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

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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2014 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 "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 (ABS (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 * ABS (y);
t = __exp1 (a1, a2, 1.9e16 * error); /* return -10 or 0 if wasn't computed exactly */
retval = (t > 0) ? t : power1 (x, y);
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 (ABS (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 */
return (k == 1) ? __ieee754_pow (-x, y) : -__ieee754_pow (-x, y);
}
/* 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 * ABS (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)
{
int i, j, 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 * ABS (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)
{
int i, j, 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 * ABS (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;
}
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