An Interval Arithmetic for Robust Error Estimation

TL;DR

This paper enhances interval arithmetic with three extensions—error intervals, movability flags, and input search—to improve robustness, validity detection, and error handling in floating-point computations, outperforming existing software in challenging scenarios.

Contribution

The paper introduces three novel extensions to interval arithmetic that address its brittleness and improve error detection, validity, and computational robustness.

Findings

Resolved 60.3% more challenging inputs

Returned 10.2x fewer indeterminate results

Avoided 64 fatal errors

Abstract

Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause domain errors. Others cause the algorithm to enter an infinite loop and fail to compute a ground truth. Plus, finding valid inputs is itself a challenge when invalid and unsamplable points make up the vast majority of the input space. These issues can make interval arithmetic brittle and temperamental. This paper introduces three extensions to interval arithmetic to address these challenges. Error intervals express rich notions of input validity and indicate whether all or some points in an interval violate implicit or explicit preconditions. Movability flags detect futile recomputations and prevent timeouts by indicating whether a higher-precision recomputation will yield a more accurate result. Andinput search restricts sampling to valid, samplable points, so they are easier to find. We compare these extensions to the state-of-the-art technical computing software Mathematica, and demonstrate that our extensions are able to resolve 60.3% more challenging inputs, return 10.2x fewer completely indeterminate results, and avoid 64 cases of fatal error.