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>How so? Because you need to be extremely careful about overflows/underflows. All operations you perform suddenly become difficult and require careful analysis

>How so? Because you need to be extremely careful about overflows/underflows. All operations you perform suddenly become difficult and require careful analysis. With fixed point numbers you need to ensure that every intermediate result of your operation gives an in range result. You can not remove that tedium by a library, since any potential application has different requirements and now you need to start out your program by defining those requirements for your library. All operations you perform need to be analysed based on your initial requirements. Also, fixed point arithmetic does not fix floats. You have an uncountable number of real numbers and you try to fit that into 32 or so bytes. It seems simple to understand that any such projectiom whatever you try has enormous drawbacks. That floats do not behave like real numbers is the consequence of their design requirements. Fixed point just means instead of being able to pretend that floating point operations are sometimes inaccurate real number operations, you have to deal with constan, domain specific renormalization and have to be extremely careful aboit choosing scaling.

Karellen 3y ago
> you need to be extremely careful about overflows/underflows. [...] you need to ensure that every intermediate result of your operation gives an in range result. How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? > Also, fixed point arithmetic does not fix floats. I wasn't expecting it to?
- constantcrying 3y ago
  >How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? It is different, because you never want to take the square root of an integer. Floats allow you to pretend you are working with sometimes inaccurate real numbers. That is the magic behind it and why fixed point arithmetic was abandoned almost immediately when floating point became fast and easily available.
- jcranmer 3y ago
  > How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? One of the key advantages of floating-point is that it is scale-independent--it doesn't matter if you're doing your calculations in meters, feet, miles, kilometers, parsecs, AUs, nanometers, Planck lengths--you'll get the same relative accuracy. If you're using fixed point (or integers), you instead have to take care to make sure that you scale things such that your units are not too large or too small. Floating-point is essentially binary scientific notation, and it should be no surprise that it's a good format if you're in a field that already uses scientific notation all the time.
  - Karellen 3y ago
    > it doesn't matter if you're doing your calculations in meters, feet, miles, kilometers, parsecs, AUs, nanometers, Planck lengths--you'll get the same relative accuracy. Yes, but if I'm working with lengths where I know micrometers are good enough, but I also want micrometer (or, at least, better-than-millimeter) precision, but I want to actually deal in meters and I know I'm not going to be dealing with lengths longer than 100km, then fixed point would be ideal for that. Simlarly, if I want to deal in dollars, but need cent precision (or, tenth of a cent precision), and need it to be precise, then fixed point would be ideal for that too. I think the disconnect might be that I wasn't considering fixed-point arithmetic as a general replacement for floating-point arithmetic, I'm considering it as a complement to floating-point, to be used in the cases where it makes sense to do so. The impression I got from the commenter I was originally replying to was "fixed point is absolutely terrible and there is never a reason to prefer it over floats". If that wasn't an intended implication, I guess I'm kind of arguing into the void.
    - constantcrying 3y ago
      >I know I'm not going to be dealing with lengths longer than 100km, then fixed point would be ideal for that. Huge fallacy. No, you can not use fixed point numbers like that, it can not work. It is irrelevant what the actual maximum/minimum scale you are dealing with is. The thing which matters is the largest/smallest intermediate value you need. You need to consider every step in all your algorithms. Imagine calculating the distance of two objects being 50km in x and 50km in y direction apart. Even though the input and output values fit within your range, the result is nonsensical if you use naive fix point arithmetic. Floating point allows you to write down the mathematical formula exactly, using fixed point arithmetic, you can not do that. Looking at the maximum and minimum resolution you need is a huge fallacy when working with fixed point arithmetic and one big reason why everyone avoids using it. You need to carefully analyze your entire algorithm to use it. >The impression I got from the commenter I was originally replying to was "fixed point is absolutely terrible and there is never a reason to prefer it over floats". My position is that sometimes there might be a situation where fixed point arithmetic could be useful. If you are willing to put in a significant amount of time and effort into analyzing your system and dataflow it can be implememted successfully. It is a far more complex and errorprone system and if you aren't careful it will bite you. In all cases floating point should be the default option and only deviated from if there are very good reasons.

Karellen 3y ago

> you need to be extremely careful about overflows/underflows. [...] you need to ensure that every intermediate result of your operation gives an in range result. How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? > Also, fixed point arithmetic does not fix floats. I wasn't expecting it to?

constantcrying 3y ago
>How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? It is different, because you never want to take the square root of an integer. Floats allow you to pretend you are working with sometimes inaccurate real numbers. That is the magic behind it and why fixed point arithmetic was abandoned almost immediately when floating point became fast and easily available.
jcranmer 3y ago
> How is that any different from ordinary integer operations/arithmetic with ints/longs/etc...? One of the key advantages of floating-point is that it is scale-independent--it doesn't matter if you're doing your calculations in meters, feet, miles, kilometers, parsecs, AUs, nanometers, Planck lengths--you'll get the same relative accuracy. If you're using fixed point (or integers), you instead have to take care to make sure that you scale things such that your units are not too large or too small. Floating-point is essentially binary scientific notation, and it should be no surprise that it's a good format if you're in a field that already uses scientific notation all the time.
- Karellen 3y ago
  > it doesn't matter if you're doing your calculations in meters, feet, miles, kilometers, parsecs, AUs, nanometers, Planck lengths--you'll get the same relative accuracy. Yes, but if I'm working with lengths where I know micrometers are good enough, but I also want micrometer (or, at least, better-than-millimeter) precision, but I want to actually deal in meters and I know I'm not going to be dealing with lengths longer than 100km, then fixed point would be ideal for that. Simlarly, if I want to deal in dollars, but need cent precision (or, tenth of a cent precision), and need it to be precise, then fixed point would be ideal for that too. I think the disconnect might be that I wasn't considering fixed-point arithmetic as a general replacement for floating-point arithmetic, I'm considering it as a complement to floating-point, to be used in the cases where it makes sense to do so. The impression I got from the commenter I was originally replying to was "fixed point is absolutely terrible and there is never a reason to prefer it over floats". If that wasn't an intended implication, I guess I'm kind of arguing into the void.
  - constantcrying 3y ago
    >I know I'm not going to be dealing with lengths longer than 100km, then fixed point would be ideal for that. Huge fallacy. No, you can not use fixed point numbers like that, it can not work. It is irrelevant what the actual maximum/minimum scale you are dealing with is. The thing which matters is the largest/smallest intermediate value you need. You need to consider every step in all your algorithms. Imagine calculating the distance of two objects being 50km in x and 50km in y direction apart. Even though the input and output values fit within your range, the result is nonsensical if you use naive fix point arithmetic. Floating point allows you to write down the mathematical formula exactly, using fixed point arithmetic, you can not do that. Looking at the maximum and minimum resolution you need is a huge fallacy when working with fixed point arithmetic and one big reason why everyone avoids using it. You need to carefully analyze your entire algorithm to use it. >The impression I got from the commenter I was originally replying to was "fixed point is absolutely terrible and there is never a reason to prefer it over floats". My position is that sometimes there might be a situation where fixed point arithmetic could be useful. If you are willing to put in a significant amount of time and effort into analyzing your system and dataflow it can be implememted successfully. It is a far more complex and errorprone system and if you aren't careful it will bite you. In all cases floating point should be the default option and only deviated from if there are very good reasons.