Twofold fast arithmetic

TL;DR

This paper introduces a method for fast arithmetic using twofold approximations to automatically verify floating-point accuracy, enabling on-the-fly error tracking during computations.

Contribution

It proposes a novel approach for real-time accuracy verification in floating-point computations based on twofold arithmetic, with implementation in C++ for Intel AVX.

Findings

Effective on-the-fly error detection

Implementation available in C++ for Intel AVX

Potential for improved computational reliability

Abstract

Can we assure math computations by automatic verifying floating-point accuracy? We define fast arithmetic (based on Dekker [1971]) over twofold approximations $z\approx z_0+z_1$, such that $z_0$ is standard result and $z_1$ assesses inaccuracy $\Delta z_0=z-z_0$. We propose on-fly tracking $z_1$, detecting if $\Delta z_0$ appears too high. We believe permanent tracking is worth its cost. C++ test code for Intel AVX available via web.