Abstract
The result of a floating-point operation is usually defined to be the floating-point number nearest to the exact real result together with a tie-breaking rule. This is called the first standard model of floating-point arithmetic, and the analysis of numerical algorithms is often solely based on that. In addition, a second standard model is used specifying the maximum relative error with respect to the computed result. In this note we take a more general perspective. For an arbitrary finite set of real numbers we identify the rounding to minimize the relative error in the first or the second standard model. The optimal “switching points” are the arithmetic or the harmonic means of adjacent floating-point numbers. Moreover, the maximum relative error of both models is minimized by taking the geometric mean. If the maximum relative error in one model is \(\alpha \), then \(\alpha /(1-\alpha )\) is the maximum relative error in the other model. Those maximal errors, that is the unit roundoff, are characteristic constants of a given finite set of reals: The floating-point model to be optimized identifies the rounding and the unit roundoff.
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Notes
For the standard formats \(\mathbb {F}\) in IEEE 754 the range could be slightly wider: For \(f\) denoting the rounded-to-nearest result in \(\mathbb {F}\) with infinite exponent range, return this \(f\) if it belongs to \(\mathbb {F}\) with the bounded exponent range. Since we are aiming on general sets \(\mathfrak {F}\), there is no notion of “exponent range”.
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Acknowledgments
Our dearest thanks go to Claude-Pierre Jeannerod from Lyon for his many detailed comments and for very helpful discussions and suggestions. Moreover, many thanks to the anonymous referees for their valuable and constructive comments.
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Communicated by Axel Ruhe.
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Rump, S.M., Lange, M. On the definition of unit roundoff. Bit Numer Math 56, 309–317 (2016). https://doi.org/10.1007/s10543-015-0554-0
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DOI: https://doi.org/10.1007/s10543-015-0554-0