Rigorous Error Analysis for Logarithmic Number Systems

Theorem proving demonstrates promising potential for verifying problems beyond the capabilities of SMT-solver-based verification tools. We explore and showcase the capability of Lean, an increasingly popular theorem-proving tool, in deriving the error bounds of table-based Logarithmic Number Systems (LNS). LNS reduces the number of bits needed to represent a high dynamic range of real numbers with finite precision and efficiently performs multiplication and division. However, in LNS, addition and subtraction become non-linear functions that must be approximated–typically using precomputed look-up tables. We provide the first rigorous analysis of LNS that covers first-order Taylor approximation, cotransformation techniques in-spired by European Logarithmic Microprocessor, and the errors introduced by fixed-point arithmetic involved in LNS implementations. By analyzing all error sources and deriving symbolic error bounds for each, then accumulating these to obtain the final error bound, we prove the correctness of these bounds using Lean and its Mathlib library. We empirically validate our analysis using an exhaustive Python implementation, demonstrating that our analytical interpolation bounds are tight, and our analytical cotransformation bounds overestimate between one and two bits.