to the linear counting estimator , because $\symPowerSeriesFunc(\symX)\approx 1$.

The new estimator can be directly translated into an estimation algorithm that does not depend on magic numbers or special cases as previous approaches. Since $\symCountVariate_\symRegVal\in\lbrace0,\ldots,\symNumReg\rbrace$, the values for $\symNumReg\,\symSmallCorrectionFunc(\symCountVariate_0/\symNumReg)$ and $\symNumReg\, \symLargeCorrectionFunc(1-\symCountVariate_{\symRegRange+1}/\symNumReg)$ can be precalculated and kept in lookup tables of size $\symNumReg+1$. In this way a complete branch-free cardinality estimation can be realized. However, on-demand calculation of $\symSmallCorrectionFunc$ and $\symLargeCorrectionFunc$ is very fast as well. The series converges quadratically for all $\symX\in[0,1)$ and its terms can be calculated recursively using elementary operations. The case $\symX=1$ needs special handling, because the series diverges and causes an infinite denominator in and therefore a vanishing cardinality estimate. As this case only occurs if all register values are in initial state ($\symCountVariate_0=\symNumReg$), this is exactly what is expected. Apart from the trivial roots at $\symX=0$ and $\symX=1$, the calculation of $\symLargeCorrectionFunc$ is slightly more expensive, because it involves square root evaluations and its series converges only linearly. Since $1-\symX^{2^{-\symRegVal}}\leq -\log(\symX) 2^{-\symRegVal}$ for $\symX\in(0,1]$ its convergence speed is comparable to a geometric series with ratio $1/8$. It is also thinkable to calculate $\symLargeCorrectionFunc$ using the approximation $\symLargeCorrectionFunc(\symX)\approx \symAlpha_\infty/\log(1/\symX)-\symSmallCorrectionFunc(\symX)$ which can be obtained from and 