A few days ago, I found this weird inverse factorial approximation for complex numbers mostly between 1 and 24. The approximation is $\sqrt{5\ln\left(x+\frac{1}{5}\right)}$.
Here are a few values with this function
$f\left(1\right)=0.9547815373$, actual value = 1
$f\left(2\right)=1.98551927763$, actual value = 2
$f\left(3\right)=2.41158745415$, actual value ≈ 2.40586998631
$f\left(6\right)=3.02038846181$, actual value = 3
$f\left(12\right)=3.53654913138$, actual value ≈ 3.52239790956
$f\left(24\right)=3.99146128201$, actual value = 4
The approximation later diverges, but would you know how this function might be derived? And if it can't, could someone explain what this has to do with factorials?