So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods:
Lagrange inversion theorem:
$$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to a}\frac{(z-\Gamma(a))^n}{\Gamma(n+1)}\frac{d^{n-1}}{dw^{n-1}}\left(\frac{w-a}{\Gamma(w)-\Gamma(a)}\right)^n$$
I also found that one could find the inverse of Stirling's approximation using the Lambert W function.
And of course, there are other ones, but which method gets the most accuracy for possibly easier calculations?