On the robustness of double-word addition algorithms

TL;DR

This work analyzes the robustness of double-word addition algorithms (sloppy add and accurate add) under moderate input overlap and various rounding modes. It establishes error bounds: the sloppy add achieves $O(u^2(|a|+|b|))$ in faithful rounding, while the accurate add attains $O(u^2)$ relative error in round-to-nearest, with extensions to directed rounding under additional conditions. The authors show that omitting normalization in double-word multiplication and using sloppy add can nearly double performance in double-word multiplication-addition (MAA), with negligible precision loss, and that error directions remain consistent under directed rounding, benefiting interval arithmetic. They provide both theoretical bounds and empirical evidence (including AVX-512 optimizations) to support substantial speedups in double-word GEMM-like computations and interval arithmetic without changing rounding modes. The findings suggest greater robustness of the two addition algorithms than previously believed and offer practical guidance for high-precision numerical libraries.

Abstract

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 $O(u^2(|a|+|b|))$ in faithful rounding. Furthermore, the accurate algorithm can achieve a relative error bound of $O(u^2)$ 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 $3u^2$ 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.

Theorems & Definitions (5)

• proof
• proof
• proof
• proof
• proof