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

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/*
* 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:uexp.c */
/* */
/* FUNCTION:uexp */
/* exp1 */
/* */
/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h */
/* */
/* An ultimate exp routine. Given an IEEE double machine number x */
/* it computes an almost correctly rounded (to nearest) value of e^x */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/***************************************************************************/
#include <math.h>
#include "endian.h"
#include "uexp.h"
#include "mydefs.h"
#include "MathLib.h"
#include "uexp.tbl"
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
#include <fenv.h>
#include <float.h>
#include "eexp.tbl"
#ifndef SECTION
# define SECTION
#endif
double
SECTION
__ieee754_exp (double x)
{
double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
double z;
mynumber junk1, junk2, binexp = {{0, 0}};
int4 i, j, m, n, ex;
int4 k;
double retval;
{
SET_RESTORE_ROUND (FE_TONEAREST);
junk1.x = x;
m = junk1.i[HIGH_HALF];
n = m & hugeint;
if (n < 0x3ff0a2b2) /* |x| < 1.03972053527832 */
{
if (n < 0x3f862e42) /* |x| < 3/2 ln 2 */
{
if (n < 0x3ed00000) /* |x| < 1/64 ln 2 */
{
if (n < 0x3e300000) /* |x| < 2^18 */
{
retval = one + junk1.x;
goto ret;
}
retval = one + junk1.x * (one + half * junk1.x);
goto ret;
}
t = junk1.x * junk1.x;
retval = junk1.x + (t * (half + junk1.x * t2) +
(t * t) * (t3 + junk1.x * t4 + t * t5));
retval = one + retval;
goto ret;
}
/* Find the multiple of 2^-6 nearest x. */
k = n >> 20;
j = (0x00100000 | (n & 0x000fffff)) >> (0x40c - k);
j = (j - 1) & ~1;
if (m < 0)
j += 134;
z = junk1.x - TBL2[j];
t = z * z;
retval = z + (t * (half + (z * t2))
+ (t * t) * (t3 + z * t4 + t * t5));
retval = TBL2[j + 1] + TBL2[j + 1] * retval;
goto ret;
}
if (n < bigint) /* && |x| >= 1.03972053527832 */
{
y = x * log2e.x + three51.x;
bexp = y - three51.x; /* multiply the result by 2**bexp */
junk1.x = y;
eps = bexp * ln_two2.x; /* x = bexp*ln(2) + t - eps */
t = x - bexp * ln_two1.x;
y = t + three33.x;
base = y - three33.x; /* t rounded to a multiple of 2**-18 */
junk2.x = y;
del = (t - base) - eps; /* x = bexp*ln(2) + base + del */
eps = del + del * del * (p3.x * del + p2.x);
binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;
i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
j = (junk2.i[LOW_HALF] & 511) << 1;
al = coar.x[i] * fine.x[j];
bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
+ coar.x[i + 1] * fine.x[j + 1]);
rem = (bet + bet * eps) + al * eps;
res = al + rem;
/* Maximum relative error is 7.8e-22 (70.1 bits).
Maximum ULP error is 0.500007. */
retval = res * binexp.x;
goto ret;
}
if (n >= badint)
{
if (n > infint)
{
retval = x + x;
goto ret;
} /* x is NaN */
if (n < infint)
{
if (x > 0)
goto ret_huge;
else
goto ret_tiny;
}
/* x is finite, cause either overflow or underflow */
if (junk1.i[LOW_HALF] != 0)
{
retval = x + x;
goto ret;
} /* x is NaN */
retval = (x > 0) ? inf.x : zero; /* |x| = inf; return either inf or 0 */
goto ret;
}
y = x * log2e.x + three51.x;
bexp = y - three51.x;
junk1.x = y;
eps = bexp * ln_two2.x;
t = x - bexp * ln_two1.x;
y = t + three33.x;
base = y - three33.x;
junk2.x = y;
del = (t - base) - eps;
eps = del + del * del * (p3.x * del + p2.x);
i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
j = (junk2.i[LOW_HALF] & 511) << 1;
al = coar.x[i] * fine.x[j];
bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
+ coar.x[i + 1] * fine.x[j + 1]);
rem = (bet + bet * eps) + al * eps;
res = al + rem;
cor = (al - res) + rem;
if (m >> 31)
{
ex = junk1.i[LOW_HALF];
if (res < 1.0)
{
res += res;
cor += cor;
ex -= 1;
}
if (ex >= -1022)
{
binexp.i[HIGH_HALF] = (1023 + ex) << 20;
/* Does not underflow: res >= 1.0, binexp >= 0x1p-1022
Maximum relative error is 7.8e-22 (70.1 bits).
