We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to
an additive error e.

For k-DNF, a multiplicative approximation is also achievable in
time polynomial in 1/e and linear in the size of f.

Previous algorithms achieved polynomial (but not linear)
dependency on the size of f and on 1/e; their dependency on k,
however, was much better than ours. Unlike previous algorithms,
our algorithm is not based on derandomization techniques, and it
is quite similar to an algorithm by Hirsch for the related problem
of solving k-SAT under the promise that an e-fraction of the
assignments are satisfying. Our analysis is different from (and
somewhat simpler than) Hirsch's.