Analytical approximations for real values of the Lambert W- function

The Lambert W is a transcendental function defined by solutions of the equation W exp(W) = x. For real values of the argument, x, the W-function has two branches, W-0 (the principal branch) and W-1 (the negative branch). A survey of the literature reveals that, in the case of the principal branch (W-0), the vast majority of W-function applications use, at any given time, only a portion of the branch viz. the parts defined by the ranges -1 less than or equal to W-0 less than or equal to 0 and 0 less than or equal to W-0. Approximations are presented for each portion of W-0, and for W-1. It is shown that the present approximations are very accurate with relative errors down to around 0.02% or smaller. The approximations can be used directly, or as starting values for iterative improvement schemes.