TITLE: Lambert function approximation $W_0$ branch
QUESTION [8 upvotes]: I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).

REPLY [2 votes]: For the range $-\frac 1e \leq x \leq 0$, using $y=\sqrt{2(1+ex)}$ (as in this post) and building Padé approximants $P_n$ around $y=0$, what is obtained is 
$$P_1=\frac{-1+\frac{2 y}{3} } {1+\frac{y}{3} }$$
$$P_2=\frac{-1+\frac{14 y}{45}+\frac{301 y^2}{1080} } {1+\frac{31 y}{45}+\frac{83 y^2}{1080} }$$
$$P_3=\frac{-1-\frac{974 y}{22659}+\frac{21865 y^2}{51792}+\frac{89131
   y^3}{1165320} } {1+\frac{23633 y}{22659}+\frac{104225 y^2}{362544}+\frac{3167
   y^3}{196560} }$$
$$P_4=\frac{-1-\frac{11637254 y}{29330279}+\frac{463636649 y^2}{1055890044}+\frac{23930361857
   y^3}{110868454620}+\frac{192684057311 y^4}{10643371643520} } {1+\frac{40967533 y}{29330279}+\frac{659231191 y^2}{1055890044}+\frac{1928737771
   y^3}{20157900840}+\frac{34384971553 y^4}{10643371643520} }$$
 The maximum errors occur for $x=0$; for the above formulae, they are $\Delta_1=-3.88\times 10^{-2}$, $\Delta_2=-1.23\times 10^{-3}$, $\Delta_3=-3.73\times 10^{-5}$ and $\Delta_4=-1.12\times 10^{-6}$.
Concerning $$S_i=\int_{-\frac 1e}^0 \left(W(x)-P_i\right)^2\,dx$$ they are $S_1=3.93\times 10^{-4}$, $S_2=2.57\times 10^{-7}$, $S_3=1.76\times 10^{-10}$ and $S_4=1.26\times 10^{-13}$.
Edit
For a "sanity" check, I also considered all the possible $[k,6-k]$ Padé approximants and obtained the following results
$$\left(
\begin{array}{ccc}
 k & \Delta_k & S_k \\
 0 & -0.108806 & 1.84 \times 10^{-3} \\
 1 & -0.007522 & 6.16 \times 10^{-6} \\
 2 & -0.000171 & 3.49 \times 10^{-9}\\
\color{red} { 3} & \color{red} {-0.000037} & \color{red} {1.76 \times 10^{-10}}\\
 4 & -0.000082 & 8.17 \times 10^{-10}\\
 5 & -0.000966 & 1.05 \times 10^{-7}\\
 6 & -0.094985 & 9.04 \times 10^{-4}
\end{array}
\right)$$