I have been interested in the Lambert W function recently and decided to work on some approximations.

https://www.desmos.com/calculator/qff548lg4g

The first line is my approximation using Newton's method. If you turn on the second line, you can compare it to the actual Lambert W function for the principal branch, though it only works for greater than -1/e.

Also, since my approximation uses Newton's method and is recursive, I made one that works with tickers instead, though rather than being able to see the shape of the function, it finds a precise value at a specific point. Just make "a" the input, start and promptly stop the ticker, and g should've become a pretty close value for W(a).

https://www.desmos.com/calculator/edycnr8l7r

I have no idea what the other branches of this function mean, so if anyone thinks they can explain it to me, I would greatly appreciate it.