Automated Roundoff Error Analysis of Probabilistic Floating-Point Computations

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 results, 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 on 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 on roundoff errors using symbolic affine arithmetic. We implement the algorithm in the PAF tool, and evaluate it on FPBench, a standard benchmark suite for the analysis of roundoff errors in small kernels. Our evaluation shows that PAF computes tighter bounds than the current state of the art on almost all benchmarks.