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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