Lambert function approximation $W_0$ branch I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).

Here is a post that I made on sci.math a while ago, regarding a method I have used.

Analysis of $we^w$
For $w\gt0$, $we^w$ increases monotonically from $0$ to $\infty$. When $w\lt0$, $we^w$ is negative.
Thus, for $x\gt0$, $\mathrm{W}(x)$ is positive and well-defined and increases monotonically.
For $w\lt0$, $we^w$ reaches a minimum of $-1/e$ at $w=-1$.  On $(-1,0)$, $w e^w$
  increases monotonically from $-1/e$ to $0$.  On $(-\infty,-1)$, $w e^w$ decreases
  monotonically from $0$ to $-1/e$.  Thus, on $(-1/e,0)$, $\mathrm{W}(x)$ can have one of two values, one in $(-1,0)$ and another in $(-\infty,-1)$.  The value in $(-1,0)$ is called the principal value.
The iteration
Using Newton's method to solve $we^w$ yields the following iterative
  step for finding $\mathrm{W}(x)$:
  $$
w_{\text{new}}=\frac{xe^{-w}+w^2}{w+1}
$$
Initial values of $w$
For the principal value, when $-1/e\le x\lt0$, and when $0\le x\le10$, use
  $w=0$.  When $x\gt10$, use $w=\log(x)-\log(\log(x))$.
For the non-principal value, when $x$ is in $[-1/e,-.1]$, use $w=-2$; and
  if $x$ is in $(-.1,0)$, use $w=\log(-x)-\log(-\log(-x))$.

This says that for the branch you want, use the iteration with an initial value of $w=0$.
