146 lines
4.2 KiB
C
146 lines
4.2 KiB
C
/* $OpenBSD: e_expl.c,v 1.3 2013/11/12 20:35:18 martynas Exp $ */
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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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/* expl.c
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*
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* Exponential function, 128-bit long double precision
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*
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, expl();
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*
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* y = expl( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns e (2.71828...) raised to the x power.
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*
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* Range reduction is accomplished by separating the argument
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* into an integer k and fraction f such that
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*
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* x k f
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* e = 2 e.
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*
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* A Pade' form of degree 2/3 is used to approximate exp(f) - 1
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* in the basic range [-0.5 ln 2, 0.5 ln 2].
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE +-MAXLOG 100,000 2.6e-34 8.6e-35
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*
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*
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* Error amplification in the exponential function can be
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* a serious matter. The error propagation involves
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* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
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* which shows that a 1 lsb error in representing X produces
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* a relative error of X times 1 lsb in the function.
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* While the routine gives an accurate result for arguments
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* that are exactly represented by a long double precision
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* computer number, the result contains amplified roundoff
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* error for large arguments not exactly represented.
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*
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*
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* ERROR MESSAGES:
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*
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* message condition value returned
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* exp underflow x < MINLOG 0.0
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* exp overflow x > MAXLOG MAXNUM
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*
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*/
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/* Exponential function */
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#include <float.h>
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#include <openlibm_math.h>
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#include "math_private.h"
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/* Pade' coefficients for exp(x) - 1
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Theoretical peak relative error = 2.2e-37,
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relative peak error spread = 9.2e-38
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*/
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static long double P[5] = {
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3.279723985560247033712687707263393506266E-10L,
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6.141506007208645008909088812338454698548E-7L,
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2.708775201978218837374512615596512792224E-4L,
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3.508710990737834361215404761139478627390E-2L,
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9.999999999999999999999999999999999998502E-1L
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};
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static long double Q[6] = {
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2.980756652081995192255342779918052538681E-12L,
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1.771372078166251484503904874657985291164E-8L,
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1.504792651814944826817779302637284053660E-5L,
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3.611828913847589925056132680618007270344E-3L,
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2.368408864814233538909747618894558968880E-1L,
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2.000000000000000000000000000000000000150E0L
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};
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/* C1 + C2 = ln 2 */
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static const long double C1 = -6.93145751953125E-1L;
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static const long double C2 = -1.428606820309417232121458176568075500134E-6L;
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static const long double LOG2EL = 1.442695040888963407359924681001892137426646L;
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static const long double MAXLOGL = 1.1356523406294143949491931077970764891253E4L;
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static const long double MINLOGL = -1.143276959615573793352782661133116431383730e4L;
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static const long double huge = 0x1p10000L;
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#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
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static const long double twom10000 = 0x1p-10000L;
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#else
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static volatile long double twom10000 = 0x1p-10000L;
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#endif
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long double
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expl(long double x)
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{
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long double px, xx;
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int n;
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if( x > MAXLOGL)
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return (huge*huge); /* overflow */
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if( x < MINLOGL )
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return (twom10000*twom10000); /* underflow */
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/* Express e**x = e**g 2**n
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* = e**g e**( n loge(2) )
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* = e**( g + n loge(2) )
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*/
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px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
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n = px;
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x += px * C1;
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x += px * C2;
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/* rational approximation for exponential
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* of the fractional part:
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* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
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*/
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xx = x * x;
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px = x * __polevll( xx, P, 4 );
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xx = __polevll( xx, Q, 5 );
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x = px/( xx - px );
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x = 1.0L + x + x;
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x = ldexpl( x, n );
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return(x);
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}
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