I remember in college when we learned about this and I had the thought, "Why don't we just store the numerator and denominator?", and threw together a little C++ class complete with (then novel, to me) operator-overloads, which implemented the concept. I felt very proud of myself. Then years later I learned that it's a thing people actually use: https://en.wikipedia.org/wiki/Rational_data_type
An other compromise in to use fixed point which is effectively a rational with a fixed denominator. Extremely popular on machines which can handle integer arithmetics but not floating point (since you can trivially do fixed-point arithmetics using integer operations, you just need to be very careful when you handle overflows). If you look at the code of old school games (including classics like Doom if memory serves) the game engine used fixed-point to work on commodity hardware without FPU.
There's also BCD (binary coded decimal) that can solve some problems by avoiding the decimal-to-binary conversions if you're mainly dealing with decimal values. For instance 0.2 can't usually be represented in binary but of course it poses no problem in BCD.
Beware that BCD, and decimal in general, accumulates roundoff error at a much higher rate than binary, if you do any inexact operations.
It is more common these days to use base-1000, instead, when you need exact decimal representations. You can fit three base-1000 "digits" in a 32-bit word, with two bits left over for sign plus any other flag you find useful. (One such use could be to make a zero in the second place indicate that the rest of the word is actually binary; then regular arithmetic works on such words.) Calculations in base-1000 are quite a lot faster than BCD.
Almost always when people think they need decimal, binary -- even binary floating-point, if the numbers are small enough -- is much, much better. Just be sure to represent everything as an integer number of the smallest unit, say pennies; and scale (*100, /100) on I/O.
Performance, correctness, and maintainability. The amount of code needed is very small, and uses native instructions for the work, which are pretty well-tested.
Fixed/floating is an interesting tradeoff for many real-time strategy games too where changes in game state are a synchronized simulation. Fixed point math in software can give more reliable and cross-platform math operations, but with a performance cost (eg: Homeworld: Deserts of Kharak). Using the CPU's floating-point hardware is faster, but you often have to ensure the correct CPU registers are set before doing calculations and those registers can be changed by other software such as a DirectX driver or the operating system (eg: Age of Empires II, Rise of Nations. etc).
I currently build deterministic multiplayer WebGL games in Unity, built via C#->IL2CPP->Emscripten->WASM. The server is the same code base running on Microsoft's .Net runtime.
The chances of being able to run deterministic floating point calculations across this stack is basically zero (even leaving aside that the games are often run on ARM chips), and so we use this library when floats are absolutely necessary (but more often just plain longs):
We also built and shipped a deterministic multiplayer WebGL game[1], but using CoffeeScript[2] + C++ -> Emscripten/dylib/DLLs to run the game in the browser and on Windows and Mac.
Our game would snapshot the entire game state every few seconds and send that back to server to detect desyncs and cheaters. Floating point math, to our astonishment, was not the source of any non-determinism.
I'm 80% sure that only source of non-determinism we encountered were from trig functions, so we just hard-coded lookup tables.
For the most part, I use longs (for instance a FixedVec is a (long,long,long) struct where 1 = 1/1000 of a meter).
However, complicated calculations or anything involving angles or other math functions quickly becomes more convenient when expressed as a Fix64, which is more or less a drop in replacement for float.
I would ideally use Fix64 everywhere, but given the torturous route the C# takes to be transformed into something that's executed on the client machines, my faith in the compiler's ability to generate good code for that is basically zero. I mentally treat long + long as a single instruction, but Fix64 + Fix64 as a function call.
That's rough, fortunately for my own projects I'm only doing Unity on desktop, so I haven't had to go this far.
Even something simple like multiplying up and dividing down quickly adds a lot of overhead, and when running on mobiles you really need all the speed you can get.
> There's also BCD (binary coded decimal) that can solve some problems by avoiding the decimal-to-binary conversions if you're mainly dealing with decimal values. For instance 0.2 can't usually be represented in binary but of course it poses no problem in BCD.
BCD is/was super common in measurement equipment for internal calculations for this reason, and also because it is trivial to format for display (LED/LCD/VFDs) or text output (bus system, printer/plotter).
Many CPUs support BCD, at least in a limited number of ways compared to their normal binary representation.
