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We analyse the forward error in the floating point summation of real numbers,from algorithms that do not require recourse to higher precision or betterhardware. We derive informative explicit expressions, and new deterministic andprobabilistic bounds for errors in three classes of algorithms: generalsummation,shifted general summation, and compensated (sequential) summation.Our probabilistic bounds for general and shifted general summation hold to allorders. For compensated summation, we also present deterministic andprobabilistic first and second order bounds, with a first order bound thatdiffers from existing ones. Numerical experiments illustrate that the boundsare informative and that among the three algorithm classes, compensatedsummation is generally the most accurate method.

Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms