Toward Accurate and Fast Summation

We introduce a new accurate summation algorithm based on the error-free summation into floating-point buckets. Our algorithm exploits ideas from Zhu and Hayes’ OnlineExactSum, but it uses a significantly smaller number of accumulators and has a better instruction-level parallelism. In the default setting, our implementation aaaSum returns a faithfully rounded floating-point approximation of the true sum. We also discuss possible modifications for the computation of reproducible, correctly rounded, and multiple precision floating-point approximations. The computational overhead for any of these modifications is kept comparably small. Numerical tests demonstrate that aaaSum performs well for very small to large problem sizes, independent of the condition number of the problem. We compare our algorithm with other accurate and high-precision summation approaches.