Algorithm for exploring a parameter space
Problem statement: You have a function $f:\mathbb{R}^2 \to \mathbb{R}^2$ where you know how to compute $f$ but not $f'$.  Given $(x,y)$, you want to find $(\alpha,\beta)$ such that $f(\alpha,\beta)=(x,y)$.

One approach is to use mathematical techniques for black-box optimization.  Define $g:\mathbb{R}^2 \to \mathbb{R}$ by

$$g(\alpha,\beta) = \|f(\alpha,\beta)-(x,y)\|^2,$$

i.e., the squared L2-norm of how far off the answer is.  Then your goal is to find $(\alpha,\beta)$ that minimizes $g$. Such a minimum will be a solution to your original problem.

The minimization problem could be solved using gradient descent, Newton's method, or some other iterative method.

The iterative methods will often require you to be able to compute the gradient of this function.  The gradient can be computed in a black-box way by using

$${\partial g \over \partial \alpha}(\alpha,\beta) = {g(\alpha+\epsilon,\beta) - g(\alpha-\epsilon,\beta) \over 2\epsilon}$$

where $\epsilon$ is a sufficiently small number, and similarly for ${\partial g \over \partial \beta}$.

This method assumes that $f$ is a "nice" function: e.g., continuous and differential and doesn't change too quickly, if you make small changes to its inputs.