Another amazing work by Hans Boehm back in 1999: a constructive real number calculator, https://www.hboehm.info/crcalc/ . In a sense, this is "as far as a computer can go" in the hierarchy of accuracy that includes floating-point, exact rational numbers, and then constructive reals.
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Yeah, C is just the field extension R[x]/(x^2 + 1).
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Computing with normal forms gets more complicated when you have multivariate polynomials but there's algorithms for it. Not to send you down a rabbit hole but this is just intro book: https://www.amazon.com/Ideals-Varieties-Algorithms-Computational-Undergraduate/dp/0387356509#reader_0387356509 …
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