I think that there is no formula. The best one can do is to estimate. Here is a simpler problem of the same sort: suppose you have a parametrization of the boundary of a simply connected region, and
suppose that 0 is inside. Consider the Riemann mapping f of this region sending 0 to 0.
The problem is to find |f'(0)|. There is no formula in any reasonable case.

Of course this is not a theorem, because one cannot define what a "formula" is.
Bothe quantities, the modulus of a ring, and |f'(0)| in the simplified problem
are solutions of certain extremal problems. So one can write a "formula" involving sup
over some class of functions.