Invert this function $y=(1-x)e^{-x}$

Question

Consider the fucntion $f:\mathcal{R}\rightarrow \mathcal{R}$ given by the rule

$ f(x)=(1-x)e^{-x} $

Now I want to invert this function(not just for fun but I have a data that seems to fit this form). I could see that $x$ can' be isolated. Taking the log on both sides doesn't help and I have tried other possibilities. So I tried the following. Using Taylor expansion $e^{-x}=1-x-\frac{x^{2}}{2}+...$ Now $ f(x)=(1-x)(1-x-\frac{x^{2}}{2})=-\frac{1}{2}x^{3}+\frac{3}{2}x^{2}-2x+1 $

Now I could solve the cubic equation and solve for $x$. But this looks ugly and I don't know how good of an approximation it is. I was looking for possible suggestions to glean some information or write approximately a reasonable function for $x$ in terms of $y$ . Thank you

Answer 1

According to WolframAlpha, the answer is $1-W\left(ey\right)$, where $W$ is the (main branch of the) Lambert W-function, which is defined as the inverse of the function $y =x e^x$. So I suspect there is no simple expression for the function you're after.

Answer 2

Here is mine ... $$ y = (1-x)\exp(-x) \\ \mathrm{e}y=(1-x)\exp(1-x) \\ W(\mathrm{e}y) = 1-x \\ x = 1-W(\mathrm{e}y) $$

Maybe Arthur typed something wrong.