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Comment by ogogmad | Hacker News Reader

> And you should understand that most of its limitations are an inherent mathematical and fundamental limitation, it is logically impossible to do better on most of its limitations

You can do exact real arithmetic. But this is only done by people who prove theorems with computers - or by the Android calculator! https://en.wikipedia.org/wiki/Computable_analysis

Other alternatives (also niche) are exact rational arithmetic, computer algebra, arbitrary precision arithmetic.

Fixed point sometimes gets used instead of floats because some operations lose no precision over them, but most operations still do.

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In my opinion, that's in the realm of "you can only delay it". Sure, you can treat real numbers via purely logical deductions like a human mathematician would, but at some point someone's going to ask, "so, where is the number on this plot?" and that's when it's time to pay the fiddler.

Same for arbitrary-precision calculations like big rationals. That just gives you as much precision as your computer can fit in memory. You will still run out of precision, just later rather then sooner.

> Same for arbitrary-precision calculations like big rationals. That just gives you as much precision as your computer can fit in memory. You will still run out of precision, later rather then sooner.

Oh, absolutely. This actually shows that floats are (in some sense) more rigorous than more idealised mathematical approaches, because they explicitly deal with finite memory.

Oh, I remembered! There's also interval arithmetic, and variants of it like affine arithmetic. At least you know when you're losing precision. Why don't these get used more? These seem more ideal, somehow.

Because the interval, on average, grows exponentially with the number of basic operations. So it quickly becomes practically useless.

If x is the interval [-1, 1], the typical implementation of IA will

evaluate x-x to [-2, 2] (instead of [0, 0], and

evaluate x*x [-1, 1] instead of [0, 1].

Therefore the intervals become too conservative to be useful.

I wouldn't call computable reals the reals. They are a subset of measure zero. Perhaps all we sentient beings can aspire to use, but still short of the glory of the completed infinities that even one arbitrary real represents.

One half : )

These are only relevant in some circumstances. For example, a calculator is typically bounded in the number of operations you can perform to a small number (humans don’t add millions of numbers). This allows for certain representations that don’t make sense elsewhere.

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