Maximum ULP error is 0.500007. */
retval = res * binexp.x;
goto ret;
}
ex = -(1022 + ex);
binexp.i[HIGH_HALF] = (1023 - ex) << 20;
res *= binexp.x;
cor *= binexp.x;
t = 1.0 + res;
y = ((1.0 - t) + res) + cor;
res = t + y;
/* Maximum ULP error is 0.5000035. */
binexp.i[HIGH_HALF] = 0x00100000;
retval = (res - 1.0) * binexp.x;
if (retval < DBL_MIN)
{
double force_underflow = tiny * tiny;
math_force_eval (force_underflow);
}
if (retval == 0)
goto ret_tiny;
goto ret;
}
else
{
binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
/* Maximum relative error is 7.8e-22 (70.1 bits).
Maximum ULP error is 0.500007. */
retval = res * binexp.x * t256.x;
if (isinf (retval))
goto ret_huge;
else
goto ret;
}
}
ret:
return retval;
ret_huge:
return hhuge * hhuge;
ret_tiny:
return tiny * tiny;
}
#ifndef __ieee754_exp
strong_alias (__ieee754_exp, __exp_finite)
#endif
/* Compute e^(x+xx). */
double
SECTION
__exp1 (double x, double xx)
{
double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
mynumber junk1, junk2, binexp = {{0, 0}};
int4 i, j, m, n, ex;
junk1.x = x;
m = junk1.i[HIGH_HALF];
n = m & hugeint; /* no sign */
/* fabs (x) > 5.551112e-17 and fabs (x) < 7.080010e+02. */
if (n > smallint && n < bigint)
{
y = x * log2e.x + three51.x;
bexp = y - three51.x; /* multiply the result by 2**bexp */
junk1.x = y;
eps = bexp * ln_two2.x; /* x = bexp*ln(2) + t - eps */
t = x - bexp * ln_two1.x;
y = t + three33.x;
base = y - three33.x; /* t rounded to a multiple of 2**-18 */
junk2.x = y;
del = (t - base) + (xx - eps); /* x = bexp*ln(2) + base + del */
eps = del + del * del * (p3.x * del + p2.x);
binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;
i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
j = (junk2.i[LOW_HALF] & 511) << 1;
al = coar.x[i] * fine.x[j];
bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
+ coar.x[i + 1] * fine.x[j + 1]);
rem = (bet + bet * eps) + al * eps;
res = al + rem;
/* Maximum relative error before rounding is 8.8e-22 (69.9 bits).
Maximum ULP error is 0.500008. */
return res * binexp.x;
}
if (n <= smallint)
return 1.0; /* if x->0 e^x=1 */
if (n >= badint)
{
if (n > infint)
return (zero / zero); /* x is NaN, return invalid */
if (n < infint)
return ((x > 0) ? (hhuge * hhuge) : (tiny * tiny));
/* x is finite, cause either overflow or underflow */
if (junk1.i[LOW_HALF] != 0)
return (zero / zero); /* x is NaN */
return ((x > 0) ? inf.x : zero); /* |x| = inf; return either inf or 0 */
}
y = x * log2e.x + three51.x;
bexp = y - three51.x;
junk1.x = y;
eps = bexp * ln_two2.x;
t = x - bexp * ln_two1.x;
y = t + three33.x;
base = y - three33.x;
junk2.x = y;
del = (t - base) + (xx - eps);
eps = del + del * del * (p3.x * del + p2.x);
i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
j = (junk2.i[LOW_HALF] & 511) << 1;
al = coar.x[i] * fine.x[j];
bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
+ coar.x[i + 1] * fine.x[j + 1]);
rem = (bet + bet * eps) + al * eps;
res = al + rem;
cor = (al - res) + rem;
if (m >> 31)
{
/* x < 0. */
ex = junk1.i[LOW_HALF];
if (res < 1.0)
{
res += res;
cor += cor;
ex -= 1;
}
if (ex >= -1022)
{
binexp.i[HIGH_HALF] = (1023 + ex) << 20;
/* Maximum ULP error is 0.500008. */
return res * binexp.x;
}
/* Denormal case - ex < -1022. */
ex = -(1022 + ex);
binexp.i[HIGH_HALF] = (1023 - ex) << 20;
res *= binexp.x;
cor *= binexp.x;
t = 1.0 + res;
y = ((1.0 - t) + res) + cor;
res = t + y;
binexp.i[HIGH_HALF] = 0x00100000;
/* Maximum ULP error is 0.500004. */
return (res - 1.0) * binexp.x;
}
else
{
binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
/* Maximum ULP error is 0.500008. */
return res * binexp.x * t256.x;
}
}
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