I figured out with some effort and several tries that the function
$$-0.0726\arctan(1.1x)+0.6317\ln(1.7x+1)$$
is a very good approximation of the Lambert-W function in the interval $[0,3]$ ( The error is less than $0.0005$ )
How can I get such approximations systematically ?
I know that there are many known methods to approximate a function, but to get a small maximal error (lets say, less then $0.001$) in a relatively long interval (lets say, length $3$ to $10$) , most methods need many terms and the expression gets long and sometimes complicated.
I am interested in a short expression like above with two or three summands containing an elementary function. I would also accept $log(log(...))$ or $(sin(cos...))$-terms, but not more nested terms. Any ideas ?
Remark : The argument within the elementary function should be a polynomial with low degree (let's say, at most $2$).
This is not an answer but it is too long for a comment.
I do not know how you arrived to such a model, but it is really good for the range you considered. In fact, starting with the same, I obtained something a little better with regard to the distribution of the residuals $$0.636428 \log (1.66855 x+1)-0.0729183 \tan ^{-1}(1.01088 x)$$ which has been tuned ($3001$ equally spaced data points). Your model gives a sum of squares equal to $1.38\times 10^{-4}$ while to the above corresponds a sum of squares equal to $6.83\times 10^{-5}$.
Now, the problem of approximation of functions is huge and I am afraid that a systematic approach could be difficult almost if you need to tune many parameters ($4$ in this case).
In any manner, be sure that the problem is very interesting to me and that I shall follow your post.