(28/48/50) Lambert W Function

[Gil]

What hawk eyes, Albert! Not only your brain works marvels, Albert, but your eyes too.
Sorry for having mixed up the post number!

More seriously: for k=-1, and a=-.36787944118 (>-1/e, about -.367879441171), how do I know that formulae post 19, more accurate, will fail (at least with the code I put in my calculator) and that formulae post 18, a wee bit less accurate, will work?

In other words: should I say, for "a" near -1/e, but just smaller — as in the example above —, I should always play safety and only use formulae in post 18?

Second question
k mappin — new k = ± k.
Does it mean that the derived formulae in posts 18 and 19 only work for k, initially = 0 or 1?

If yes, are there other pitfalls to look at for initial k≠{0, +-1} when "a" is near 0 or -1/e?

Regards