Mercurial > octave-nkf
changeset 15697:9811b32b645e
maint: merge in Faddeeva functions and relatives
| author | Jordi Gutiérrez Hermoso <jordigh@octave.org> |
|---|---|
| date | Thu, 29 Nov 2012 10:40:32 -0500 |
| parents | 310db14aa605 (current diff) 2fac72a256ce (diff) |
| children | ea2788117ee3 |
| files | |
| diffstat | 22 files changed, 2737 insertions(+), 108 deletions(-) [+] |
line wrap: on
line diff
--- a/doc/interpreter/arith.txi Wed Nov 28 14:05:20 2012 -0500 +++ b/doc/interpreter/arith.txi Thu Nov 29 10:40:32 2012 -0500 @@ -291,6 +291,10 @@ @DOCSTRING(erfcx) +@DOCSTRING(erfi) + +@DOCSTRING(dawson) + @DOCSTRING(erfinv) @DOCSTRING(erfcinv)
--- a/libinterp/corefcn/mappers.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/corefcn/mappers.cc Thu Nov 29 10:40:32 2012 -0500 @@ -547,7 +547,7 @@ @end example\n\ \n\ @end ifnottex\n\ -@seealso{erfc, erfcx, erfinv, erfcinv}\n\ +@seealso{erfc, erfcx, erfi, dawson, erfinv, erfcinv}\n\ @end deftypefn") { octave_value retval; @@ -584,6 +584,13 @@ %! assert (erfc (x), 1-v, 1.e-6); %! assert (erfinv (v), x, 1.e-6); +%!test +%! x = [1+2i,-1+2i,1e-6+2e-6i,0+2i]; +%! v = [-0.53664356577857-5.04914370344703i, 0.536643565778565-5.04914370344703i, 0.112837916709965e-5+0.225675833419178e-5i, 18.5648024145755526i]; +%! assert (erf (x), v, -1.e-10); +%! assert (erf (-x), -v, -1.e-10); +%! assert (erfc (x), 1-v, -1.e-10); + %!error erf () %!error erf (1, 2) */ @@ -596,7 +603,7 @@ @example\n\ erf (@var{y}) == @var{x}\n\ @end example\n\ -@seealso{erf, erfc, erfcx, erfcinv}\n\ +@seealso{erf, erfc, erfcx, erfi, dawson, erfcinv}\n\ @end deftypefn") { octave_value retval; @@ -634,7 +641,7 @@ @example\n\ erfc (@var{y}) == @var{x}\n\ @end example\n\ -@seealso{erfc, erf, erfcx, erfinv}\n\ +@seealso{erfc, erf, erfcx, erfi, dawson, erfinv}\n\ @end deftypefn") { octave_value retval; @@ -674,7 +681,7 @@ @ifnottex\n\ @w{@code{1 - erf (@var{z})}}.\n\ @end ifnottex\n\ -@seealso{erfcinv, erfcx, erf, erfinv}\n\ +@seealso{erfcinv, erfcx, erfi, dawson, erf, erfinv}\n\ @end deftypefn") { octave_value retval; @@ -707,11 +714,11 @@ @ifnottex\n\ \n\ @example\n\ -exp (z^2) * erfc (x)\n\ +exp (z^2) * erfc (z)\n\ @end example\n\ \n\ @end ifnottex\n\ -@seealso{erfc, erf, erfinv, erfcinv}\n\ +@seealso{erfc, erf, erfi, dawson, erfinv, erfcinv}\n\ @end deftypefn") { octave_value retval; @@ -724,12 +731,98 @@ } /* -## FIXME: Need a test for erfcx + +%!test +%! x = [1+2i,-1+2i,1e-6+2e-6i,0+2i]; +%! assert (erfcx (x), exp (x.^2) .* erfc(x), -1.e-10); + +%!test +%! x = [100, 100+20i]; +%! v = [0.0056416137829894329, 0.0054246791754558-0.00108483153786434i]; +%! assert (erfcx (x), v, -1.e-10); %!error erfcx () %!error erfcx (1, 2) */ +DEFUN (erfi, args, , + "-*- texinfo -*-\n\ +@deftypefn {Mapping Function} {} erfi (@var{z})\n\ +Compute the imaginary error function,\n\ +@tex\n\ +$$\n\ + -i {\\rm erf} (iz) \n\ +$$\n\ +@end tex\n\ +@ifnottex\n\ +\n\ +@example\n\ +-i * erf (i*z)\n\ +@end example\n\ +\n\ +@end ifnottex\n\ +@seealso{erfc, erf, erfcx, dawson, erfinv, erfcinv}\n\ +@end deftypefn") +{ + octave_value retval; + if (args.length () == 1) + retval = args(0).erfi (); + else + print_usage (); + + return retval; +} + +/* + +%!test +%! x = [-0.1, 0.1, 1, 1+2i,-1+2i,1e-6+2e-6i,0+2i]; +%! assert (erfi (x), -i * erf(i*x), -1.e-10); + +%!error erfi () +%!error erfi (1, 2) +*/ + +DEFUN (dawson, args, , + "-*- texinfo -*-\n\ +@deftypefn {Mapping Function} {} dawson (@var{z})\n\ +Compute the Dawson (scaled imaginary error) function,\n\ +@tex\n\ +$$\n\ + {\\sqrt{\\pi} \\over 2} e^{-z^2} {\\rm erfi} (z) \\equiv -i {\\sqrt{\\pi} \\over 2} e^{-z^2} {\\rm erf} (iz)\n\ +$$\n\ +@end tex\n\ +@ifnottex\n\ +\n\ +@example\n\ +(sqrt (pi) / 2) * exp (-z^2) * erfi (z)\n\ +@end example\n\ +\n\ +@end ifnottex\n\ +@seealso{erfc, erf, erfcx, erfi, erfinv, erfcinv}\n\ +@end deftypefn") +{ + octave_value retval; + if (args.length () == 1) + retval = args(0).dawson (); + else + print_usage (); + + return retval; +} + +/* + +%!test +%! x = [0.1, 1, 1+2i,-1+2i,1e-4+2e-4i,0+2i]; +%! v = [0.099335992397852861, 0.53807950691, -13.38892731648-11.828715104i, 13.38892731648-11.828715104i, 0.0001000000073333+0.000200000001333i, 48.160012114291i]; +%! assert (dawson (x), v, -1.e-10); +%! assert (dawson (-x), -v, -1.e-10); + +%!error dawson () +%!error dawson (1, 2) +*/ + DEFUN (exp, args, , "-*- texinfo -*-\n\ @deftypefn {Mapping Function} {} exp (@var{x})\n\
--- a/libinterp/octave-value/ov-base.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-base.cc Thu Nov 29 10:40:32 2012 -0500 @@ -1194,6 +1194,8 @@ "erfcinv", "erfc", "erfcx", + "erfi", + "dawson", "exp", "expm1", "finite",
--- a/libinterp/octave-value/ov-base.h Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-base.h Thu Nov 29 10:40:32 2012 -0500 @@ -689,6 +689,8 @@ umap_erfcinv, umap_erfc, umap_erfcx, + umap_erfi, + umap_dawson, umap_exp, umap_expm1, umap_finite,
--- a/libinterp/octave-value/ov-complex.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-complex.cc Thu Nov 29 10:40:32 2012 -0500 @@ -431,6 +431,11 @@ SCALAR_MAPPER (asinh, ::asinh); SCALAR_MAPPER (atan, ::atan); SCALAR_MAPPER (atanh, ::atanh); + SCALAR_MAPPER (erf, ::erf); + SCALAR_MAPPER (erfc, ::erfc); + SCALAR_MAPPER (erfcx, ::erfcx); + SCALAR_MAPPER (erfi, ::erfi); + SCALAR_MAPPER (dawson, ::dawson); SCALAR_MAPPER (ceil, ::ceil); SCALAR_MAPPER (conj, std::conj); SCALAR_MAPPER (cos, std::cos);
--- a/libinterp/octave-value/ov-cx-mat.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-cx-mat.cc Thu Nov 29 10:40:32 2012 -0500 @@ -777,6 +777,11 @@ ARRAY_MAPPER (asinh, Complex, ::asinh); ARRAY_MAPPER (atan, Complex, ::atan); ARRAY_MAPPER (atanh, Complex, ::atanh); + ARRAY_MAPPER (erf, Complex, ::erf); + ARRAY_MAPPER (erfc, Complex, ::erfc); + ARRAY_MAPPER (erfcx, Complex, ::erfcx); + ARRAY_MAPPER (erfi, Complex, ::erfi); + ARRAY_MAPPER (dawson, Complex, ::dawson); ARRAY_MAPPER (ceil, Complex, ::ceil); ARRAY_MAPPER (cos, Complex, std::cos); ARRAY_MAPPER (cosh, Complex, std::cosh);
--- a/libinterp/octave-value/ov-cx-sparse.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-cx-sparse.cc Thu Nov 29 10:40:32 2012 -0500 @@ -903,6 +903,11 @@ ARRAY_MAPPER (asinh, Complex, ::asinh); ARRAY_MAPPER (atan, Complex, ::atan); ARRAY_MAPPER (atanh, Complex, ::atanh); + ARRAY_MAPPER (erf, Complex, ::erf); + ARRAY_MAPPER (erfc, Complex, ::erfc); + ARRAY_MAPPER (erfcx, Complex, ::erfcx); + ARRAY_MAPPER (erfi, Complex, ::erfi); + ARRAY_MAPPER (dawson, Complex, ::dawson); ARRAY_MAPPER (ceil, Complex, ::ceil); ARRAY_MAPPER (conj, Complex, std::conj<double>); ARRAY_MAPPER (cos, Complex, std::cos);
--- a/libinterp/octave-value/ov-float.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-float.cc Thu Nov 29 10:40:32 2012 -0500 @@ -295,6 +295,8 @@ SCALAR_MAPPER (erfcinv, ::erfcinv); SCALAR_MAPPER (erfc, ::erfcf); SCALAR_MAPPER (erfcx, ::erfcx); + SCALAR_MAPPER (erfi, ::erfi); + SCALAR_MAPPER (dawson, ::dawson); SCALAR_MAPPER (gamma, xgamma); SCALAR_MAPPER (lgamma, rc_lgamma); SCALAR_MAPPER (cbrt, ::cbrtf);
--- a/libinterp/octave-value/ov-flt-complex.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-flt-complex.cc Thu Nov 29 10:40:32 2012 -0500 @@ -416,6 +416,11 @@ SCALAR_MAPPER (asinh, ::asinh); SCALAR_MAPPER (atan, ::atan); SCALAR_MAPPER (atanh, ::atanh); + SCALAR_MAPPER (erf, ::erf); + SCALAR_MAPPER (erfc, ::erfc); + SCALAR_MAPPER (erfcx, ::erfcx); + SCALAR_MAPPER (erfi, ::erfi); + SCALAR_MAPPER (dawson, ::dawson); SCALAR_MAPPER (ceil, ::ceil); SCALAR_MAPPER (conj, std::conj); SCALAR_MAPPER (cos, std::cos);
--- a/libinterp/octave-value/ov-flt-cx-mat.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-flt-cx-mat.cc Thu Nov 29 10:40:32 2012 -0500 @@ -744,6 +744,11 @@ ARRAY_MAPPER (asinh, FloatComplex, ::asinh); ARRAY_MAPPER (atan, FloatComplex, ::atan); ARRAY_MAPPER (atanh, FloatComplex, ::atanh); + ARRAY_MAPPER (erf, FloatComplex, ::erf); + ARRAY_MAPPER (erfc, FloatComplex, ::erfc); + ARRAY_MAPPER (erfcx, FloatComplex, ::erfcx); + ARRAY_MAPPER (erfi, FloatComplex, ::erfi); + ARRAY_MAPPER (dawson, FloatComplex, ::dawson); ARRAY_MAPPER (ceil, FloatComplex, ::ceil); ARRAY_MAPPER (cos, FloatComplex, std::cos); ARRAY_MAPPER (cosh, FloatComplex, std::cosh);
--- a/libinterp/octave-value/ov-flt-re-mat.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-flt-re-mat.cc Thu Nov 29 10:40:32 2012 -0500 @@ -783,6 +783,8 @@ ARRAY_MAPPER (erfcinv, float, ::erfcinv); ARRAY_MAPPER (erfc, float, ::erfcf); ARRAY_MAPPER (erfcx, float, ::erfcx); + ARRAY_MAPPER (erfi, float, ::erfi); + ARRAY_MAPPER (dawson, float, ::dawson); ARRAY_MAPPER (gamma, float, xgamma); RC_ARRAY_MAPPER (lgamma, FloatComplex, rc_lgamma); ARRAY_MAPPER (cbrt, float, ::cbrtf);
--- a/libinterp/octave-value/ov-re-mat.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-re-mat.cc Thu Nov 29 10:40:32 2012 -0500 @@ -906,6 +906,8 @@ ARRAY_MAPPER (erfcinv, double, ::erfcinv); ARRAY_MAPPER (erfc, double, ::erfc); ARRAY_MAPPER (erfcx, double, ::erfcx); + ARRAY_MAPPER (erfi, double, ::erfi); + ARRAY_MAPPER (dawson, double, ::dawson); ARRAY_MAPPER (gamma, double, xgamma); RC_ARRAY_MAPPER (lgamma, Complex, rc_lgamma); ARRAY_MAPPER (cbrt, double, ::cbrt);
--- a/libinterp/octave-value/ov-re-sparse.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-re-sparse.cc Thu Nov 29 10:40:32 2012 -0500 @@ -912,6 +912,9 @@ ARRAY_MAPPER (erfinv, double, ::erfinv); ARRAY_MAPPER (erfcinv, double, ::erfcinv); ARRAY_MAPPER (erfc, double, ::erfc); + ARRAY_MAPPER (erfcx, double, ::erfcx); + ARRAY_MAPPER (erfi, double, ::erfi); + ARRAY_MAPPER (dawson, double, ::dawson); ARRAY_MAPPER (gamma, double, xgamma); ARRAY_MAPPER (lgamma, Complex, rc_lgamma); ARRAY_MAPPER (cbrt, double, ::cbrt);
--- a/libinterp/octave-value/ov-scalar.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov-scalar.cc Thu Nov 29 10:40:32 2012 -0500 @@ -311,6 +311,8 @@ SCALAR_MAPPER (erfcinv, ::erfcinv); SCALAR_MAPPER (erfc, ::erfc); SCALAR_MAPPER (erfcx, ::erfcx); + SCALAR_MAPPER (erfi, ::erfi); + SCALAR_MAPPER (dawson, ::dawson); SCALAR_MAPPER (gamma, xgamma); SCALAR_MAPPER (lgamma, rc_lgamma); SCALAR_MAPPER (cbrt, ::cbrt);
--- a/libinterp/octave-value/ov.h Wed Nov 28 14:05:20 2012 -0500 +++ b/libinterp/octave-value/ov.h Thu Nov 29 10:40:32 2012 -0500 @@ -1126,6 +1126,8 @@ MAPPER_FORWARD (erfcinv) MAPPER_FORWARD (erfc) MAPPER_FORWARD (erfcx) + MAPPER_FORWARD (erfi) + MAPPER_FORWARD (dawson) MAPPER_FORWARD (exp) MAPPER_FORWARD (expm1) MAPPER_FORWARD (finite)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/liboctave/cruft/Faddeeva/Faddeeva.cc Thu Nov 29 10:40:32 2012 -0500 @@ -0,0 +1,2429 @@ +/* Copyright (c) 2012 Massachusetts Institute of Technology + * + * 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. + */ + +/* (Note that this file can be compiled with either C++, in which + case it uses C++ std::complex<double>, or C, in which case it + uses C99 double complex.) */ + +/* Available at: http://ab-initio.mit.edu/Faddeeva + + Computes various error functions (erf, erfc, erfi, erfcx), + including the Dawson integral, in the complex plane, based + on algorithms for the computation of the Faddeeva function + w(z) = exp(-z^2) * erfc(-i*z). + Given w(z), the error functions are mostly straightforward + to compute, except for certain regions where we have to + switch to Taylor expansions to avoid cancellation errors + [e.g. near the origin for erf(z)]. + + To compute the Faddeeva function, we use a combination of two + algorithms: + + For sufficiently large |z|, we use a continued-fraction expansion + for w(z) similar to those described in: + + Walter Gautschi, "Efficient computation of the complex error + function," SIAM J. Numer. Anal. 7(1), pp. 187-198 (1970) + + G. P. M. Poppe and C. M. J. Wijers, "More efficient computation + of the complex error function," ACM Trans. Math. Soft. 16(1), + pp. 38-46 (1990). + + Unlike those papers, however, we switch to a completely different + algorithm for smaller |z|: + + Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the + Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38(2), 15 + (2011). + + (I initially used this algorithm for all z, but it turned out to be + significantly slower than the continued-fraction expansion for + larger |z|. On the other hand, it is competitive for smaller |z|, + and is significantly more accurate than the Poppe & Wijers code + in some regions, e.g. in the vicinity of z=1+1i.) + + Note that this is an INDEPENDENT RE-IMPLEMENTATION of these algorithms, + based on the description in the papers ONLY. In particular, I did + not refer to the authors' Fortran or Matlab implementations, respectively, + (which are under restrictive ACM copyright terms and therefore unusable + in free/open-source software). + + Steven G. Johnson, Massachusetts Institute of Technology + http://math.mit.edu/~stevenj + October 2012. + + -- Note that Algorithm 916 assumes that the erfc(x) function, + or rather the scaled function erfcx(x) = exp(x*x)*erfc(x), + is supplied for REAL arguments x. I originally used an + erfcx routine derived from DERFC in SLATEC, but I have + since replaced it with a much faster routine written by + me which uses a combination of continued-fraction expansions + and a lookup table of Chebyshev polynomials. For speed, + I implemented a similar algorithm for Im[w(x)] of real x, + since this comes up frequently in the other error functions. + + A small test program is included the end, which checks + the w(z) etc. results against several known values. To compile + the test function, compile with -DTEST_FADDEEVA (that is, + #define TEST_FADDEEVA). + + If HAVE_CONFIG_H is #defined (e.g. by compiling with -DHAVE_CONFIG_H), + then we #include "config.h", which is assumed to be a GNU autoconf-style + header defining HAVE_* macros to indicate the presence of features. In + particular, if HAVE_ISNAN and HAVE_ISINF are #defined, we use those + functions in math.h instead of defining our own, and if HAVE_ERF and/or + HAVE_ERFC are defined we use those functions from <cmath> for erf and + erfc of real arguments, respectively, instead of defining our own. + + REVISION HISTORY: + 4 October 2012: Initial public release (SGJ) + 5 October 2012: Revised (SGJ) to fix spelling error, + start summation for large x at round(x/a) (> 1) + rather than ceil(x/a) as in the original + paper, which should slightly improve performance + (and, apparently, slightly improves accuracy) + 19 October 2012: Revised (SGJ) to fix bugs for large x, large -y, + and 15<x<26. Performance improvements. Prototype + now supplies default value for relerr. + 24 October 2012: Switch to continued-fraction expansion for + sufficiently large z, for performance reasons. + Also, avoid spurious overflow for |z| > 1e154. + Set relerr argument to min(relerr,0.1). + 27 October 2012: Enhance accuracy in Re[w(z)] taken by itself, + by switching to Alg. 916 in a region near + the real-z axis where continued fractions + have poor relative accuracy in Re[w(z)]. Thanks + to M. Zaghloul for the tip. + 29 October 2012: Replace SLATEC-derived erfcx routine with + completely rewritten code by me, using a very + different algorithm which is much faster. + 30 October 2012: Implemented special-case code for real z + (where real part is exp(-x^2) and imag part is + Dawson integral), using algorithm similar to erfx. + Export ImFaddeeva_w function to make Dawson's + integral directly accessible. + 3 November 2012: Provide implementations of erf, erfc, erfcx, + and Dawson functions in Faddeeva:: namespace, + in addition to Faddeeva::w. Provide header + file Faddeeva.hh. + 4 November 2012: Slightly faster erf for real arguments. + Updated MATLAB and Octave plugins. + 27 November 2012: Support compilation with either C++ or + plain C (using C99 complex numbers). + For real x, use standard-library erf(x) + and erfc(x) if available (for C99 or C++11). + #include "config.h" if HAVE_CONFIG_H is #defined. +*/ + +///////////////////////////////////////////////////////////////////////// +/* If this file is compiled as a part of a larger project, + support using an autoconf-style config.h header file + (with various "HAVE_*" #defines to indicate features) + if HAVE_CONFIG_H is #defined (in GNU autotools style). */ + +#ifdef HAVE_CONFIG_H +# include "config.h" +#endif + +///////////////////////////////////////////////////////////////////////// +// basic math stuff, C++ & C99 compatibility + +#define Inf (1./0.) // infinity +#define NaN (0./0.) // NaN + +#ifdef __cplusplus + +# include "Faddeeva.hh" + +# include <cfloat> +# include <cmath> +using namespace std; + +typedef complex<double> cmplx; + +// Use C-like complex syntax, since the C syntax is more restrictive +# define cexp(z) exp(z) +# define creal(z) real(z) +# define cimag(z) imag(z) +# define cpolar(r,t) polar(r,t) + +# define CMPLX(a,b) cmplx(a,b) + +# define FADDEEVA(name) Faddeeva::name +# define FADDEEVA_RE(name) Faddeeva::name + +// isnan/isinf were introduced in C++11 +# if (__cplusplus < 201103L) && (!defined(HAVE_ISNAN) || !defined(HAVE_ISINF)) +static inline bool my_isnan(double x) { return x != x; } +# define isnan my_isnan +static inline bool my_isinf(double x) { return 1/x == 0.; } +# define isinf my_isinf +# endif + +#else // !