206 lines
4.6 KiB
C
206 lines
4.6 KiB
C
/* $OpenBSD: e_log10l.c,v 1.2 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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/* log10l.c
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*
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* Common 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, log10l();
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*
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* y = log10l( 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 10 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 30000 9.0e-20 2.6e-20
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* IEEE exp(+-10000) 30000 6.0e-20 2.3e-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 MINLOG
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* log domain: x < 0; returns MINLOG
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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 = 6.2e-22
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*/
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static long double P[] = {
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4.9962495940332550844739E-1L,
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1.0767376367209449010438E1L,
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7.7671073698359539859595E1L,
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2.5620629828144409632571E2L,
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4.2401812743503691187826E2L,
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3.4258224542413922935104E2L,
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1.0747524399916215149070E2L,
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};
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static long double Q[] = {
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/* 1.0000000000000000000000E0,*/
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2.3479774160285863271658E1L,
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1.9444210022760132894510E2L,
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7.7952888181207260646090E2L,
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1.6911722418503949084863E3L,
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2.0307734695595183428202E3L,
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1.2695660352705325274404E3L,
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3.2242573199748645407652E2L,
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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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/* log10(2) */
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#define L102A 0.3125L
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#define L102B -1.1470004336018804786261e-2L
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/* log10(e) */
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#define L10EA 0.5L
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#define L10EB -6.5705518096748172348871e-2L
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#define SQRTH 0.70710678118654752440L
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long double
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log10l(long double x)
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{
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long double y;
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volatile long double z;
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int e;
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if( isnan(x) )
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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 (-1.0L / (x - x));
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else
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return (x - x) / (x - x);
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}
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if( x == INFINITY )
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return(INFINITY);
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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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y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
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goto done;
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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, 7 ) );
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y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
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done:
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/* Multiply log of fraction by log10(e)
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* and base 2 exponent by log10(2).
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*
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* ***CAUTION***
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*
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* This sequence of operations is critical and it may
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* be horribly defeated by some compiler optimizers.
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*/
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z = y * (L10EB);
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z += x * (L10EB);
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z += e * (L102B);
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z += y * (L10EA);
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z += x * (L10EA);
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z += e * (L102A);
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return( z );
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}
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