Primitive Floats in Coq

Abstract

16 Some mathematical proofs involve intensive computations, for instance: the four-color theorem, Hales’ 17 theorem on sphere packing (formerly known as the Kepler conjecture) or interval arithmetic. For 18 numerical computations, floating-point arithmetic enjoys widespread usage thanks to its efficiency, 19 despite the introduction of rounding errors. 20 Formal guarantees can be obtained on floating-point algorithms based on the IEEE 754 standard, 21 which precisely specifies floating-point arithmetic and its rounding modes, and a proof assistant 22 such as Coq, that enjoys efficient computation capabilities. Coq offers machine integers, however 23 floating-point arithmetic still needed to be emulated using these integers. 24 A modified version of Coq is presented that enables using the machine floating-point operators. 25 The main obstacles to such an implementation and its soundness are discussed. Benchmarks show 26 potential performance gains of two orders of magnitude. 27 2012 ACM Subject Classification Theory of computation→ Type theory; Mathematics of computing 28 → Numerical analysis; General and reference → Performance 29