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/* Exponential function.
Copyright (C) 2011-2021 Free Software Foundation, Inc.
This file 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 3 of the
License, or (at your option) any later version.
This file 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 program. If not, see <https://www.gnu.org/licenses/>. */
#include <config.h>
/* Specification. */
#include <math.h>
#if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
long double
expl (long double x)
{
return exp (x);
}
#else
# include <float.h>
/* gl_expl_table[i] = exp((i - 128) * log(2)/256). */
extern const long double gl_expl_table[257];
/* A value slightly larger than log(2). */
#define LOG2_PLUS_EPSILON 0.6931471805599454L
/* Best possible approximation of log(2) as a 'long double'. */
#define LOG2 0.693147180559945309417232121458176568075L
/* Best possible approximation of 1/log(2) as a 'long double'. */
#define LOG2_INVERSE 1.44269504088896340735992468100189213743L
/* Best possible approximation of log(2)/256 as a 'long double'. */
#define LOG2_BY_256 0.00270760617406228636491106297444600221904L
/* Best possible approximation of 256/log(2) as a 'long double'. */
#define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181L
/* The upper 32 bits of log(2)/256. */
#define LOG2_BY_256_HI_PART 0.0027076061733168899081647396087646484375L
/* log(2)/256 - LOG2_HI_PART. */
#define LOG2_BY_256_LO_PART \
0.000000000000745396456746323365681353781544922399845L
long double
expl (long double x)
{
if (isnanl (x))
return x;
if (x >= (long double) LDBL_MAX_EXP * LOG2_PLUS_EPSILON)
/* x > LDBL_MAX_EXP * log(2)
hence exp(x) > 2^LDBL_MAX_EXP, overflows to Infinity. */
return HUGE_VALL;
if (x <= (long double) (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG) * LOG2_PLUS_EPSILON)
/* x < (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG) * log(2)
hence exp(x) < 2^(LDBL_MIN_EXP-1-LDBL_MANT_DIG),
underflows to zero. */
return 0.0L;
/* Decompose x into
x = n * log(2) + m * log(2)/256 + y
where
n is an integer,
m is an integer, -128 <= m <= 128,
y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
Then
exp(x) = 2^n * exp(m * log(2)/256) * exp(y)
The first factor is an ldexpl() call.
The second factor is a table lookup.
The third factor is computed
- either as sinh(y) + cosh(y)
where sinh(y) is computed through the power series:
sinh(y) = y + y^3/3! + y^5/5! + ...
and cosh(y) is computed as hypot(1, sinh(y)),
- or as exp(2*z) = (1 + tanh(z)) / (1 - tanh(z))
where z = y/2
and tanh(z) is computed through its power series:
tanh(z) = z
- 1/3 * z^3
+ 2/15 * z^5
- 17/315 * z^7
+ 62/2835 * z^9
- 1382/155925 * z^11
+ 21844/6081075 * z^13
- 929569/638512875 * z^15
+ ...
Since |z| <= log(2)/1024 < 0.0007, the relative contribution of the
z^13 term is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
can truncate the series after the z^11 term.
Given the usual bounds LDBL_MAX_EXP <= 16384, LDBL_MIN_EXP >= -16381,
LDBL_MANT_DIG <= 120, we can estimate x: -11440 <= x <= 11357.
This means, when dividing x by log(2), where we want x mod log(2)
to be precise to LDBL_MANT_DIG bits, we have to use an approximation
to log(2) that has 14+LDBL_MANT_DIG bits. */
{
long double nm = roundl (x * LOG2_BY_256_INVERSE); /* = 256 * n + m */
/* n has at most 15 bits, nm therefore has at most 23 bits, therefore
n * LOG2_HI_PART is computed exactly, and n * LOG2_LO_PART is computed
with an absolute error < 2^15 * 2e-10 * 2^-LDBL_MANT_DIG. */
long double y_tmp = x - nm * LOG2_BY_256_HI_PART;
long double y = y_tmp - nm * LOG2_BY_256_LO_PART;
long double z = 0.5L * y;
/* Coefficients of the power series for tanh(z). */
#define TANH_COEFF_1 1.0L
#define TANH_COEFF_3 -0.333333333333333333333333333333333333334L
#define TANH_COEFF_5 0.133333333333333333333333333333333333334L
#define TANH_COEFF_7 -0.053968253968253968253968253968253968254L
#define TANH_COEFF_9 0.0218694885361552028218694885361552028218L
#define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
#define TANH_COEFF_13 0.00359212803657248101692546136990581435026L
#define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
long double z2 = z * z;
long double tanh_z =
(((((TANH_COEFF_11
* z2 + TANH_COEFF_9)
* z2 + TANH_COEFF_7)
* z2 + TANH_COEFF_5)
* z2 + TANH_COEFF_3)
* z2 + TANH_COEFF_1)
* z;
long double exp_y = (1.0L + tanh_z) / (1.0L - tanh_z);
int n = (int) roundl (nm * (1.0L / 256.0L));
int m = (int) nm - 256 * n;
return ldexpl (gl_expl_table[128 + m] * exp_y, n);
}
}
#endif
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