  We describe algorithms to compute elliptic functions and their relatives
(Jacobi theta functions, modular forms, elliptic integrals, and the
arithmetic-geometric mean) numerically to arbitrary precision with rigorous
error bounds for arbitrary complex variables. Implementations in ball
arithmetic are available in the open source Arb library. We discuss the
algorithms from a concrete implementation point of view, with focus on
performance at tens to thousands of digits of precision.