The 8086 (and its descendants, of course) supports BCD by having instructions to adjust the result after the basic add/sub/mul/div instructions, though only one byte at a time.
The 6502's add and subtract instructions would operate on, and output, BCD values if the special purpose "decimal" flag was set. Again only in 8-bit (two digit) chunks but that is to be expected as it was an 8-bit chip generally.
It's actually in use in many places, for things like handling currency and money, and for when you get funny corner cases involving rounding such numbers and pooling the change.
Whenever I see someone handling currency in floats, something inside me wither and die a small death.
> Whenever I see someone handling currency in floats, something inside me wither and die a small death.
Meh. When used correctly in the right circumstances it is acceptable to use floats.
Here's an example. Suppose you are pricing bonds, annuities, or derivatives. All the intermediate calculations make essential use of floating point operation. The Black–Scholes model for example requires the logarithm, the exponential, the square root, and the CDF of the normal distribution. None of that is doable without floats.
Even for simpler examples it is sometimes okay to use floats. If you only ever need to store an exact number of cents, you can totally store the number of cents in a double. Integer operations are exact using IEEE-754 double operations when they are smaller than 2^53-1 or so. There's usually no benefit of doing so, but hey it's possible.
Currency handling is almost never done with rationals (numerator and denominator) and is frequently (and correctly so!) done with fixed or floating point decimal types.
I develop accounting software for banks, brokerage houses and likes.
Currency, taxes, rebates, etc. handling is NEVER done with floating point.
Whatever you do with money you need predictable, reproducible results. It is norm that calculations are checked by software at two companies on both sides of transaction. Any discrepancies are alarms, bug reports, unhappy customers.
Every significant operation is exactly specified with rounding rules, etc.
For card payments and especially on terminals usually BCD is used.
For everything else usually some kind of arbitrary length decimal library (BigInteger, BigDecimal).
> Currency, taxes, rebates, etc. handling is NEVER done with floating point.
Nonsense. I’ve seen real banking code at reputable banks that uses floats.
> Whatever you do with money you need predictable, reproducible results.
Floats aren’t random. They’re perfectly deterministic, predictable and reproducible. If you do the same operation in two places you get the same result.
I write real banking code. There is definitely a banking code that uses floats, e.g. valuation of financial instruments. The parent comment talks about software that does transactions and “simpler” calculations, like taxes and fees etc.
When people talk about non-determinism of floating point, what they usually mean is non-associativity, that is (x+y)+z may not be exactly equal to x+(y+z).
> When people talk about non-determinism of floating point, what they usually mean is non-associativity, that is (x+y)+z may not be exactly equal to x+(y+z).
Every single time you run those two statements, you’ll get the same result. Yes they're non-associative. But that's specified and documented. That's not the same thing as non-deterministic in any way.
The same code might be optimised in different ways by different compilers, though (or the same compiler with different flags). This might lead to different results for the same code. In that sense, it's non-deterministic.
What is and what is not optimization and what changes are allowed or not depends on the application.
MP3 is an optimization of WAV, yet it changes the result.
Some applications are ok with reducing precision of calculations because they are not sensitive enough to small inaccuracies or they take effort to control inaccuarcies.
For example, graphics applications are typically heavy in FP calculations and yet they tend to not care much about precision and much more about performance. For those applications reducing accuracy for slight performance increase is likely win.
> Floats aren’t random. They’re perfectly deterministic, predictable and reproducible. If you do the same operation in two places you get the same result.
That's not exactly true in real hardware, or at least it wasn't until ~10 years ago. With the x87 FPU, internal precision was 80 bits, while the x86 registers were at most 64 bits. So, depending on the way the program would transfer data between the CPU and FPU your could get different results. It is very likely that different compilers and different optimization decisions could change the way these operations were implemented, so you would get slight differences between different versions of the software.
There are/were also several global FP flags that could get changed by other programs running on the same CPU/FPU that could impact the result of calculations. So, if you want 100% reproducible FP, you would have to either audit all software running on the same machine to ensure it doesn't touch those flags, or set the flags yourself for every FP calculation in your your program.
It's not false - strictfp mandates deterministic FP. If you use that your program will always run all floating point calculations in exactly the same way, full stop.
