$\symPowerSeriesFunc(\symX)\approx 1$. The advantage is that the calculation of $\symSmallCorrectionFunc$ and the additional logarithm is slightly faster than the direct calculation of $\symLargeCorrectionFunc$ using . However, for arguments close to 1 both terms may become very large compared to their difference, which requires special care to avoid numerical cancellation. The calculation of $\symLargeCorrectionFunc$ can be omitted at all, if the parameters are chosen such that $2^{\symPrecision+\symRegRange}$ is much larger than the expected cardinality. The number of saturated registers is negligible in this case ($\symCountVariate_{\symRegRange+1}\approx 0$) and therefore $\symLargeCorrectionFunc(1 - \symCountVariate_{\symRegRange+1}/\symNumReg)\approx\symLargeCorrectionFunc(1) = 0$.

Results
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![Relative error of the improved estimator for $\symPrecision = 12$ and $\symRegRange=20$.[]{data-label="fig:raw_corrected_estimation_error_12_20"}](raw_corrected_estimate_12_20){width="1\columnwidth"}

![Standard deviations of the relative error of different cardinality estimators for $\symPrecision = 12$ and $\symRegRange=20$.[]{data-label="fig:stdev_comparison"}](stdev_comparison){width="1\columnwidth"}

\[sec:corrected\_raw\_estimation\_error\] \[fig:raw\_corrected\_estimation\_error\_12\_20\] shows the relative estimation error of the new improved estimator, again based on randomly generated sketches with parameters $\symPrecision=12$ and $\symRegRange=20$. The experimental results show that new estimator is unbiased up to cardinalities of 10 billions which is a clear improvement over the raw estimator (compare \[fig:raw\_estimate\]). The improved estimator beats the precision of existing methods that apply bias correction on the raw estimator [@Heule2013; @Rhodes2015; @Sanfilippo2014]. Based on the simulated data we have empirically determined 