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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?

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    $\begingroup$ Do you know anything more about F and G? E.g. are they both strictly increasing? $\endgroup$ – Peter Taylor Aug 10 at 15:07
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    $\begingroup$ Without more info, this problem is not answerable. In particular, if $g(x)=0$, then the problem becomes degenerate. $\endgroup$ – Don Thousand Aug 10 at 15:10
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    $\begingroup$ Oh yeah they are both increasing. thx $\endgroup$ – FFjet Aug 10 at 15:10
  • $\begingroup$ why $F(x)>x-1$ ? $\endgroup$ – G Cab Aug 10 at 15:19
  • $\begingroup$ well I have just modified the conditions $\endgroup$ – FFjet Aug 10 at 15:23

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