Secondly, on mainstream implementations, strictfp is already documented the same as default! They're planning to remove it anyway as it's a no-op in almost all cases.
> It's not false - strictfp mandates deterministic FP. If you use that your program will always run all floating point calculations in exactly the same way, full stop.
If you use it. Which is not the default. Your original claim remains false.
It does not matter. When you are doing accounting you are supposed to be able to sum large collections of numbers and get the same result regardless of the order.
That's something FP does not provide and it makes it completely unusable for accounting.
It seems you have no idea what double-side accounting is.
Each operation is accounted on two opposite sides of various account in a way that always keeps sides balanced (ie. they must sum up to the same value).
When you go to your bank account, for example, you have various sums on both sides of your account. Yet when you sum them up they MUST agree or you will be crying blood and suing your bank.
True, that's a good point. I was thinking of C & C++, but you're right, newer languages do a much better job of specifying and controlling this behavior.
It's not about IEEE754, it's about the precision that the FP co-processor offers. The results you get are correct per IEEE754, it's just that they may have even less error than required by IEEE754 in some cases. But, this is enough to make the results non-deterministic between different compilation options.
Also, changes applied to the FP co-processor by other processes on the machine could impact your process, regardless of your own compilation settings.
That's ancient history. Compilers don't use that instruction set any more in normal operation.
GCC, Java, LLVM, etc, will normally emit SSE2 in order to be standards compliant. They will only relax this if you tell them to, then it's your problem.
Yes, I was explicitly talking about the x87, and did mention that it has stopped being relevant for at least 10 years.
I believe there is still quite a bit of cautionary discussion of floating point numbers that was written in the age of the x87, so it's important to understand that people were not just misunderstanding IEEE754, even though their concerns are no longer applicable to modern hardware.
I did not say floats are random. But when you do accounting you need to be able to sum large sets of numbers and compare results with another sum of different numbers and the sum must match. This just does not work with FP.
Poor souls that use FP for accounting are scourge of the industry and source of jokes.
That's what I used to think, then I met these banking types, and they told me 'no we understand their semantics and we use them correctly and we know it is safe for our programs.' These teams have compiler experts on them - they aren't ignorant.
I started working on accounting software in 2002 and right now work for Citi. Compiler experts in accounting? If you are doing HFT you are not doing accounting. Accounting is what happens later when all those transactions need to actually be accounted for and balance calculated
I don't see any problem with it if it's decimal. Here's an accepted answer on stack overflow with hundreds of upvotes recommending the use of `decimal` to store currency amounts in C#. That's a decimal floating point type.
There are different implementations, and BCD is only one of them. Another popular one is a mantissa and exponent, but the exponent is for a 10-based shift rather than the typical floating point.
Tbey mean radix-10 floating point, as compared to the radix-2 floating point you are thinking of. The packing of the decimal fractional digits in the significand of a radix-10 FP number need not be in BCD, it can use other encodings (e.g., DPD or something else).
0.3 is exactly representable in radix-10 floating point but not radix-2 FP (would be rounded to a maximum of 0.5 ulp error as seen in the title), for instance, just as 1/3 = 0.3333... is exactly representable in radix-3 floating point but neither radix-2 or radix-10 FP, etc.
Right, it is not correct. But many programs do it wrong. If you just do a couple of additions the problem will never be noticed. It's easy to write a program that sums up 0.01 until the result is not equal to n * 0.01. Not at my computer now, so I can't do it again. I remember n was bigger to be relevant for any supermarket cashier. But of course applications exist where it matters.
> It's easy to write a program that sums up 0.01 until the result is not equal to n * 0.01.
It's not easy to do that if you use a floating point decimal type, like I recommended. For instance, using C#'s decimal, that will take you somewhere in the neighborhood of 10 to the 26 iterations. With a binary floating point number, it's less than 10.
Of course with a decimal type there is no rounding issue. That's not what 0.30000000000000004 is about.
Many languages have no decimal support built in or at least it is not the default type. With a binary type the rounding becomes already visible after 10959 additions of 1 cent.
