What would be useful is a method that generates (i.e., a generator in
the Python sense) the (continued fraction) convergents to a rational.
People wanting specific constraints on a rational approximation
(including, but not limited to, the two you identified) can easily
build them on top of such a generator.

By "useful" I don't mean lots of people will use it ;-)  I mean /some/
people will use it -- a way to generate the sequence of convergents is
a fundamental tool that can be used for all sorts of stuff, by those
with advanced applications.
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