__cplusplus, i.e. pure C (requires C99 features) + +# include "Faddeeva.h" + +# include <float.h> +# include <math.h> + +typedef double complex cmplx; + +# define CMPLX(a,b) ((a) + I*(b)) + +static inline cmplx cpolar(double r, double t) +{ + if (r == 0.0 && !isnan(t)) + return 0.0; + else + return CMPLX(r * cos(t), r * sin(t)); +} + +# define FADDEEVA(name) Faddeeva_ ## name +# define FADDEEVA_RE(name) Faddeeva_ ## name ## _re + +#endif + +///////////////////////////////////////////////////////////////////////// +// Auxiliary routines to compute other special functions based on w(z) + +// compute erfcx(z) = exp(z^2) erfz(z) +cmplx FADDEEVA(erfcx)(cmplx z, double relerr) +{ + return FADDEEVA(w)(CMPLX(-cimag(z), creal(z)), relerr); +} + +// compute the error function erf(x) +double FADDEEVA_RE(erf)(double x) +{ +#if !defined(__cplusplus) + return erf(x); // C99 supplies erf in math.h +#elif (__cplusplus >= 201103L) || defined(HAVE_ERF) + return ::erf(x); // C++11 supplies std::erf in cmath +#else + double mx2 = -x*x; + if (mx2 < -750) // underflow + return (x >= 0 ? 1.0 : -1.0); + + if (x >= 0) { + if (x < 5e-3) goto taylor; + return 1.0 - exp(mx2) * FADDEEVA_RE(erfcx)(x); + } + else { // x < 0 + if (x > -5e-3) goto taylor; + return exp(mx2) * FADDEEVA_RE(erfcx)(-x) - 1.0; + } + + // Use Taylor series for small |x|, to avoid cancellation inaccuracy + // erf(x) = 2/sqrt(pi) * x * (1 - x^2/3 + x^4/10 - ...) + taylor: + return x * (1.1283791670955125739 + + mx2 * (0.37612638903183752464 + + mx2 * 0.11283791670955125739)); +#endif +} + +// compute the error function erf(z) +cmplx FADDEEVA(erf)(cmplx z, double relerr) +{ + double x = creal(z), y = cimag(z); + + if (y == 0) + return CMPLX(FADDEEVA_RE(erf)(x), + y); // preserve sign of 0 + if (x == 0) // handle separately for speed & handling of y = Inf or NaN + return CMPLX(x, // preserve sign of 0 + /* handle y -> Inf limit manually, since + exp(y^2) -> Inf but Im[w(y)] -> 0, so + IEEE will give us a NaN when it should be Inf */ + y*y > 720 ? (y > 0 ? Inf : -Inf) + : exp(y*y) * FADDEEVA(w_im)(y)); + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2*x*y; // Im(-z^2) + if (mRe_z2 < -750) // underflow + return (x >= 0 ? 1.0 : -1.0); + + /* Handle positive and negative x via different formulas, + using the mirror symmetries of w, to avoid overflow/underflow + problems from multiplying exponentially large and small quantities. */ + if (x >= 0) { + if (x < 5e-3) { + if (fabs(y) < 5e-3) + goto taylor; + else if (fabs(mIm_z2) < 5e-3) + goto taylor_erfi; + } + /* don't use complex exp function, since that will produce spurious NaN + values when multiplying w in an overflow situation. */ + return 1.0 - exp(mRe_z2) * + (CMPLX(cos(mIm_z2), sin(mIm_z2)) + * FADDEEVA(w)(CMPLX(-y,x), relerr)); + } + else { // x < 0 + if (x > -5e-3) { // duplicate from above to avoid fabs(x) call + if (fabs(y) < 5e-3) + goto taylor; + else if (fabs(mIm_z2) < 5e-3) + goto taylor_erfi; + } + else if (isnan(x)) + return CMPLX(NaN, y == 0 ? 0 : NaN); + /* don't use complex exp function, since that will produce spurious NaN + values when multiplying w in an overflow situation. */ + return exp(mRe_z2) * + (CMPLX(cos(mIm_z2), sin(mIm_z2)) + * FADDEEVA(w)(CMPLX(y,-x), relerr)) - 1.0; + } + + // Use Taylor series for small |z|, to avoid cancellation inaccuracy + // erf(z) = 2/sqrt(pi) * z * (1 - z^2/3 + z^4/10 - ...) + taylor: + { + cmplx mz2 = CMPLX(mRe_z2, mIm_z2); // -z^2 + return z * (1.1283791670955125739 + + mz2 * (0.37612638903183752464 + + mz2 * 0.11283791670955125739)); + } + + /* for small |x| and small |xy|, + use Taylor series to avoid cancellation inaccuracy: + erf(x+iy) = erf(iy) + + 2*exp(y^2)/sqrt(pi) * + [ x * (1 - x^2 * (1+2y^2)/3 + x^4 * (3+12y^2+4y^4)/30 + ... + - i * x^2 * y * (1 - x^2 * (3+2y^2)/6 + ...) ] + where: + erf(iy) = exp(y^2) * Im[w(y)] + */ + taylor_erfi: + { + double x2 = x*x, y2 = y*y; + double expy2 = exp(y2); + return CMPLX + (expy2 * x * (1.1283791670955125739 + - x2 * (0.37612638903183752464 + + 0.75225277806367504925*y2) + + x2*x2 * (0.11283791670955125739 + + y2 * (0.45135166683820502956 + + 0.15045055561273500986*y2))), + expy2 * (FADDEEVA(w_im)(y) + - x2*y * (1.1283791670955125739 + - x2 * (0.56418958354775628695 + + 0.37612638903183752464*y2)))); + } +} + +// erfi(z) = -i erf(iz) +cmplx FADDEEVA(erfi)(cmplx z, double relerr) +{ + cmplx e = FADDEEVA(erf)(CMPLX(-cimag(z),creal(z)), relerr); + return CMPLX(cimag(e), -creal(e)); +} + +// erfi(x) = -i erf(ix) +double FADDEEVA_RE(erfi)(double x) +{ + return x*x > 720 ? (x > 0 ? Inf : -Inf) + : exp(x*x) * FADDEEVA(w_im)(x); +} + +// erfc(x) = 1 - erf(x) +double FADDEEVA_RE(erfc)(double x) +{ +#if !defined(__cplusplus) + return erfc(x); // C99 supplies erfc in math.h +#elif (__cplusplus >= 201103L) || defined(HAVE_ERFC) + return ::erfc(x); // C++11 supplies std::erfc in cmath +#else + if (x*x > 750) // underflow + return (x >= 0 ? 0.0 : 2.0); + return x >= 0 ? exp(-x*x) * FADDEEVA_RE(erfcx)(x) + : 2. - exp(-x*x) * FADDEEVA_RE(erfcx)(-x); +#endif +} + +// erfc(z) = 1 - erf(z) +cmplx FADDEEVA(erfc)(cmplx z, double relerr) +{ + double x = creal(z), y = cimag(z); + + if (x == 0.) + return CMPLX(1, + /* handle y -> Inf limit manually, since + exp(y^2) -> Inf but Im[w(y)] -> 0, so + IEEE will give us a NaN when it should be Inf */ + y*y > 720 ? (y > 0 ? -Inf : Inf) + : -exp(y*y) * FADDEEVA(w_im)(y)); + if (y == 0.) { + if (x*x > 750) // underflow + return CMPLX(x >= 0 ? 0.0 : 2.0, + -y); // preserve sign of 0 + return CMPLX(x >= 0 ? exp(-x*x) * FADDEEVA_RE(erfcx)(x) + : 2. - exp(-x*x) * FADDEEVA_RE(erfcx)(-x), + -y); // preserve sign of zero + } + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2*x*y; // Im(-z^2) + if (mRe_z2 < -750) // underflow + return (x >= 0 ? 0.0 : 2.0); + + if (x >= 0) + return cexp(CMPLX(mRe_z2, mIm_z2)) + * FADDEEVA(w)(CMPLX(-y,x), relerr); + else + return 2.0 - cexp(CMPLX(mRe_z2, mIm_z2)) + * FADDEEVA(w)(CMPLX(y,-x), relerr); +} + +// compute Dawson(x) = sqrt(pi)/2 * exp(-x^2) * erfi(x) +double FADDEEVA_RE(Dawson)(double x) +{ + const double spi2 = 0.8862269254527580136490837416705725913990; // sqrt(pi)/2 + return spi2 * FADDEEVA(w_im)(x); +} + +// compute Dawson(z) = sqrt(pi)/2 * exp(-z^2) * erfi(z) +cmplx FADDEEVA(Dawson)(cmplx z, double relerr) +{ + const double spi2 = 0.8862269254527580136490837416705725913990; // sqrt(pi)/2 + double x = creal(z), y = cimag(z); + + // handle axes separately for speed & proper handling of x or y = Inf or NaN + if (y == 0) + return CMPLX(spi2 * FADDEEVA(w_im)(x), + -y); // preserve sign of 0 + if (x == 0) { + double y2 = y*y; + if (y2 < 2.5e-5) { // Taylor expansion + return CMPLX(x, // preserve sign of 0 + y * (1. + + y2 * (0.6666666666666666666666666666666666666667 + + y2 * 0.2666666666666666666666666666666666666667))); + } + return CMPLX(x, // preserve sign of 0 + spi2 * (y >= 0 + ? exp(y2) - FADDEEVA_RE(erfcx)(y) + : FADDEEVA_RE(erfcx)(-y) - exp(y2))); + } + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2*x*y; // Im(-z^2) + cmplx mz2 = CMPLX(mRe_z2, mIm_z2); // -z^2 + + /* Handle positive and negative x via different formulas, + using the mirror symmetries of w, to avoid overflow/underflow + problems from multiplying exponentially large and small quantities. */ + if (y >= 0) { + if (y < 5e-3) { + if (fabs(x) < 5e-3) + goto taylor; + else if (fabs(mIm_z2) < 5e-3) + goto taylor_realaxis; + } + cmplx res = cexp(mz2) - FADDEEVA(w)(z, relerr); + return spi2 * CMPLX(-cimag(res), creal(res)); + } + else { // y < 0 + if (y > -5e-3) { // duplicate from above to avoid fabs(x) call + if (fabs(x) < 5e-3) + goto taylor; + else if (fabs(mIm_z2) < 5e-3) + goto taylor_realaxis; + } + else if (isnan(y)) + return CMPLX(x == 0 ? 0 : NaN, NaN); + cmplx res = FADDEEVA(w)(-z, relerr) - cexp(mz2); + return spi2 * CMPLX(-cimag(res), creal(res)); + } + + // Use Taylor series for small |z|, to avoid cancellation inaccuracy + // dawson(z) = z - 2/3 z^3 + 4/15 z^5 + ... + taylor: + return z * (1. + + mz2 * (0.6666666666666666666666666666666666666667 + + mz2 * 0.2666666666666666666666666666666666666667)); + + /* for small |y| and small |xy|, + use Taylor series to avoid cancellation inaccuracy: + dawson(x + iy) + = D + y^2 (D + x - 2Dx^2) + + y^4 (D/2 + 5x/6 - 2Dx^2 - x^3/3 + 2Dx^4/3) + + iy [ (1-2Dx) + 2/3 y^2 (1 - 3Dx - x^2 + 2Dx^3) + + y^4/15 (4 - 15Dx - 9x^2 + 20Dx^3 + 2x^4 - 4Dx^5) ] + ... + where D = dawson(x) + + However, for large |x|, 2Dx -> 1 which gives cancellation problems in + this series (many of the leading terms cancel). So, for large |x|, + we need to substitute a continued-fraction expansion for D. + + dawson(x) = 0.5 / (x-0.5/(x-1/(x-1.5/(x-2/(x-2.5/(x...)))))) + + The 6 terms shown here seems to be the minimum needed to be + accurate as soon as the simpler Taylor expansion above starts + breaking down. Using this 6-term expansion, factoring out the + denominator, and simplifying with Maple, we obtain: + + Re dawson(x + iy) * (-15 + 90x^2 - 60x^4 + 8x^6) / x + = 33 - 28x^2 + 4x^4 + y^2 (18 - 4x^2) + 4 y^4 + Im dawson(x + iy) * (-15 + 90x^2 - 60x^4 + 8x^6) / y + = -15 + 24x^2 - 4x^4 + 2/3 y^2 (6x^2 - 15) - 4 y^4 + + Finally, for |x| > 5e7, we can use a simpler 1-term continued-fraction + expansion for the real part, and a 2-term expansion for the imaginary + part. (This avoids overflow problems for huge |x|.) This yields: + + Re dawson(x + iy) = [1 + y^2 (1 + y^2/2 - (xy)^2/3)] / (2x) + Im dawson(x + iy) = y [ -1 - 2/3 y^2 + y^4/15 (2x^2 - 4) ] / (2x^2 - 1) + + */ + taylor_realaxis: + { + double x2 = x*x; + if (x2 > 1600) { // |x| > 40 + double y2 = y*y; + if (x2 > 25e14) {// |x| > 5e7 + double xy2 = (x*y)*(x*y); + return CMPLX((0.5 + y2 * (0.5 + 0.25*y2 + - 0.16666666666666666667*xy2)) / x, + y * (-1 + y2 * (-0.66666666666666666667 + + 0.13333333333333333333*xy2 + - 0.26666666666666666667*y2)) + / (2*x2 - 1)); + } + return (1. / (-15 + x2*(90 + x2*(-60 + 8*x2)))) * + CMPLX(x * (33 + x2 * (-28 + 4*x2) + + y2 * (18 - 4*x2 + 4*y2)), + y * (-15 + x2 * (24 - 4*x2) + + y2 * (4*x2 - 10 - 4*y2))); + } + else { + double D = spi2 * FADDEEVA(w_im)(x); + double x2 = x*x, y2 = y*y; + return CMPLX + (D + y2 * (D + x - 2*D*x2) + + y2*y2 * (D * (0.5 - x2 * (2 - 0.66666666666666666667*x2)) + + x * (0.83333333333333333333 + - 0.33333333333333333333 * x2)), + y * (1 - 2*D*x + + y2 * 0.66666666666666666667 * (1 - x2 - D*x * (3 - 2*x2)) + + y2*y2 * (0.26666666666666666667 - + x2 * (0.6 - 0.13333333333333333333 * x2) + - D*x * (1 - x2 * (1.3333333333333333333 + - 0.26666666666666666667 * x2))))); + } + } +} + +///////////////////////////////////////////////////////////////////////// + +// return sinc(x) = sin(x)/x, given both x and sin(x) +// [since we only use this in cases where sin(x) has already been computed] +static inline double sinc(double x, double sinx) { + return fabs(x) < 1e-4 ? 1 - (0.1666666666666666666667)*x*x : sinx / x; +} + +// sinh(x) via Taylor series, accurate to machine precision for |x| < 1e-2 +static inline double sinh_taylor(double x) { + return x * (1 + (x*x) * (0.1666666666666666666667 + + 0.00833333333333333333333 * (x*x))); +} + +static inline double sqr(double x) { return x*x; } + +// precomputed table of expa2n2[n-1] = exp(-a2*n*n) +// for double-precision a2 = 0.26865... in FADDEEVA(w), below. +static const double expa2n2[] = { + 7.64405281671221563e-01, + 3.41424527166548425e-01, + 8.91072646929412548e-02, + 1.35887299055460086e-02, + 1.21085455253437481e-03, + 6.30452613933449404e-05, + 1.91805156577114683e-06, + 3.40969447714832381e-08, + 3.54175089099469393e-10, + 2.14965079583260682e-12, + 7.62368911833724354e-15, + 1.57982797110681093e-17, + 1.91294189103582677e-20, + 1.35344656764205340e-23, + 5.59535712428588720e-27, + 1.35164257972401769e-30, + 1.90784582843501167e-34, + 1.57351920291442930e-38, + 7.58312432328032845e-43, + 2.13536275438697082e-47, + 3.51352063787195769e-52, + 3.37800830266396920e-57, + 1.89769439468301000e-62, + 6.22929926072668851e-68, + 1.19481172006938722e-73, + 1.33908181133005953e-79, + 8.76924303483223939e-86, + 3.35555576166254986e-92, + 7.50264110688173024e-99, + 9.80192200745410268e-106, + 7.48265412822268959e-113, + 3.33770122566809425e-120, + 8.69934598159861140e-128, + 1.32486951484088852e-135, + 1.17898144201315253e-143, + 6.13039120236180012e-152, + 1.86258785950822098e-160, + 3.30668408201432783e-169, + 3.43017280887946235e-178, + 2.07915397775808219e-187, + 7.36384545323984966e-197, + 1.52394760394085741e-206, + 1.84281935046532100e-216, + 1.30209553802992923e-226, + 5.37588903521080531e-237, + 1.29689584599763145e-247, + 1.82813078022866562e-258, + 1.50576355348684241e-269, + 7.24692320799294194e-281, + 2.03797051314726829e-292, + 3.34880215927873807e-304, + 0.0 // underflow (also prevents reads past array end, below) +}; + +///////////////////////////////////////////////////////////////////////// + +cmplx FADDEEVA(w)(cmplx z, double relerr) +{ + if (creal(z) == 0.0) + return CMPLX(FADDEEVA_RE(erfcx)(cimag(z)), + creal(z)); // give correct sign of 0 in cimag(w) + else if (cimag(z) == 0) + return CMPLX(exp(-sqr(creal(z))), + FADDEEVA(w_im)(creal(z))); + + double a, a2, c; + if (relerr <= DBL_EPSILON) { + relerr = DBL_EPSILON; + a = 0.518321480430085929872; // pi / sqrt(-log(eps*0.5)) + c = 0.329973702884629072537; // (2/pi) * a; + a2 = 0.268657157075235951582; // a^2 + } + else { + const double pi = 3.14159265358979323846264338327950288419716939937510582; + if (relerr > 0.1) relerr = 0.1; // not sensible to compute < 1 digit + a = pi / sqrt(-log(relerr*0.5)); + c = (2/pi)*a; + a2 = a*a; + } + const double x = fabs(creal(z)); + const double y = cimag(z), ya = fabs(y); + + cmplx ret = 0.; // return value + + double sum1 = 0, sum2 = 0, sum3 = 0, sum4 = 0, sum5 = 0; + +#define USE_CONTINUED_FRACTION 1 // 1 to use continued fraction for large |z| + +#if USE_CONTINUED_FRACTION + if (ya > 7 || (x > 6 // continued fraction is faster + /* As pointed out by M. Zaghloul, the continued + fraction seems to give a large relative error in + Re w(z) for |x| ~ 6 and small |y|, so use + algorithm 816 in this region: */ + && (ya > 0.1 || (x > 8 && ya > 1e-10) || x > 28))) { + + /* Poppe & Wijers suggest using a number of terms + nu = 3 + 1442 / (26*rho + 77) + where rho = sqrt((x/x0)^2 + (y/y0)^2) where x0=6.3, y0=4.4. + (They only use this expansion for rho >= 1, but rho a little less + than 1 seems okay too.) + Instead, I did my own fit to a slightly different function + that avoids the hypotenuse calculation, using NLopt to minimize + the sum of the squares of the errors in nu with the constraint + that the estimated nu be >= minimum nu to attain machine precision. + I also separate the regions where nu == 2 and nu == 1. */ + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + double xs = y < 0 ? -creal(z) : creal(z); // compute for -z if y < 0 + if (x + ya > 4000) { // nu <= 2 + if (x + ya > 1e7) { // nu == 1, w(z) = i/sqrt(pi) / z + // scale to avoid overflow + if (x > ya) { + double yax = ya / xs; + double denom = ispi / (xs + yax*ya); + ret = CMPLX(denom*yax, denom); + } + else if (isinf(ya)) + return ((isnan(x) || y < 0) + ? CMPLX(NaN,NaN) : CMPLX(0,0)); + else { + double xya = xs / ya; + double denom = ispi / (xya*xs + ya); + ret = CMPLX(denom, denom*xya); + } + } + else { // nu == 2, w(z) = i/sqrt(pi) * z / (z*z - 0.5) + double dr = xs*xs - ya*ya - 0.5, di = 2*xs*ya; + double denom = ispi / (dr*dr + di*di); + ret = CMPLX(denom * (xs*di-ya*dr), denom * (xs*dr+ya*di)); + } + } + else { // compute nu(z) estimate and do general continued fraction + const double c0=3.9, c1=11.398, c2=0.08254, c3=0.1421, c4=0.2023; // fit + double nu = floor(c0 + c1 / (c2*x + c3*ya + c4)); + double wr = xs, wi = ya; + for (nu = 0.5 * (nu - 1); nu > 0.4; nu -= 0.5) { + // w <- z - nu/w: + double denom = nu / (wr*wr + wi*wi); + wr = xs - wr * denom; + wi = ya + wi * denom; + } + { // w(z) = i/sqrt(pi) / w: + double denom = ispi / (wr*wr + wi*wi); + ret = CMPLX(denom*wi, denom*wr); + } + } + if (y < 0) { + // use w(z) = 2.0*exp(-z*z) - w(-z), + // but be careful of overflow in exp(-z*z) + // = exp(-(xs*xs-ya*ya) -2*i*xs*ya) + return 2.0*cexp(CMPLX((ya-xs)*(xs+ya), 