log10l.c

/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log10l.c */
/*
 * 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.
 */
/*
 *      Common logarithm, long double precision
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, log10l();
 *
 * y = log10l( x );
 *
 *
 * DESCRIPTION:
 *
 * Returns the base 10 logarithm of x.
 *
 * The argument 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      0.5, 2.0     30000      9.0e-20     2.6e-20
 *    IEEE     exp(+-10000)  30000      6.0e-20     2.3e-20
 *
 * In the tests over the interval exp(+-10000), the logarithms
 * of the random arguments were uniformly distributed over
 * [-10000, +10000].
 *
 * ERROR MESSAGES:
 *
 * log singularity:  x = 0; returns MINLOG
 * log domain:       x < 0; returns MINLOG
 */

#include "libm.h"

#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
long double log10l(long double x)
{
	return log10(x);
}
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
/* 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 = 6.2e-22
 */
static long double P[] = {
 4.9962495940332550844739E-1L,
 1.0767376367209449010438E1L,
 7.7671073698359539859595E1L,
 2.5620629828144409632571E2L,
 4.2401812743503691187826E2L,
 3.4258224542413922935104E2L,
 1.0747524399916215149070E2L,
};
static long double Q[] = {
/* 1.0000000000000000000000E0,*/
 2.3479774160285863271658E1L,
 1.9444210022760132894510E2L,
 7.7952888181207260646090E2L,
 1.6911722418503949084863E3L,
 2.0307734695595183428202E3L,
 1.2695660352705325274404E3L,
 3.2242573199748645407652E2L,
};

/* 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,
};
/* log10(2) */
#define L102A 0.3125L
#define L102B -1.1470004336018804786261e-2L
/* log10(e) */
#define L10EA 0.5L
#define L10EB -6.5705518096748172348871e-2L

#define SQRTH 0.70710678118654752440L

long double log10l(long double x)
{
	long double y;
	volatile long double z;
	int e;

	if (isnan(x))
		return x;
	if(x <= 0.0L) {
		if(x == 0.0L)
			return -1.0L / (x - x);
		return (x - x) / (x - x);
	}
	if (x == INFINITY)
		return INFINITY;
	/* separate mantissa from exponent */
	/* Note, frexp is used 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;
		y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
		goto done;
	}

	/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
	if (x < SQRTH) {
		e -= 1;
		x = ldexpl(x, 1) - 1.0L; /*  2x - 1  */
	} else {
		x = x - 1.0L;
	}
	z = x*x;
	y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
	y = y - ldexpl(z, -1);   /* -0.5x^2 + ... */

done:
	/* Multiply log of fraction by log10(e)
	 * and base 2 exponent by log10(2).
	 *
	 * ***CAUTION***
	 *
	 * This sequence of operations is critical and it may
	 * be horribly defeated by some compiler optimizers.
	 */
	z = y * (L10EB);
	z += x * (L10EB);
	z += e * (L102B);
	z += y * (L10EA);
	z += x * (L10EA);
	z += e * (L102A);
	return z;
}
#endif