Let's assume we're talking about real functions purely for clarity purposes, but my question is more general.
We say that a function $f$ is computable, if for a given $x$, a turing machine can output $y$ such that $f(x)=y$.
However, I'm wondering whether there is some concept of "computability for the expresssion of a function". For example, let's say I have some implicitly defined function $f$, and let's say that this function is equivalent to $g(x)=\sin(x)$, but we don't know this yet. Then what we would want is to compute the fact that $f(x)=\sin (x)$, but we don't necessarily need to compute that $f(0.45)=0.434965534...$
Is there a concept that rigorously captures this idea of the expression of a function being computable, rather than the function itself being computable? In other words, it may be that a function is incomputible, but that there is at least a finite number of computations we can do to derive that it has a certain closed form expression.