132 lines
3.4 KiB
C
132 lines
3.4 KiB
C
/* $OpenBSD: e_expl.c,v 1.3 2013/11/12 20:35:19 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, 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 +-10000 50000 1.12e-19 2.81e-20
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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 <openlibm_math.h>
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#include "math_private.h"
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static long double P[3] = {
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1.2617719307481059087798E-4L,
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3.0299440770744196129956E-2L,
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9.9999999999999999991025E-1L,
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};
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static long double Q[4] = {
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3.0019850513866445504159E-6L,
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2.5244834034968410419224E-3L,
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2.2726554820815502876593E-1L,
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2.0000000000000000000897E0L,
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};
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static const long double C1 = 6.9314575195312500000000E-1L;
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static const long double C2 = 1.4286068203094172321215E-6L;
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static const long double MAXLOGL = 1.1356523406294143949492E4L;
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static const long double MINLOGL = -1.13994985314888605586758E4L;
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static const long double LOG2EL = 1.4426950408889634073599E0L;
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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( isnan(x) )
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return(x);
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if( x > MAXLOGL)
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return( INFINITY );
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if( x < MINLOGL )
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return(0.0L);
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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, 2 );
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x = px/( __polevll( xx, Q, 3 ) - px );
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x = 1.0L + ldexpl( x, 1 );
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x = ldexpl( x, n );
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return(x);
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}
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