$\symPrecision = 12$ and $\symRegRange=20$.[]{data-label="fig:original_estimate"}](original_estimate){width="1\columnwidth"}

The large range correction formula is not satisfying either as it does not reduce the estimation error but makes it even worse. Instead of underestimating cardinalities, they are now overestimated. Another indication for the incorrectness of the proposed large range correction is the fact that it is not even defined for all possible states. For instance, consider a $(\symPrecision,\symRegRange)$- sketch with $\symPrecision+\symRegRange=32$ for which all registers are equal to the maximum possible value $\symRegRange+1$. The raw estimate would be $\symCardinalityRawEstimate = \symAlpha_\symNumReg 2^{33}$, which is greater than $2^{32}$ and outside of the domain of the large range correction formula. A simple approach to avoid the need of any large range corrections is to extend the operating range of the raw estimator to larger cardinalities. This can be easily accomplished by increasing $\symPrecision+\symRegRange$, which corresponds to using hash values with more bits. Each additional bit doubles the operating range which scales like $2^{\symPrecision+\symRegRange}$. However, in case $\symRegRange \geq 31$ the number of possible register values, which are $\lbrace 0, 1, \ldots, \symRegRange+1\rbrace$, exceeds 32 and is no longer representable by 5 bits. Therefore, it was proposed to use 6 bits per register in combination with 64-bit 