Finite-precision floating point arithmetic unavoidably introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be extremely rare events in practice. Here we develop a probabilistic model of rounding errors with which it becomes possible to estimate the likelihood that the rounding error of an algorithm lies within a given interval. Given an input distribution, we show how to compute the distribution of rounding errors. We do this exactly for low precision arithmetic, for high precision arithmetic we derive a simple approximation. The model is then entirely compositional: given a numerical program written in a simple imperative programming language we can recursively compute the distribution of rounding errors at each step of the computation and propagate it through each program instruction. This is done by applying a formalism originally developed by Kozen to formalize the semantics of probabilistic programs. We then discuss an implementation of the model and use it to perform probabilistic range analyses on some benchmarks. arXiv:1912.00867v2 [math.NA] 10 Dec 2019 Assumption 0. The probability density function is constant at the scale of the intervals between floating point numbers, i.e.for all values of t such that Eq. (8) holds. Assumption 1. Writing z(s, k) for z(e min , s, k) assume that: 0≤k<2 p s∈{0,1}

No abstract

No 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.

No abstract