FLoPS: Semantics, Operations, and Properties of P3109 Floating-Point Representations in Lean

TL;DR

This paper introduces FLoPS, a formal Lean model of the P3109 floating-point standard, enabling rigorous analysis and verification of low-precision formats and algorithms for future hardware and software.

Contribution

It provides the first comprehensive formalization of P3109 in Lean, addressing verification gaps and analyzing key algorithms' properties.

Findings

FastTwoSum computes exact overflow error under saturation with any rounding mode.

ExtractScalar properties fail for formats with one bit of precision.

The Lean model is open source and facilitates future formal verification.

Abstract

The upcoming IEEE-P3109 standard for low-precision floating-point arithmetic can become the foundation of future machine learning hardware and software. Unlike IEEE-754, P3109 introduces a parametric framework defined by bitwidth, precision, signedness, and domain. This flexibility results in a vast combinatorial space of formats -- some with as little as one bit of precision -- alongside novel features such as stochastic rounding and saturation arithmetic. These deviations create a unique verification gap that this paper intends to address. This paper presents FLoPS, Formalization in Lean of the P3109 Standard, which is a comprehensive formal model of P3109 in Lean. Our work serves as a rigorous, machine-checked specification that facilitates deep analysis of the standard. We demonstrate the model's utility by verifying foundational properties and analyzing key algorithms within the P3109 context. Specifically, we reveal that FastTwoSum exhibits a novel property of computing exact "overflow error" under saturation using any rounding mode, whereas previously established properties of the ExtractScalar algorithm fail for formats with one bit of precision. This work provides a verified foundation for reasoning about P3109 and enables formal verification of future numerical software. Our Lean development is open source and publicly available.