Computations with effective real numbers

A real number x is said to be effective if there exists an algorithm which, given a required tolerance ɛ∈Z2Z, returns a binary approximation for x with . Effective real numbers are interesting in areas of numerical analysis where numerical instability is a major problem.One key problem with effective real numbers is to perform intermediate computations at the smallest precision which is sufficient to guarantee an exact end-result. In this paper we first review two classical techniques to achieve this: a priori error estimates and interval analysis. We next present two new techniques: “relaxed evaluations” reduce the amount of re-evaluations at larger precisions and “balanced error estimates” automatically provide good tolerances for intermediate computations.