  Let $\RR_S$ denote the expansion of the real ordered field by a family of
real-valued functions $S$, where each function in $S$ is defined on a compact
box and is a member of some quasianalytic class which is closed under the
operations of function composition, division by variables, and extraction of
implicitly defined functions. It is shown that if the family $S$ is generic
(which is a certain technically defined transcendence condition), then the
theory of $\RR_S$ is decidable if and only if $S$ is computably $C^\infty$
(which means that all the partial derivatives of the functions in $S$ may be
effectively approximated). It is also shown that, in a certain topological
sense, many generic, computably $C^\infty$ families $S$ exist.
