It seems Newton's method can be used to calculate the value we need for the lambda function. What I call "Za". It seems to be much faster.
https://math.stackexchange.com/a/4503144/1082376
I think that value is a root of the nth composition that is the computation of n iteration of z²+c.
This could be an alternative implementation. We could keep both for the record or just remove the old one if this is better.