  In 1985, van den Dries showed that the theory of the reals with a predicate
for the integer powers of two admits quantifier elimination in an expanded
language, and is hence decidable. He gave a model-theoretic argument, which
provides no apparent bounds on the complexity of a decision procedure. We
provide a syntactic argument that yields a procedure that is primitive
recursive, although not elementary. In particular, we show that it is possible
to eliminate a single block of existential quantifiers in time $2^0_{O(n)}$,
where $n$ is the length of the input formula and $2_k^x$ denotes $k$-fold
iterated exponentiation.
