Well, the paper explicitly takes the principal branch to solve this.

So it isn't exploiting the branching for computation.

I agree, as the sibling comment there are two different things that are named "branches". Anyway, to get the principal branch in the microprocessor it's necessary to implement "atan2" that has a lot of special cases.

For example, IIRC ln( -inf.0 + y * i ) = ´+inf.0 + pi * sign(y)

Different sense of “branching”

Yep.