2*xs*y)) - ret; + } + else + return ret; + } +#else // !USE_CONTINUED_FRACTION + if (x + ya > 1e7) { // w(z) = i/sqrt(pi) / z, to machine precision + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + double xs = y < 0 ? -creal(z) : creal(z); // compute for -z if y < 0 + // scale to avoid overflow + if (x > ya) { + double yax = ya / xs; + double denom = ispi / (xs + yax*ya); + ret = CMPLX(denom*yax, denom); + } + else { + double xya = xs / ya; + double denom = ispi / (xya*xs + ya); + ret = CMPLX(denom, denom*xya); + } + if (y < 0) { + // use w(z) = 2.0*exp(-z*z) - w(-z), + // but be careful of overflow in exp(-z*z) + // = exp(-(xs*xs-ya*ya) -2*i*xs*ya) + return 2.0*cexp(CMPLX((ya-xs)*(xs+ya), 2*xs*y)) - ret; + } + else + return ret; + } +#endif // !USE_CONTINUED_FRACTION + + /* Note: The test that seems to be suggested in the paper is x < + sqrt(-log(DBL_MIN)), about 26.6, since otherwise exp(-x^2) + underflows to zero and sum1,sum2,sum4 are zero. However, long + before this occurs, the sum1,sum2,sum4 contributions are + negligible in double precision; I find that this happens for x > + about 6, for all y. On the other hand, I find that the case + where we compute all of the sums is faster (at least with the + precomputed expa2n2 table) until about x=10. Furthermore, if we + try to compute all of the sums for x > 20, I find that we + sometimes run into numerical problems because underflow/overflow + problems start to appear in the various coefficients of the sums, + below. Therefore, we use x < 10 here. */ + else if (x < 10) { + double prod2ax = 1, prodm2ax = 1; + double expx2; + + if (isnan(y)) + return CMPLX(y,y); + + /* Somewhat ugly copy-and-paste duplication here, but I see significant + speedups from using the special-case code with the precomputed + exponential, and the x < 5e-4 special case is needed for accuracy. */ + + if (relerr == DBL_EPSILON) { // use precomputed exp(-a2*(n*n)) table + if (x < 5e-4) { // compute sum4 and sum5 together as sum5-sum4 + const double x2 = x*x; + expx2 = 1 - x2 * (1 - 0.5*x2); // exp(-x*x) via Taylor + // compute exp(2*a*x) and exp(-2*a*x) via Taylor, to double precision + const double ax2 = 1.036642960860171859744*x; // 2*a*x + const double exp2ax = + 1 + ax2 * (1 + ax2 * (0.5 + 0.166666666666666666667*ax2)); + const double expm2ax = + 1 - ax2 * (1 - ax2 * (0.5 - 0.166666666666666666667*ax2)); + for (int n = 1; 1; ++n) { + const double coef = expa2n2[n-1] * expx2 / (a2*(n*n) + y*y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum3 += coef * prod2ax; + + // really = sum5 - sum4 + sum5 += coef * (2*a) * n * sinh_taylor((2*a)*n*x); + + // test convergence via sum3 + if (coef * prod2ax < relerr * sum3) break; + } + } + else { // x > 5e-4, compute sum4 and sum5 separately + expx2 = exp(-x*x); + const double exp2ax = exp((2*a)*x), expm2ax = 1 / exp2ax; + for (int n = 1; 1; ++n) { + const double coef = expa2n2[n-1] * expx2 / (a2*(n*n) + y*y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum4 += (coef * prodm2ax) * (a*n); + sum3 += coef * prod2ax; + sum5 += (coef * prod2ax) * (a*n); + // test convergence via sum5, since this sum has the slowest decay + if ((coef * prod2ax) * (a*n) < relerr * sum5) break; + } + } + } + else { // relerr != DBL_EPSILON, compute exp(-a2*(n*n)) on the fly + const double exp2ax = exp((2*a)*x), expm2ax = 1 / exp2ax; + if (x < 5e-4) { // compute sum4 and sum5 together as sum5-sum4 + const double x2 = x*x; + expx2 = 1 - x2 * (1 - 0.5*x2); // exp(-x*x) via Taylor + for (int n = 1; 1; ++n) { + const double coef = exp(-a2*(n*n)) * expx2 / (a2*(n*n) + y*y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum3 += coef * prod2ax; + + // really = sum5 - sum4 + sum5 += coef * (2*a) * n * sinh_taylor((2*a)*n*x); + + // test convergence via sum3 + if (coef * prod2ax < relerr * sum3) break; + } + } + else { // x > 5e-4, compute sum4 and sum5 separately + expx2 = exp(-x*x); + for (int n = 1; 1; ++n) { + const double coef = exp(-a2*(n*n)) * expx2 / (a2*(n*n) + y*y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum4 += (coef * prodm2ax) * (a*n); + sum3 += coef * prod2ax; + sum5 += (coef * prod2ax) * (a*n); + // test convergence via sum5, since this sum has the slowest decay + if ((coef * prod2ax) * (a*n) < relerr * sum5) break; + } + } + } + const double expx2erfcxy = // avoid spurious overflow for large negative y + y > -6 // for y < -6, erfcx(y) = 2*exp(y*y) to double precision + ? expx2*FADDEEVA_RE(erfcx)(y) : 2*exp(y*y-x*x); + if (y > 5) { // imaginary terms cancel + const double sinxy = sin(x*y); + ret = (expx2erfcxy - c*y*sum1) * cos(2*x*y) + + (c*x*expx2) * sinxy * sinc(x*y, sinxy); + } + else { + double xs = creal(z); + const double sinxy = sin(xs*y); + const double sin2xy = sin(2*xs*y), cos2xy = cos(2*xs*y); + const double coef1 = expx2erfcxy - c*y*sum1; + const double coef2 = c*xs*expx2; + ret = CMPLX(coef1 * cos2xy + coef2 * sinxy * sinc(xs*y, sinxy), + coef2 * sinc(2*xs*y, sin2xy) - coef1 * sin2xy); + } + } + else { // x large: only sum3 & sum5 contribute (see above note) + if (isnan(x)) + return CMPLX(x,x); + if (isnan(y)) + return CMPLX(y,y); + +#if USE_CONTINUED_FRACTION + ret = exp(-x*x); // |y| < 1e-10, so we only need exp(-x*x) term +#else + if (y < 0) { + /* erfcx(y) ~ 2*exp(y*y) + (< 1) if y < 0, so + erfcx(y)*exp(-x*x) ~ 2*exp(y*y-x*x) term may not be negligible + if y*y - x*x > -36 or so. So, compute this term just in case. + We also need the -exp(-x*x) term to compute Re[w] accurately + in the case where y is very small. */ + ret = cpolar(2*exp(y*y-x*x) - exp(-x*x), -2*creal(z)*y); + } + else + ret = exp(-x*x); // not negligible in real part if y very small +#endif + // (round instead of ceil as in original paper; note that x/a > 1 here) + double n0 = floor(x/a + 0.5); // sum in both directions, starting at n0 + double dx = a*n0 - x; + sum3 = exp(-dx*dx) / (a2*(n0*n0) + y*y); + sum5 = a*n0 * sum3; + double exp1 = exp(4*a*dx), exp1dn = 1; + int dn; + for (dn = 1; n0 - dn > 0; ++dn) { // loop over n0-dn and n0+dn terms + double np = n0 + dn, nm = n0 - dn; + double tp = exp(-sqr(a*dn+dx)); + double tm = tp * (exp1dn *= exp1); // trick to get tm from tp + tp /= (a2*(np*np) + y*y); + tm /= (a2*(nm*nm) + y*y); + sum3 += tp + tm; + sum5 += a * (np * tp + nm * tm); + if (a * (np * tp + nm * tm) < relerr * sum5) goto finish; + } + while (1) { // loop over n0+dn terms only (since n0-dn <= 0) + double np = n0 + dn++; + double tp = exp(-sqr(a*dn+dx)) / (a2*(np*np) + y*y); + sum3 += tp; + sum5 += a * np * tp; + if (a * np * tp < relerr * sum5) goto finish; + } + } + finish: + return ret + CMPLX((0.5*c)*y*(sum2+sum3), + (0.5*c)*copysign(sum5-sum4, creal(z))); +} + +///////////////////////////////////////////////////////////////////////// + +/* erfcx(x) = exp(x^2) erfc(x) function, for real x, written by + Steven G. Johnson, October 2012. + + This function combines a few different ideas. + + First, for x > 50, it uses a continued-fraction expansion (same as + for the Faddeeva function, but with algebraic simplifications for z=i*x). + + Second, for 0 <= x <= 50, it uses Chebyshev polynomial approximations, + but with two twists: + + a) It maps x to y = 4 / (4+x) in [0,1]. This simple transformation, + inspired by a similar transformation in the octave-forge/specfun + erfcx by Soren Hauberg, results in much faster Chebyshev convergence + than other simple transformations I have examined. + + b) Instead of using a single Chebyshev polynomial for the entire + [0,1] y interval, we break the interval up into 100 equal + subintervals, with a switch/lookup table, and use much lower + degree Chebyshev polynomials in each subinterval. This greatly + improves performance in my tests. + + For x < 0, we use the relationship erfcx(-x) = 2 exp(x^2) - erfc(x), + with the usual checks for overflow etcetera. + + Performance-wise, it seems to be substantially faster than either + the SLATEC DERFC function [or an erfcx function derived therefrom] + or Cody's CALERF function (from netlib.org/specfun), while + retaining near machine precision in accuracy. */ + +/* Given y100=100*y, where y = 4/(4+x) for x >= 0, compute erfc(x). + + Uses a look-up table of 100 different Chebyshev polynomials + for y intervals [0,0.01], [0.01,0.02], ...., [0.99,1], generated + with the help of Maple and a little shell script. This allows + the Chebyshev polynomials to be of significantly lower degree (about 1/4) + compared to fitting the whole [0,1] interval with a single polynomial. */ +static double erfcx_y100(double y100) +{ + switch ((int) y100) { +case 0: { +double t = 2*y100 - 1; +return 0.70878032454106438663e-3 + (0.71234091047026302958e-3 + (0.35779077297597742384e-5 + (0.17403143962587937815e-7 + (0.81710660047307788845e-10 + (0.36885022360434957634e-12 + 0.15917038551111111111e-14 * t) * t) * t) * t) * t) * t; +} +case 1: { +double t = 2*y100 - 3; +return 0.21479143208285144230e-2 + (0.72686402367379996033e-3 + (0.36843175430938995552e-5 + (0.18071841272149201685e-7 + (0.85496449296040325555e-10 + (0.38852037518534291510e-12 + 0.16868473576888888889e-14 * t) * t) * t) * t) * t) * t; +} +case 2: { +double t = 2*y100 - 5; +return 0.36165255935630175090e-2 + (0.74182092323555510862e-3 + (0.37948319957528242260e-5 + (0.18771627021793087350e-7 + (0.89484715122415089123e-10 + (0.40935858517772440862e-12 + 0.17872061464888888889e-14 * t) * t) * t) * t) * t) * t; +} +case 3: { +double t = 2*y100 - 7; +return 0.51154983860031979264e-2 + (0.75722840734791660540e-3 + (0.39096425726735703941e-5 + (0.19504168704300468210e-7 + (0.93687503063178993915e-10 + (0.43143925959079664747e-12 + 0.18939926435555555556e-14 * t) * t) * t) * t) * t) * t; +} +case 4: { +double t = 2*y100 - 9; +return 0.66457513172673049824e-2 + (0.77310406054447454920e-3 + (0.40289510589399439385e-5 + (0.20271233238288381092e-7 + (0.98117631321709100264e-10 + (0.45484207406017752971e-12 + 0.20076352213333333333e-14 * t) * t) * t) * t) * t) * t; +} +case 5: { +double t = 2*y100 - 11; +return 0.82082389970241207883e-2 + (0.78946629611881710721e-3 + (0.41529701552622656574e-5 + (0.21074693344544655714e-7 + (0.10278874108587317989e-9 + (0.47965201390613339638e-12 + 0.21285907413333333333e-14 * t) * t) * t) * t) * t) * t; +} +case 6: { +double t = 2*y100 - 13; +return 0.98039537275352193165e-2 + (0.80633440108342840956e-3 + (0.42819241329736982942e-5 + (0.21916534346907168612e-7 + (0.10771535136565470914e-9 + (0.50595972623692822410e-12 + 0.22573462684444444444e-14 * t) * t) * t) * t) * t) * t; +} +case 7: { +double t = 2*y100 - 15; +return 0.11433927298290302370e-1 + (0.82372858383196561209e-3 + (0.44160495311765438816e-5 + (0.22798861426211986056e-7 + (0.11291291745879239736e-9 + (0.53386189365816880454e-12 + 0.23944209546666666667e-14 * t) * t) * t) * t) * t) * t; +} +case 8: { +double t = 2*y100 - 17; +return 0.13099232878814653979e-1 + (0.84167002467906968214e-3 + (0.45555958988457506002e-5 + (0.23723907357214175198e-7 + (0.11839789326602695603e-9 + (0.56346163067550237877e-12 + 0.25403679644444444444e-14 * t) * t) * t) * t) * t) * t; +} +case 9: { +double t = 2*y100 - 19; +return 0.14800987015587535621e-1 + (0.86018092946345943214e-3 + (0.47008265848816866105e-5 + (0.24694040760197315333e-7 + (0.12418779768752299093e-9 + (0.59486890370320261949e-12 + 0.26957764568888888889e-14 * t) * t) * t) * t) * t) * t; +} +case 10: { +double t = 2*y100 - 21; +return 0.16540351739394069380e-1 + (0.87928458641241463952e-3 + (0.48520195793001753903e-5 + (0.25711774900881709176e-7 + (0.13030128534230822419e-9 + (0.62820097586874779402e-12 + 0.28612737351111111111e-14 * t) * t) * t) * t) * t) * t; +} +case 11: { +double t = 2*y100 - 23; +return 0.18318536789842392647e-1 + (0.89900542647891721692e-3 + (0.50094684089553365810e-5 + (0.26779777074218070482e-7 + (0.13675822186304615566e-9 + (0.66358287745352705725e-12 + 0.30375273884444444444e-14 * t) * t) * t) * t) * t) * t; +} +case 12: { +double t = 2*y100 - 25; +return 0.20136801964214276775e-1 + (0.91936908737673676012e-3 + (0.51734830914104276820e-5 + (0.27900878609710432673e-7 + (0.14357976402809042257e-9 + (0.70114790311043728387e-12 + 0.32252476000000000000e-14 * t) * t) * t) * t) * t) * t; +} +case 13: { +double t = 2*y100 - 27; +return 0.21996459598282740954e-1 + (0.94040248155366777784e-3 + (0.53443911508041164739e-5 + (0.29078085538049374673e-7 + (0.15078844500329731137e-9 + (0.74103813647499204269e-12 + 0.34251892320000000000e-14 * t) * t) * t) * t) * t) * t; +} +case 14: { +double t = 2*y100 - 29; +return 0.23898877187226319502e-1 + (0.96213386835900177540e-3 + (0.55225386998049012752e-5 + (0.30314589961047687059e-7 + (0.15840826497296335264e-9 + (0.78340500472414454395e-12 + 0.36381553564444444445e-14 * t) * t) * t) * t) * t) * t; +} +case 15: { +double t = 2*y100 - 31; +return 0.25845480155298518485e-1 + (0.98459293067820123389e-3 + (0.57082915920051843672e-5 + (0.31613782169164830118e-7 + (0.16646478745529630813e-9 + (0.82840985928785407942e-12 + 0.38649975768888888890e-14 * t) * t) * t) * t) * t) * t; +} +case 16: { +double t = 2*y100 - 33; +return 0.27837754783474696598e-1 + (0.10078108563256892757e-2 + (0.59020366493792212221e-5 + (0.32979263553246520417e-7 + (0.17498524159268458073e-9 + (0.87622459124842525110e-12 + 0.41066206488888888890e-14 * t) * t) * t) * t) * t) * t; +} +case 17: { +double t = 2*y100 - 35; +return 0.29877251304899307550e-1 + (0.10318204245057349310e-2 + (0.61041829697162055093e-5 + (0.34414860359542720579e-7 + (0.18399863072934089607e-9 + (0.92703227366365046533e-12 + 0.43639844053333333334e-14 * t) * t) * t) * t) * t) * t; +} +case 18: { +double t = 2*y100 - 37; +return 0.31965587178596443475e-1 + (0.10566560976716574401e-2 + (0.63151633192414586770e-5 + (0.35924638339521924242e-7 + (0.19353584758781174038e-9 + (0.98102783859889264382e-12 + 0.46381060817777777779e-14 * t) * t) * t) * t) * t) * t; +} +case 19: { +double t = 2*y100 - 39; +return 0.34104450552588334840e-1 + (0.10823541191350532574e-2 + (0.65354356159553934436e-5 + (0.37512918348533521149e-7 + (0.20362979635817883229e-9 + (0.10384187833037282363e-11 + 0.49300625262222222221e-14 * t) * t) * t) * t) * t) * t; +} +case 20: { +double t = 2*y100 - 41; +return 0.36295603928292425716e-1 + (0.11089526167995268200e-2 + (0.67654845095518363577e-5 + (0.39184292949913591646e-7 + (0.21431552202133775150e-9 + (0.10994259106646731797e-11 + 0.52409949102222222221e-14 * t) * t) * t) * t) * t) * t; +} +case 21: { +double t = 2*y100 - 43; +return 0.38540888038840509795e-1 + (0.11364917134175420009e-2 + (0.70058230641246312003e-5 + (0.40943644083718586939e-7 + (0.22563034723692881631e-9 + (0.11642841011361992885e-11 + 0.55721092871111111110e-14 * t) * t) * t) * t) * t) * t; +} +case 22: { +double t = 2*y100 - 45; +return 0.40842225954785960651e-1 + (0.11650136437945673891e-2 + (0.72569945502343006619e-5 + (0.42796161861855042273e-7 + (0.23761401711005024162e-9 + (0.12332431172381557035e-11 + 0.59246802364444444445e-14 * t) * t) * t) * t) * t) * t; +} +case 23: { +double t = 2*y100 - 47; +return 0.43201627431540222422e-1 + (0.11945628793917272199e-2 + (0.75195743532849206263e-5 + (0.44747364553960993492e-7 + (0.25030885216472953674e-9 + (0.13065684400300476484e-11 + 0.63000532853333333334e-14 * t) * t) * t) * t) * t) * t; +} +case 24: { +double t = 2*y100 - 49; +return 0.45621193513810471438e-1 + (0.12251862608067529503e-2 + (0.77941720055551920319e-5 + (0.46803119830954460212e-7 + (0.26375990983978426273e-9 + (0.13845421370977119765e-11 + 0.66996477404444444445e-14 * t) * t) * t) * t) * t) * t; +} +case 25: { +double t = 2*y100 - 51; +return 0.48103121413299865517e-1 + (0.12569331386432195113e-2 + (0.80814333496367673980e-5 + (0.48969667335682018324e-7 + (0.27801515481905748484e-9 + (0.14674637611609884208e-11 + 0.71249589351111111110e-14 * t) * t) * t) * t) * t) * t; +} +case 26: { +double t = 2*y100 - 53; +return 0.50649709676983338501e-1 + (0.12898555233099055810e-2 + (0.83820428414568799654e-5 + (0.51253642652551838659e-7 + (0.29312563849675507232e-9 + (0.15556512782814827846e-11 + 0.75775607822222222221e-14 * t) * t) * t) * t) * t) * t; +} +case 27: { +double t = 2*y100 - 55; +return 0.53263363664388864181e-1 + (0.13240082443256975769e-2 + (0.86967260015007658418e-5 + (0.53662102750396795566e-7 + (0.30914568786634796807e-9 + (0.16494420240828493176e-11 + 0.80591079644444444445e-14 * t) * t) * t) * t) * t) * t; +} +case 28: { +double t = 2*y100 - 57; +return 0.55946601353500013794e-1 + (0.13594491197408190706e-2 + (0.90262520233016380987e-5 + (0.56202552975056695376e-7 + (0.32613310410503135996e-9 + (0.17491936862246367398e-11 + 0.85713381688888888890e-14 * t) * t) * t) * t) * t) * t; +} +case 29: { +double t = 2*y100 - 59; +return 0.58702059496154081813e-1 + (0.13962391363223647892e-2 + (0.93714365487312784270e-5 + (0.58882975670265286526e-7 + (0.34414937110591753387e-9 + (0.18552853109751857859e-11 + 0.91160736711111111110e-14 * t) * t) * t) * t) * t) * t; +} +case 30: { +double t = 2*y100 - 61; +return 0.61532500145144778048e-1 + (0.14344426411912015247e-2 + (0.97331446201016809696e-5 + (0.61711860507347175097e-7 + (0.36325987418295300221e-9 + (0.19681183310134518232e-11 + 0.96952238400000000000e-14 * t) * t) * t) * t) * t) * t; +} +case 31: { +double t = 2*y100 - 63; +return 0.64440817576653297993e-1 + (0.14741275456383131151e-2 + (0.10112293819576437838e-4 + (0.64698236605933246196e-7 + (0.38353412915303665586e-9 + (0.20881176114385120186e-11 + 0.10310784480000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 32: { +double t = 2*y100 - 65; +return 0.67430045633130393282e-1 + (0.15153655418916540370e-2 + (0.10509857606888328667e-4 + (0.67851706529363332855e-7 + (0.40504602194811140006e-9 + (0.22157325110542534469e-11 + 0.10964842115555555556e-13 * t) * t) * t) * t) * t) * t; +} +case 33: { +double t = 2*y100 - 67; +return 0.70503365513338850709e-1 + (0.15582323336495709827e-2 + (0.10926868866865231089e-4 + (0.71182482239613507542e-7 + (0.42787405890153386710e-9 + (0.23514379522274416437e-11 + 0.11659571751111111111e-13 * t) * t) * t) * t) * t) * t; +} +case 34: { +double t = 2*y100 - 69; +return 0.73664114037944596353e-1 + (0.16028078812438820413e-2 + (0.11364423678778207991e-4 + (0.74701423097423182009e-7 + (0.45210162777476488324e-9 + (0.24957355004088569134e-11 + 0.12397238257777777778e-13 * t) * t) * t) * t) * t) * t; +} +case 35: { +double t = 2*y100 - 71; +return 0.76915792420819562379e-1 + (0.16491766623447889354e-2 + (0.11823685320041302169e-4 + (0.78420075993781544386e-7 + (0.47781726956916478925e-9 + (0.26491544403815724749e-11 + 0.13180196462222222222e-13 * t) * t) * t) * t) * t) * t; +} +case 36: { +double t = 2*y100 - 73; +return 0.80262075578094612819e-1 + (0.16974279491709504117e-2 + (0.12305888517309891674e-4 + (0.82350717698979042290e-7 + (0.50511496109857113929e-9 + (0.28122528497626897696e-11 + 0.14010889635555555556e-13 * t) * t) * t) * t) * t) * t; +} +case 37: { +double t = 2*y100 - 75; +return 0.83706822008980357446e-1 + (0.17476561032212656962e-2 + (0.12812343958540763368e-4 + (0.86506399515036435592e-7 + (0.53409440823869467453e-9 + (0.29856186620887555043e-11 + 0.14891851591111111111e-13 * t) * t) * t) * t) * t) * t; +} +case 38: { +double t = 2*y100 - 77; +return 0.87254084284461718231e-1 + (0.17999608886001962327e-2 + (0.13344443080089492218e-4 + (0.90900994316429008631e-7 + (0.56486134972616465316e-9 + (0.31698707080033956934e-11 + 0.15825697795555555556e-13 * t) * t) * t) * t) * t) * t; +} +case 39: { +double t = 2*y100 - 79; +return 0.90908120182172748487e-1 + (0.18544478050657699758e-2 + (0.13903663143426120077e-4 + (0.95549246062549906177e-7 + (0.59752787125242054315e-9 + (0.33656597366099099413e-11 + 0.16815130613333333333e-13 * t) * t) * t) * t) * t) * t; +} +case 40: { +double t = 2*y100 - 81; +return 0.94673404508075481121e-1 + (0.19112284419887303347e-2 + (0.14491572616545004930e-4 + (0.10046682186333613697e-6 + (0.63221272959791000515e-9 + (0.35736693975589130818e-11 + 0.17862931591111111111e-13 * t) * t) * t) * t) * t) * t; +} +case 41: { +double t = 2*y100 - 83; +return 0.98554641648004456555e-1 + (0.19704208544725622126e-2 + (0.15109836875625443935e-4 + (0.10567036667675984067e-6 + (0.66904168640019354565e-9 + (0.37946171850824333014e-11 + 0.18971959040000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 42: { +double t = 2*y100 - 85; +return 0.10255677889470089531e0 + (0.20321499629472857418e-2 + (0.15760224242962179564e-4 + (0.11117756071353507391e-6 + (0.70814785110097658502e-9 + (0.40292553276632563925e-11 + 0.20145143075555555556e-13 * t) * t) * t) * t) * t) * t; +} +case 43: { +double t = 2*y100 - 87; +return 0.10668502059865093318e0 + (0.20965479776148731610e-2 + (0.16444612377624983565e-4 + (0.11700717962026152749e-6 + (0.74967203250938418991e-9 + (0.42783716186085922176e-11 + 0.21385479360000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 44: { +double t = 2*y100 - 89; +return 0.11094484319386444474e0 + (0.21637548491908170841e-2 + (0.17164995035719657111e-4 + (0.12317915750735938089e-6 + (0.79376309831499633734e-9 + (0.45427901763106353914e-11 + 0.22696025653333333333e-13 * t) * t) * t) * t) * t) * t; +} +case 45: { +double t = 2*y100 - 91; +return 0.11534201115268804714e0 + (0.22339187474546420375e-2 + (0.17923489217504226813e-4 + (0.12971465288245997681e-6 + (0.84057834180389073587e-9 + (0.48233721206418027227e-11 + 0.24079890062222222222e-13 * t) * t) * t) * t) * t) * t; +} +case 46: { +double t = 2*y100 - 93; +return 0.11988259392684094740e0 + (0.23071965691918689601e-2 + (0.18722342718958935446e-4 + (0.13663611754337957520e-6 + (0.89028385488493287005e-9 + (0.51210161569225846701e-11 + 0.25540227111111111111e-13 * t) * t) * t) * t) * t) * t; +} +case 47: { +double t = 2*y100 - 95; +return 0.12457298393509812907e0 + (0.23837544771809575380e-2 + (0.19563942105711612475e-4 + (0.14396736847739470782e-6 + (0.94305490646459247016e-9 + (0.54366590583134218096e-11 + 0.27080225920000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 48: { +double t = 2*y100 - 97; +return 0.12941991566142438816e0 + (0.24637684719508859484e-2 + (0.20450821127475879816e-4 + (0.15173366280523906622e-6 + (0.99907632506389027739e-9 + (0.57712760311351625221e-11 + 0.28703099555555555556e-13 * t) * t) * t) * t) * t) * t; +} +case 49: { +double t = 2*y100 - 99; +return 0.13443048593088696613e0 + (0.25474249981080823877e-2 + (0.21385669591362915223e-4 + (0.15996177579900443030e-6 + (0.10585428844575134013e-8 + (0.61258809536787882989e-11 + 0.30412080142222222222e-13 * t) * t) * t) * t) * t) * t; +} +case 50: { +double t = 2*y100 - 101; +return 0.13961217543434561353e0 + (0.26349215871051761416e-2 + (0.22371342712572567744e-4 + (0.16868008199296822247e-6 + (0.11216596910444996246e-8 + (0.65015264753090890662e-11 + 0.32210394506666666666e-13 * t) * t) * t) * t) * t) * t; +} +case 51: { +double t = 2*y100 - 103; +return 0.14497287157673800690e0 + (0.27264675383982439814e-2 + (0.23410870961050950197e-4 + (0.17791863939526376477e-6 + (0.11886425714330958106e-8 + (0.68993039665054288034e-11 + 0.34101266222222222221e-13 * t) * t) * t) * t) * t) * t; +} +case 52: { +double t = 2*y100 - 105; +return 0.15052089272774618151e0 + (0.28222846410136238008e-2 + (0.24507470422713397006e-4 + (0.18770927679626136909e-6 + (0.12597184587583370712e-8 + (0.73203433049229821618e-11 + 0.36087889048888888890e-13 * t) * t) * t) * t) * t) * t; +} +case 53: { +double t = 2*y100 - 107; +return 0.15626501395774612325e0 + (0.29226079376196624949e-2 + (0.25664553693768450545e-4 + (0.19808568415654461964e-6 + (0.13351257759815557897e-8 + (0.77658124891046760667e-11 + 0.38173420035555555555e-13 * t) * t) * t) * t) * t) * t; +} +case 54: { +double t = 2*y100 - 109; +return 0.16221449434620737567e0 + (0.30276865332726475672e-2 + (0.26885741326534564336e-4 + (0.20908350604346384143e-6 + (0.14151148144240728728e-8 + (0.82369170665974313027e-11 + 0.40360957457777777779e-13 * t) * t) * t) * t) * t) * t; +} +case 55: { +double t = 2*y100 - 111; +return 0.16837910595412130659e0 + (0.31377844510793082301e-2 + (0.28174873844911175026e-4 + (0.22074043807045782387e-6 + (0.14999481055996090039e-8 + (0.87348993661930809254e-11 + 0.42653528977777777779e-13 * t) * t) * t) * t) * t) * t; +} +case 56: { +double t = 2*y100 - 113; +return 0.17476916455659369953e0 + (0.32531815370903068316e-2 + (0.29536024347344364074e-4 + (0.23309632627767074202e-6 + (0.15899007843582444846e-8 + (0.92610375235427359475e-11 + 0.45054073102222222221e-13 * t) * t) * t) * t) * t) * t; +} +case 57: { +double t = 2*y100 - 115; +return 0.18139556223643701364e0 + (0.33741744168096996041e-2 + (0.30973511714709500836e-4 + (0.24619326937592290996e-6 + (0.16852609412267750744e-8 + (0.98166442942854895573e-11 + 0.47565418097777777779e-13 * t) * t) * t) * t) * t) * t; +} +case 58: { +double t = 2*y100 - 117; +return 0.18826980194443664549e0 + (0.35010775057740317997e-2 + (0.32491914440014267480e-4 + (0.26007572375886319028e-6 + (0.17863299617388376116e-8 + (0.10403065638343878679e-10 + 0.50190265831111111110e-13 * t) * t) * t) * t) * t) * t; +} +case 59: { +double t = 2*y100 - 119; +return 0.19540403413693967350e0 + (0.36342240767211326315e-2 + (0.34096085096200907289e-4 + (0.27479061117017637474e-6 + (0.18934228504790032826e-8 + (0.11021679075323598664e-10 + 0.52931171733333333334e-13 * t) * t) * t) * t) * t) * t; +} +case 60: { +double t = 2*y100 - 121; +return 0.20281109560651886959e0 + (0.37739673859323597060e-2 + (0.35791165457592409054e-4 + (0.29038742889416172404e-6 + (0.20068685374849001770e-8 + (0.11673891799578381999e-10 + 0.55790523093333333334e-13 * t) * t) * t) * t) * t) * t; +} +case 61: { +double t = 2*y100 - 123; +return 0.21050455062669334978e0 + (0.39206818613925652425e-2 + (0.37582602289680101704e-4 + (0.30691836231886877385e-6 + (0.21270101645763677824e-8 + (0.12361138551062899455e-10 + 0.58770520160000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 62: { +double t = 2*y100 - 125; +return 0.21849873453703332479e0 + (0.40747643554689586041e-2 + (0.39476163820986711501e-4 + (0.32443839970139918836e-6 + (0.22542053491518680200e-8 + (0.13084879235290858490e-10 + 0.61873153262222222221e-13 * t) * t) * t) * t) * t) * t; +} +case 63: { +double t = 2*y100 - 127; +return 0.22680879990043229327e0 + (0.42366354648628516935e-2 + (0.41477956909656896779e-4 + (0.34300544894502810002e-6 + (0.23888264229264067658e-8 + (0.13846596292818514601e-10 + 0.65100183751111111110e-13 * t) * t) * t) * t) * t) * t; +} +case 64: { +double t = 2*y100 - 129; +return 0.23545076536988703937e0 + (0.44067409206365170888e-2 + (0.43594444916224700881e-4 + (0.36268045617760415178e-6 + (0.25312606430853202748e-8 + (0.14647791812837903061e-10 + 0.68453122631111111110e-13 * t) * t) * t) * t) * t) * t; +} +case 65: { +double t = 2*y100 - 131; +return 0.24444156740777432838e0 + (0.45855530511605787178e-2 + (0.45832466292683085475e-4 + (0.38352752590033030472e-6 + (0.26819103733055603460e-8 + (0.15489984390884756993e-10 + 0.71933206364444444445e-13 * t) * t) * t) * t) * t) * t; +} +case 66: { +double t = 2*y100 - 133; +return 0.25379911500634264643e0 + (0.47735723208650032167e-2 + (0.48199253896534185372e-4 + (0.40561404245564732314e-6 + (0.28411932320871165585e-8 + (0.16374705736458320149e-10 + 0.75541379822222222221e-13 * t) * t) * t) * t) * t) * t; +} +case 67: { +double t = 2*y100 - 135; +return 0.26354234756393613032e0 + (0.49713289477083781266e-2 + (0.50702455036930367504e-4 + (0.42901079254268185722e-6 + (0.30095422058900481753e-8 + (0.17303497025347342498e-10 + 0.79278273368888888890e-13 * t) * t) * t) * t) * t) * t; +} +case 68: { +double t = 2*y100 - 137; +return 0.27369129607732343398e0 + (0.51793846023052643767e-2 + (0.53350152258326602629e-4 + (0.45379208848865015485e-6 + (0.31874057245814381257e-8 + (0.18277905010245111046e-10 + 0.83144182364444444445e-13 * t) * t) * t) * t) * t) * t; +} +case 69: { +double t = 2*y100 - 139; +return 0.28426714781640316172e0 + (0.53983341916695141966e-2 + (0.56150884865255810638e-4 + (0.48003589196494734238e-6 + (0.33752476967570796349e-8 + (0.19299477888083469086e-10 + 0.87139049137777777779e-13 * t) * t) * t) * t) * t) * t; +} +case 70: { +double t = 2*y100 - 141; +return 0.29529231465348519920e0 + (0.56288077305420795663e-2 + (0.59113671189913307427e-4 + (0.50782393781744840482e-6 + (0.35735475025851713168e-8 + (0.20369760937017070382e-10 + 0.91262442613333333334e-13 * t) * t) * t) * t) * t) * t; +} +case 71: { +double t = 2*y100 - 143; +return 0.30679050522528838613e0 + (0.58714723032745403331e-2 + (0.62248031602197686791e-4 + (0.53724185766200945789e-6 + (0.37827999418960232678e-8 + (0.21490291930444538307e-10 + 0.95513539182222222221e-13 * t) * t) * t) * t) * t) * t; +} +case 72: { +double t = 2*y100 - 145; +return 0.31878680111173319425e0 + (0.61270341192339103514e-2 + (0.65564012259707640976e-4 + (0.56837930287837738996e-6 + (0.40035151353392378882e-8 + (0.22662596341239294792e-10 + 0.99891109760000000000e-13 * t) * t) * t) * t) * t) * t; +} +case 73: { +double t = 2*y100 - 147; +return 0.33130773722152622027e0 + (0.63962406646798080903e-2 + (0.69072209592942396666e-4 + (0.60133006661885941812e-6 + (0.42362183765883466691e-8 + (0.23888182347073698382e-10 + 0.10439349811555555556e-12 * t) * t) * t) * t) * t) * t; +} +case 74: { +double t = 2*y100 - 149; +return 0.34438138658041336523e0 + (0.66798829540414007258e-2 + (0.72783795518603561144e-4 + (0.63619220443228800680e-6 + (0.44814499336514453364e-8 + (0.25168535651285475274e-10 + 0.10901861383111111111e-12 * t) * t) * t) * t) * t) * t; +} +case 75: { +double t = 2*y100 - 151; +return 0.35803744972380175583e0 + (0.69787978834882685031e-2 + (0.76710543371454822497e-4 + (0.67306815308917386747e-6 + (0.47397647975845228205e-8 + (0.26505114141143050509e-10 + 0.11376390933333333333e-12 * t) * t) * t) * t) * t) * t; +} +case 76: { +double t = 2*y100 - 153; +return 0.37230734890119724188e0 + (0.72938706896461381003e-2 + (0.80864854542670714092e-4 + (0.71206484718062688779e-6 + (0.50117323769745883805e-8 + (0.27899342394100074165e-10 + 0.11862637614222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 77: { +double t = 2*y100 - 155; +return 0.38722432730555448223e0 + (0.76260375162549802745e-2 + (0.85259785810004603848e-4 + (0.75329383305171327677e-6 + (0.52979361368388119355e-8 + (0.29352606054164086709e-10 + 0.12360253370666666667e-12 * t) * t) * t) * t) * t) * t; +} +case 78: { +double t = 2*y100 - 157; +return 0.40282355354616940667e0 + (0.79762880915029728079e-2 + (0.89909077342438246452e-4 + (0.79687137961956194579e-6 + (0.55989731807360403195e-8 + (0.30866246101464869050e-10 + 0.12868841946666666667e-12 * t) * t) * t) * t) * t) * t; +} +case 79: { +double t = 2*y100 - 159; +return 0.41914223158913787649e0 + (0.83456685186950463538e-2 + (0.94827181359250161335e-4 + (0.84291858561783141014e-6 + (0.59154537751083485684e-8 + (0.32441553034347469291e-10 + 0.13387957943111111111e-12 * t) * t) * t) * t) * t) * t; +} +case 80: { +double t = 2*y100 - 161; +return 0.43621971639463786896e0 + (0.87352841828289495773e-2 + (0.10002929142066799966e-3 + (0.89156148280219880024e-6 + (0.62480008150788597147e-8 + (0.34079760983458878910e-10 + 0.13917107176888888889e-12 * t) * t) * t) * t) * t) * t; +} +case 81: { +double t = 2*y100 - 163; +return 0.45409763548534330981e0 + (0.91463027755548240654e-2 + (0.10553137232446167258e-3 + (0.94293113464638623798e-6 + (0.65972492312219959885e-8 + (0.35782041795476563662e-10 + 0.14455745872000000000e-12 * t) * t) * t) * t) * t) * t; +} +case 82: { +double t = 2*y100 - 165; +return 0.47282001668512331468e0 + (0.95799574408860463394e-2 + (0.11135019058000067469e-3 + (0.99716373005509038080e-6 + (0.69638453369956970347e-8 + (0.37549499088161345850e-10 + 0.15003280712888888889e-12 * t) * t) * t) * t) * t) * t; +} +case 83: { +double t = 2*y100 - 167; +return 0.49243342227179841649e0 + (0.10037550043909497071e-1 + (0.11750334542845234952e-3 + (0.10544006716188967172e-5 + (0.73484461168242224872e-8 + (0.39383162326435752965e-10 + 0.15559069118222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 84: { +double t = 2*y100 - 169; +return 0.51298708979209258326e0 + (0.10520454564612427224e-1 + (0.12400930037494996655e-3 + (0.11147886579371265246e-5 + (0.77517184550568711454e-8 + (0.41283980931872622611e-10 + 0.16122419680000000000e-12 * t) * t) * t) * t) * t) * t; +} +case 85: { +double t = 2*y100 - 171; +return 0.53453307979101369843e0 + (0.11030120618800726938e-1 + (0.13088741519572269581e-3 + (0.11784797595374515432e-5 + (0.81743383063044825400e-8 + (0.43252818449517081051e-10 + 0.16692592640000000000e-12 * t) * t) * t) * t) * t) * t; +} +case 86: { +double t = 2*y100 - 173; +return 0.55712643071169299478e0 + (0.11568077107929735233e-1 + (0.13815797838036651289e-3 + (0.12456314879260904558e-5 + (0.86169898078969313597e-8 + (0.45290446811539652525e-10 + 0.17268801084444444444e-12 * t) * t) * t) * t) * t) * t; +} +case 87: { +double t = 2*y100 - 175; +return 0.58082532122519320968e0 + (0.12135935999503877077e-1 + (0.14584223996665838559e-3 + (0.13164068573095710742e-5 + (0.90803643355106020163e-8 + (0.47397540713124619155e-10 + 0.17850211608888888889e-12 * t) * t) * t) * t) * t) * t; +} +case 88: { +double t = 2*y100 - 177; +return 0.60569124025293375554e0 + (0.12735396239525550361e-1 + (0.15396244472258863344e-3 + (0.13909744385382818253e-5 + (0.95651595032306228245e-8 + (0.49574672127669041550e-10 + 0.18435945564444444444e-12 * t) * t) * t) * t) * t) * t; +} +case 89: { +double t = 2*y100 - 179; +return 0.63178916494715716894e0 + (0.13368247798287030927e-1 + (0.16254186562762076141e-3 + (0.14695084048334056083e-5 + (0.10072078109604152350e-7 + (0.51822304995680707483e-10 + 0.19025081422222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 90: { +double t = 2*y100 - 181; +return 0.65918774689725319200e0 + (0.14036375850601992063e-1 + (0.17160483760259706354e-3 + (0.15521885688723188371e-5 + (0.10601827031535280590e-7 + (0.54140790105837520499e-10 + 0.19616655146666666667e-12 * t) * t) * t) * t) * t) * t; +} +case 91: { +double t = 2*y100 - 183; +return 0.68795950683174433822e0 + (0.14741765091365869084e-1 + (0.18117679143520433835e-3 + (0.16392004108230585213e-5 + (0.11155116068018043001e-7 + (0.56530360194925690374e-10 + 0.20209663662222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 92: { +double t = 2*y100 - 185; +return 0.71818103808729967036e0 + (0.15486504187117112279e-1 + (0.19128428784550923217e-3 + (0.17307350969359975848e-5 + (0.11732656736113607751e-7 + (0.58991125287563833603e-10 + 0.20803065333333333333e-12 * t) * t) * t) * t) * t) * t; +} +case 93: { +double t = 2*y100 - 187; +return 0.74993321911726254661e0 + (0.16272790364044783382e-1 + (0.20195505163377912645e-3 + (0.18269894883203346953e-5 + (0.12335161021630225535e-7 + (0.61523068312169087227e-10 + 0.21395783431111111111e-12 * t) * t) * t) * t) * t) * t; +} +case 94: { +double t = 2*y100 - 189; +return 0.78330143531283492729e0 + (0.17102934132652429240e-1 + (0.21321800585063327041e-3 + (0.19281661395543913713e-5 + (0.12963340087354341574e-7 + (0.64126040998066348872e-10 + 0.21986708942222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 95: { +double t = 2*y100 - 191; +return 0.81837581041023811832e0 + (0.17979364149044223802e-1 + (0.22510330592753129006e-3 + (0.20344732868018175389e-5 + (0.13617902941839949718e-7 + (0.66799760083972474642e-10 + 0.22574701262222222222e-12 * t) * t) * t) * t) * t) * t; +} +case 96: { +double t = 2*y100 - 193; +return 0.85525144775685126237e0 + (0.18904632212547561026e-1 + (0.23764237370371255638e-3 + (0.21461248251306387979e-5 + (0.14299555071870523786e-7 + (0.69543803864694171934e-10 + 0.23158593688888888889e-12 * t) * t) * t) * t) * t) * t; +} +case 97: { +double t = 2*y100 - 195; +return 0.89402868170849933734e0 + (0.19881418399127202569e-1 + (0.25086793128395995798e-3 + (0.22633402747585233180e-5 + (0.15008997042116532283e-7 + (0.72357609075043941261e-10 + 0.23737194737777777778e-12 * t) * t) * t) * t) * t) * t; +} +case 98: { +double t = 2*y100 - 197; +return 0.93481333942870796363e0 + (0.20912536329780368893e-1 + (0.26481403465998477969e-3 + (0.23863447359754921676e-5 + (0.15746923065472184451e-7 + (0.75240468141720143653e-10 + 0.24309291271111111111e-12 * t) * t) * t) * t) * t) * t; +} +case 99: { +double t = 2*y100 - 199; +return 0.97771701335885035464e0 + (0.22000938572830479551e-1 + (0.27951610702682383001e-3 + (0.25153688325245314530e-5 + (0.16514019547822821453e-7 + (0.78191526829368231251e-10 + 0.24873652355555555556e-12 * t) * t) * t) * t) * t) * t; +} + } + // we only get here if y = 1, i.e. |x| < 4*eps, in which case + // erfcx is within 1e-15 of 1.. + return 1.0; +} + +double FADDEEVA_RE(erfcx)(double x) +{ + if (x >= 0) { + if (x > 50) { // continued-fraction expansion is faster + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x > 5e7) // 1-term expansion, important to avoid overflow + return ispi / x; + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x+0.5/(x+1/(x+1.5/(x+2/x)))) */ + return ispi*((x*x) * (x*x+4.5) + 2) / (x * ((x*x) * (x*x+5) + 3.75)); + } + return erfcx_y100(400/(4+x)); + } + else + return x < -26.7 ? HUGE_VAL : (x < -6.1 ? 2*exp(x*x) + : 2*exp(x*x) - erfcx_y100(400/(4-x))); +} + +///////////////////////////////////////////////////////////////////////// +/* Compute a scaled Dawson integral + FADDEEVA(w_im)(x) = 2*Dawson(x)/sqrt(pi) + equivalent to the imaginary part w(x) for real x. + + Uses methods similar to the erfcx calculation above: continued fractions + for large |x|, a lookup table of Chebyshev polynomials for smaller |x|, + and finally a Taylor expansion for |x|<0.01. + + Steven G. Johnson, October 2012. */ + +/* Given y100=100*y, where y = 1/(1+x) for x >= 0, compute w_im(x). + + Uses a look-up table of 100 different Chebyshev polynomials + for y intervals [0,0.01], [0.01,0.02], ...., [0.99,1], generated + with the help of Maple and a little shell script. This allows + the Chebyshev polynomials to be of significantly lower degree (about 1/30) + compared to fitting the whole [0,1] interval with a single polynomial. */ +static double w_im_y100(double y100, double x) { + switch ((int) y100) { + case 0: { + double t = 2*y100 - 1; + return 0.28351593328822191546e-2 + (0.28494783221378400759e-2 + (0.14427470563276734183e-4 + (0.10939723080231588129e-6 + (0.92474307943275042045e-9 + (0.89128907666450075245e-11 + 0.92974121935111111110e-13 * t) * t) * t) * t) * t) * t; + } + case 1: { + double t = 2*y100 - 3; + return 0.85927161243940350562e-2 + (0.29085312941641339862e-2 + (0.15106783707725582090e-4 + (0.11716709978531327367e-6 + (0.10197387816021040024e-8 + (0.10122678863073360769e-10 + 0.10917479678400000000e-12 * t) * t) * t) * t) * t) * t; + } + case 2: { + double t = 2*y100 - 5; + return 0.14471159831187703054e-1 + (0.29703978970263836210e-2 + (0.15835096760173030976e-4 + (0.12574803383199211596e-6 + (0.11278672159518415848e-8 + (0.11547462300333495797e-10 + 0.12894535335111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 3: { + double t = 2*y100 - 7; + return 0.20476320420324610618e-1 + (0.30352843012898665856e-2 + (0.16617609387003727409e-4 + (0.13525429711163116103e-6 + (0.12515095552507169013e-8 + (0.13235687543603382345e-10 + 0.15326595042666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 4: { + double t = 2*y100 - 9; + return 0.26614461952489004566e-1 + (0.31034189276234947088e-2 + (0.17460268109986214274e-4 + (0.14582130824485709573e-6 + (0.13935959083809746345e-8 + (0.15249438072998932900e-10 + 0.18344741882133333333e-12 * t) * t) * t) * t) * t) * t; + } + case 5: { + double t = 2*y100 - 11; + return 0.32892330248093586215e-1 + (0.31750557067975068584e-2 + (0.18369907582308672632e-4 + (0.15761063702089457882e-6 + (0.15577638230480894382e-8 + (0.17663868462699097951e-10 + (0.22126732680711111111e-12 + 0.30273474177737853668e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 6: { + double t = 2*y100 - 13; + return 0.39317207681134336024e-1 + (0.32504779701937539333e-2 + (0.19354426046513400534e-4 + (0.17081646971321290539e-6 + (0.17485733959327106250e-8 + (0.20593687304921961410e-10 + (0.26917401949155555556e-12 + 0.38562123837725712270e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 7: { + double t = 2*y100 - 15; + return 0.45896976511367738235e-1 + (0.33300031273110976165e-2 + (0.20423005398039037313e-4 + (0.18567412470376467303e-6 + (0.19718038363586588213e-8 + (0.24175006536781219807e-10 + (0.33059982791466666666e-12 + 0.49756574284439426165e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 8: { + double t = 2*y100 - 17; + return 0.52640192524848962855e-1 + (0.34139883358846720806e-2 + (0.21586390240603337337e-4 + (0.20247136501568904646e-6 + (0.22348696948197102935e-8 + (0.28597516301950162548e-10 + (0.41045502119111111110e-12 + 0.65151614515238361946e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 9: { + double t = 2*y100 - 19; + return 0.59556171228656770456e-1 + (0.35028374386648914444e-2 + (0.22857246150998562824e-4 + (0.22156372146525190679e-6 + (0.25474171590893813583e-8 + (0.34122390890697400584e-10 + (0.51593189879111111110e-12 + 0.86775076853908006938e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 10: { + double t = 2*y100 - 21; + return 0.66655089485108212551e-1 + (0.35970095381271285568e-2 + (0.24250626164318672928e-4 + (0.24339561521785040536e-6 + (0.29221990406518411415e-8 + (0.41117013527967776467e-10 + (0.65786450716444444445e-12 + 0.11791885745450623331e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 11: { + double t = 2*y100 - 23; + return 0.73948106345519174661e-1 + (0.36970297216569341748e-2 + (0.25784588137312868792e-4 + (0.26853012002366752770e-6 + (0.33763958861206729592e-8 + (0.50111549981376976397e-10 + (0.85313857496888888890e-12 + 0.16417079927706899860e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 12: { + double t = 2*y100 - 25; + return 0.81447508065002963203e-1 + (0.38035026606492705117e-2 + (0.27481027572231851896e-4 + (0.29769200731832331364e-6 + (0.39336816287457655076e-8 + (0.61895471132038157624e-10 + (0.11292303213511111111e-11 + 0.23558532213703884304e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 13: { + double t = 2*y100 - 27; + return 0.89166884027582716628e-1 + (0.39171301322438946014e-2 + (0.29366827260422311668e-4 + (0.33183204390350724895e-6 + (0.46276006281647330524e-8 + (0.77692631378169813324e-10 + (0.15335153258844444444e-11 + 0.35183103415916026911e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 14: { + double t = 2*y100 - 29; + return 0.97121342888032322019e-1 + (0.40387340353207909514e-2 + (0.31475490395950776930e-4 + (0.37222714227125135042e-6 + (0.55074373178613809996e-8 + (0.99509175283990337944e-10 + (0.21552645758222222222e-11 + 0.55728651431872687605e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 15: { + double t = 2*y100 - 31; + return 0.10532778218603311137e0 + (0.41692873614065380607e-2 + (0.33849549774889456984e-4 + (0.42064596193692630143e-6 + (0.66494579697622432987e-8 + (0.13094103581931802337e-9 + (0.31896187409777777778e-11 + 0.97271974184476560742e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 16: { + double t = 2*y100 - 33; + return 0.11380523107427108222e0 + (0.43099572287871821013e-2 + (0.36544324341565929930e-4 + (0.47965044028581857764e-6 + (0.81819034238463698796e-8 + (0.17934133239549647357e-9 + (0.50956666166186293627e-11 + (0.18850487318190638010e-12 + 0.79697813173519853340e-14 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 17: { + double t = 2*y100 - 35; + return 0.12257529703447467345e0 + (0.44621675710026986366e-2 + (0.39634304721292440285e-4 + (0.55321553769873381819e-6 + (0.10343619428848520870e-7 + (0.26033830170470368088e-9 + (0.87743837749108025357e-11 + (0.34427092430230063401e-12 + 0.10205506615709843189e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 18: { + double t = 2*y100 - 37; + return 0.13166276955656699478e0 + (0.46276970481783001803e-2 + (0.43225026380496399310e-4 + (0.64799164020016902656e-6 + (0.13580082794704641782e-7 + (0.39839800853954313927e-9 + (0.14431142411840000000e-10 + 0.42193457308830027541e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 19: { + double t = 2*y100 - 39; + return 0.14109647869803356475e0 + (0.48088424418545347758e-2 + (0.47474504753352150205e-4 + (0.77509866468724360352e-6 + (0.18536851570794291724e-7 + (0.60146623257887570439e-9 + (0.18533978397305276318e-10 + (0.41033845938901048380e-13 - 0.46160680279304825485e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 20: { + double t = 2*y100 - 41; + return 0.15091057940548936603e0 + (0.50086864672004685703e-2 + (0.52622482832192230762e-4 + (0.95034664722040355212e-6 + (0.25614261331144718769e-7 + (0.80183196716888606252e-9 + (0.12282524750534352272e-10 + (-0.10531774117332273617e-11 - 0.86157181395039646412e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 21: { + double t = 2*y100 - 43; + return 0.16114648116017010770e0 + (0.52314661581655369795e-2 + (0.59005534545908331315e-4 + (0.11885518333915387760e-5 + (0.33975801443239949256e-7 + (0.82111547144080388610e-9 + (-0.12357674017312854138e-10 + (-0.24355112256914479176e-11 - 0.75155506863572930844e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 22: { + double t = 2*y100 - 45; + return 0.17185551279680451144e0 + (0.54829002967599420860e-2 + (0.67013226658738082118e-4 + (0.14897400671425088807e-5 + (0.40690283917126153701e-7 + (0.44060872913473778318e-9 + (-0.52641873433280000000e-10 - 0.30940587864543343124e-11 * t) * t) * t) * t) * t) * t) * t; + } + case 23: { + double t = 2*y100 - 47; + return 0.18310194559815257381e0 + (0.57701559375966953174e-2 + (0.76948789401735193483e-4 + (0.18227569842290822512e-5 + (0.41092208344387212276e-7 + (-0.44009499965694442143e-9 + (-0.92195414685628803451e-10 + (-0.22657389705721753299e-11 + 0.10004784908106839254e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 24: { + double t = 2*y100 - 49; + return 0.19496527191546630345e0 + (0.61010853144364724856e-2 + (0.88812881056342004864e-4 + (0.21180686746360261031e-5 + (0.30652145555130049203e-7 + (-0.16841328574105890409e-8 + (-0.11008129460612823934e-9 + (-0.12180794204544515779e-12 + 0.15703325634590334097e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 25: { + double t = 2*y100 - 51; + return 0.20754006813966575720e0 + (0.64825787724922073908e-2 + (0.10209599627522311893e-3 + (0.22785233392557600468e-5 + (0.73495224449907568402e-8 + (-0.29442705974150112783e-8 + (-0.94082603434315016546e-10 + (0.23609990400179321267e-11 + 0.14141908654269023788e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 26: { + double t = 2*y100 - 53; + return 0.22093185554845172146e0 + (0.69182878150187964499e-2 + (0.11568723331156335712e-3 + (0.22060577946323627739e-5 + (-0.26929730679360840096e-7 + (-0.38176506152362058013e-8 + (-0.47399503861054459243e-10 + (0.40953700187172127264e-11 + 0.69157730376118511127e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 27: { + double t = 2*y100 - 55; + return 0.23524827304057813918e0 + (0.74063350762008734520e-2 + (0.12796333874615790348e-3 + (0.18327267316171054273e-5 + (-0.66742910737957100098e-7 + (-0.40204740975496797870e-8 + (0.14515984139495745330e-10 + (0.44921608954536047975e-11 - 0.18583341338983776219e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 28: { + double t = 2*y100 - 57; + return 0.25058626331812744775e0 + (0.79377285151602061328e-2 + (0.13704268650417478346e-3 + (0.11427511739544695861e-5 + (-0.10485442447768377485e-6 + (-0.34850364756499369763e-8 + (0.72656453829502179208e-10 + (0.36195460197779299406e-11 - 0.84882136022200714710e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 29: { + double t = 2*y100 - 59; + return 0.26701724900280689785e0 + (0.84959936119625864274e-2 + (0.14112359443938883232e-3 + (0.17800427288596909634e-6 + (-0.13443492107643109071e-6 + (-0.23512456315677680293e-8 + (0.11245846264695936769e-9 + (0.19850501334649565404e-11 - 0.11284666134635050832e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 30: { + double t = 2*y100 - 61; + return 0.28457293586253654144e0 + (0.90581563892650431899e-2 + (0.13880520331140646738e-3 + (-0.97262302362522896157e-6 + (-0.15077100040254187366e-6 + (-0.88574317464577116689e-9 + (0.12760311125637474581e-9 + (0.20155151018282695055e-12 - 0.10514169375181734921e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 31: { + double t = 2*y100 - 63; + return 0.30323425595617385705e0 + (0.95968346790597422934e-2 + (0.12931067776725883939e-3 + (-0.21938741702795543986e-5 + (-0.15202888584907373963e-6 + (0.61788350541116331411e-9 + (0.11957835742791248256e-9 + (-0.12598179834007710908e-11 - 0.75151817129574614194e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 32: { + double t = 2*y100 - 65; + return 0.32292521181517384379e0 + (0.10082957727001199408e-1 + (0.11257589426154962226e-3 + (-0.33670890319327881129e-5 + (-0.13910529040004008158e-6 + (0.19170714373047512945e-8 + (0.94840222377720494290e-10 + (-0.21650018351795353201e-11 - 0.37875211678024922689e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 33: { + double t = 2*y100 - 67; + return 0.34351233557911753862e0 + (0.10488575435572745309e-1 + (0.89209444197248726614e-4 + (-0.43893459576483345364e-5 + (-0.11488595830450424419e-6 + (0.28599494117122464806e-8 + (0.61537542799857777779e-10 - 0.24935749227658002212e-11 * t) * t) * t) * t) * t) * t) * t; + } + case 34: { + double t = 2*y100 - 69; + return 0.36480946642143669093e0 + (0.10789304203431861366e-1 + (0.60357993745283076834e-4 + (-0.51855862174130669389e-5 + (-0.83291664087289801313e-7 + (0.33898011178582671546e-8 + (0.27082948188277716482e-10 + (-0.23603379397408694974e-11 + 0.19328087692252869842e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 35: { + double t = 2*y100 - 71; + return 0.38658679935694939199e0 + (0.10966119158288804999e-1 + (0.27521612041849561426e-4 + (-0.57132774537670953638e-5 + (-0.48404772799207914899e-7 + (0.35268354132474570493e-8 + (-0.32383477652514618094e-11 + (-0.19334202915190442501e-11 + 0.32333189861286460270e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 36: { + double t = 