In a setting where we have intervals for the values of floating-point variables x, a, and b, we are interested in improving these intervals when the floating-point equality x ⊕ a = b holds. This problem is common in constraint propagation, and called the inverse projection of the addition. It also appears in abstract interpretation for the analysis of programs containing IEEE 754 operations. We propose floating-point theorems that provide optimal bounds for all the intervals. Fast loop-free algorithms compute these optimal bounds using only floating-point computations at the target precision.
Digital lters are small iterative algorithms, used as basic bricks in signal processing (lters) and control theory (controllers). They receive as input a stream of values, and output another stream of values, computed from their internal states and from the previous inputs. These systems can be found in communication, aeronautics, automotive, robotics, etc. As the application domain may be critical, we aim at providing a formal guarantee of the good behavior of these algorithms. In particular, we formally proved in Coq some error analysis theorems about digital lters, namely the Worst-Case Peak Gain theorem and the existence of a lter characterizing the dierence between the exact lter and the implemented one. Moreover, the digital signal processing literature provides us with many equivalent algorithms, called realizations. We formally dened and proved the equivalence of several realizations (Direct Forms and State-Space).
Dot products (also called sums of products) are ubiquitous in matrix computations, for instance in signal processing. We are especially interested in digital filters, where they are the core operation. We therefore focus on fixed-point arithmetic, used in embedded systems for time and energy efficiency. Common dot product algorithms ensure faithful rounding. For the sake of accuracy and reproducibility, we want to ensure correct rounding. This article describes an algorithm that computes a correctly-rounded sum of products from inputs whose format is known in advance. This algorithm relies on odd rounding (that is easily implemented in hardware) and comes with a careful proof and some cost analysis.
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