: We present efficient representations and new algorithms for evaluating the sign of an algebraic expression and computing the roots of a polynomial equation. We use an extended precision floating-point representation based on automatic forward error propagation. Based on this representation, we present a new root finding algorithm that accurately computes all the roots of a univariate polynomial. These algorithms make no assumption related to the size of algebraic predicates or the size of their coefficients. They have been implemented as part of a library, called PRECISE, and have been used for several applications: accurate root-finding for high degree polynomials, root isolation using Sturm sequences, computing the arrangement of algebraic curves, and in sign of determinant computations. 1 Introduction Many geometric algorithms make decisions based on signs of geometric predicates. In most cases, these predicates correspond to algebraic functions of input parameters. It is import..