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/* Library libcerf:
* Compute complex error functions, based on a new implementation of
* Faddeeva's w_of_z. Also provide Dawson and Voigt functions.
*
* File erfcx.c:
* Compute erfcx(x) = exp(x^2) erfc(x) function, for real x,
* using a novel algorithm that is much faster than DERFC of SLATEC.
* This function is used in the computation of Faddeeva, Dawson, and
* other complex error functions.
*
* Copyright:
* (C) 2012 Massachusetts Institute of Technology
* (C) 2013, 2025 Forschungszentrum Jülich GmbH
*
* Licence:
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
* LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
* OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
* WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*
* Authors:
* Steven G. Johnson, Massachusetts Institute of Technology, 2012
* Joachim Wuttke, Forschungszentrum Jülich, 2013, 2025
*
* Website:
* http://apps.jcns.fz-juelich.de/libcerf
*
* Revision history:
* libcerf-2.5, April 2025:
* - For large |x|, continuous fractions replaced with asymptotic expansions.
* - Expanded the small |x| range where Taylor series is used.
* - Code for intermediate |x| entirely rewritten, using methods described in:
* Joachim Wuttke and Alexander Kleinsorge:
* "Code generation for piecewise Chebyshev approximation."
*
* See also ../CHANGELOG
*
* Manual page:
* man 3 erfcx
*/
#include "cerf.h"
#include "defs.h" // defines _cerf_cmplx, NaN, C, cexp, ...
#include <math.h>
#include <stdalign.h>
#include <stdio.h>
#include <stdlib.h>
#include "auto_cheb_erfcx.c"
//! Returns polynomial approximation to f(x), using auto-tabulated expansion coefficients.
//! Code taken from https://jugit.fz-juelich.de/mlz/ppapp.
static double chebInterpolant(double x)
{
// Application-specific constants:
static const int loff = (ppapp_j0 + 1) * (1 << ppapp_M) + ppapp_l0; // precomputed offset
// For given x, obtain mantissa xm and exponent je:
int je; // will be set in next line
const double xm = frexp2(x, &je); // sets xm and je
// Integer arithmetics to obtain reduced coordinate t:
const int ip = (int)((1 << (ppapp_M+1)) * xm); // index in octave + 2^M
const int lij = je * (1 << ppapp_M) + ip - loff; // index in lookup table
const double t = (1 << (ppapp_M+2)) * xm - (1 + 2*ip);
const double *const P = ppapp_Coeffs0 + lij*8;
const double *const Q = ppapp_Coeffs1 + lij*2;
return ((((((((P[0] * t
+ P[1]) * t
+ P[2]) * t
+ P[3]) * t
+ P[4]) * t
+ P[5]) * t
+ P[6]) * t
+ P[7]) * t
+ Q[0]) * t
+ Q[1];
}
/******************************************************************************/
/* Library function erfcx */
/******************************************************************************/
double erfcx(double x) {
// Uses the following methods:
// - asymptotic expansion for large positive x,
// - Chebyshev polynomials for medium positive x,
// - Taylor (Maclaurin) series for small |x|,
// - 2*exp(x^2)-erfcx(-x) for medium negative x,
// - Asymptote 2exp(x^2) for large negative x.
double ax = fabs(x);
if (ax < .125) {
// Use Taylor expansion
return ((((((((((((((+1.9841269841269841e-04) * x -
5.3440090793734269e-04) * x +
1.3888888888888889e-03) * x -
3.4736059015927274e-03) * x +
8.3333333333333332e-03) * x -
1.9104832458760001e-02) * x +
4.1666666666666664e-02) * x -
8.5971746064419999e-02) * x +
1.6666666666666666e-01) * x -
3.0090111122547003e-01) * x +
5.0000000000000000e-01) * x -
7.5225277806367508e-01) * x +
1.0000000000000000e+00) * x -
1.1283791670955126e+00) * x +
1.0000000000000000e+00;
}
if (x < 0) {
if (x < -26.7)
return HUGE_VAL;
if (x < -6.1)
return 2 * exp(x * x);
return 2 * exp(x * x) - chebInterpolant(-x);
}
if (x < 12)
return chebInterpolant(x);
/* else */ {
// Use asymptotic expansion
//
// Coefficient are a_0 = 1/sqrt(pi), a_N = (2N-1)!!/2^N/sqrt(pi).
const double r = 1 / x;
if (x < 150) {
if (x < 23.2)
return (((((((((((+3.6073371500083758e+05) * (r * r) -
3.7971970000088164e+04) * (r * r) +
4.4672905882456671e+03) * (r * r) -
5.9563874509942218e+02) * (r * r) +
9.1636730015295726e+01) * (r * r) -
1.6661223639144676e+01) * (r * r) +
3.7024941420321507e+00) * (r * r) -
1.0578554691520430e+00) * (r * r) +
4.2314218766081724e-01) * (r * r) -
2.8209479177387814e-01) * (r * r) +
5.6418958354775628e-01) * r;
return (((((((+9.1636730015295726e+01) * (r * r) -
1.6661223639144676e+01) * (r * r) +
3.7024941420321507e+00) * (r * r) -
1.0578554691520430e+00) * (r * r) +
4.2314218766081724e-01) * (r * r) -
2.8209479177387814e-01) * (r * r) +
5.6418958354775628e-01) * r;
}
if (x < 6.9e7)
return ((((-1.0578554691520430e+00) * (r * r) +
4.2314218766081724e-01) * (r * r) -
2.8209479177387814e-01) * (r * r) +
5.6418958354775628e-01) * r;
// 1-term expansion, important to avoid overflow
return 0.56418958354775629 * r;
}
} // erfcx
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