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

2026-02-17

AI Summary

This work addresses the lack of formal verification foundations for the IEEE P3109 low-precision floating-point standard, whose flexible format and novel features—such as stochastic rounding and saturating arithmetic—pose unique challenges. We present the first complete, parameterized formal model of P3109 in the Lean theorem prover, enabling machine-checkable analysis of its semantics, operations, and key algorithms. Our contributions include the first mechanically verified specification of P3109, a proof that FastTwoSum precisely captures overflow error under saturating arithmetic, and the discovery that ExtractScalar fails at 1-bit precision. The accompanying open-source formal library provides a reusable foundation for the reliable verification of low-precision numerical software.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Hardware-aware ML
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

The upcoming IEEE-P3109 standard for low-precision floating-point arithmetic can become the foundation of future machine learning hardware and software. Unlike the fixed types of 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.

Problem

• IEEE-P3109
• floating-point arithmetic
• formal verification
• low-precision
• saturation arithmetic

Innovation

• formal verification
• IEEE-P3109
• low-precision floating-point
• Lean theorem prover
• saturation arithmetic