We consider the fundamental derandomization problem of deterministically finding a satisfying assignment to a CNF formula that has many satisfying assignments. We give a deterministic algorithm which, given an n-variable poly(n)-clause CNF formula F that has $\mid F^{-1}\left(1\right)\mid \ge \epsilon 2^n$, runs in time $n^{\tilde{O}(\log \log n{)}^2}$ for $\epsilon \ge 1/\backslash polylog\left(n\right)$ and outputs a satisfying assignment of F. Prior to our work the fastest known algorithm for this problem was simply to enumerate over all seeds of a pseudorandom generator for CNFs; using the best known PRGs for CNFs [DETT10], this takes time $n^{\tilde{\Omega }\left(\log n\right)}$ even for constant $\epsilon$. Joint work with Rocco Servedio.