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According to List of compilable functions Erf and Erfc are compilable functions.

However, I want to make a compiled version of the PDF of a VoightDistribution to use in a NonlinearModelFit and it doesn't seem that the Erfc of a complex value will compile.

funcReal = 
 Compile[{{x, _Real}}, Erfc[x I], CompilationTarget -> "C", 
  RuntimeOptions -> "Speed"]
funcComplex = 
 Compile[{{x, _Complex}}, Erfc[x I], CompilationTarget -> "C", 
  RuntimeOptions -> "Speed"]

CompilePrint[funcReal] (*same as funcComplex*)

    1 argument
    2 Real registers
    4 Complex registers
    Underflow checking off
    Overflow checking off
    Integer overflow checking off
    RuntimeAttributes -> {}

    R0 = A1
    C0 = 0. + 1. I
    R1 = 0.
    Result = C3

1   C1 = R0 + R1 I
2   C1 = C1 * C0
3   C2 = R0 + R1 I
4   C2 = C2 * C0
5   C3 = MainEvaluate[ Hold[Erfc][ C2]]
6   Return

Note the call to MainEvalulate

Erfc[I] // N

1. - 1.65043 I

1. - 1.65043 I

1. - 1.65043 I

All the functions work but because of the MainEvalulate they offer no performance benefit. How can I compile this function? Is this possible? Is there an alternative formula I could use?

Removing the CompilationTarget doesn't solve the problem either.

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They compile for real values, but do not appear to compile for complex arguments. –  asim Feb 20 '13 at 17:56
Look at e.g. Plot3D[Arg@Erf[x + I y], {x, -5, 5}, {y, -5, 5}]. This is not a function you would probably wish to try to find an alternative formula for, even approximately. –  Oleksandr R. Feb 20 '13 at 18:25
On the other hand, regarding the PDF of the Voigt distribution: dx.doi.org/10.1016/0368-2048(94)02189-7 –  Oleksandr R. Feb 20 '13 at 18:46
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1 Answer

up vote 2 down vote accepted

From Mathematica Documentation, we have a Pseudo Voigt Distribution that might be used as an approximation. This might be useful as a basis for making a compilable function.

In[1]:= PseudoVoigtDistribution[de_, si_] := 
 Block[{g = (de^5 + si^5 + 2.69296 si^4 de + 2.42843 si^3 de^2 + 
      4.47163 si^2 de^3 + 0.07842 si de^4)^(1/5), eta},
  eta = de/g;
  eta = eta*(1.36603 - 0.47719 eta + 0.11116 eta^2);
  MixtureDistribution[{1 - eta, eta}, {NormalDistribution[0, g], 
    CauchyDistribution[0, g]}]
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