Accurate Sum and Dot Product with New Instruction for High-Precision Computing on ARMv8 Processor
Kaisen Xie, LU Qing-feng, Hao Jiang, Hongxia Wang · Mathematics · 2025
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 7n−5 and 10n−5 to 4n−2 and 7n−2, compared with the classic compensated precision algorithms. It has been proven that our accurate summation and dot algorithms’ error bounds are γn−1γncond+u and γnγn+1cond+u, where ‘cond’ denotes the condition number, γn=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.