Propagation of Roundoff Errors in Finite Precision Computations: A Semantics Approach

Abstract

We introduce a concrete semantics for floating-point operations which describes the propagation of roundoff errors throughout a computation. This semantics is used to assert the correctness of an abstract interpretation which can be straightforwardly derived from it. In our model, every elementary operation introduces a new first order error term, which is later 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.

Dynamic floating-point cancellation detection

“…More recently, static program analysis has provided another way to conduct error analysis [12,6,13,14]. This approach characterizes the error of mathematical operations using a set of static inference rules, allowing a compile-time analysis to determine the worst-case precision of a final result.…”

Static Analysis of Finite Precision Computations

“…The underlying idea of the concrete model, first sketched in [11], then further described in more details in [18], and meanwhile implemented in a first version of the Fluctuat static analyzer [12], is to describe the difference of behaviour between the execution of a program in real numbers and in floating-point numbers. For that, the concrete value of a program variable is a triplet (f x , r x , e x ), where f x ∈ F is the value of the variable if the program is executed with a finite-precision semantics, r…”

“…Another idea of [11,18,12], also developed here, is that it could be of interest for a static analysis to decompose the error term e x along its provenance in the source code of the analyzed program, in order to point out the main sources of numerical discrepancy. For that, depending on the level of detail required, control points, blocks, or functions of a program can be annotated by a label , which will be used to identify the errors introduced during a computation.…”

“…A word of length n thus identifies an order n error, which originates from several points in the program, and as for labels, we can choose different levels of abstraction of multiple sources of errors. We refer to [18] for details about this. Our concrete model also models integers.…”

Asserting the Precision of Floating-Point Computations: A Simple Abstract Interpreter

“…Although this approximation is accurate enough for most applications, there are some cases where results become irrelevant because of the precision lost at some stages of the computation, even when the underlying numerical scheme is stable. In this paper, we present a tool for studying the propagation of rounding errors in floating-point computations, that carries out some ideas proposed in [3], [7]. Its aim is to detect automatically a possible catastrophic loss of precision, and its source.…”