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mpq_t variables represent rationals using an
mpz_t numerator and
denominator (see Integer Internals).
The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1.
It’s believed that casting out common factors at each stage of a calculation
is best in general. A GCD is an O(N^2) operation so it’s better to do
a few small ones immediately than to delay and have to do a big one later.
Knowing the numerator and denominator have no common factors can be used for
mpq_mul to make only two cross GCDs necessary, not four.
This general approach to common factors is badly sub-optimal in the presence
of simple factorizations or little prospect for cancellation, but GMP has no
way to know when this will occur. As per Efficiency, that’s left to
mpq_t framework might still suit, with
mpq_denref for direct access to the numerator and
denominator, or of course
mpz_t variables can be used directly.