Confidence Intervals for Stochastic Arithmetic

TL;DR

This paper introduces a comprehensive statistical framework for analyzing stochastic arithmetic methods used in floating-point error estimation, unifying existing approaches and providing confidence intervals, demonstrated on industrial codes.

Contribution

It unifies existing definitions of significant digits in stochastic arithmetic and introduces a new measure for accuracy, with validated confidence intervals for various distributions.

Findings

Unified framework for stochastic arithmetic analysis

New measure for digits contributing to accuracy

Validated confidence intervals on industrial codes

Abstract

Quantifying errors and losses due to the use of Floating-Point (FP) calculations in industrial scientific computing codes is an important part of the Verification, Validation and Uncertainty Quantification (VVUQ) process. Stochastic Arithmetic is one way to model and estimate FP losses of accuracy, which scales well to large, industrial codes. It exists in different flavors, such as CESTAC or MCA, implemented in various tools such as CADNA, Verificarlo or Verrou. These methodologies and tools are based on the idea that FP losses of accuracy can be modeled via randomness. Therefore, they share the same need to perform a statistical analysis of programs results in order to estimate the significance of the results. In this paper, we propose a framework to perform a solid statistical analysis of Stochastic Arithmetic. This framework unifies all existing definitions of the number of significant digits (CESTAC and MCA), and also proposes a new quantity of interest: the number of digits contributing to the accuracy of the results. Sound confidence intervals are provided for all estimators, both in the case of normally distributed results, and in the general case. The use of this framework is demonstrated by two case studies of large, industrial codes: Europlexus and code_aster.