Scientific and engineering applications depend on floating point arithmetic to approximate real arithmetic. This approximation introduces rounding error, which can accumulate to produce unacceptable results. While the numerical methods literature provides techniques to mitigate rounding error, applying these techniques requires manually rearranging expressions and understanding the finer details of floating point arithmetic. We introduce Herbie, a tool which automatically discovers the rewrites experts perform to improve accuracy. Herbie's heuristic search estimates and localizes rounding error using sampled points (rather than static error analysis), applies a database of rules to generate improvements, takes series expansions, and combines improvements for different input regions. We evaluated Herbie on examples from a classic numerical methods textbook, and found that Herbie was able to improve accuracy on each example, some by up to 60 bits, while imposing a median performance overhead of 40%. Colleagues in machine learning have used Herbie to significantly improve the results of a clustering algorithm, and a mathematical library has accepted two patches generated using Herbie.

Scientific and engineering applications depend on floating point arithmetic to approximate real arithmetic. This approximation introduces rounding error, which can accumulate to produce unacceptable results. While the numerical methods literature provides techniques to mitigate rounding error, applying these techniques requires manually rearranging expressions and understanding the finer details of floating point arithmetic.We introduce Herbie, a tool which automatically discovers the rewrites experts perform to improve accuracy. Herbie's heuristic search estimates and localizes rounding error using sampled points (rather than static error analysis), applies a database of rules to generate improvements, takes series expansions, and combines improvements for different input regions. We evaluated Herbie on examples from a classic numerical methods textbook, and found that Herbie was able to improve accuracy on each example, some by up to 60 bits, while imposing a median performance overhead of 40%. Colleagues in machine learning have used Herbie to significantly improve the results of a clustering algorithm, and a mathematical library has accepted two patches generated using Herbie.

Floating-point arithmetic plays a central role in science, engineering, and finance by enabling developers to approximate real arithmetic. To address numerical issues in large floating-point applications, developers must identify root causes, which is difficult because floating-point errors are generally non-local, non-compositional, and non-uniform.This paper presents Herbgrind, a tool to help developers identify and address root causes in numerical code written in low-level languages like C/C++ and Fortran. Herbgrind dynamically tracks dependencies between operations and program outputs to avoid false positives and abstracts erroneous computations to simplified program fragments whose improvement can reduce output error. We perform several case studies applying Herbgrind to large, expert-crafted numerical programs and show that it scales to applications spanning hundreds of thousands of lines, correctly handling the low-level details of modern floating point hardware and mathematical libraries and tracking error across function boundaries and through the heap.CCS Concepts • Software and its engineering → Software maintenance tools;

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Foundational verification allows programmers to build software which has been empirically shown to have high levels of assurance in a variety of important domains. However, the cost of producing foundationally verified software remains prohibitively high for most projects, as it requires significant manual effort by highly trained experts. In this paper we present Proverbot9001, a proof search system using machine learning techniques to produce proofs of software correctness in interactive theorem provers. We demonstrate Prover-bot9001 on the proof obligations from a large practical proof project, the CompCert verified C compiler, and show that it can effectively automate what were previously manual proofs, automatically producing proofs for 28% of theorem statements in our test dataset, when combined with solver-based tooling. Without any additional solvers, we exhibit a proof completion rate that is a 4X improvement over prior state-of-the-art machine learning models for generating proofs in Coq.