Numerical solution w/o derivative

I know the function y = f(x).

I have observed the true y-value and want to find the x that it implies.

It is impossible to invert the function, i.e. to analytically solve for x=g(y).

The derivative of the function is unavailable, so I cannot use the Newton-Raphson method.

It isn't difficult to come up with a way of trial and error which work. But I want to check with you if there is some intelligent way to deal with such iterations in order to make the procedure reasonably efficient?

While it is *possible* to formulate the derivative of y=f(x), it is *impractical *to use it because it is a formula which fills an entire page! (According to Mathematica) I need to implement a way to solve this problem in c++.