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/* $OpenBSD: e_expl.c,v 1.4 2016/09/12 19:47:02 guenther Exp $ */
/*
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
*
* Permission to use, copy, modify, and distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
*/
/* expl.c
*
* Exponential function, 128-bit long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, expl();
*
* y = expl( x );
*
*
*
* DESCRIPTION:
*
* Returns e (2.71828...) raised to the x power.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
*
* x k f
* e = 2 e.
*
* A Pade' form of degree 2/3 is used to approximate exp(f) - 1
* in the basic range [-0.5 ln 2, 0.5 ln 2].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-MAXLOG 100,000 2.6e-34 8.6e-35
*
*
* Error amplification in the exponential function can be
* a serious matter. The error propagation involves
* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
* which shows that a 1 lsb error in representing X produces
* a relative error of X times 1 lsb in the function.
* While the routine gives an accurate result for arguments
* that are exactly represented by a long double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < MINLOG 0.0
* exp overflow x > MAXLOG MAXNUM
*
*/
/* Exponential function */
#include <float.h>
#include <math.h>
#include "math_private.h"
/* Pade' coefficients for exp(x) - 1
Theoretical peak relative error = 2.2e-37,
relative peak error spread = 9.2e-38
*/
static long double P[5] = {
3.279723985560247033712687707263393506266E-10L,
6.141506007208645008909088812338454698548E-7L,
2.708775201978218837374512615596512792224E-4L,
3.508710990737834361215404761139478627390E-2L,
9.999999999999999999999999999999999998502E-1L
};
static long double Q[6] = {
2.980756652081995192255342779918052538681E-12L,
1.771372078166251484503904874657985291164E-8L,
1.504792651814944826817779302637284053660E-5L,
3.611828913847589925056132680618007270344E-3L,
2.368408864814233538909747618894558968880E-1L,
2.000000000000000000000000000000000000150E0L
};
/* C1 + C2 = ln 2 */
static const long double C1 = -6.93145751953125E-1L;
static const long double C2 = -1.428606820309417232121458176568075500134E-6L;
static const long double LOG2EL = 1.442695040888963407359924681001892137426646L;
static const long double MAXLOGL = 1.1356523406294143949491931077970764891253E4L;
static const long double MINLOGL = -1.143276959615573793352782661133116431383730e4L;
static const long double huge = 0x1p10000L;
#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
static const long double twom10000 = 0x1p-10000L;
#else
static volatile long double twom10000 = 0x1p-10000L;
#endif
long double
expl(long double x)
{
long double px, xx;
int n;
if( x > MAXLOGL)
return (huge*huge); /* overflow */
if( x < MINLOGL )
return (twom10000*twom10000); /* underflow */
/* Express e**x = e**g 2**n
* = e**g e**( n loge(2) )
* = e**( g + n loge(2) )
*/
px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
n = px;
x += px * C1;
x += px * C2;
/* rational approximation for exponential
* of the fractional part:
* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*/
xx = x * x;
px = x * __polevll( xx, P, 4 );
xx = __polevll( xx, Q, 5 );
x = px/( xx - px );
x = 1.0L + x + x;
x = ldexpl( x, n );
return(x);
}
DEF_STD(expl);
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