2024/04/09 by Yuanyuan Yang, Yang, Yuanyuan, XinYu Lyu +9 · 1 voice
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Low-power high-performance VLSI design #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Parallel Computing and Optimization Techniques #math.NA
paper · pdf · doi:10.48550/arxiv.2404.05948
openalex publication_date 2024/04/09 · arxiv published 2024/04/09 · arxiv updated 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We demonstrate that, even when there are moderate overlaps in the inputs of sloppy or accurate double-word addition algorithms in the QD library, these algorithms still guarantee error bounds of O(u2(|a|+|b|)) in faithful rounding. Furthermore, the accurate algorithm can achieve a relative error bound of O(u2) in the presence of moderate overlaps in the inputs when rounding function is round-to-nearest. The relative error bound also holds in directed rounding, but certain additional conditions are required. Consequently, in double-word multiplication and addition operations, we can safely omit the normalization step of double-word multiplication and replace the accurate addition algorithm with the sloppy one. Numerical experiments confirm that this approach nearly doubles the performance of double-word multiplication and addition operations, with negligible precision costs. Moreover, in directed rounding mode, the signs of the errors of the two algorithms are consistent with the rounding direction, even in the presence of input overlap. This allows us to avoid changing the rounding mode in interval arithmetic. We also prove that the relative error bound of the sloppy addition algorithm exceeds 3u2 if and only if the input meets the condition of Sterbenz's Lemma when rounding to nearest. These findings suggest that the two addition algorithms are more robust than previously believed.