CForm not getting exp(), pow, log() functions

ClearAll[foo]
foo = RawBoxes[Replace[ToBoxes@#, InterpretationBox[a_, b_, c___] :>
      With[{aa = StringReplace[a, 
            { "Sqrt" -> "sqrt", "Power(E," -> "exp(", "Power" -> "pow"}]}, aa], 
      {0, Infinity}]] &;

foo@CForm[D[f, M]]

I would unprotect Power and use Format:

Unprotect[Power];
Format[Power[E, a_], CForm] := exp[a]
Format[Power[a_, 1/2], CForm] := sqrt[a]
Format[Power[a_, b_], CForm] := pow[a, b]
Protect[Power];

Then:

D[f, M] //CForm

-((exp((mu - sqrt(pow(M,2) + pow(x,2)))/T)*M)/(T*sqrt(pow(M,2) + pow(x,2))))

I've you're compiling with C, then you can #include the mdefs.h that shipped with your version of Mathematica.

You can find it by using:

FileNames["*",FileNameJoin[{$InstallationDirectory,"SystemFiles","IncludeFiles","C"}]]

{C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\dllexport.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\extern.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\mdefs.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframCompileLibrary.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframImageLibrary.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframIOLibraryFunctions.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframLibrary.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframRawArrayLibrary.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframRTL.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframSparseLibrary.h,C:\Program Files\Wolfram Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\WolframStreamsLibrary.h}

On my system, it is the 3rd entry in the preceding list:

mdefs = FileNames["*",FileNameJoin[{$InstallationDirectory,"SystemFiles","IncludeFiles","C"}]][[3]]

"C:\Program Files\Wolfram \ Research\Mathematica\11.2\SystemFiles\IncludeFiles\C\mdefs.h"

You can look at the file with:

FilePrint[mdefs]

Output:

/*************************************************************************

                        Mathematica source file

        Copyright 1986 through 1999 by Wolfram Research Inc.


*************************************************************************/

/* C language definitions for use with Mathematica output */
#define Power(x, y) (pow((double)(x), (double)(y)))
#define Sqrt(x)     (sqrt((double)(x)))

#define Abs(x)      (fabs((double)(x)))

#define Exp(x)      (exp((double)(x)))
#define Log(x)      (log((double)(x)))
#define Sin(x)      (sin((double)(x)))
#define Cos(x)      (cos((double)(x)))
#define Tan(x)      (tan((double)(x)))

#define ArcSin(x)       (asin((double)(x)))
#define ArcCos(x)       (acos((double)(x)))
#define ArcTan(x)       (atan((double)(x)))

#define Sinh(x)          (sinh((double)(x)))
#define Cosh(x)          (cosh((double)(x)))
#define Tanh(x)          (tanh((double)(x)))


#define E       2.71828182845904523536029
#define Pi      3.14159265358979323846264
#define Degree      0.01745329251994329576924

/** Could add definitions for Random(), SeedRandom(), etc. **/

These #define statements will allow you to using the direct output of CForm.