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/*-
* SPDX-License-Identifier: BSD-3-Clause
*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/* EXP(X)
* RETURN THE EXPONENTIAL OF X
* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
*
* Required system supported functions:
* ldexp(x,n)
* copysign(x,y)
* isfinite(x)
*
* Method:
* 1. Argument Reduction: given the input x, find r and integer k such
* that
* x = k*ln2 + r, |r| <= 0.5*ln2.
* r will be represented as r := z+c for better accuracy.
*
* 2. Compute exp(r) by
*
* exp(r) = 1 + r + r*R1/(2-R1),
* where
* R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
*
* 3. exp(x) = 2^k * exp(r) .
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF)= 0;
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* exp(x) returns the exponential of x nearly rounded. In a test run
* with 1,156,000 random arguments on a VAX, the maximum observed
* error was 0.869 ulps (units in the last place).
*/
static const double
p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */
p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */
p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */
p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */
p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */
static const double
ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */
ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */
static const double
lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */
lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */
invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */
/* returns exp(r = x + c) for |c| < |x| with no overlap. */
static double
__exp__D(double x, double c)
{
double hi, lo, z;
int k;
if (x != x) /* x is NaN. */
return(x);
if (x <= lnhuge) {
if (x >= lntiny) {
/* argument reduction: x --> x - k*ln2 */
z = invln2 * x;
k = z + copysign(0.5, x);
/*
* Express (x + c) - k * ln2 as hi - lo.
* Let x = hi - lo rounded.
*/
hi = x - k * ln2hi; /* Exact. */
lo = k * ln2lo - c;
x = hi - lo;
/* Return 2^k*[1+x+x*c/(2+c)] */
z = x * x;
c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 +
z * p5))));
c = (x * c) / (2 - c);
return (ldexp(1 + (hi - (lo - c)), k));
} else {
/* exp(-INF) is 0. exp(-big) underflows to 0. */
return (isfinite(x) ? ldexp(1., -5000) : 0);
}
} else
/* exp(INF) is INF, exp(+big#) overflows to INF */
return (isfinite(x) ? ldexp(1., 5000) : x);
}
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