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- /* $OpenBSD: e_expl.c,v 1.3 2013/11/12 20:35:18 martynas Exp $ */
- /*
- * Copyright (c) 2008 Stephen L. Moshier <[email protected]>
- *
- * 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 <openlibm_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);
- }
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