There's two functions, called $F(x)$ and $G(x)$, where $F(x)>x,1<G(x) < x$ ,$F'(x)>0, G'(x)>0$ on $(2,\infty)$, and $F(x),G(x)\in(2,\infty)$. Given $x, y\in (2,\infty)$, and now I want to find the best way to approximate $y$ in finite steps, that is, $F(G(G(F(x)...)))\sim y$.
For example, given $F(x)=\Gamma(x+1),G(x)=\sqrt{x}, x=8, y = 2.2$, and set the maximum steps to $3$. It can be proved that the best approximation is $$ \sqrt{\Gamma \left(\sqrt{8}+1\right)}=2.20394... $$
It could be easily done with BFS, however I think there should be a faster way to give the result.
Meanwhile, I haven't found any results about the best approximation (e.g. the estimation of the error of the best approximation)
Does anyone have ideas?