lambertw, all branches

[Gil]

We accept...
You decided to accept that the real part in question is indeed zero.
But according to your other examples, for much more tinier results in the real part output you did not round — and did correctly — that real part to zero.
The rules should be set up before the "game" starts.
I understand that tiny is tiny, but sometimes we round it to zero, sometimes not, after checking the results. In a way, it's not very fair.

But I understand that, in that case, we cannot do much better.
The exception being for the previous case W0/W-1 (-pi/2), where we can "cheat" in advance in the program — if we want to, of course — fixing up the issue for a<=0, and get the the correct answer, as I developed in a previous post.