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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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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One way of looking at this is that you can compose a bunch of numbers and operations and choose how much precision you want at the very end, while other approaches force you to choose a level of precision for individual parts of your calculation.
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It's a lot like vector graphics vs raster graphics. With vector graphics, you can combine a bunch of different shapes and render to concrete pixels at the end, while with raster graphics you might have components rendered at different resolutions that won't combine nicely.
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