If you want to do without a parameter, the geometric mean IS
the natural split that follows the roughly log distribution of
floating-point numbers. For us it would mean splitting off
zero& infinity until one is wholly within the positives or
negatives. That is, until we have either 0< inf(X)<= sup(X)
< +oo or -oo< inf(X)<= sup(X)< 0. From that point on,
split(X) = +/-sqrt(inf(X)*sup(X)) as the case may be.
Since this is equivalent to
log|split(X)| = (log|inf(X)| + log|sup(X)|)/2
it is logarithmically distributed just like floating-point
numbers. Which also has the property that it is nearly the
arithmetic mean when inf(X)& sup(X) are near one another&
away from zero.
I hesitated to propose it because of all the special case
splitting of zero& infinity that must be done before the
split kicked in. But if the parameter is the problem, it
is the natural solution.