Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

TL;DR

This paper develops a rigorous probabilistic model for floating-point roundoff errors, providing tight bounds and dependencies analysis, implemented in the PAF tool and evaluated on standard benchmarks.

Contribution

It introduces a novel probabilistic error analysis model for floating-point computations, with an algorithm that tracks dependencies and provides tight bounds using symbolic affine arithmetic.

Findings

PAF computes tighter bounds than existing methods on most benchmarks.

The model accurately captures the distribution of roundoff errors in probabilistic computations.

The approach effectively manages complex dependencies with an SMT solver.

Abstract

We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are generally close to being uncorrelated with their generating distribution. Based on these theoretical advances, we propose a model of IEEE floating-point arithmetic for numerical expressions with probabilistic inputs and an algorithm for evaluating this model. Our algorithm provides rigorous bounds to the output and error distributions of arithmetic expressions over random variables, evaluated in the presence of roundoff errors. It keeps track of complex dependencies between random variables using an SMT solver, and is capable of providing sound but tight probabilistic bounds to roundoff errors using symbolic affine arithmetic. We implemented the algorithm in the PAF tool, and evaluated it on FPBench, a standard benchmark suite for the analysis of roundoff errors. Our evaluation shows that PAF computes tighter bounds than current state-of-the-art on almost all benchmarks.