Twofold fast summation

Abstract: Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum is more accurate than value alone, so error can estimate deviation between value and its exact target. Fast summation algorithm, that provides twofold sum of x₁+...+x_N or dot product x₁y₁+...+x_Ny_N, can be same fast as direct summation sometimes if leveraging processor underused potential. This way, we can hit three goals: improve precision, track inaccuracy, and do this with little if any loss in performance.

2013 (C) Evgeny Latkin, Free for non-commercial and academic use

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Text also uploaded to arXiv: http://arxiv.org/abs/1401.0248