Type classes for efficient exact real arithmetic in Coq

TL;DR

This paper presents an extended Coq library for exact real arithmetic using type classes, enabling efficient computation of functions like sine and cosine, with significant speed improvements and broader applicability to undecidable structures.

Contribution

The authors extend their previous Coq implementation by adding new functions, alternative dense set representations, and a hierarchy for undecidable structures, improving efficiency and generality.

Findings

Achieved a 100x speed-up in basic operations.

Successfully implemented and verified sine and cosine functions.

Extended hierarchy to include order on undecidable structures.

Abstract

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.