  The information presented in this paper defines LogLog-Beta. LogLog-Beta is a
new algorithm for estimating cardinalities based on LogLog counting. The new
algorithm uses only one formula and needs no additional bias corrections for
the entire range of cardinalities, therefore, it is more efficient and simpler
to implement. Our simulations show that the accuracy provided by the new
algorithm is as good as or better than the accuracy provided by either of
HyperLogLog or HyperLogLog++. In addition to LogLog-Beta we also provide
another one-formula estimator for cardinalities based on order statistics, a
modification of an algorithm developed by Lumbroso.