2*y100 - 73; + return 0.40858275583808707870e0 + (0.11006378016848466550e-1 + (-0.76396376685213286033e-5 + (-0.59609835484245791439e-5 + (-0.13834610033859313213e-7 + (0.33406952974861448790e-8 + (-0.26474915974296612559e-10 + (-0.13750229270354351983e-11 + 0.36169366979417390637e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 37: { + double t = 2*y100 - 75; + return 0.43051714914006682977e0 + (0.10904106549500816155e-1 + (-0.43477527256787216909e-4 + (-0.59429739547798343948e-5 + (0.17639200194091885949e-7 + (0.29235991689639918688e-8 + (-0.41718791216277812879e-10 + (-0.81023337739508049606e-12 + 0.33618915934461994428e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 38: { + double t = 2*y100 - 77; + return 0.45210428135559607406e0 + (0.10659670756384400554e-1 + (-0.78488639913256978087e-4 + (-0.56919860886214735936e-5 + (0.44181850467477733407e-7 + (0.23694306174312688151e-8 + (-0.49492621596685443247e-10 + (-0.31827275712126287222e-12 + 0.27494438742721623654e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 39: { + double t = 2*y100 - 79; + return 0.47306491195005224077e0 + (0.10279006119745977570e-1 + (-0.11140268171830478306e-3 + (-0.52518035247451432069e-5 + (0.64846898158889479518e-7 + (0.17603624837787337662e-8 + (-0.51129481592926104316e-10 + (0.62674584974141049511e-13 + 0.20055478560829935356e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 40: { + double t = 2*y100 - 81; + return 0.49313638965719857647e0 + (0.97725799114772017662e-2 + (-0.14122854267291533334e-3 + (-0.46707252568834951907e-5 + (0.79421347979319449524e-7 + (0.11603027184324708643e-8 + (-0.48269605844397175946e-10 + (0.32477251431748571219e-12 + 0.12831052634143527985e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 41: { + double t = 2*y100 - 83; + return 0.51208057433416004042e0 + (0.91542422354009224951e-2 + (-0.16726530230228647275e-3 + (-0.39964621752527649409e-5 + (0.88232252903213171454e-7 + (0.61343113364949928501e-9 + (-0.42516755603130443051e-10 + (0.47910437172240209262e-12 + 0.66784341874437478953e-14 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 42: { + double t = 2*y100 - 85; + return 0.52968945458607484524e0 + (0.84400880445116786088e-2 + (-0.18908729783854258774e-3 + (-0.32725905467782951931e-5 + (0.91956190588652090659e-7 + (0.14593989152420122909e-9 + (-0.35239490687644444445e-10 + 0.54613829888448694898e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 43: { + double t = 2*y100 - 87; + return 0.54578857454330070965e0 + (0.76474155195880295311e-2 + (-0.20651230590808213884e-3 + (-0.25364339140543131706e-5 + (0.91455367999510681979e-7 + (-0.23061359005297528898e-9 + (-0.27512928625244444444e-10 + 0.54895806008493285579e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 44: { + double t = 2*y100 - 89; + return 0.56023851910298493910e0 + (0.67938321739997196804e-2 + (-0.21956066613331411760e-3 + (-0.18181127670443266395e-5 + (0.87650335075416845987e-7 + (-0.51548062050366615977e-9 + (-0.20068462174044444444e-10 + 0.50912654909758187264e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 45: { + double t = 2*y100 - 91; + return 0.57293478057455721150e0 + (0.58965321010394044087e-2 + (-0.22841145229276575597e-3 + (-0.11404605562013443659e-5 + (0.81430290992322326296e-7 + (-0.71512447242755357629e-9 + (-0.13372664928000000000e-10 + 0.44461498336689298148e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 46: { + double t = 2*y100 - 93; + return 0.58380635448407827360e0 + (0.49717469530842831182e-2 + (-0.23336001540009645365e-3 + (-0.51952064448608850822e-6 + (0.73596577815411080511e-7 + (-0.84020916763091566035e-9 + (-0.76700972702222222221e-11 + 0.36914462807972467044e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 47: { + double t = 2*y100 - 95; + return 0.59281340237769489597e0 + (0.40343592069379730568e-2 + (-0.23477963738658326185e-3 + (0.34615944987790224234e-7 + (0.64832803248395814574e-7 + (-0.90329163587627007971e-9 + (-0.30421940400000000000e-11 + 0.29237386653743536669e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 48: { + double t = 2*y100 - 97; + return 0.59994428743114271918e0 + (0.30976579788271744329e-2 + (-0.23308875765700082835e-3 + (0.51681681023846925160e-6 + (0.55694594264948268169e-7 + (-0.91719117313243464652e-9 + (0.53982743680000000000e-12 + 0.22050829296187771142e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 49: { + double t = 2*y100 - 99; + return 0.60521224471819875444e0 + (0.21732138012345456060e-2 + (-0.22872428969625997456e-3 + (0.92588959922653404233e-6 + (0.46612665806531930684e-7 + (-0.89393722514414153351e-9 + (0.31718550353777777778e-11 + 0.15705458816080549117e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 50: { + double t = 2*y100 - 101; + return 0.60865189969791123620e0 + (0.12708480848877451719e-2 + (-0.22212090111534847166e-3 + (0.12636236031532793467e-5 + (0.37904037100232937574e-7 + (-0.84417089968101223519e-9 + (0.49843180828444444445e-11 + 0.10355439441049048273e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 51: { + double t = 2*y100 - 103; + return 0.61031580103499200191e0 + (0.39867436055861038223e-3 + (-0.21369573439579869291e-3 + (0.15339402129026183670e-5 + (0.29787479206646594442e-7 + (-0.77687792914228632974e-9 + (0.61192452741333333334e-11 + 0.60216691829459295780e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 52: { + double t = 2*y100 - 105; + return 0.61027109047879835868e0 + (-0.43680904508059878254e-3 + (-0.20383783788303894442e-3 + (0.17421743090883439959e-5 + (0.22400425572175715576e-7 + (-0.69934719320045128997e-9 + (0.67152759655111111110e-11 + 0.26419960042578359995e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 53: { + double t = 2*y100 - 107; + return 0.60859639489217430521e0 + (-0.12305921390962936873e-2 + (-0.19290150253894682629e-3 + (0.18944904654478310128e-5 + (0.15815530398618149110e-7 + (-0.61726850580964876070e-9 + 0.68987888999111111110e-11 * t) * t) * t) * t) * t) * t; + } + case 54: { + double t = 2*y100 - 109; + return 0.60537899426486075181e0 + (-0.19790062241395705751e-2 + (-0.18120271393047062253e-3 + (0.19974264162313241405e-5 + (0.10055795094298172492e-7 + (-0.53491997919318263593e-9 + (0.67794550295111111110e-11 - 0.17059208095741511603e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 55: { + double t = 2*y100 - 111; + return 0.60071229457904110537e0 + (-0.26795676776166354354e-2 + (-0.16901799553627508781e-3 + (0.20575498324332621581e-5 + (0.51077165074461745053e-8 + (-0.45536079828057221858e-9 + (0.64488005516444444445e-11 - 0.29311677573152766338e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 56: { + double t = 2*y100 - 113; + return 0.59469361520112714738e0 + (-0.33308208190600993470e-2 + (-0.15658501295912405679e-3 + (0.20812116912895417272e-5 + (0.93227468760614182021e-9 + (-0.38066673740116080415e-9 + (0.59806790359111111110e-11 - 0.36887077278950440597e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 57: { + double t = 2*y100 - 115; + return 0.58742228631775388268e0 + (-0.39321858196059227251e-2 + (-0.14410441141450122535e-3 + (0.20743790018404020716e-5 + (-0.25261903811221913762e-8 + (-0.31212416519526924318e-9 + (0.54328422462222222221e-11 - 0.40864152484979815972e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 58: { + double t = 2*y100 - 117; + return 0.57899804200033018447e0 + (-0.44838157005618913447e-2 + (-0.13174245966501437965e-3 + (0.20425306888294362674e-5 + (-0.53330296023875447782e-8 + (-0.25041289435539821014e-9 + (0.48490437205333333334e-11 - 0.42162206939169045177e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 59: { + double t = 2*y100 - 119; + return 0.56951968796931245974e0 + (-0.49864649488074868952e-2 + (-0.11963416583477567125e-3 + (0.19906021780991036425e-5 + (-0.75580140299436494248e-8 + (-0.19576060961919820491e-9 + (0.42613011928888888890e-11 - 0.41539443304115604377e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 60: { + double t = 2*y100 - 121; + return 0.55908401930063918964e0 + (-0.54413711036826877753e-2 + (-0.10788661102511914628e-3 + (0.19229663322982839331e-5 + (-0.92714731195118129616e-8 + (-0.14807038677197394186e-9 + (0.36920870298666666666e-11 - 0.39603726688419162617e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 61: { + double t = 2*y100 - 123; + return 0.54778496152925675315e0 + (-0.58501497933213396670e-2 + (-0.96582314317855227421e-4 + (0.18434405235069270228e-5 + (-0.10541580254317078711e-7 + (-0.10702303407788943498e-9 + (0.31563175582222222222e-11 - 0.36829748079110481422e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 62: { + double t = 2*y100 - 125; + return 0.53571290831682823999e0 + (-0.62147030670760791791e-2 + (-0.85782497917111760790e-4 + (0.17553116363443470478e-5 + (-0.11432547349815541084e-7 + (-0.72157091369041330520e-10 + (0.26630811607111111111e-11 - 0.33578660425893164084e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 63: { + double t = 2*y100 - 127; + return 0.52295422962048434978e0 + (-0.65371404367776320720e-2 + (-0.75530164941473343780e-4 + (0.16613725797181276790e-5 + (-0.12003521296598910761e-7 + (-0.42929753689181106171e-10 + (0.22170894940444444444e-11 - 0.30117697501065110505e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 64: { + double t = 2*y100 - 129; + return 0.50959092577577886140e0 + (-0.68197117603118591766e-2 + (-0.65852936198953623307e-4 + (0.15639654113906716939e-5 + (-0.12308007991056524902e-7 + (-0.18761997536910939570e-10 + (0.18198628922666666667e-11 - 0.26638355362285200932e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 65: { + double t = 2*y100 - 131; + return 0.49570040481823167970e0 + (-0.70647509397614398066e-2 + (-0.56765617728962588218e-4 + (0.14650274449141448497e-5 + (-0.12393681471984051132e-7 + (0.92904351801168955424e-12 + (0.14706755960177777778e-11 - 0.23272455351266325318e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 66: { + double t = 2*y100 - 133; + return 0.48135536250935238066e0 + (-0.72746293327402359783e-2 + (-0.48272489495730030780e-4 + (0.13661377309113939689e-5 + (-0.12302464447599382189e-7 + (0.16707760028737074907e-10 + (0.11672928324444444444e-11 - 0.20105801424709924499e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 67: { + double t = 2*y100 - 135; + return 0.46662374675511439448e0 + (-0.74517177649528487002e-2 + (-0.40369318744279128718e-4 + (0.12685621118898535407e-5 + (-0.12070791463315156250e-7 + (0.29105507892605823871e-10 + (0.90653314645333333334e-12 - 0.17189503312102982646e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 68: { + double t = 2*y100 - 137; + return 0.45156879030168268778e0 + (-0.75983560650033817497e-2 + (-0.33045110380705139759e-4 + (0.11732956732035040896e-5 + (-0.11729986947158201869e-7 + (0.38611905704166441308e-10 + (0.68468768305777777779e-12 - 0.14549134330396754575e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 69: { + double t = 2*y100 - 139; + return 0.43624909769330896904e0 + (-0.77168291040309554679e-2 + (-0.26283612321339907756e-4 + (0.10811018836893550820e-5 + (-0.11306707563739851552e-7 + (0.45670446788529607380e-10 + (0.49782492549333333334e-12 - 0.12191983967561779442e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 70: { + double t = 2*y100 - 141; + return 0.42071877443548481181e0 + (-0.78093484015052730097e-2 + (-0.20064596897224934705e-4 + (0.99254806680671890766e-6 + (-0.10823412088884741451e-7 + (0.50677203326904716247e-10 + (0.34200547594666666666e-12 - 0.10112698698356194618e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 71: { + double t = 2*y100 - 143; + return 0.40502758809710844280e0 + (-0.78780384460872937555e-2 + (-0.14364940764532853112e-4 + (0.90803709228265217384e-6 + (-0.10298832847014466907e-7 + (0.53981671221969478551e-10 + (0.21342751381333333333e-12 - 0.82975901848387729274e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 72: { + double t = 2*y100 - 145; + return 0.38922115269731446690e0 + (-0.79249269708242064120e-2 + (-0.91595258799106970453e-5 + (0.82783535102217576495e-6 + (-0.97484311059617744437e-8 + (0.55889029041660225629e-10 + (0.10851981336888888889e-12 - 0.67278553237853459757e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 73: { + double t = 2*y100 - 147; + return 0.37334112915460307335e0 + (-0.79519385109223148791e-2 + (-0.44219833548840469752e-5 + (0.75209719038240314732e-6 + (-0.91848251458553190451e-8 + (0.56663266668051433844e-10 + (0.23995894257777777778e-13 - 0.53819475285389344313e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 74: { + double t = 2*y100 - 149; + return 0.35742543583374223085e0 + (-0.79608906571527956177e-2 + (-0.12530071050975781198e-6 + (0.68088605744900552505e-6 + (-0.86181844090844164075e-8 + (0.56530784203816176153e-10 + (-0.43120012248888888890e-13 - 0.42372603392496813810e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 75: { + double t = 2*y100 - 151; + return 0.34150846431979618536e0 + (-0.79534924968773806029e-2 + (0.37576885610891515813e-5 + (0.61419263633090524326e-6 + (-0.80565865409945960125e-8 + (0.55684175248749269411e-10 + (-0.95486860764444444445e-13 - 0.32712946432984510595e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 76: { + double t = 2*y100 - 153; + return 0.32562129649136346824e0 + (-0.79313448067948884309e-2 + (0.72539159933545300034e-5 + (0.55195028297415503083e-6 + (-0.75063365335570475258e-8 + (0.54281686749699595941e-10 - 0.13545424295111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 77: { + double t = 2*y100 - 155; + return 0.30979191977078391864e0 + (-0.78959416264207333695e-2 + (0.10389774377677210794e-4 + (0.49404804463196316464e-6 + (-0.69722488229411164685e-8 + (0.52469254655951393842e-10 - 0.16507860650666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 78: { + double t = 2*y100 - 157; + return 0.29404543811214459904e0 + (-0.78486728990364155356e-2 + (0.13190885683106990459e-4 + (0.44034158861387909694e-6 + (-0.64578942561562616481e-8 + (0.50354306498006928984e-10 - 0.18614473550222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 79: { + double t = 2*y100 - 159; + return 0.27840427686253660515e0 + (-0.77908279176252742013e-2 + (0.15681928798708548349e-4 + (0.39066226205099807573e-6 + (-0.59658144820660420814e-8 + (0.48030086420373141763e-10 - 0.20018995173333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 80: { + double t = 2*y100 - 161; + return 0.26288838011163800908e0 + (-0.77235993576119469018e-2 + (0.17886516796198660969e-4 + (0.34482457073472497720e-6 + (-0.54977066551955420066e-8 + (0.45572749379147269213e-10 - 0.20852924954666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 81: { + double t = 2*y100 - 163; + return 0.24751539954181029717e0 + (-0.76480877165290370975e-2 + (0.19827114835033977049e-4 + (0.30263228619976332110e-6 + (-0.50545814570120129947e-8 + (0.43043879374212005966e-10 - 0.21228012028444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 82: { + double t = 2*y100 - 165; + return 0.23230087411688914593e0 + (-0.75653060136384041587e-2 + (0.21524991113020016415e-4 + (0.26388338542539382413e-6 + (-0.46368974069671446622e-8 + (0.40492715758206515307e-10 - 0.21238627815111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 83: { + double t = 2*y100 - 167; + return 0.21725840021297341931e0 + (-0.74761846305979730439e-2 + (0.23000194404129495243e-4 + (0.22837400135642906796e-6 + (-0.42446743058417541277e-8 + (0.37958104071765923728e-10 - 0.20963978568888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 84: { + double t = 2*y100 - 169; + return 0.20239979200788191491e0 + (-0.73815761980493466516e-2 + (0.24271552727631854013e-4 + (0.19590154043390012843e-6 + (-0.38775884642456551753e-8 + (0.35470192372162901168e-10 - 0.20470131678222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 85: { + double t = 2*y100 - 171; + return 0.18773523211558098962e0 + (-0.72822604530339834448e-2 + (0.25356688567841293697e-4 + (0.16626710297744290016e-6 + (-0.35350521468015310830e-8 + (0.33051896213898864306e-10 - 0.19811844544000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 86: { + double t = 2*y100 - 173; + return 0.17327341258479649442e0 + (-0.71789490089142761950e-2 + (0.26272046822383820476e-4 + (0.13927732375657362345e-6 + (-0.32162794266956859603e-8 + (0.30720156036105652035e-10 - 0.19034196304000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 87: { + double t = 2*y100 - 175; + return 0.15902166648328672043e0 + (-0.70722899934245504034e-2 + (0.27032932310132226025e-4 + (0.11474573347816568279e-6 + (-0.29203404091754665063e-8 + (0.28487010262547971859e-10 - 0.18174029063111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 88: { + double t = 2*y100 - 177; + return 0.14498609036610283865e0 + (-0.69628725220045029273e-2 + (0.27653554229160596221e-4 + (0.92493727167393036470e-7 + (-0.26462055548683583849e-8 + (0.26360506250989943739e-10 - 0.17261211260444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 89: { + double t = 2*y100 - 179; + return 0.13117165798208050667e0 + (-0.68512309830281084723e-2 + (0.28147075431133863774e-4 + (0.72351212437979583441e-7 + (-0.23927816200314358570e-8 + (0.24345469651209833155e-10 - 0.16319736960000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 90: { + double t = 2*y100 - 181; + return 0.11758232561160626306e0 + (-0.67378491192463392927e-2 + (0.28525664781722907847e-4 + (0.54156999310046790024e-7 + (-0.21589405340123827823e-8 + (0.22444150951727334619e-10 - 0.15368675584000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 91: { + double t = 2*y100 - 183; + return 0.10422112945361673560e0 + (-0.66231638959845581564e-2 + (0.28800551216363918088e-4 + (0.37758983397952149613e-7 + (-0.19435423557038933431e-8 + (0.20656766125421362458e-10 - 0.14422990012444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 92: { + double t = 2*y100 - 185; + return 0.91090275493541084785e-1 + (-0.65075691516115160062e-2 + (0.28982078385527224867e-4 + (0.23014165807643012781e-7 + (-0.17454532910249875958e-8 + (0.18981946442680092373e-10 - 0.13494234691555555556e-12 * t) * t) * t) * t) * t) * t; + } + case 93: { + double t = 2*y100 - 187; + return 0.78191222288771379358e-1 + (-0.63914190297303976434e-2 + (0.29079759021299682675e-4 + (0.97885458059415717014e-8 + (-0.15635596116134296819e-8 + (0.17417110744051331974e-10 - 0.12591151763555555556e-12 * t) * t) * t) * t) * t) * t; + } + case 94: { + double t = 2*y100 - 189; + return 0.65524757106147402224e-1 + (-0.62750311956082444159e-2 + (0.29102328354323449795e-4 + (-0.20430838882727954582e-8 + (-0.13967781903855367270e-8 + (0.15958771833747057569e-10 - 0.11720175765333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 95: { + double t = 2*y100 - 191; + return 0.53091065838453612773e-1 + (-0.61586898417077043662e-2 + (0.29057796072960100710e-4 + (-0.12597414620517987536e-7 + (-0.12440642607426861943e-8 + (0.14602787128447932137e-10 - 0.10885859114666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 96: { + double t = 2*y100 - 193; + return 0.40889797115352738582e-1 + (-0.60426484889413678200e-2 + (0.28953496450191694606e-4 + (-0.21982952021823718400e-7 + (-0.11044169117553026211e-8 + (0.13344562332430552171e-10 - 0.10091231402844444444e-12 * t) * t) * t) * t) * t) * t; + } + case 97: { + double t = 2*y100 - 195; + return 0.28920121009594899986e-1 + (-0.59271325915413781788e-2 + (0.28796136372768177423e-4 + (-0.30300382596279568642e-7 + (-0.97688275022802329749e-9 + (0.12179215701512592356e-10 - 0.93380988481777777779e-13 * t) * t) * t) * t) * t) * t; + } + case 98: { + double t = 2*y100 - 197; + return 0.17180782722617876655e-1 + (-0.58123419543161127769e-2 + (0.28591841095380959666e-4 + (-0.37642963496443667043e-7 + (-0.86055809047367300024e-9 + (0.11101709356762665578e-10 - 0.86272947493333333334e-13 * t) * t) * t) * t) * t) * t; + } + case 99: case 100: { // use Taylor expansion for small x (|x| <= 0.010101...) + // (2/sqrt(pi)) * (x - 2/3 x^3 + 4/15 x^5 - 8/105 x^7) + double x2 = x*x; + return x * (1.1283791670955125739 + - x2 * (0.75225277806367504925 + - x2 * (0.30090111122547001970 + - x2 * 0.085971746064420005629))); + } + } + /* Since 0 <= y100 < 101, this is only reached if x is NaN, + in which case we should return NaN. */ + return NaN; +} + +double FADDEEVA(w_im)(double x) +{ + if (x >= 0) { + if (x > 45) { // continued-fraction expansion is faster + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x > 5e7) // 1-term expansion, important to avoid overflow + return ispi / x; + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x-0.5/(x-1/(x-1.5/(x-2/x)))) */ + return ispi*((x*x) * (x*x-4.5) + 2) / (x * ((x*x) * (x*x-5) + 3.75)); + } + return w_im_y100(100/(1+x), x); + } + else { // = -FADDEEVA(w_im)(-x) + if (x < -45) { // continued-fraction expansion is faster + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x < -5e7) // 1-term expansion, important to avoid overflow + return ispi / x; + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x-0.5/(x-1/(x-1.5/(x-2/x)))) */ + return ispi*((x*x) * (x*x-4.5) + 2) / (x * ((x*x) * (x*x-5) + 3.75)); + } + return -w_im_y100(100/(1-x), -x); + } +} + +///////////////////////////////////////////////////////////////////////// + +// Compile with -DTEST_FADDEEVA to compile a little test program +#ifdef TEST_FADDEEVA + +#ifdef __cplusplus +# include <cstdio> +#else +# include <stdio.h> +#endif + +// compute relative error |b-a|/|a|, handling case of NaN and Inf, +static double relerr(double a, double b) { + if (isnan(a) || isnan(b) || isinf(a) || isinf(b)) { + if ((isnan(a) && !isnan(b)) || (!isnan(a) && isnan(b)) || + (isinf(a) && !isinf(b)) || (!isinf(a) && isinf(b)) || + (isinf(a) && isinf(b) && a*b < 0)) + return Inf; // "infinite" error + return 0; // matching infinity/nan results counted as zero error + } + if (a == 0) + return b == 0 ? 0 : Inf; + else + return fabs((b-a) / a); +} + +int main(void) { + double errmax_all = 0; + { + printf("############# w(z) tests #############\n"); +#define NTST 57 // define instead of const for C compatibility + cmplx z[NTST] = { + CMPLX(624.2,-0.26123), + CMPLX(-0.4,3.), + CMPLX(0.6,2.), + CMPLX(-1.,1.), + CMPLX(-1.,-9.), + CMPLX(-1.,9.), + CMPLX(-0.0000000234545,1.1234), + CMPLX(-3.,5.1), + CMPLX(-53,30.1), + CMPLX(0.0,0.12345), + CMPLX(11,1), + CMPLX(-22,-2), + CMPLX(9,-28), + CMPLX(21,-33), + CMPLX(1e5,1e5), + CMPLX(1e14,1e14), + CMPLX(-3001,-1000), + CMPLX(1e160,-1e159), + CMPLX(-6.01,0.01), + CMPLX(-0.7,-0.7), + CMPLX(2.611780000000000e+01, 4.540909610972489e+03), + CMPLX(0.8e7,0.3e7), + CMPLX(-20,-19.8081), + CMPLX(1e-16,-1.1e-16), + CMPLX(2.3e-8,1.3e-8), + CMPLX(6.3,-1e-13), + CMPLX(6.3,1e-20), + CMPLX(1e-20,6.3), + CMPLX(1e-20,16.3), + CMPLX(9,1e-300), + CMPLX(6.01,0.11), + CMPLX(8.01,1.01e-10), + CMPLX(28.01,1e-300), + CMPLX(10.01,1e-200), + CMPLX(10.01,-1e-200), + CMPLX(10.01,0.99e-10), + CMPLX(10.01,-0.99e-10), + CMPLX(1e-20,7.01), + CMPLX(-1,7.01), + CMPLX(5.99,7.01), + CMPLX(1,0), + CMPLX(55,0), + CMPLX(-0.1,0), + CMPLX(1e-20,0), + CMPLX(0,5e-14), + CMPLX(0,51), + CMPLX(Inf,0), + CMPLX(-Inf,0), + CMPLX(0,Inf), + CMPLX(0,-Inf), + CMPLX(Inf,Inf), + CMPLX(Inf,-Inf), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(0,NaN), + CMPLX(NaN,Inf), + CMPLX(Inf,NaN) + }; + cmplx w[NTST] = { /* w(z), computed with WolframAlpha + ... note that WolframAlpha is problematic + some of the above inputs, so I had to + use the continued-fraction expansion + in WolframAlpha in some cases, or switch + to Maple */ + CMPLX(-3.78270245518980507452677445620103199303131110e-7, + 0.000903861276433172057331093754199933411710053155), + CMPLX(0.1764906227004816847297495349730234591778719532788, + -0.02146550539468457616788719893991501311573031095617), + CMPLX(0.2410250715772692146133539023007113781272362309451, + 0.06087579663428089745895459735240964093522265589350), + CMPLX(0.30474420525691259245713884106959496013413834051768, + -0.20821893820283162728743734725471561394145872072738), + CMPLX(7.317131068972378096865595229600561710140617977e34, + 8.321873499714402777186848353320412813066170427e34), + CMPLX(0.0615698507236323685519612934241429530190806818395, + -0.00676005783716575013073036218018565206070072304635), + CMPLX(0.3960793007699874918961319170187598400134746631, + -5.593152259116644920546186222529802777409274656e-9), + CMPLX(0.08217199226739447943295069917990417630675021771804, + -0.04701291087643609891018366143118110965272615832184), + CMPLX(0.00457246000350281640952328010227885008541748668738, + -0.00804900791411691821818731763401840373998654987934), + CMPLX(0.8746342859608052666092782112565360755791467973338452, + 0.), + CMPLX(0.00468190164965444174367477874864366058339647648741, + 0.0510735563901306197993676329845149741675029197050), + CMPLX(-0.0023193175200187620902125853834909543869428763219, + -0.025460054739731556004902057663500272721780776336), + CMPLX(9.11463368405637174660562096516414499772662584e304, + 3.97101807145263333769664875189354358563218932e305), + CMPLX(-4.4927207857715598976165541011143706155432296e281, + -2.8019591213423077494444700357168707775769028e281), + CMPLX(2.820947917809305132678577516325951485807107151e-6, + 2.820947917668257736791638444590253942253354058e-6), + CMPLX(2.82094791773878143474039725787438662716372268e-15, + 2.82094791773878143474039725773333923127678361e-15), + CMPLX(-0.0000563851289696244350147899376081488003110150498, + -0.000169211755126812174631861529808288295454992688), + CMPLX(-5.586035480670854326218608431294778077663867e-162, + 5.586035480670854326218608431294778077663867e-161), + CMPLX(0.00016318325137140451888255634399123461580248456, + -0.095232456573009287370728788146686162555021209999), + CMPLX(0.69504753678406939989115375989939096800793577783885, + -1.8916411171103639136680830887017670616339912024317), + CMPLX(0.0001242418269653279656612334210746733213167234822, + 7.145975826320186888508563111992099992116786763e-7), + CMPLX(2.318587329648353318615800865959225429377529825e-8, + 6.182899545728857485721417893323317843200933380e-8), + CMPLX(-0.0133426877243506022053521927604277115767311800303, + -0.0148087097143220769493341484176979826888871576145), + CMPLX(1.00000000000000012412170838050638522857747934, + 1.12837916709551279389615890312156495593616433e-16), + CMPLX(0.9999999853310704677583504063775310832036830015, + 2.595272024519678881897196435157270184030360773e-8), + CMPLX(-1.4731421795638279504242963027196663601154624e-15, + 0.090727659684127365236479098488823462473074709), + CMPLX(5.79246077884410284575834156425396800754409308e-18, + 0.0907276596841273652364790985059772809093822374), + CMPLX(0.0884658993528521953466533278764830881245144368, + 1.37088352495749125283269718778582613192166760e-22), + CMPLX(0.0345480845419190424370085249304184266813447878, + 2.11161102895179044968099038990446187626075258e-23), + CMPLX(6.63967719958073440070225527042829242391918213e-36, + 0.0630820900592582863713653132559743161572639353), + CMPLX(0.00179435233208702644891092397579091030658500743634, + 0.0951983814805270647939647438459699953990788064762), + CMPLX(9.09760377102097999924241322094863528771095448e-13, + 0.0709979210725138550986782242355007611074966717), + CMPLX(7.2049510279742166460047102593255688682910274423e-304, + 0.0201552956479526953866611812593266285000876784321), + CMPLX(3.04543604652250734193622967873276113872279682e-44, + 0.0566481651760675042930042117726713294607499165), + CMPLX(3.04543604652250734193622967873276113872279682e-44, + 0.0566481651760675042930042117726713294607499165), + CMPLX(0.5659928732065273429286988428080855057102069081e-12, + 0.056648165176067504292998527162143030538756683302), + CMPLX(-0.56599287320652734292869884280802459698927645e-12, + 0.0566481651760675042929985271621430305387566833029), + CMPLX(0.0796884251721652215687859778119964009569455462, + 1.11474461817561675017794941973556302717225126e-22), + CMPLX(0.07817195821247357458545539935996687005781943386550, + -0.01093913670103576690766705513142246633056714279654), + CMPLX(0.04670032980990449912809326141164730850466208439937, + 0.03944038961933534137558064191650437353429669886545), + CMPLX(0.36787944117144232159552377016146086744581113103176, + 0.60715770584139372911503823580074492116122092866515), + CMPLX(0, + 0.010259688805536830986089913987516716056946786526145), + CMPLX(0.99004983374916805357390597718003655777207908125383, + -0.11208866436449538036721343053869621153527769495574), + CMPLX(0.99999999999999999999999999999999999999990000, + 1.12837916709551257389615890312154517168802603e-20), + CMPLX(0.999999999999943581041645226871305192054749891144158, + 0), + CMPLX(0.0110604154853277201542582159216317923453996211744250, + 0), + CMPLX(0,0), + CMPLX(0,0), + CMPLX(0,0), + CMPLX(Inf,0), + CMPLX(0,0), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN) + }; + double errmax = 0; + for (int i = 0; i < NTST; ++i) { + cmplx fw = FADDEEVA(w)(z[i],0.); + double re_err = relerr(creal(w[i]), creal(fw)); + double im_err = relerr(cimag(w[i]), cimag(fw)); + printf("w(%g%+gi) = %g%+gi (vs. %g%+gi), re/im rel. err. = %0.2g/%0.2g)\n", + creal(z[i]),cimag(z[i]), creal(fw),cimag(fw), creal(w[i]),cimag(w[i]), + re_err, im_err); + if (re_err > errmax) errmax = re_err; + if (im_err > errmax) errmax = im_err; + } + if (errmax > 1e-13) { + printf("FAILURE -- relative error %g too large!\n", errmax); + return 1; + } + printf("SUCCESS (max relative error = %g)\n", errmax); + if (errmax > errmax_all) errmax_all = errmax; + } + { +#undef NTST +#define NTST 33 // define instead of const for C compatibility + cmplx z[NTST] = { + CMPLX(1,2), + CMPLX(-1,2), + CMPLX(1,-2), + CMPLX(-1,-2), + CMPLX(9,-28), + CMPLX(21,-33), + CMPLX(1e3,1e3), + CMPLX(-3001,-1000), + CMPLX(1e160,-1e159), + CMPLX(5.1e-3, 1e-8), + CMPLX(-4.9e-3, 4.95e-3), + CMPLX(4.9e-3, 0.5), + CMPLX(4.9e-4, -0.5e1), + CMPLX(-4.9e-5, -0.5e2), + CMPLX(5.1e-3, 0.5), + CMPLX(5.1e-4, -0.5e1), + CMPLX(-5.1e-5, -0.5e2), + CMPLX(1e-6,2e-6), + CMPLX(0,2e-6), + CMPLX(0,2), + CMPLX(0,20), + CMPLX(0,200), + CMPLX(Inf,0), + CMPLX(-Inf,0), + CMPLX(0,Inf), + CMPLX(0,-Inf), + CMPLX(Inf,Inf), + CMPLX(Inf,-Inf), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(0,NaN), + CMPLX(NaN,Inf), + CMPLX(Inf,NaN) + }; + cmplx w[NTST] = { // erf(z[i]), evaluated with Maple + CMPLX(-0.5366435657785650339917955593141927494421, + -5.049143703447034669543036958614140565553), + CMPLX(0.5366435657785650339917955593141927494421, + -5.049143703447034669543036958614140565553), + CMPLX(-0.5366435657785650339917955593141927494421, + 5.049143703447034669543036958614140565553), + CMPLX(0.5366435657785650339917955593141927494421, + 5.049143703447034669543036958614140565553), + CMPLX(0.3359473673830576996788000505817956637777e304, + -0.1999896139679880888755589794455069208455e304), + CMPLX(0.3584459971462946066523939204836760283645e278, + 0.3818954885257184373734213077678011282505e280), + CMPLX(0.9996020422657148639102150147542224526887, + 0.00002801044116908227889681753993542916894856), + CMPLX(-1, 0), + CMPLX(1, 0), + CMPLX(0.005754683859034800134412990541076554934877, + 0.1128349818335058741511924929801267822634e-7), + CMPLX(-0.005529149142341821193633460286828381876955, + 0.005585388387864706679609092447916333443570), + CMPLX(0.007099365669981359632319829148438283865814, + 0.6149347012854211635026981277569074001219), + CMPLX(0.3981176338702323417718189922039863062440e8, + -0.8298176341665249121085423917575122140650e10), + CMPLX(-Inf, + -Inf), + CMPLX(0.007389128308257135427153919483147229573895, + 0.6149332524601658796226417164791221815139), + CMPLX(0.4143671923267934479245651547534414976991e8, + -0.8298168216818314211557046346850921446950e10), + CMPLX(-Inf, + -Inf), + CMPLX(0.1128379167099649964175513742247082845155e-5, + 0.2256758334191777400570377193451519478895e-5), + CMPLX(0, + 0.2256758334194034158904576117253481476197e-5), + CMPLX(0, + 18.56480241457555259870429191324101719886), + CMPLX(0, + 0.1474797539628786202447733153131835124599e173), + CMPLX(0, + Inf), + CMPLX(1,0), + CMPLX(-1,0), + CMPLX(0,Inf), + CMPLX(0,-Inf), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(0,NaN), + CMPLX(NaN,NaN), + CMPLX(NaN,NaN) + }; +#define TST(f,isc) \ + printf("############# " #f "(z) tests #############\n"); \ + double errmax = 0; \ + for (int i = 0; i < NTST; ++i) { \ + cmplx fw = FADDEEVA(f)(z[i],0.); \ + double re_err = relerr(creal(w[i]), creal(fw)); \ + double im_err = relerr(cimag(w[i]), cimag(fw)); \ + printf(#f "(%g%+gi) = %g%+gi (vs. %g%+gi), re/im rel. err. = %0.2g/%0.2g)\n", \ + creal(z[i]),cimag(z[i]), creal(fw),cimag(fw), creal(w[i]),cimag(w[i]), \ + re_err, im_err); \ + if (re_err > errmax) errmax = re_err; \ + if (im_err > errmax) errmax = im_err; \ + } \ + if (errmax > 1e-13) { \ + printf("FAILURE -- relative error %g too large!\n", errmax); \ + return 1; \ + } \ + printf("Checking " #f "(x) special case...\n"); \ + for (int i = 0; i < 10000; ++i) { \ + double x = pow(10., -300. + i * 600. / (10000 - 1)); \ + double re_err = relerr(FADDEEVA_RE(f)(x), \ + creal(FADDEEVA(f)(CMPLX(x,x*isc),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(-x), \ + creal(FADDEEVA(f)(CMPLX(-x,x*isc),0.))); \ + if (re_err > errmax) errmax = re_err; \ + } \ + { \ + double re_err = relerr(FADDEEVA_RE(f)(Inf), \ + creal(FADDEEVA(f)(CMPLX(Inf,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(-Inf), \ + creal(FADDEEVA(f)(CMPLX(-Inf,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(NaN), \ + creal(FADDEEVA(f)(CMPLX(NaN,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + } \ + if (errmax > 1e-13) { \ + printf("FAILURE -- relative error %g too large!