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> > No, you're generally not throwing out precision when you round in this case. You're throwing out error. > That's not always true from a numerical analysis

> > No, you're generally not throwing out precision when you round in this case. You're throwing out error. > That's not always true from a numerical analysis standpoint. This isn't about numerical analysis, which is what you seem to not be getting. It's about the medium have specific attributes, and floats being incapable of perfectly representing the value. With respect to the currency being tracked, any difference smaller than one hundredth of a full unit (depending on currency) that results from simple addition or subtraction of accurate values is an error because it's not possible in reality. > 3. The cumulative error of a chain of a million operations, each with an error in the 9th digit place, can at most only affect the 3rd decimal digit. The 2nd decimal will always be correct Cumulative error is irrelevant. It only takes a single error that causes a value to be less than the correct value by a very small amount and then if there's any place where it exits the system without rounding, that error may be increased to a full minimum difference of the medium ($0.01 in this case). > If introducing extraneous rounding to a much lower precision magically 'fixed' errors, then how could long floating point calculations themselves be inaccurate? By nature of the actual thing being represented by a the floating point value. There is no point talking about floating point without talking about what it's representing. In this case, it's currency, which has very specific characteristics. You can argue that a sphere is the best container shape because it maximizes volume to surface area all you like, that doesn't mean it's the best container shape when what you are storing is shoe boxes. > Rounding is often appropriate if the rounded value is the actual source of truth Yes, nobody is arguing that you shouldn't round floats in this case. I'm arguing you shouldn't use floats so you don't have to round at all. > Excel's a pretty common tool that they use for basically everything right? It must be sporting some fancy arbitrary precision or decimal machinery under the hood, right? ... Nope, just floats all the way down! Excel is a system to represent values of all types. Their requirements necessitate an amount of flexibility that makes Floats a good choice. Even then, they will be handling all the rounding and representation automatically for you. When you use excel, you aren't using floats, you're using an excel numeric type which is implemented underneath using floats and some very specific behavior. The fact that it automatically deals with floating point errors and rounding is what makes it not a float. In the case we are discussing, the same requirements are not present. We don't have to worry about representing any conceivable value, just the ones allowed by the type. A float provides more flexibility than we need, and at the cost of error that needs to be cleaned up at all the edges of the system. Again, I ask you, why should we choose an underlying type that doesn't fit the needs as well as another option? In all aspects, integer containers either have a less problematic error case or do not suffer the same problem.

smallnamespace 8y ago
> This isn't about numerical analysis, which is what you seem to not be getting. Of course it is. The point of the computation is to get the correct result. Numerical analysis hints at whether your computation will give you the right answer or not. You're asserting that it's trivial to make sure that your fixnums won't overflow, but in most normal cases where you'll fit into a 64-bit int range, you'll also fit into the 52-bit precision range in a double. If you can prove that ints are OK, then you can just as easily prove that doubles are safe to use. And unlike floating point rounding errors, where the answers often have some hint about the correct result, undetected overflows silently give you a totally garbage result. Your system does not magically become 'safe by design' simply because you use ints everywhere, you still need to put in the extra work to make sure your numeric range is always sufficient—at which point you're doing numerical analysis, whether you choose to call it that or not. Honestly, I'm surprised that you will denounce the usage of floating point numbers, repeatedly make untrue claims about FP calculations, then wave off the entire subbranch of CS that studies how numbers are represented on computers and how those calculations can go wrong as being entirely irrelevant. > It only takes a single error that causes a value to be less than the correct value by a very small amount and then if there's any place where it exits the system without rounding, that error may be increased to a full minimum difference of the medium ($0.01 in this case). What? The only case where a number rounds wrongly is when the final value is more than half a cent away from the correct value. If you can't be bothered to understand how FP calculation works, at least please stop making demonstrably false claims and misleading others. > By nature of the actual thing being represented by a the floating point value. As soon as you actually do any 'complicated' math (take a square root or a log), your result is no longer exactly representable, because your results aren't decimals or even rationals, but irrationals. Even an arbitrary precision number type doesn't help you here—your only solution is an arbitrary precision math package—but that isn't what you're suggesting It sounds like you think using fixnums everywhere guarantees you the Platonic exact answer, but they can't actually do that: - If you're doing anything complicated, you will unavoidably be making approximations and rounding. Adding a bunch of extra casts back to fixnums don't allow you to avoid approximation, just a false sense of security. - OTOH, if you're not doing anything complicated, just totaling small numbers multiplied by round constants, then doubles are provably sufficient at getting an exact answer after rounding. > When you use excel, you aren't using floats, you're using an excel numeric type which is implemented underneath using floats and some very specific behavior. The fact that it automatically deals with floating point errors and rounding is what makes it not a float. You're contradicting yourself here. All Excel does is 1) compute using floats (doubles) everywhere and 2) round the result before showing it. That's exactly what I'm advocating. Whether you choose to wrap it into a special Numeric or Money class doesn't change your answer. Look, here's C#'s decimal class: https://docs.microsoft.com/en-us/dotnet/csharp/language-reference/keywords/decimal https://docs.microsoft.com/en-us/dotnet/csharp/language-refe... Must use some fancy arbitrary precision arithmetic right? No, it's just a floating point number—just a particularly wide (128-bit) type.

