245 lines
6.3 KiB
C
245 lines
6.3 KiB
C
/* log1pl.c
|
|
*
|
|
* Relative error logarithm
|
|
* Natural logarithm of 1+x, 128-bit 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(w-1)/(w+1),
|
|
*
|
|
* log(w) = z + z^3 P(z)/Q(z).
|
|
*
|
|
*
|
|
*
|
|
* ACCURACY:
|
|
*
|
|
* Relative error:
|
|
* arithmetic domain # trials peak rms
|
|
* IEEE -1, 8 100000 1.9e-34 4.3e-35
|
|
*/
|
|
|
|
/* Copyright 2001 by Stephen L. Moshier
|
|
|
|
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, write to the Free Software
|
|
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
|
|
|
|
|
|
#include "quadmath-imp.h"
|
|
|
|
/* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
|
|
* 1/sqrt(2) <= 1+x < sqrt(2)
|
|
* Theoretical peak relative error = 5.3e-37,
|
|
* relative peak error spread = 2.3e-14
|
|
*/
|
|
static const __float128
|
|
P12 = 1.538612243596254322971797716843006400388E-6Q,
|
|
P11 = 4.998469661968096229986658302195402690910E-1Q,
|
|
P10 = 2.321125933898420063925789532045674660756E1Q,
|
|
P9 = 4.114517881637811823002128927449878962058E2Q,
|
|
P8 = 3.824952356185897735160588078446136783779E3Q,
|
|
P7 = 2.128857716871515081352991964243375186031E4Q,
|
|
P6 = 7.594356839258970405033155585486712125861E4Q,
|
|
P5 = 1.797628303815655343403735250238293741397E5Q,
|
|
P4 = 2.854829159639697837788887080758954924001E5Q,
|
|
P3 = 3.007007295140399532324943111654767187848E5Q,
|
|
P2 = 2.014652742082537582487669938141683759923E5Q,
|
|
P1 = 7.771154681358524243729929227226708890930E4Q,
|
|
P0 = 1.313572404063446165910279910527789794488E4Q,
|
|
/* Q12 = 1.000000000000000000000000000000000000000E0Q, */
|
|
Q11 = 4.839208193348159620282142911143429644326E1Q,
|
|
Q10 = 9.104928120962988414618126155557301584078E2Q,
|
|
Q9 = 9.147150349299596453976674231612674085381E3Q,
|
|
Q8 = 5.605842085972455027590989944010492125825E4Q,
|
|
Q7 = 2.248234257620569139969141618556349415120E5Q,
|
|
Q6 = 6.132189329546557743179177159925690841200E5Q,
|
|
Q5 = 1.158019977462989115839826904108208787040E6Q,
|
|
Q4 = 1.514882452993549494932585972882995548426E6Q,
|
|
Q3 = 1.347518538384329112529391120390701166528E6Q,
|
|
Q2 = 7.777690340007566932935753241556479363645E5Q,
|
|
Q1 = 2.626900195321832660448791748036714883242E5Q,
|
|
Q0 = 3.940717212190338497730839731583397586124E4Q;
|
|
|
|
/* 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
|
|
R5 = -8.828896441624934385266096344596648080902E-1Q,
|
|
R4 = 8.057002716646055371965756206836056074715E1Q,
|
|
R3 = -2.024301798136027039250415126250455056397E3Q,
|
|
R2 = 2.048819892795278657810231591630928516206E4Q,
|
|
R1 = -8.977257995689735303686582344659576526998E4Q,
|
|
R0 = 1.418134209872192732479751274970992665513E5Q,
|
|
/* S6 = 1.000000000000000000000000000000000000000E0Q, */
|
|
S5 = -1.186359407982897997337150403816839480438E2Q,
|
|
S4 = 3.998526750980007367835804959888064681098E3Q,
|
|
S3 = -5.748542087379434595104154610899551484314E4Q,
|
|
S2 = 4.001557694070773974936904547424676279307E5Q,
|
|
S1 = -1.332535117259762928288745111081235577029E6Q,
|
|
S0 = 1.701761051846631278975701529965589676574E6Q;
|
|
|
|
/* C1 + C2 = ln 2 */
|
|
static const __float128 C1 = 6.93145751953125E-1Q;
|
|
static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q;
|
|
|
|
static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q;
|
|
static const __float128 zero = 0.0Q;
|
|
|
|
|
|
__float128
|
|
log1pq (__float128 xm1)
|
|
{
|
|
__float128 x, y, z, r, s;
|
|
ieee854_float128 u;
|
|
int32_t hx;
|
|
int e;
|
|
|
|
/* Test for NaN or infinity input. */
|
|
u.value = xm1;
|
|
hx = u.words32.w0;
|
|
if (hx >= 0x7fff0000)
|
|
return xm1;
|
|
|
|
/* log1p(+- 0) = +- 0. */
|
|
if (((hx & 0x7fffffff) == 0)
|
|
&& (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
|
|
return xm1;
|
|
|
|
x = xm1 + 1.0Q;
|
|
|
|
/* log1p(-1) = -inf */
|
|
if (x <= 0.0Q)
|
|
{
|
|
if (x == 0.0Q)
|
|
return (-1.0Q / (x - x));
|
|
else
|
|
return (zero / (x - x));
|
|
}
|
|
|
|
/* Separate mantissa from exponent. */
|
|
|
|
/* Use frexp used so that denormal numbers will be handled properly. */
|
|
x = frexpq (x, &e);
|
|
|
|
/* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
|
|
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;
|
|
r = ((((R5 * z
|
|
+ R4) * z
|
|
+ R3) * z
|
|
+ R2) * z
|
|
+ R1) * z
|
|
+ R0;
|
|
s = (((((z
|
|
+ S5) * z
|
|
+ S4) * z
|
|
+ S3) * z
|
|
+ S2) * z
|
|
+ S1) * z
|
|
+ S0;
|
|
z = x * (z * r / s);
|
|
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 = 2.0Q * x - 1.0Q; /* 2x - 1 */
|
|
else
|
|
x = xm1;
|
|
}
|
|
else
|
|
{
|
|
if (e != 0)
|
|
x = x - 1.0Q;
|
|
else
|
|
x = xm1;
|
|
}
|
|
z = x * x;
|
|
r = (((((((((((P12 * x
|
|
+ P11) * x
|
|
+ P10) * x
|
|
+ P9) * x
|
|
+ P8) * x
|
|
+ P7) * x
|
|
+ P6) * x
|
|
+ P5) * x
|
|
+ P4) * x
|
|
+ P3) * x
|
|
+ P2) * x
|
|
+ P1) * x
|
|
+ P0;
|
|
s = (((((((((((x
|
|
+ Q11) * x
|
|
+ Q10) * x
|
|
+ Q9) * x
|
|
+ Q8) * x
|
|
+ Q7) * x
|
|
+ Q6) * x
|
|
+ Q5) * x
|
|
+ Q4) * x
|
|
+ Q3) * x
|
|
+ Q2) * x
|
|
+ Q1) * x
|
|
+ Q0;
|
|
y = x * (z * r / s);
|
|
y = y + e * C2;
|
|
z = y - 0.5Q * z;
|
|
z = z + x;
|
|
z = z + e * C1;
|
|
return (z);
|
|
}
|