In the following equation $$10^{10^z} = 10^{10^x}+10^{10^y}$$ I want to find an algorithm that computes $z$ in a floating point accurate manner given any values of $x$ and $y$ (e.g. $x=y=2000$). The algorithm should thus avoid evaluating $10^x$ or $10^y$ directly otherwise a memory usage error happens. What would be a good method?
I tried looking through the literature but was unable to find anything relevant to this double exponentiation problem.