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Comment by Spivak | Hacker News Reader

> floats cannot correctly represent any non-contrived calculation

Like adjusting all your financial calculations to use microcents and partitoning instead of division to keep the result representable by integers? Neither can represent 1/3 even shifted. When you want to do exact calculations with floats, and you can, you just have to set yourself up so that the result is exactly representable, it's not as intractable as you make it seem.

> A `float` on the other hand, approximates the real numbers

Okay so that's not at all what they do, they represent subsets of the reals, just like how integers represent a subset of the reals. Even arbitrary precision libraries can only represent a subset of the rationals.

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> they represent subsets of the reals

Sure. My point is that this subset is useless. Because trying to add, subtract, multiply, or divide members of this set will result in a number outside the set.

> When you want to do exact calculations with floats, and you can, you just have to set yourself up so that the result is exactly representable, it's not as intractable as you make it seem.

In the general case, you absolutely cannot. Lets look at some examples.

In forex trading, you need 9 digits after the decimal place in the price. So right off the bat, a valid price like 1000000.000000001 cannot be represented by a float. If the exchange sends your system that price, your system is guaranteed to be wrong.

Lets say you start at a representable price, like 1000000.0 and want to tick it up or down by the tick size, say 0.025 . The result of that addition / subtraction is not representable, so you cannot calculate and round prices correctly.

If you don't have control of your inputs, and you need precision, floats will never work.

I think you're getting the impression that my stance is "floats are usable for all problem domains" when it's really "floating point arithmetic is not the same as approximate calculations."

Nearly all mathematical calculations cause the result to be outside the range of integers. You can't do much else other than subtract without accounting for edge cases. No matter what tool you use you must work with your chosen representation and around its limitations and make sure your domain can be modeled exactly. For example Python's base random function chooses a floating point uniformly in the range [0, 1) but it achieves this by requiring that the result be a multiple of 2^-52 which is exactly representable so rounding doesn't introduce bias.

> If you don't have control of your inputs

Well you clearly do to some degree because you're sure you can model anything you might might receive with fixed sized fixed precision integers. I'm not saying this means you can just switch to floats but that you're doing the same thing, mapping the real life problem domain exactly to a subset of the reals that is closed under the operations you want to perform.

> When you want to do exact calculations with floats, and you can, you just have to set yourself up so that the result is exactly representable, it's not as intractable as you make it seem.

Can you expand on what you mean by that? Whenever I have dealt with calculation errors (either in fixed or floating point) "set yourself up so that the result is exactly representable" has been the key problem to prevent errors accumulating.

I obviously disagree on some other points, but circular discussions go nowhere!

Edit to add: Essentially I'm fishing for tactics. A big one in graphics development is detailed here: https://developer.nvidia.com/content/depth-precision-visuali... - but in fixed point I got very used to working out how to premultiply variables depending on their expected ranges.

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