OpenLibm/ld80/s_log1pl.c

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/* $OpenBSD: s_log1pl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $ */
/*
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
*
* 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.
*/
/* log1pl.c
*
* Relative error logarithm
* Natural logarithm of 1+x, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, log1pl();
*
* y = log1pl( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of 1+x.
*
* The argument 1+x is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the logarithm
* of the fraction is approximated by
*
* log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
*
* Otherwise, setting z = 2(x-1)/x+1),
*
* log(x) = z + z^3 P(z)/Q(z).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20
*
* ERROR MESSAGES:
*
* log singularity: x-1 = 0; returns -INFINITY
* log domain: x-1 < 0; returns NAN
*/
#include <openlibm_math.h>
#include "math_private.h"
/* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 2.32e-20
*/
static long double P[] = {
4.5270000862445199635215E-5L,
4.9854102823193375972212E-1L,
6.5787325942061044846969E0L,
2.9911919328553073277375E1L,
6.0949667980987787057556E1L,
5.7112963590585538103336E1L,
2.0039553499201281259648E1L,
};
static long double Q[] = {
/* 1.0000000000000000000000E0,*/
1.5062909083469192043167E1L,
8.3047565967967209469434E1L,
2.2176239823732856465394E2L,
3.0909872225312059774938E2L,
2.1642788614495947685003E2L,
6.0118660497603843919306E1L,
};
/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
* where z = 2(x-1)/(x+1)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 6.16e-22
*/
static long double R[4] = {
1.9757429581415468984296E-3L,
-7.1990767473014147232598E-1L,
1.0777257190312272158094E1L,
-3.5717684488096787370998E1L,
};
static long double S[4] = {
/* 1.00000000000000000000E0L,*/
-2.6201045551331104417768E1L,
1.9361891836232102174846E2L,
-4.2861221385716144629696E2L,
};
static const long double C1 = 6.9314575195312500000000E-1L;
static const long double C2 = 1.4286068203094172321215E-6L;
#define SQRTH 0.70710678118654752440L
long double
log1pl(long double xm1)
{
long double x, y, z;
int e;
if( isnan(xm1) )
return(xm1);
if( xm1 == INFINITY )
return(xm1);
if(xm1 == 0.0)
return(xm1);
x = xm1 + 1.0L;
/* Test for domain errors. */
if( x <= 0.0L )
{
if( x == 0.0L )
return( -INFINITY );
else
return( NAN );
}
/* Separate mantissa from exponent.
Use frexp so that denormal numbers will be handled properly. */
x = frexpl( x, &e );
/* logarithm using log(x) = z + z^3 P(z)/Q(z),
where z = 2(x-1)/x+1) */
if( (e > 2) || (e < -2) )
{
if( x < SQRTH )
{ /* 2( 2x-1 )/( 2x+1 ) */
e -= 1;
z = x - 0.5L;
y = 0.5L * z + 0.5L;
}
else
{ /* 2 (x-1)/(x+1) */
z = x - 0.5L;
z -= 0.5L;
y = 0.5L * x + 0.5L;
}
x = z / y;
z = x*x;
z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
z = z + e * C2;
z = z + x;
z = z + e * C1;
return( z );
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if( x < SQRTH )
{
e -= 1;
if (e != 0)
x = 2.0 * x - 1.0L;
else
x = xm1;
}
else
{
if (e != 0)
x = x - 1.0L;
else
x = xm1;
}
z = x*x;
y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) );
y = y + e * C2;
z = y - 0.5 * z;
z = z + x;
z = z + e * C1;
return( z );
}