"I’m also not aware of any calculator (or programming language) that currently works this way." [represent fractions exactly wherever reasonable to do so]

Author here. The paraphrase makes this sound wrong, but what I actually meant is that I don’t know of any calculator that uses rationals while the numerator and denominator can be represented as integer floating point doubles, and then falls back to a single floating point double if the numerator or denominator overflow.

I am aware of many systems that use arbitrary precision rationals, and I mentioned this in the post:

Rational numbers where the numerator and denominator are allowed to grow arbitrarily large are used by computer algebra systems (CAS) like Mathematica and Maple (and also some high-end handheld graphing calculators).