Hacker News Reader

Comment by duped | Hacker News Reader

> C/C++ compilers give you a way out of this with the -fast-math

People should really use the individual optimization flags they want (no signed zeros, no trapping math, associative math, reciprocal math) and not -ffast-math because the other optimizations it enables leads to surprising code (for example, isinf and isnan may become noops, which will break production code).

Basically never use an optimization flag that changes the semantics of your code without understanding exactly what that means. I have had to fix this in a number of codebases because someone thought that flag was as innocuous as -O3.

And if you have to enable flush to zero/denormals are zero it should be explicit in your code and scoped.

View on HN

Yes. In fact, even if you don't ask for semantics like the -fast-math flag, the floating point types are worth some extra time to understand before relying on them.

They're much stranger than the machine integers. The machine integers are basically like the Integers you were taught in school, except for overflow. That's not nothing but it's a complexity you can ignore entirely so long as you never overflow. In Rust you can have the language keep you safe - if an overflow occurs we'll panic and we're done. However the floating point types are a weird thing entirely invented for the convenience of the machine. They're too often introduced as if, like the machine integers, they're almost familiar numbers from school. Some languages even call these types "real" - but they very much are not actually the Real numbers, not even the approximation that the machine integers were to actual Integers. The programming language can't help you cope today. You can use software like "Herbie" to help you a bit, but today's languages just leave you with it.

https://herbie.uwplse.org/

Here's an easy example you saw in school, a tenth, written 0.1 in decimal. The floating point types cannot represent this number. When you ask for the 32-bit floating point value 0.1 in a language like C or Rust, you actually get exactly 0.100000001490116119384765625 because that was a number the type can represent and it was deemed "close enough".

I mean I get that novice programmers might get tripped up on floating point representation but if you don't know "f32 can't represent 0.1 exactly" then you shouldn't yet be worried about the nuance of relaxing IEEE 754 compliance for the purposes of performance.

They're not real, they're rationals (well rationals are real but you get it).

If you want "simple" rationals, you can use the numerator/denominator scheme. This has its own problems but if you avoid overflows, they are the rationals you were taught in school, just like the integers.

The problem is that people (and languages) default to floating-point without understanding the consequences. Many times it does not matter and then floating point are indeed better (if you know how to use them, e.g. not comparing for equality), but sometimes it does.

> The problem is that people (and languages) default to floating-point without understanding the consequences.

If you were to insist on there being only one numeric data type in a language, then floating-point turns out to be the best compromise, especially because someone who doesn't understand the pitfalls of floating-point are going to be less likely to have it blow up in their face than other options. Fixed-point has a problem when the numbers have very different scales. Rational numbers don't let you do basic things like "measure the distance between two points" (because functions like sqrt or exp aren't defined on rational numbers).

> Rational numbers don't let you do basic things like "measure the distance between two points" (because functions like sqrt or exp aren't defined on rational numbers).

However the floating point types are just binary rationals, so if we took this "can't do basic things" at face value we couldn't do these operations on the floating point types either.

The reason they're so weird is a convenience to the implementation. I am not an EE so I can't tell you how much that saved, but it was a choice, obviously we can't implement the Reals because Almost All Reals aren't even Computable, but I think most software engineers really don't have an appropriate understanding of the floating point types and the result is buggy software.

The main selling point that people tout for using rational numbers over floating-point is their exactness, and the main thing people complain about floating-point is their approximate nature. Introduce the ability to do sqrt on rational numbers, and you somehow have to invent approximation for rational numbers. Choose the nearest representable rational number, and oops, nearly every operation on that rational number could overflow the numerator or denominator, so change that and oops now you've basically reinvented floating-point but worse.

Which is sort of the thing of floating-point: they may be weird, but it kind of turns out that they're ultimately less weird than all of their alternatives, if you try to do anything other than toy examples on them. It's almost like floating-point was designed by people who needed to do a lot of numerical work in computers instead of students in Programming 101!

As somebody who has implemented Hans Boehm's "Towards an API for the Real Numbers" paper I cannot agree that the floating point types are "less weird" than alternatives. They're much faster of course because they're implemented in hardware, but that's hardly surprising.

In my (unpublished) toy calculator I added a mechanism to bring an f32 or f64 into existence and convert it to my Rational type and so that's why I had the exact value of the f32 we call 0.1 to hand earlier. But of course the API goes much further than that. Want to multiply the square root of 40 by the square root of 1000? No problem that's exactly 20. A bright teenager can figure that out, and so can a properly designed API for numbers. The ordinary human activities you talked about of course work just fine in this API, it has historically been how the calculator in an Android phone works for example.

Symbolic approaches also have value in this space.

In both cases, what you do not get is the problem where the programmer mistakenly believes these types will behave like the numbers in school but actually they behave in an entirely different way. You have to go really far off piste, by which point likely the programmer has appreciated that they do not understand what's going on, before you trip on a hazard we cannot manage with this approach - whereas in traditional floating point that happens without them realising.

Yes, I was being a bit concise: Individual optimisations should be turned on as determined by profiling, application semantics, etc. My point was more that if you want to get as close as possible between integer and floating-point, then there is a flag that does it. That doesn't mean that you should do it, however...

Reply on news.ycombinator.com