Floating-point arithmetic is a well-known and extremely efficient way of performing approximate computations over the real numbers. Although it requires some careful considerations, floating-point numbers are nowadays routinely used to prove mathematical theorems. Numerical computations have been applied in the context of formal proofs too, as illustrated by the CoqInterval library. But these computations do not benefit from the powerful floating-point units available in modern processors, since they are emulated inside the logic of the formal system. This paper experiments with the use of hardware floating-point numbers for numerically intensive proofs verified by the Coq proof assistant. This gives rise to various questions regarding the formalization, the implementation, the usability, and the level of trust. This approach has been applied to the CoqInterval and ValidSDP libraries, which demonstrates a speedup of at least one order of magnitude.

浮点运算是一种众所周知且极其有效的对实数执行近似计算的方法。尽管需要一些仔细的考虑，浮点数现在通常用于证明数学定理。数值计算也已应用于形式证明中，如 CoqInterval 库所示。但这些计算并没有受益于现代处理器中可用的强大浮点单元，因为它们是在形式系统的逻辑内部进行模拟的。本文尝试使用硬件浮点数进行由 Coq 证明助手验证的数值密集型证明。这引发了有关形式化、实施、可用性和信任级别的各种问题。