/* 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, see
<http://www.gnu.org/licenses/>. */
#include <float.h>
#include <math.h>
#include <math_private.h>
#include <math-underflow.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 = L(1.538612243596254322971797716843006400388E-6),
P11 = L(4.998469661968096229986658302195402690910E-1),
P10 = L(2.321125933898420063925789532045674660756E1),
P9 = L(4.114517881637811823002128927449878962058E2),
P8 = L(3.824952356185897735160588078446136783779E3),
P7 = L(2.128857716871515081352991964243375186031E4),
P6 = L(7.594356839258970405033155585486712125861E4),
P5 = L(1.797628303815655343403735250238293741397E5),
P4 = L(2.854829159639697837788887080758954924001E5),
P3 = L(3.007007295140399532324943111654767187848E5),
P2 = L(2.014652742082537582487669938141683759923E5),
P1 = L(7.771154681358524243729929227226708890930E4),
P0 = L(1.313572404063446165910279910527789794488E4),
/* Q12 = 1.000000000000000000000000000000000000000E0L, */
Q11 = L(4.839208193348159620282142911143429644326E1),
Q10 = L(9.104928120962988414618126155557301584078E2),
Q9 = L(9.147150349299596453976674231612674085381E3),
Q8 = L(5.605842085972455027590989944010492125825E4),
Q7 = L(2.248234257620569139969141618556349415120E5),
Q6 = L(6.132189329546557743179177159925690841200E5),
Q5 = L(1.158019977462989115839826904108208787040E6),
Q4 = L(1.514882452993549494932585972882995548426E6),
Q3 = L(1.347518538384329112529391120390701166528E6),
Q2 = L(7.777690340007566932935753241556479363645E5),
Q1 = L(2.626900195321832660448791748036714883242E5),
Q0 = L(3.940717212190338497730839731583397586124E4);
/* 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 = L(-8.828896441624934385266096344596648080902E-1),
R4 = L(8.057002716646055371965756206836056074715E1),
R3 = L(-2.024301798136027039250415126250455056397E3),
R2 = L(2.048819892795278657810231591630928516206E4),
R1 = L(-8.977257995689735303686582344659576526998E4),
R0 = L(1.418134209872192732479751274970992665513E5),
/* S6 = 1.000000000000000000000000000000000000000E0L, */
S5 = L(-1.186359407982897997337150403816839480438E2),
S4 = L(3.998526750980007367835804959888064681098E3),
S3 = L(-5.748542087379434595104154610899551484314E4),
S2 = L(4.001557694070773974936904547424676279307E5),
S1 = L(-1.332535117259762928288745111081235577029E6),
S0 = L(1.701761051846631278975701529965589676574E6);
/* C1 + C2 = ln 2 */
static const _Float128 C1 = L(6.93145751953125E-1);
static const _Float128 C2 = L(1.428606820309417232121458176568075500134E-6);
static const _Float128 sqrth = L(0.7071067811865475244008443621048490392848);
/* ln (2^16384 * (1 - 2^-113)) */
static const _Float128 zero = 0;
_Float128
__log1pl (_Float128 xm1)
{
_Float128 x, y, z, r, s;
ieee854_long_double_shape_type u;
int32_t hx;
int e;
/* Test for NaN or infinity input. */
u.value = xm1;
hx = u.parts32.w0;
if ((hx & 0x7fffffff) >= 0x7fff0000)
return xm1 + fabsl (xm1);
/* log1p(+- 0) = +- 0. */
if (((hx & 0x7fffffff) == 0)
&& (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
return xm1;
if ((hx & 0x7fffffff) < 0x3f8e0000)
{
math_check_force_underflow (xm1);
if ((int) xm1 == 0)
return xm1;
}
if (xm1 >= L(0x1p113))
x = xm1;
else
x = xm1 + 1;
/* log1p(-1) = -inf */
if (x <= 0)
{
if (x == 0)
return (-1 / zero); /* log1p(-1) = -inf */
else
return (zero / (x - x));
}
/* Separate mantissa from exponent. */
/* Use frexp used so that denormal numbers will be handled properly. */
x = __frexpl (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 - L(0.5);
y = L(0.5) * z + L(0.5);
}
else
{ /* 2 (x-1)/(x+1) */
z = x - L(0.5);
z -= L(0.5);
y = L(0.5) * x + L(0.5);
}
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 * x - 1; /* 2x - 1 */
else
x = xm1;
}
else
{
if (e != 0)
x = x - 1;
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 - L(0.5) * z;
z = z + x;
z = z + e * C1;
return (z);
}