\n", errmax); \ + return 1; \ + } \ + printf("SUCCESS (max relative error = %g)\n", errmax); \ + if (errmax > errmax_all) errmax_all = errmax + + TST(erf, 1e-20); + } + { + // since erfi just calls through to erf, just one test should + // be sufficient to make sure I didn't screw up the signs or something +#undef NTST +#define NTST 1 // define instead of const for C compatibility + cmplx z[NTST] = { CMPLX(1.234,0.5678) }; + cmplx w[NTST] = { // erfi(z[i]), computed with Maple + CMPLX(1.081032284405373149432716643834106923212, + 1.926775520840916645838949402886591180834) + }; + TST(erfi, 0); + } + { + // since erfcx just calls through to w, just one test should + // be sufficient to make sure I didn't screw up the signs or something +#undef NTST +#define NTST 1 // define instead of const for C compatibility + cmplx z[NTST] = { CMPLX(1.234,0.5678) }; + cmplx w[NTST] = { // erfcx(z[i]), computed with Maple + CMPLX(0.3382187479799972294747793561190487832579, + -0.1116077470811648467464927471872945833154) + }; + TST(erfcx, 0); + } + { +#undef NTST +#define NTST 30 // define instead of const for C compatibility + cmplx z[NTST] = { + CMPLX(1,2), + CMPLX(-1,2), + CMPLX(1,-2), + CMPLX(-1,-2), + CMPLX(9,-28), + CMPLX(21,-33), + CMPLX(1e3,1e3), + CMPLX(-3001,-1000), + CMPLX(1e160,-1e159), + CMPLX(5.1e-3, 1e-8), + CMPLX(0,2e-6), + CMPLX(0,2), + CMPLX(0,20), + CMPLX(0,200), + CMPLX(2e-6,0), + CMPLX(2,0), + CMPLX(20,0), + CMPLX(200,0), + CMPLX(Inf,0), + CMPLX(-Inf,0), + CMPLX(0,Inf), + CMPLX(0,-Inf), + CMPLX(Inf,Inf), + CMPLX(Inf,-Inf), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(0,NaN), + CMPLX(NaN,Inf), + CMPLX(Inf,NaN), + CMPLX(88,0) + }; + cmplx w[NTST] = { // erfc(z[i]), evaluated with Maple + CMPLX(1.536643565778565033991795559314192749442, + 5.049143703447034669543036958614140565553), + CMPLX(0.4633564342214349660082044406858072505579, + 5.049143703447034669543036958614140565553), + CMPLX(1.536643565778565033991795559314192749442, + -5.049143703447034669543036958614140565553), + CMPLX(0.4633564342214349660082044406858072505579, + -5.049143703447034669543036958614140565553), + CMPLX(-0.3359473673830576996788000505817956637777e304, + 0.1999896139679880888755589794455069208455e304), + CMPLX(-0.3584459971462946066523939204836760283645e278, + -0.3818954885257184373734213077678011282505e280), + CMPLX(0.0003979577342851360897849852457775473112748, + -0.00002801044116908227889681753993542916894856), + CMPLX(2, 0), + CMPLX(0, 0), + CMPLX(0.9942453161409651998655870094589234450651, + -0.1128349818335058741511924929801267822634e-7), + CMPLX(1, + -0.2256758334194034158904576117253481476197e-5), + CMPLX(1, + -18.56480241457555259870429191324101719886), + CMPLX(1, + -0.1474797539628786202447733153131835124599e173), + CMPLX(1, -Inf), + CMPLX(0.9999977432416658119838633199332831406314, + 0), + CMPLX(0.004677734981047265837930743632747071389108, + 0), + CMPLX(0.5395865611607900928934999167905345604088e-175, + 0), + CMPLX(0, 0), + CMPLX(0, 0), + CMPLX(2, 0), + CMPLX(1, -Inf), + CMPLX(1, Inf), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, 0), + CMPLX(1, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(0,0) + }; + TST(erfc, 1e-20); + } + { +#undef NTST +#define NTST 47 // define instead of const for C compatibility + cmplx z[NTST] = { + CMPLX(2,1), + CMPLX(-2,1), + CMPLX(2,-1), + CMPLX(-2,-1), + CMPLX(-28,9), + CMPLX(33,-21), + CMPLX(1e3,1e3), + CMPLX(-1000,-3001), + CMPLX(1e-8, 5.1e-3), + CMPLX(4.95e-3, -4.9e-3), + CMPLX(0.5, 4.9e-3), + CMPLX(-0.5e1, 4.9e-4), + CMPLX(-0.5e2, -4.9e-5), + CMPLX(0.5e3, 4.9e-6), + CMPLX(0.5, 5.1e-3), + CMPLX(-0.5e1, 5.1e-4), + CMPLX(-0.5e2, -5.1e-5), + CMPLX(1e-6,2e-6), + CMPLX(2e-6,0), + CMPLX(2,0), + CMPLX(20,0), + CMPLX(200,0), + CMPLX(0,4.9e-3), + CMPLX(0,-5.1e-3), + CMPLX(0,2e-6), + CMPLX(0,-2), + CMPLX(0,20), + CMPLX(0,-200), + CMPLX(Inf,0), + CMPLX(-Inf,0), + CMPLX(0,Inf), + CMPLX(0,-Inf), + CMPLX(Inf,Inf), + CMPLX(Inf,-Inf), + CMPLX(NaN,NaN), + CMPLX(NaN,0), + CMPLX(0,NaN), + CMPLX(NaN,Inf), + CMPLX(Inf,NaN), + CMPLX(39, 6.4e-5), + CMPLX(41, 6.09e-5), + CMPLX(4.9e7, 5e-11), + CMPLX(5.1e7, 4.8e-11), + CMPLX(1e9, 2.4e-12), + CMPLX(1e11, 2.4e-14), + CMPLX(1e13, 2.4e-16), + CMPLX(1e300, 2.4e-303) + }; + cmplx w[NTST] = { // dawson(z[i]), evaluated with Maple + CMPLX(0.1635394094345355614904345232875688576839, + -0.1531245755371229803585918112683241066853), + CMPLX(-0.1635394094345355614904345232875688576839, + -0.1531245755371229803585918112683241066853), + CMPLX(0.1635394094345355614904345232875688576839, + 0.1531245755371229803585918112683241066853), + CMPLX(-0.1635394094345355614904345232875688576839, + 0.1531245755371229803585918112683241066853), + CMPLX(-0.01619082256681596362895875232699626384420, + -0.005210224203359059109181555401330902819419), + CMPLX(0.01078377080978103125464543240346760257008, + 0.006866888783433775382193630944275682670599), + CMPLX(-0.5808616819196736225612296471081337245459, + 0.6688593905505562263387760667171706325749), + CMPLX(Inf, + -Inf), + CMPLX(0.1000052020902036118082966385855563526705e-7, + 0.005100088434920073153418834680320146441685), + CMPLX(0.004950156837581592745389973960217444687524, + -0.004899838305155226382584756154100963570500), + CMPLX(0.4244534840871830045021143490355372016428, + 0.002820278933186814021399602648373095266538), + CMPLX(-0.1021340733271046543881236523269967674156, + -0.00001045696456072005761498961861088944159916), + CMPLX(-0.01000200120119206748855061636187197886859, + 0.9805885888237419500266621041508714123763e-8), + CMPLX(0.001000002000012000023960527532953151819595, + -0.9800058800588007290937355024646722133204e-11), + CMPLX(0.4244549085628511778373438768121222815752, + 0.002935393851311701428647152230552122898291), + CMPLX(-0.1021340732357117208743299813648493928105, + -0.00001088377943049851799938998805451564893540), + CMPLX(-0.01000200120119126652710792390331206563616, + 0.1020612612857282306892368985525393707486e-7), + CMPLX(0.1000000000007333333333344266666666664457e-5, + 0.2000000000001333333333323199999999978819e-5), + CMPLX(0.1999999999994666666666675199999999990248e-5, + 0), + CMPLX(0.3013403889237919660346644392864226952119, + 0), + CMPLX(0.02503136792640367194699495234782353186858, + 0), + CMPLX(0.002500031251171948248596912483183760683918, + 0), + CMPLX(0,0.004900078433419939164774792850907128053308), + CMPLX(0,-0.005100088434920074173454208832365950009419), + CMPLX(0,0.2000000000005333333333341866666666676419e-5), + CMPLX(0,-48.16001211429122974789822893525016528191), + CMPLX(0,0.4627407029504443513654142715903005954668e174), + CMPLX(0,-Inf), + CMPLX(0,0), + CMPLX(-0,0), + CMPLX(0, Inf), + CMPLX(0, -Inf), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, 0), + CMPLX(0, NaN), + CMPLX(NaN, NaN), + CMPLX(NaN, NaN), + CMPLX(0.01282473148489433743567240624939698290584, + -0.2105957276516618621447832572909153498104e-7), + CMPLX(0.01219875253423634378984109995893708152885, + -0.1813040560401824664088425926165834355953e-7), + CMPLX(0.1020408163265306334945473399689037886997e-7, + -0.1041232819658476285651490827866174985330e-25), + CMPLX(0.9803921568627452865036825956835185367356e-8, + -0.9227220299884665067601095648451913375754e-26), + CMPLX(0.5000000000000000002500000000000000003750e-9, + -0.1200000000000000001800000188712838420241e-29), + CMPLX(5.00000000000000000000025000000000000000000003e-12, + -1.20000000000000000000018000000000000000000004e-36), + CMPLX(5.00000000000000000000000002500000000000000000e-14, + -1.20000000000000000000000001800000000000000000e-42), + CMPLX(5e-301, 0) + }; + TST(Dawson, 1e-20); + } + printf("#####################################\n"); + printf("SUCCESS (max relative error = %g)\n", errmax_all); +} + +#endif
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/liboctave/cruft/Faddeeva/Faddeeva.hh Thu Nov 29 10:40:32 2012 -0500 @@ -0,0 +1,62 @@ +/* Copyright (c) 2012 Massachusetts Institute of Technology + * + * 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. + */ + +/* Available at: http://ab-initio.mit.edu/Faddeeva + + Header file for Faddeeva.cc; see that file for more information. */ + +#ifndef FADDEEVA_HH +#define FADDEEVA_HH 1 + +#include <complex> + +namespace Faddeeva { + +// compute w(z) = exp(-z^2) erfc(-iz) [ Faddeeva / scaled complex error func ] +extern std::complex<double> w(std::complex<double> z,double relerr=0); +extern double w_im(double x); // special-case code for Im[w(x)] of real x + +// Various functions that we can compute with the help of w(z) + +// compute erfcx(z) = exp(z^2) erfc(z) +extern std::complex<double> erfcx(std::complex<double> z, double relerr=0); +extern double erfcx(double x); // special case for real x + +// compute erf(z), the error function of complex arguments +extern std::complex<double> erf(std::complex<double> z, double relerr=0); +extern double erf(double x); // special case for real x + +// compute erfi(z) = -i erf(iz), the imaginary error function +extern std::complex<double> erfi(std::complex<double> z, double relerr=0); +extern double erfi(double x); // special case for real x + +// compute erfc(z) = 1 - erf(z), the complementary error function +extern std::complex<double> erfc(std::complex<double> z, double relerr=0); +extern double erfc(double x); // special case for real x + +// compute Dawson(z) = sqrt(pi)/2 * exp(-z^2) * erfi(z) +extern std::complex<double> Dawson(std::complex<double> z, double relerr=0); +extern double Dawson(double x); // special case for real x + +} // namespace Faddeeva + +#endif // FADDEEVA_HH
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/liboctave/cruft/Faddeeva/module.mk Thu Nov 29 10:40:32 2012 -0500 @@ -0,0 +1,3 @@ +EXTRA_DIST += Faddeeva/module.mk + +libcruft_la_SOURCES += Faddeeva/Faddeeva.cc Faddeeva/Faddeeva.hh
--- a/liboctave/cruft/Makefile.am Wed Nov 28 14:05:20 2012 -0500 +++ b/liboctave/cruft/Makefile.am Thu Nov 29 10:40:32 2012 -0500 @@ -47,6 +47,7 @@ include daspk/module.mk include dasrt/module.mk include dassl/module.mk +include Faddeeva/module.mk include fftpack/module.mk include lapack-xtra/module.mk include misc/module.mk
--- a/liboctave/numeric/lo-specfun.cc Wed Nov 28 14:05:20 2012 -0500 +++ b/liboctave/numeric/lo-specfun.cc Thu Nov 29 10:40:32 2012 -0500 @@ -50,6 +50,8 @@ #define M_PI 3.14159265358979323846 #endif +#include "Faddeeva.hh" + extern "C" { F77_RET_T @@ -287,6 +289,82 @@ } #endif +// Complex error function from the Faddeeva package +Complex +erf (const Complex& x) +{ + return Faddeeva::erf (x); +} +FloatComplex +erf (const FloatComplex& x) +{ + Complex xd (real (x), imag (x)); + Complex ret = Faddeeva::erf (xd, std::numeric_limits<float>::epsilon ()); + return FloatComplex (real (ret), imag (ret)); +} + +// Complex complementary error function from the Faddeeva package +Complex +erfc (const Complex& x) +{ + return Faddeeva::erfc (x); +} +FloatComplex +erfc (const FloatComplex& x) +{ + Complex xd (real (x), imag (x)); + Complex ret = Faddeeva::erfc (xd, std::numeric_limits<float>::epsilon ()); + return FloatComplex (real (ret), imag (ret)); +} + +// Real and complex scaled complementary error function from Faddeeva package +float erfcx (float x) { return Faddeeva::erfcx(x); } +double erfcx (double x) { return Faddeeva::erfcx(x); } +Complex +erfcx (const Complex& x) +{ + return Faddeeva::erfcx (x); +} +FloatComplex +erfcx (const FloatComplex& x) +{ + Complex xd (real (x), imag (x)); + Complex ret = Faddeeva::erfcx (xd, std::numeric_limits<float>::epsilon ()); + return FloatComplex (real (ret), imag (ret)); +} + +// Real and complex imaginary error function from Faddeeva package +float erfi (float x) { return Faddeeva::erfi(x); } +double erfi (double x) { return Faddeeva::erfi(x); } +Complex +erfi (const Complex& x) +{ + return Faddeeva::erfi (x); +} +FloatComplex +erfi (const FloatComplex& x) +{ + Complex xd (real (x), imag (x)); + Complex ret = Faddeeva::erfi (xd, std::numeric_limits<float>::epsilon ()); + return FloatComplex (real (ret), imag (ret)); +} + +// Real and complex Dawson function (= scaled erfi) from Faddeeva package +float dawson (float x) { return Faddeeva::Dawson(x); } +double dawson (double x) { return Faddeeva::Dawson(x); } +Complex +dawson (const Complex& x) +{ + return Faddeeva::Dawson (x); +} +FloatComplex +dawson (const FloatComplex& x) +{ + Complex xd (real (x), imag (x)); + Complex ret = Faddeeva::Dawson (xd, std::numeric_limits<float>::epsilon ()); + return FloatComplex (real (ret), imag (ret)); +} + double xgamma (double x) { @@ -3029,106 +3107,6 @@ return do_erfcinv (x, false); } -// Implementation based on the Fortran code by W.J.Cody -// see http://www.netlib.org/specfun/erf. -// Templatized and simplified workflow. - -// FIXME: Maybe this should be globally visible. -static inline float erfc (float x) { return erfcf (x); } - -template <class T> -static T -erfcx_impl (T x) -{ - static const T c[] = - { - 5.64188496988670089e-1,8.88314979438837594, - 6.61191906371416295e+1,2.98635138197400131e+2, - 8.81952221241769090e+2,1.71204761263407058e+3, - 2.05107837782607147e+3,1.23033935479799725e+3, - 2.15311535474403846e-8 - }; - - static const T d[] = - { - 1.57449261107098347e+1,1.17693950891312499e+2, - 5.37181101862009858e+2,1.62138957456669019e+3, - 3.29079923573345963e+3,4.36261909014324716e+3, - 3.43936767414372164e+3,1.23033935480374942e+3 - }; - - static const T p[] = - { - 3.05326634961232344e-1,3.60344899949804439e-1, - 1.25781726111229246e-1,1.60837851487422766e-2, - 6.58749161529837803e-4,1.63153871373020978e-2 - }; - - static const T q[] = - { - 2.56852019228982242,1.87295284992346047, - 5.27905102951428412e-1,6.05183413124413191e-2, - 2.33520497626869185e-3 - }; - - static const T sqrpi = 5.6418958354775628695e-1; - static const T xhuge = sqrt (1.0 / std::numeric_limits<T>::epsilon ()); - static const T xneg = -sqrt (log (std::numeric_limits<T>::max ()/2.0)); - - double y = fabs (x), result; - if (x < xneg) - result = octave_Inf; - else if (y <= 0.46875) - result = std::exp (x*x) * erfc (x); - else - { - if (y <= 4.0) - { - double xnum = c[8]*y, xden = y; - for (int i = 0; i < 7; i++) - { - xnum = (xnum + c[i]) * y; - xden = (xden + d[i]) * y; - } - - result = (xnum + c[7]) / (xden + d[7]); - } - else if (y <= xhuge) - { - double y2 = 1/(y*y), xnum = p[5]*y2, xden = y2; - for (int i = 0; i < 4; i++) - { - xnum = (xnum + p[i]) * y2; - xden = (xden + q[i]) * y2; - } - - result = y2 * (xnum + p[4]) / (xden + q[4]); - result = (sqrpi - result) / y; - } - else - result = sqrpi / y; - - // Fix up negative argument. - if (x < 0) - { - double y2 = ceil (x / 16.0) * 16.0, del = (x-y2)*(x+y2); - result = 2*(std::exp (y2*y2) * std::exp (del)) - result; - } - } - - return result; -} - -double erfcx (double x) -{ - return erfcx_impl (x); -} - -float erfcx (float x) -{ - return erfcx_impl (x); -} - // // Incomplete Beta function ratio //
--- a/liboctave/numeric/lo-specfun.h Wed Nov 28 14:05:20 2012 -0500 +++ b/liboctave/numeric/lo-specfun.h Thu Nov 29 10:40:32 2012 -0500 @@ -56,10 +56,14 @@ #if !defined (HAVE_ERF) extern OCTAVE_API double erf (double); #endif +extern OCTAVE_API Complex erf (const Complex& x); +extern OCTAVE_API FloatComplex erf (const FloatComplex& x); #if !defined (HAVE_ERFC) extern OCTAVE_API double erfc (double); #endif +extern OCTAVE_API Complex erfc (const Complex& x); +extern OCTAVE_API FloatComplex erfc (const FloatComplex& x); #if !defined (HAVE_ACOSHF) extern OCTAVE_API float acoshf (float); @@ -578,8 +582,20 @@ extern OCTAVE_API double erfcinv (double x); extern OCTAVE_API float erfcinv (float x); +extern OCTAVE_API float erfcx (float x); extern OCTAVE_API double erfcx (double x); -extern OCTAVE_API float erfcx (float x); +extern OCTAVE_API Complex erfcx (const Complex& x); +extern OCTAVE_API FloatComplex erfcx (const FloatComplex& x); + +extern OCTAVE_API float erfi (float x); +extern OCTAVE_API double erfi (double x); +extern OCTAVE_API Complex erfi (const Complex& x); +extern OCTAVE_API FloatComplex erfi (const FloatComplex& x); + +extern OCTAVE_API float dawson (float x); +extern OCTAVE_API double dawson (double x); +extern OCTAVE_API Complex dawson (const Complex& x); +extern OCTAVE_API FloatComplex dawson (const FloatComplex& x); extern OCTAVE_API double betaincinv (double x, double a, double b); extern OCTAVE_API Array<double> betaincinv (double x, double a, const Array<double>& b);
--- a/liboctave/numeric/module.mk Wed Nov 28 14:05:20 2012 -0500 +++ b/liboctave/numeric/module.mk Thu Nov 29 10:40:32 2012 -0500 @@ -179,6 +179,7 @@ numeric_libnumeric_la_CPPFLAGS = \ $(liboctave_la_CPPFLAGS) \ $(ARPACK_CPPFLAGS) \ + -I$(srcdir)/cruft/Faddeeva \ $(FFTW_XCPPFLAGS) \ $(SPARSE_XCPPFLAGS)