.c

/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2016 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                                                        */
/*             power1                                                      */
/*             my_log2                                                     */
/*             log1                                                        */
/*             checkint                                                    */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h                             */
/*               halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c       */
/*                          uexp.c  upow.c				   */
/*               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.h>

#ifndef SECTION
# define SECTION
#endif

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

double __exp1 (double x, double xx, double error);
static double log1 (double x, double *delta, double *error);
static double my_log2 (double x, double *delta, double *error);
double __slowpow (double x, double y, double z);
static double power1 (double x, double y);
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, error, 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))
	return 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, &error);	/* 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;
	error = error * fabs (y);
	t = __exp1 (a1, a2, 1.9e16 * error);	/* return -10 or 0 if wasn't computed exactly */
	retval = (t > 0) ? t : power1 (x, y);
      }

      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;
      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;
  if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))	/* NaN */
    return x == 1.0 ? 1.0 : 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 x^y using more accurate but more slow log routine.  */
static double
SECTION
power1 (double x, double y)
{
  double z, a, aa, error, t, a1, a2, y1, y2;
  z = my_log2 (x, &aa, &error);
  t = y * CN;
  y1 = t - (t - y);
  y2 = y - y1;
  t = z * CN;
  a1 = t - (t - z);
  a2 = z - a1;
  a = y * z;
  aa = ((y1 * a1 - a) + y1 * a2 + y2 * a1) + y2 * a2 + aa * y;
  a1 = a + aa;
  a2 = (a - a1) + aa;
  error = error * fabs (y);
  t = __exp1 (a1, a2, 1.9e16 * error);
  return (t >= 0) ? t : __slowpow (x, y, z);
}

/* Compute log(x) (x is left argument). The result is the returned double + the
   parameter DELTA.  The result is bounded by ERROR.  */
static double
SECTION
log1 (double x, double *delta, double *error)
{
  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];
  *error = 0;
  *delta = 0;
  if (m < 0x00100000)		/*  1<x<2^-1007 */
    {
      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)	/* nx = 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;
	  *error = 1.0e-21 * fabs (t);
	  *delta = (e1 - res) + e2;
	  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;
	  *error = 1.0e-24;
	  *delta = (t1 - res) + t2;
	  return res;
	}
    }				/* nx = 0 */
  else				/* nx != 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;
      *error = 1.0e-21;
      *delta = (t1 - res) + t2;
      return res;
    }				/* nx != 0   */
}

/* Slower but more accurate routine of log.  The returned result is double +
   DELTA.  The result is bounded by ERROR.  */
static double
SECTION
my_log2 (double x, double *delta, double *error)
{
  unsigned int i, j;
  int m;
  double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
  double ou1, ou2, lu1, lu2, ov, lv1, lv2, a, a1, a2;
  double y, yy, z, zz, j1, j2, j7, j8;
#ifndef DLA_FMS
  double j3, j4, j5, j6;
#endif
  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];
  *error = 0;
  *delta = 0;
  add = 0;
  if (m < 0x00100000)
    {				/* x < 2^-1022 */
      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;
  /*------------------------------------- |x-1| < 2**-11-------------------------------  */
  if ((two52.i[LOW_HALF] == 1023) && (i == 1200))
    {
      t = x - 1.0;
      EMULV (t, s3, y, yy, j1, j2, j3, j4, j5);
      ADD2 (-0.5, 0, y, yy, z, zz, j1, j2);
      MUL2 (t, 0, z, zz, y, yy, j1, j2, j3, j4, j5, j6, j7, j8);
      MUL2 (t, 0, y, yy, z, zz, j1, j2, j3, j4, j5, j6, j7, j8);

      e1 = t + z;
      e2 = ((((t - e1) + z) + zz) + t * t * t
	    * (ss3 + t * (s4 + t * (s5 + t * (s6 + t * (s7 + t * s8))))));
      res = e1 + e2;
      *error = 1.0e-25 * fabs (t);
      *delta = (e1 - res) + e2;
      return res;
    }
  /*----------------------------- |x-1| > 2**-11  --------------------------  */
  else
    {				/*Computing log(x) according to log table                        */
      nx = (two52.x - two52e.x) + add;
      ou1 = ui.x[i];
      ou2 = ui.x[i + 1];
      lu1 = ui.x[i + 2];
      lu2 = ui.x[i + 3];
      v.x = u.x * (ou1 + ou2) + bigv.x;
      vv = v.x - bigv.x;
      j = v.i[LOW_HALF] & 0x0007ffff;
      j = j + j + j;
      eps = u.x - uu * vv;
      ov = vj.x[j];
      lv1 = vj.x[j + 1];
      lv2 = vj.x[j + 2];
      a = (ou1 + ou2) * (1.0 + ov);
      a1 = (a + 1.0e10) - 1.0e10;
      a2 = a * (1.0 - a1 * uu * vv);
      e1 = eps * a1;
      e2 = eps * a2;
      e = e1 + e2;
      e2 = (e1 - e) + e2;
      t = nx * ln2a.x + lu1 + lv1;
      t1 = t + e;
      t2 = ((((t - t1) + e) + (lu2 + lv2 + nx * ln2b.x + e2)) + e * e
	    * (p2 + e * (p3 + e * p4)));
      res = t1 + t2;
      *error = 1.0e-27;
      *delta = (t1 - res) + t2;
      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, 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))
	return 0;		/* if not integer */
      return (n << (k - 21)) ? -1 : 1;
    }
  if (n)
    return 0;			/*if  not integer */
  if (k == 20)
    return (m & 1) ? -1 : 1;
  if (m << (k + 12))
    return 0;
  return (m << (k + 11)) ? -1 : 1;
}