FloatLib: Verified Floating-Point Arithmetic in Lean

2026-09-16

AI Summary

FloatLib使用Lean 4开发了一个验证过的任意精度浮点数算术库，统一了多种格式和舍入规则，通过验证的软件后端保证正确性和效率。

Abstract

We present FloatLib, a verified arbitrary-precision floating-point arithmetic library in Lean 4 that combines broad format coverage, machine-checked correctness, and efficient certified execution. To our knowledge, FloatLib is the first Lean library to unify IEEE binary and decimal arithmetic, arbitrary-width posits, P3109, and user-defined formats and rounding rules behind interchangeable certified software backends. Every certified backend is proved equal to a complete encoded specification, preserving signed zeros and exceptional values, while numerical theorems connect execution to real rounding, error bounds, and exactness. FloatLib combines exhaustive certified tables for small formats with verified word and limb kernels based on guard-and-sticky invariants and independently checked quotient candidates. Its posit development additionally proves standard rounding thresholds and exact quire accumulation within capacity for arbitrary widths. Across matched workloads, FloatLib achieves speedups of up to 1.46x over FLoPS and 116x over Universal, while remaining slower in some regimes such as binary arithmetic against MPFR. Independent conformance testing includes more than 102 million TestFloat evaluations with zero differences under the tested relation. We release the library, proofs, benchmarks, evaluation data, and guide as open source.

Problem

• verified floating-point arithmetic
• arbitrary-precision
• Lean 4
• IEEE binary and decimal arithmetic
• posits

Innovation

• arbitrary-precision floating-point arithmetic
• certified software backends
• unified IEEE binary and decimal arithmetic
• verified word and limb kernels
• standard rounding thresholds