I have a function defined by a combination of basic math functions (abs, cosh, sinh, exp, ...).
I was wondering if it makes a difference (in speed) to use, for example,
numpy.abs() instead of abs()?
You should use numpy function to deal with numpy's types and use regular python function to deal with regular python types.
Worst performance usually occurs when mixing python builtins with numpy, because of types conversion. Those type conversion have been optimized lately, but it's still often better to not use them. Of course, your mileage may vary, so use profiling tools to figure out.
Also consider the use of programs like cython or making a C module if you want to optimize further your program. Or consider not to use python when performances matters.
but, when your data has been put into a numpy array, then numpy can be really fast at computing bunch of data.
Are NumPy's math functions faster than Python's?