  In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form
whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and
Stark-Heegner points. Here we describe how this construction gives rise to a
practical algorithm for explicitly computing these logarithms to specified
precision, and how to recover the exact values of the Gross-Stark units and
Stark-Heegner points from them. Key tools are overconvergent modular forms,
reduction theory of quadratic forms and Newton polygons. As an application, we
tabulate Brumer-Stark units in narrow Hilbert class fields of real quadratic
fields with discriminants up to $10000$, for primes less than $20$, as well as
Stark-Heegner points on elliptic curves.
