, Volume 19, Issue 1, pp 7-30

Semantics of roundoff error propagation in finite precision calculations

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Abstract

We introduce a concrete semantics for floating-point operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it.

In our model, every elementary operation introduces a new first order error term, which is later propagated and combined with other error terms, yielding higher order error terms. The semantics is parameterized by the maximal order of error to be examined and verifies whether higher order errors actually are negligible. We consider also coarser semantics computing the contribution, to the final error, of the errors due to some intermediate computations. As a result, we obtain a family of semantics and we show that the less precise ones are abstractions of the more precise ones.