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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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 don't do equality testing on reals. If you think about how you might write a ray-tracer, say, using constructive reals, then I feel like many of the things that are awkward to do turn out to nudge you towards realistic physics.
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Eg. you can't do a hard equality test to see if A is in front of B. As they become close, it's hard to use constructive reals to tell which is in front, so instead you want to make A and B slightly transparent and blend, which is what a really thin real-world material looks like.
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This approach seems very close to how on-demand arbitrary-precision calculations are implemented in Wolfram Language (Mathematica). But it also can do symbolic simplifications that handle cases like sqrt(2)*sqrt(2), and some (but not all) much more trickier ones.
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When presented with an equality a==b it tries to disprove it (reduce to False) using arbitrary-precision computations, and at the same time tries to prove it (reduce to True) using symbolic methods. If neither succeeds, the expression is left in an unevaluated form.
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