Sound Probabilistic Numerical Error Analysis

Numerical software uses floating-point arithmetic to implement real-valued algorithms which inevitably introduces roundoff errors. Additionally, in an effort to reduce energy consumption, approximate hardware introduces further errors. As errors are propagated through a computation, the result of the approximated floating-point program can be vastly different from the real-valued ideal one. Previous work on soundly bounding (roundoff) errors has focused on worst-case absolute error analysis. However, not all inputs and not all errors are equally likely such that these methods can lead to overly pessimistic error bounds.

In this paper, we present a sound probabilistic static analysis which takes into account the probability distributions of inputs and propagates roundoff and approximation errors probabilistically through the program. We observe that the computed probability distributions of errors are hard to interpret, and propose an alternative metric and computation of refined error bounds which are valid with some probability.