# Revealing Floating-Point Accumulation Orders in Software/Hardware Implementations

arXiv: https://arxiv.org/abs/2411.00442 · HTML: https://mlsanity.com/paper/2411.00442/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2024.

## Built on by (most cited first, 4 of 4)

- [TrainVerify: Equivalence-Based Verification for Distributed LLM Training](https://mlsanity.com/paper/2506.15961/) (2025)
- [TAO: Tolerance-Aware Optimistic Verification for Floating-Point Neural Networks](https://mlsanity.com/paper/2510.16028/) (2025)
- [Bit-Accurate Modeling of GPU Matrix Multiply-Accumulate Units: Demystifying Numerical Discrepancy and Accuracy](https://mlsanity.com/paper/2511.10909/) (2025)
- [Accelerating the Mitigation of LLM Inference Nondeterminism Across GPU Architectures](https://mlsanity.com/paper/2609.25624/) (2026)

## Abstract (arXiv)

Accumulation-based operations, such as summation and matrix multiplication, are fundamental to numerous computational domains. However, their accumulation orders are often undocumented in existing software and hardware implementations, making it difficult for developers to ensure consistent results across systems. To address this issue, we introduce FPRev, a diagnostic tool designed to reveal the accumulation order in the software and hardware implementations through numerical testing. With FPRev, developers can identify and compare accumulation orders, enabling developers to create reproducible software and verify implementation equivalence. FPRev is a testing-based tool that non-intrusively reveals the accumulation order by analyzing the outputs of the tested implementation for distinct specially designed inputs. Employing FPRev, we showcase the accumulation orders of popular libraries (such as NumPy and PyTorch) on CPUs and GPUs (including GPUs with specialized matrix accelerators such as Tensor Cores). We also validate the efficiency of FPRev through extensive experiments. FPRev exhibits a lower time complexity compared to the basic solution. FPRev is open-sourced at https://github.com/peichenxie/FPRev.