We assume that a real square-free polynomial $A$ has a degree $d$, a maximum coefficient bitsize $\tau$ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of $t=2^{-L}$. The algorithm has Boolean complexity $O_B~(d^2 \tau + d L )$. This substantially decreases the known bound $O_B~(d^3 +d^2L)$ and is optimal up to a polylogarithmic factor. Furthermore we readily extend our algorithm to support the same upper bound on the complexity of the refinement of $r$ real roots, for any $r\le d$, by incorporating the known efficient algorithms for multipoint polynomial evaluation. The main ingredient for the latter is an efficient algorithm for (approximate) polynomial division; we present a variation based on structured matrix computation with quasi-optimal Boolean complexity.