191 lines
4.4 KiB
C
191 lines
4.4 KiB
C
/* $OpenBSD: e_logl.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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/* logl.c
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*
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* Natural logarithm, 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, logl();
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*
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* y = logl( 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 the base e (2.718...) logarithm of x.
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*
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* The argument is separated into its exponent and fractional
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* parts. If the exponent is between -1 and +1, the logarithm
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* of the fraction is approximated by
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*
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* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
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*
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* Otherwise, setting z = 2(x-1)/x+1),
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*
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* log(x) = z + z**3 P(z)/Q(z).
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*
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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 0.5, 2.0 150000 8.71e-20 2.75e-20
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* IEEE exp(+-10000) 100000 5.39e-20 2.34e-20
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*
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* In the tests over the interval exp(+-10000), the logarithms
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* of the random arguments were uniformly distributed over
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* [-10000, +10000].
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*
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* ERROR MESSAGES:
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*
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* log singularity: x = 0; returns -INFINITY
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* log domain: x < 0; returns NAN
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*/
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#include <openlibm_math.h>
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#include "math_private.h"
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/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
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* 1/sqrt(2) <= x < sqrt(2)
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* Theoretical peak relative error = 2.32e-20
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*/
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static long double P[] = {
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4.5270000862445199635215E-5L,
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4.9854102823193375972212E-1L,
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6.5787325942061044846969E0L,
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2.9911919328553073277375E1L,
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6.0949667980987787057556E1L,
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5.7112963590585538103336E1L,
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2.0039553499201281259648E1L,
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};
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static long double Q[] = {
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/* 1.0000000000000000000000E0,*/
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1.5062909083469192043167E1L,
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8.3047565967967209469434E1L,
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2.2176239823732856465394E2L,
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3.0909872225312059774938E2L,
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2.1642788614495947685003E2L,
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6.0118660497603843919306E1L,
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};
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/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
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* where z = 2(x-1)/(x+1)
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* 1/sqrt(2) <= x < sqrt(2)
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* Theoretical peak relative error = 6.16e-22
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*/
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static long double R[4] = {
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1.9757429581415468984296E-3L,
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-7.1990767473014147232598E-1L,
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1.0777257190312272158094E1L,
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-3.5717684488096787370998E1L,
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};
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static long double S[4] = {
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/* 1.00000000000000000000E0L,*/
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-2.6201045551331104417768E1L,
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1.9361891836232102174846E2L,
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-4.2861221385716144629696E2L,
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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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#define SQRTH 0.70710678118654752440L
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long double
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logl(long double x)
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{
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long double y, z;
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int e;
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if( isnan(x) )
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return(x);
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if( x == INFINITY )
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return(x);
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/* Test for domain */
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if( x <= 0.0L )
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{
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if( x == 0.0L )
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return( -INFINITY );
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else
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return( NAN );
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}
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/* separate mantissa from exponent */
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/* Note, frexp is used so that denormal numbers
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* will be handled properly.
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*/
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x = frexpl( x, &e );
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/* logarithm using log(x) = z + z**3 P(z)/Q(z),
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* where z = 2(x-1)/x+1)
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*/
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if( (e > 2) || (e < -2) )
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{
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if( x < SQRTH )
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{ /* 2( 2x-1 )/( 2x+1 ) */
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e -= 1;
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z = x - 0.5L;
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y = 0.5L * z + 0.5L;
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}
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else
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{ /* 2 (x-1)/(x+1) */
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z = x - 0.5L;
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z -= 0.5L;
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y = 0.5L * x + 0.5L;
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}
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x = z / y;
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z = x*x;
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z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
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z = z + e * C2;
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z = z + x;
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z = z + e * C1;
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return( z );
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}
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/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
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if( x < SQRTH )
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{
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e -= 1;
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x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
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}
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else
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{
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x = x - 1.0L;
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}
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z = x*x;
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y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) );
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y = y + e * C2;
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z = y - ldexpl( z, -1 ); /* y - 0.5 * z */
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/* Note, the sum of above terms does not exceed x/4,
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* so it contributes at most about 1/4 lsb to the error.
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*/
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z = z + x;
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z = z + e * C1; /* This sum has an error of 1/2 lsb. */
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return( z );
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}
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