Formalization of Double-Word Arithmetic, and Comments on “Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic”

Recently, a complete set of algorithms for manipulating double-word numbers (some classical, some new) was analyzed [16]. We have formally proven all the theorems given in that article, using the Coq proof assistant. The formal proof work led us to: (i) locate mistakes in some of the original paper proofs (mistakes that, however, do not hinder the validity of the algorithms), (ii) significantly improve some error bounds, and (iii) generalize some results by showing that they are still valid if we slightly change the rounding mode. The consequence is that the algorithms presented in [16] can be used with high confidence, and that some of them are even more accurate than what was believed before. This illustrates what formal proof can bring to computer arithmetic: beyond mere (yet extremely useful) verification, correction, and consolidation of already known results, it can help to find new properties. All our formal proofs are freely available.