A real number is said to be effective if there exists an algorithm which, given a required tolerance , returns a binary approximation for 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.

Keywords: effective real number, algorithm, interval analysis, error estimates

A.M.S. subject classification: 68W25, 65G20, 65G40, 26E40

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