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.

We propose an optimization for space-efficient implementations of explicit model-predictive controllers (MPC) for robust control of linear time-invariant (LTI) systems on embedded platforms. We obtain an explicit-form robust model-predictive controller as a solution to a multi-parametric linear programming problem. The structure of the controller is a polyhedral decomposition of the control domain, with an affine map for each domain. While explicit MPC is suited for embedded devices with low computational power, the memory requirements for such controllers can be high. We provide an optimization algorithm for a mixed-precision implementation of the controller, where the deviation of the implemented controller from the original one is within the robustness margin of the robust control problem. The core of the mixed-precision optimization is an iterative static analysis that co-designs a robust controller and a low-bitwidth approximation that is statically guaranteed to always be within the robustness margin of the original controller. We have implemented our algorithm and show on a set of benchmarks that our optimization can reduce space requirements by up to 20.9% and on average by 12.6% compared to a minimal uniform precision implementation of the original controller.