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Improved estimates for error in floating point representation analysis. (English) Zbl 1104.65048

The authors consider floating point representations \(fl(x)\) of numbers \(x\) in base \(\beta\). In order to obtain \(fl(x)\) they use two ways of roundings: rounding to nearest and rounding by chopping. Taking into account the first non-zero digit \(d_1\) of \(x\) they estimate the relative error \(r_x= (x- fl(x))/x\). Moreover, using \(d_1\) they present a bound for the error \(|x- x^*|\), where \(x^*\) denotes an approximation of \(x\) which is correct to \(r\) significant digits. Based on their results they estimate the error of the sum and the difference of two numbers indicating by examples that their bounds are better by one-order of magnitude than well-known estimates from the literature.

MSC:

65G50 Roundoff error
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