  Jacobi's $\theta$ function has numerous applications in mathematics and
computer science; a naive algorithm allows the computation of $\theta(z,\tau)$,
for $z, \tau$ verifying certain conditions, with precision $P$ in
$O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the
number of operations needed to multiply two complex $P$-bit numbers. We
generalize an algorithm which computes specific values of the $\theta$ function
(the \textit{theta-constants}) in asymptotically faster time; this gives us an
algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P)
\log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using
the quasi-periodicity of $\theta$.
