We give a deterministic white-box algorithm to estimate the expectation of a read-once branching program of length $n$ and width $w$ in space
$$\tilde{O}\left(\log n+\sqrt{\log n}\cdot\log w\right).$$
In particular, we obtain an almost optimal space $\tilde{O}(\log n)$ derandomization of programs up to width $w=2^{\sqrt{\log n}}$.
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