On Accurate Floating-Point Summation The accumulation of floating-point sums is considered on a computer which performs t-digit base B floating-point addition with exponents in the range -m to M. An algorithm is given for accurately summing N t-digit floating-point numbers. Each of these N numbers is split into q parts, forming qN t-digit floating-point numbers. Each of these is then added to the appropriate one of n auxiliary t-digit accumulators. Finally, the accumulators are added together to yield the computed sum. In all, qN+n-1 t-digit floating-point additions are performed. Under usual conditions, the relative error in the computed sum is at most [(t+1)/v]B^(1-t) for some v. Further, with an additional q+n-1 t-digit additions, the computed sum can be corrected to full t-digit accuracy. For example, for the IBM/360 (B=16, t=14, M=63, m=64), typical values for q and n are q=2 and n=32. In this case, (*) becomes N <= 32,768, and we have [(t+1)/v]B^(1-t) = 4x16^-13. CACM November, 1971 Malcolm, M. A. floating-point summation, error analysis 5.11 5.19 CA711105 JB February 2, 1978 10:48 AM 1328 4 2144 1333 4 2144 2144 4 2144 1052 5 2144 2144 5 2144 2144 5 2144 2144 5 2144