relibc_openlibm/ld80/s_log1pl.c

/*	$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 );
}