Computability of an expression of a function rather than a function itself? 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.
I assume that a closed form of a function is a finite expression involving function symbols from some fixed natural finite list together with some real[1] parameters. Since we know what it means to compute a real number, and since we know what it means to compute finite tuples of stuff we know how to compute with, we obtain in a direct way a notion of what it means to compute a closed form of a function.

Then in particular we can ask questions such as "given these parameters for the differential equation, can we compute a closed form of the solution".

Note however that all usual continuous functions are computable. Given a closed form of a function $f$ and some input $x \in \mathbb{R}$, we can compute $f(x)$. However, the problem "given a continuous function that has a closed form, find that closed form" is not computable. 

So asking for closed forms is asking for more, not asking for less.

[1] If you don't like computing with real numbers, go for parameters from the algebraic closure of $\mathbb{Q}[\pi,e]$ instead.