Want to point out that there are far superior methods:

http://math.eretrandre.org/tetrationforu...70#pid7470

http://math.eretrandre.org/tetrationforu...12#pid7512

and below.

This code calculates arbitrary branches of the Lambert W function for arbitrary inputs and is extremely fast (I can get thousands of digits of precision in a fraction of a second on my machine) by using Newton's method.

http://math.eretrandre.org/tetrationforu...70#pid7470

http://math.eretrandre.org/tetrationforu...12#pid7512

and below.

This code calculates arbitrary branches of the Lambert W function for arbitrary inputs and is extremely fast (I can get thousands of digits of precision in a fraction of a second on my machine) by using Newton's method.