Minimax rational approximation of $W(x)/\ln(1+x)$ with polynomial degree $1$

Question

I need to compute the minimax rational approximation of $W(x)/\ln(1+x)$ on the range $(1/e,e]$, with numerator and denominator of degree not larger $1$, where $W$ is the Lambert W Function.

If this is not possible, then I don't mind extending the range $(1/e,e]$, but of course, keeping it as tight as possible for the sake of accuracy of the approximation.

Any ideas how to do this?

Thank you!

Answer 1

For $x \in (1/e,e]$

$$\frac{W(x)}{\log (x+1)}=\frac{1.14777 x+0.996697}{1.62101 x+1}$$ gives errors smaller than $0.0001$