compile - Compiling the VoigtDistribution PDF

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 VoigtDistribution 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"]

Needs["CompiledFunctionTools`"];
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 MainEvaluate:

Erfc[I] // N

funcReal[1]
funcComplex[1]

1. - 1.65043 I

1. - 1.65043 I

1. - 1.65043 I

All the functions work, but because of the MainEvaluate, 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.

Answer

From the 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.

PseudoVoigtDistribution[δ_, σ_] := 
 Block[{g = (δ^5 + σ^5 + 2.69296 σ^4 δ + 2.42843 σ^3 δ^2 + 
        4.47163 σ^2 δ^3 + 0.07842 σ δ^4)^(1/5), η},
       η = δ/g;
       η = η*(1.36603 - 0.47719 η + 0.11116 η^2);
       MixtureDistribution[{1 - η, η}, {NormalDistribution[0, g], 
                           CauchyDistribution[0, g]}]

       ]