/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_logl.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.
*/
/*
* Natural logarithm, long double precision
*
*
* SYNOPSIS:
*
* long double x, y, logl();
*
* y = logl( x );
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) 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 150000 8.71e-20 2.75e-20
* IEEE exp(+-10000) 100000 5.39e-20 2.34e-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 -INFINITY
* log domain: x < 0; returns NAN
*/
#include "libm.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
long double logl(long double x)
{
return log(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 = 2.32e-20
*/
static const long double P[] = {
4.5270000862445199635215E-5L,
4.9854102823193375972212E-1L,
6.5787325942061044846969E0L,
2.9911919328553073277375E1L,
6.0949667980987787057556E1L,
5.7112963590585538103336E1L,
2.0039553499201281259648E1L,
};
static const 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 const long double R[4] = {
1.9757429581415468984296E-3L,
-7.1990767473014147232598E-1L,
1.0777257190312272158094E1L,
-3.5717684488096787370998E1L,
};
static const 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 logl(long double x)
{
long double y, z;
int e;
if (isnan(x))
return x;
if (x == INFINITY)
return x;
if (x <= 0.0L) {
if (x == 0.0L)
return -INFINITY;
return NAN;
}
/* 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;
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;
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, 6));
y = y + e * C2;
z = y - ldexpl(z, -1); /* y - 0.5 * z */
/* Note, the sum of above terms does not exceed x/4,
* so it contributes at most about 1/4 lsb to the error.
*/
z = z + x;
z = z + e * C1; /* This sum has an error of 1/2 lsb. */
return z;
}
#endif
