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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Where this becomes clever is compositionality: the library exposes functions that take constructive reals as input and refine their output on demand. With this, we can do almost anything! BUT...
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Some seemingly trivial tests like sqrt(2)*sqrt(2)==2 may loop infinitely because they refine a lower and upper bound that approach 2 on both sides without ever reaching it. So it's not a completely usable construct.
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In a grand scheme of programming language design, I'd love to have a subtype hierarchy with int64 (64-bit integers) <= int (arbitrary precision integers) <= rational (arbitrary precision rational numbers) <= real (constructive reals).
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Just pray you never have to compute x == y :)
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It would be interesting to explore: is there a useful subclass of computable reals that goes beyond rational numbers while retaining computable equality? The quadratic reals are one example, but how far can this go?
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Is it very different from the exact computation described in https://www.doc.ic.ac.uk/exact-computation/ … ? Well.. I guess I'll check. I've always wanted exact computation as an option into my programming languages.
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Sounds like Ziv's method from 91: "Fast evaluation of elementary mathematical functions with correctly rounded last bit"
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