Compiling a function depending on an external function

Question

Consider the simple case:

g[x] := x^3;
f[x] := x^2 + g[x];
cf = Compile[{{x, _Real}}, x^2 + g[x]]

Evaluating the two functions, Mathematica complains that the second depends on the function g[x], saying:

CompiledFunction::cfex: Could not complete external evaluation at instruction 2; proceeding with uncompiled evaluation.

Which is the correct way to compile the function f in this case?

Answer 1

Long time ago I wrote a macro, which expands global DownValues - based definitions, called withGlobalFunctions. It can be found at the end of this post. With it, all you need to do is wrap the Compile call like this:

g[x] := x^3;
f[x] := x^2 + g[x];
cf = withGlobalFunctions @ Compile[{{x, _Real}}, f[x]]

This has the advantage over Evaluate advice in that you can't leak a global value for x in, even if it exists. And it has an advantage over "InlineExternalDefinitions" -> True advice in that it expands arbitrary long chains of calls.

The limitation of this approach is that patterns in function definitions you may want to inline / expand in this way, better be very simple, involving blanks but not much else. This is because what this does is a kind of a macro-expansion, without actual evaluation involved. So that expansion will get stuck if patterns do any non-trivial checks.

withGlobalFunctions can trivially be extended to expand definitions based on other ...Values. As written, it only expands definitions from Global` context, but that restriction can be removed or lifted as well.