smallnamespace 8y ago

> This isn't about numerical analysis, which is what you seem to not be getting. Of course it is. The point of the computation is to get the correct result. Numerical analysis hints at whether your computation will give you the right answer or not. You're asserting that it's trivial to make sure that your fixnums won't overflow, but in most normal cases where you'll fit into a 64-bit int range, you'll also fit into the 52-bit precision range in a double. If you can prove that ints are OK, then you can just as easily prove that doubles are safe to use. And unlike floating point rounding errors, where the answers often have some hint about the correct result, undetected overflows silently give you a totally garbage result. Your system does not magically become 'safe by design' simply because you use ints everywhere, you still need to put in the extra work to make sure your numeric range is always sufficient—at which point you're doing numerical analysis, whether you choose to call it that or not. Honestly, I'm surprised that you will denounce the usage of floating point numbers, repeatedly make untrue claims about FP calculations, then wave off the entire subbranch of CS that studies how numbers are represented on computers and how those calculations can go wrong as being entirely irrelevant. > It only takes a single error that causes a value to be less than the correct value by a very small amount and then if there's any place where it exits the system without rounding, that error may be increased to a full minimum difference of the medium ($0.01 in this case). What? The only case where a number rounds wrongly is when the final value is more than half a cent away from the correct value. If you can't be bothered to understand how FP calculation works, at least please stop making demonstrably false claims and misleading others. > By nature of the actual thing being represented by a the floating point value. As soon as you actually do any 'complicated' math (take a square root or a log), your result is no longer exactly representable, because your results aren't decimals or even rationals, but irrationals. Even an arbitrary precision number type doesn't help you here—your only solution is an arbitrary precision math package—but that isn't what you're suggesting It sounds like you think using fixnums everywhere guarantees you the Platonic exact answer, but they can't actually do that: - If you're doing anything complicated, you will unavoidably be making approximations and rounding. Adding a bunch of extra casts back to fixnums don't allow you to avoid approximation, just a false sense of security. - OTOH, if you're not doing anything complicated, just totaling small numbers multiplied by round constants, then doubles are provably sufficient at getting an exact answer after rounding. > When you use excel, you aren't using floats, you're using an excel numeric type which is implemented underneath using floats and some very specific behavior. The fact that it automatically deals with floating point errors and rounding is what makes it not a float. You're contradicting yourself here. All Excel does is 1) compute using floats (doubles) everywhere and 2) round the result before showing it. That's exactly what I'm advocating. Whether you choose to wrap it into a special Numeric or Money class doesn't change your answer. Look, here's C#'s decimal class: https://docs.microsoft.com/en-us/dotnet/csharp/language-reference/keywords/decimal https://docs.microsoft.com/en-us/dotnet/csharp/language-refe... Must use some fancy arbitrary precision arithmetic right? No, it's just a floating point number—just a particularly wide (128-bit) type.