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| -rw-r--r-- | doc/algorithms.tex | 17 |
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diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 7bd24a2..defc437 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1645,6 +1645,23 @@ on the real part and of \] on the imaginary part of the result. +If we further assume that $(n-1) 2^{-p} \leq 1$, then +$\left( (1 + 2^{-p})^{n-1} - 1 \right) \leq 2 (n-1) 2^{-p}$, +because $(1+\varepsilon)^m-1 = \exp(m \log(1+\varepsilon)) - 1 +\leq \exp(\varepsilon m) - 1 \leq 2 \varepsilon m$ as long as +$\varepsilon m \leq 1$. This gives the simplified bounds +\[ +\left( 2 + 2^{\Exp (\Im \appro x_k) - \Exp (\Re \appro x_k) + 2} \right) +(n-1) \Ulp (\Re \appro x_k) +\] +on the real part and of +\[ +\left( 2 + 2^{\Exp (\Re \appro x_k) - \Exp (\Im \appro x_k) + 2} \right) +(n-1) \Ulp (\Im \appro x_k) +\] +on the imaginary part. + + \bibliographystyle{acm} \bibliography{algorithms} |