what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000



best approximation to the LambertW(x) or exp(LambertW(x))
**Extended answer**


The approximation described below is original, explicit (in some sense), and very accurate.
It is closely related to [this question](https://mathoverflow.net/questions/45390/another-special-property-of-the-exponential-function) (second paragraph). 


So, you want to approximate the solution $W(x)$ of $x=W(x)e^{W(x)}$, for large $x$ (the order of $x$ does not play a very significant role in what follows). 
First, define
$$
\varphi (x,r) = 1 + \sum\limits\_{k = 1}^{\left\lceil r \right\rceil } {\frac{{x^k [r - (k - 1)]^k }}{{k!}}} .
$$
Now, consider the following series of approximations, where $r$ is assumed sufficiently large. The first one is
$$
\tilde W^1 (x,r) = \frac{1}{r}\ln \varphi (x,r).
$$
Subsequent approximations are defined recursively by 
$$
\tilde W^{n + 1} (x,r) = \frac{1}{r}\ln \bigg[\frac{{\tilde W^n (1 + \tilde W^n )}}{x}\varphi (x,r)\bigg].
$$


Example. For $x=2000$, even $r$ as low as $80$ gives quite accurate results:
$$
\tilde W^5 (2000,80) \approx 5.83673149492073
$$
and
$$
\tilde W^6 (2000,80) \approx 5.836731494908671,
$$
while the exact solution (according to Wims Function Calculator) is 
$$
W(2000) = 5.836731494908178747....
$$
Based on many numerical results, this approximation seems quite interesting. Here is one further example.
The [Omega constant](http://en.wikipedia.org/wiki/Omega_constant) $\Omega$ is the value of $W(1)$: 
$$
\Omega = W(1) \approx 0.5671432904097838729999686622.
$$
With $r$ as low as $30$, we already get the following impressive approximations:
$$
\tilde W^1 (1,30) \approx 0.5710729200334063, 
$$
$$
\tilde W^2 (1,30) \approx 0.5674569334624368,
$$
$$ 
\tilde W^3 (1,30) \approx 0.5671683899602143, 
$$
$$
\tilde W^4 (1,30) \approx 0.5671452994467842,
$$
$$ 
\tilde W^5 (1,30) \approx 0.5671434512213455,
$$
$$
\tilde W^6 (1,30) \approx 0.5671433032818183,
$$
$$
\tilde W^7 (1,30) \approx 0.5671432914401158,
$$
$$
\tilde W^8 (1,30) \approx 0.567143290492256,
$$
$$
\tilde W^9 (1,30) \approx 0.5671432904163853,
$$
$$
\tilde W^{10} (1,30) \approx 0.5671432904103123,
$$
$$
\tilde W^{11} (1,30) \approx 0.5671432904098261,
$$
$$
\tilde W^{12} (1,30) \approx 0.5671432904097873.
$$
So, $\tilde W^{12} (1,30) - W(1) \approx 3 \times 10^{-15}$.
It is interesting to compare this sophisticated approximation with the standard one obtained from the converging sequence $\Omega\_n \to \Omega$ defined by $\Omega\_{n+1} = e^{-\Omega\_n}$ (with initial value $\Omega\_0$). For example, with $\Omega\_0 = 0.5$, we only get 
$$
\Omega\_1 \approx 0.6065306597126334,
$$
$$
\Omega\_2 \approx 0.545239211892605,
$$
$$
\Omega\_3 \approx 0.5797030948780683,
$$
$$
\Omega\_4 \approx 0.5600646279389019,
$$
$$
\Omega\_5 \approx 0.5711721489772151,
$$
$$
\Omega\_6 \approx 0.5648629469803235,
$$
$$
\Omega\_7 \approx 0.5684380475700662,
$$
$$
\Omega\_8 \approx 0.5664094527469208,
$$
$$
\Omega\_9 \approx 0.5675596342622424,
$$
$$
\Omega\_{10} \approx 0.5669072129354714,
$$
$$
\Omega\_{11} \approx 0.5672771959707785,
$$
$$
\Omega\_{12} \approx 0.5670673518537281.
$$


