In the case of Mandelbrot, there is a self similar renormalization process, so you can obtain such a formula. For the "fixed points" of the renormalization proc

In the case of Mandelbrot, there is a self similar renormalization process, so you can obtain such a formula. For the "fixed points" of the renormalization process, the formula is super simple; for other points, you might need more computations, but it's nevertheless an efficient method. There is a paper of Bartholdi where he explains this in terms of automata.