I was working on a problem about carnot heat pump and came across this equation $y=-W(-e^{-x-1})$ and when i plotted it, it seems to me that for large value of x it started becoming linear so i was wondering if there is a way I could approximate this for large value of x.dotted line represents $y=x+5$ and the black line represent $y=-W(-e^{-x-1})$
I've tried some approximation already. I tried using the lagrange inversion theorem to expand the Lambert's function and after some approximation i ended up with $y=e^{-x-1}(e^{-x-1}+1)$, then after i've plotted it (dotted line is after the approximation) it seems like it's working for only small values of y.
Chatzigeorgiou proved bounds on the exact expression you have in 2013: for $x>0$ $$1+\sqrt{2x}+\frac23x<-W_{-1}(-e^{-x-1})<1+\sqrt{2x}+x$$ Tighter upper bounds are possible if $x$ is known to be bounded; if $x<1$, $\frac34x$ can replace $x$ in the upper bound.
Another approximation is given in the first section of the paper. Let $a=0.3205$, then $$-W_{-1}(-e^{-x-1})\approx1+x+\frac2a\left(1-\frac1{1+a\sqrt{x/2}}\right)$$
Ioannis Chatzigeorgiou (2013), "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation", IEEE Communications Letters 17 (8), pp. 1505–1508