/* log2l.c
* Base 2 logarithm, 128-bit long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, log2l();
*
* y = log2l( x );
*
*
*
* DESCRIPTION:
*
* Returns the base 2 logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the (natural)
* 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, 2.0 100,000 2.6e-34 4.9e-35
* IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35
*
* In the tests over the interval exp(+-10000), the logarithms
* of the random arguments were uniformly distributed over
* [-10000, +10000].
*
*/
/*
Cephes Math Library Release 2.2: January, 1991
Copyright 1984, 1991 by Stephen L. Moshier
Adapted for glibc November, 2001
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see <
http://www.gnu.org/licenses/>.
*/
#include "quadmath-imp.h"
/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 5.3e-37,
* relative peak error spread = 2.3e-14
*/
static const __float128 P[13] =
{
1.313572404063446165910279910527789794488E4Q,
7.771154681358524243729929227226708890930E4Q,
2.014652742082537582487669938141683759923E5Q,
3.007007295140399532324943111654767187848E5Q,
2.854829159639697837788887080758954924001E5Q,
1.797628303815655343403735250238293741397E5Q,
7.594356839258970405033155585486712125861E4Q,
2.128857716871515081352991964243375186031E4Q,
3.824952356185897735160588078446136783779E3Q,
4.114517881637811823002128927449878962058E2Q,
2.321125933898420063925789532045674660756E1Q,
4.998469661968096229986658302195402690910E-1Q,
1.538612243596254322971797716843006400388E-6Q
};
static const __float128 Q[12] =
{
3.940717212190338497730839731583397586124E4Q,
2.626900195321832660448791748036714883242E5Q,
7.777690340007566932935753241556479363645E5Q,
1.347518538384329112529391120390701166528E6Q,
1.514882452993549494932585972882995548426E6Q,
1.158019977462989115839826904108208787040E6Q,
6.132189329546557743179177159925690841200E5Q,
2.248234257620569139969141618556349415120E5Q,
5.605842085972455027590989944010492125825E4Q,
9.147150349299596453976674231612674085381E3Q,
9.104928120962988414618126155557301584078E2Q,
4.839208193348159620282142911143429644326E1Q
/* 1.000000000000000000000000000000000000000E0L, */
};
/* 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 = 1.1e-35,
* relative peak error spread 1.1e-9
*/
static const __float128 R[6] =
{
1.418134209872192732479751274970992665513E5Q,
-8.977257995689735303686582344659576526998E4Q,
2.048819892795278657810231591630928516206E4Q,
-2.024301798136027039250415126250455056397E3Q,
8.057002716646055371965756206836056074715E1Q,
-8.828896441624934385266096344596648080902E-1Q
};
static const __float128 S[6] =
{
1.701761051846631278975701529965589676574E6Q,
-1.332535117259762928288745111081235577029E6Q,
4.001557694070773974936904547424676279307E5Q,
-5.748542087379434595104154610899551484314E4Q,
3.998526750980007367835804959888064681098E3Q,
-1.186359407982897997337150403816839480438E2Q
/* 1.000000000000000000000000000000000000000E0L, */
};
static const __float128
/* log2(e) - 1 */
LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q,
/* sqrt(2)/2 */
SQRTH = 7.071067811865475244008443621048490392848359E-1Q;
/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
static __float128
neval (__float128 x, const __float128 *p, int n)
{
__float128 y;
p += n;
y = *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}
/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
static __float128
deval (__float128 x, const __float128 *p, int n)
{
__float128 y;
p += n;
y = x + *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}
__float128
log2q (__float128 x)
{
__float128 z;
__float128 y;
int e;
int64_t hx, lx;
/* Test for domain */
GET_FLT128_WORDS64 (hx, lx, x);
if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
return (-1 / fabsq (x)); /* log2l(+-0)=-inf */
if (hx < 0)
return (x - x) / (x - x);
if (hx >= 0x7fff000000000000LL)
return (x + x);
if (x == 1)
return 0;
/* separate mantissa from exponent */
/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpq (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 - 0.5Q;
y = 0.5Q * z + 0.5Q;
}
else
{ /* 2 (x-1)/(x+1) */
z = x - 0.5Q;
z -= 0.5Q;
y = 0.5Q * x + 0.5Q;
}
x = z / y;
z = x * x;
y = x * (z * neval (z, R, 5) / deval (z, S, 5));
goto done;
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if (x < SQRTH)
{
e -= 1;
x = 2.0 * x - 1; /* 2x - 1 */
}
else
{
x = x - 1;
}
z = x * x;
y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
y = y - 0.5 * z;
done:
/* Multiply log of fraction by log2(e)
* and base 2 exponent by 1
*/
z = y * LOG2EA;
z += x * LOG2EA;
z += y;
z += x;
z += e;
return (z);
}
