  The Continued Logarithm Algorithm - CL for short- introduced by Gosper in
1978 computes the gcd of two integers; it seems very efficient, as it only
performs shifts and subtractions. Shallit has studied its worst-case complexity
in 2016 and showed it to be linear. We here perform the average-case analysis
of the algorithm: we study its main parameters (number of iterations, total
number of shifts) and obtain precise asymptotics for their mean values. Our
'dynamical' analysis involves the dynamical system underlying the algorithm,
that produces continued fraction expansions whose quotients are powers of 2.
Even though this CL system has already been studied by Chan (around 2005), the
presence of powers of 2 in the quotients ingrains into the central parameters a
dyadic flavour that cannot be grasped solely by studying the CL system. We thus
introduce a dyadic component and deal with a two-component system. With this
new mixed system at hand, we then provide a complete average-case analysis of
the CL algorithm, with explicit constants.
