  Finite precision approximations of discrete probability distributions are
considered, applicable for distribution synthesis, e.g., probabilistic shaping.
Two algorithms are presented that find the optimal $M$-type approximation $Q$
of a distribution $P$ in terms of the variational distance $| Q-P|_1$ and the
informational divergence $\mathbb{D}(Q| P)$. Bounds on the approximation errors
are derived and shown to be asymptotically tight. Several examples illustrate
that the variational distance optimal approximation can be quite different from
the informational divergence optimal approximation.
