Accurate Sum and Dot Product with New Instruction for High-Precision Computing on ARMv8 Processor

Kaisen Xie, Qingfeng Lu, Hao Jiang () and Hongxia Wang
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Kaisen Xie: College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China
Qingfeng Lu: College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China
Hao Jiang: College of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China
Hongxia Wang: College of Science, National University of Defense Technology, Changsha 410073, China

Mathematics, 2025, vol. 13, issue 2, 1-19

Abstract: The accumulation of rounding errors can lead to unreliable results. Therefore, accurate and efficient algorithms are required. A processor from the ARMv8 architecture has introduced new instructions for high-precision computation. We have redesigned and implemented accurate summation and the accurate dot product. The number of floating-point operations has been reduced from 7 n − 5 and 10 n − 5 to 4 n − 2 and 7 n − 2 , compared with the classic compensated precision algorithms. It has been proven that our accurate summation and dot algorithms’ error bounds are γ n − 1 γ n cond + u and γ n γ n + 1 cond + u , where ‘cond’ denotes the condition number, γ n = n · u / ( 1 − n · u ) , and u denotes the relative rounding error unit. Our accurate summation and dot product achieved a 1.69× speedup and a 1.14× speedup, respectively, on a simulation platform. Numerical experiments also illustrate that, under round-towards-zero mode, our algorithms are as accurate as the classic compensated precision algorithms.

Keywords: compensated precision; accurate summation; accurate dot product; error analysis; error-free transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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