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The problem of locating and computing with certainty all the simple roots of a twice continuously differentiable function $f\colon [a,b] \subset $ ${\R } \to $ ${\R }$ is studied when some additional information on the distribution of the roots in the interval is available. The framework is the one proposed by [SIAM J. Sci. Comput., 17 (1996), pp. 1232--1248], where only the uniform case was examined. This paper settles some of the problems posed there and generalizes some of its results by considering an arbitrary distribution of the roots in [a,b]. The theoretical results are accompanied by simulations in a number of problems of various size.