That is simply not true. The C# decimal type doesn't accumulate errors when adding, unless you exceed its ~28 digits of precision. E.g. see here: https://rextester.com/RMHNNF58645
It is correct! Using decimal types is the widely recommended way of solving this problem. That includes fixed and floating point types. The problem is using base-2 floating point types, since those are subject to the kinds of rounding errors in the OP. But decimal floating point types are not subject to these kinds of rounding errors.
But they still can't precisely represent quantities like 1/3 or pi.
> Using decimal types is the widely recommended way of solving this problem.
No, it's not. The widely recommended way of solving this problem is to use fixed-point numbers. Or, if one's language/platform does not support fixed-point numbers, then the widely recommended way of solving this problem is to emulate fixed-point numbers with integers.
There is zero legitimate reason to use floating-point numbers in this context, regardless of whether those numbers are in base-2 or base-10 or base-pi or whatever. The absolute smallest unit of currency any (US) financial institution is ever likely to use is the mill (one tenth of a cent), and you can represent 9,223,372,036,854,775,807 of them in a 64-bit signed integer. That's more than $9 quadrillion, which is 121-ish times the current gross world product; if you're really at the point where you need to represent such massive amounts of money (and/or do arithmetic on them), then you can probably afford to design and fabricate your own 128-bit computer to do those calculations instead of even shoehorning it onto a 64-bit CPU, let alone resorting to floating-point.
Regardless of all that, my actual point (pun intended) is that there are plenty of big ERP systems (e.g. NetSuite) that use binary floating point numbers for monetary values, and that's phenomenally bad.
In this case, it's both. Decimal floating point types do not lose precision with base-10 numbers, unless using trig, square roots, arbitrary division and the like.
It's not always terrible. I've seen doubles appropriately used in cases where performance was paramount, and floating point error was either not relevant or less important.
That said, yeah, when working with money in situations where money matters, some sort of decimal or rational datatype should be the rule, not the exception.
Storing money in floating point is always terrible. If speed is an issue, store it in integer types representing the smallest unit in the currency, e.g. pennies.
Unless you’re doing, what, massively parallel GPU algos on batches of independent amounts? But even then you could use the float as an int in that way... Honestly when is float ever actually good for money? Not for speed, not for correctness, ...
I think you mean that storing money in floating point is always terrible for accounting. Not all of finance is accounting.
Imagine you work at a hedge fund, and you have a model that predicts the true value of some option. Assume the option is trading for $3.00. You do not really care if your model spits out $3.5 or $3.5000000001, you are going to buy either way. And your model probably involves a bunch of transcendental functions or maybe even non-deterministic machine learning, so it's not really meaningful to expect it to be “exact” to some decimal or even rational value.
Even more saliently, you probably don't care whether your model outputs 2.9999999 or 3.000000 or 3.000001, either, because in any of those cases the actual correct interpretation is “we’re just not sure whether to buy or not”.
I think a good first-order characterization of domains where floating point can safely be used is “when the difference between < and <= is not very meaningful” (in calculus terms: when “how meaningful is a difference of `x`” is a continuous function of `x`).
I think the "floating point are bad for storing currencies" is one of the most common misconception about floating point.
Most people don't realize that the IEEE-754 single precision floating point represent real numbers with 9 decimal digits (or 23 binary digits). The double, on the other hand, represents the real numbers with 17 decimal digits.
This means that the double error UPPER BOUND is (0.00000000000000001)/2 per operation. But in reality the error is lower because of the rounding operations.
Also, it is posssible to extend the range using denormals, but most (all?) compilers disable them when compiling with anything other than O0 to avoid performance degradation.
The overheads associate with dealing with non-float types for most applications might not be worth it the cost and risk. If course, if the language are working with provides a currency type, go for it. But if doesn't , there is no need to worry.
> Most people don't realize that the IEEE-754 single precision floating point represent real numbers with 9 decimal digits (or 23 binary digits). The double, on the other hand, represents the real numbers with 17 decimal digits.
No, they don't. They merely can be converted back to decimal with those numbers of significant digits without loss of information.
That is important because (a) if this matters, you have to make sure you actually control the number of significant digits when converting to decimal, or you might end up with a different decimal, and (b) the operations that you do on the floats do not reliably behave as if there was the supposedly represented decimal number stored in them.
Now, sure, you can use floats for currency, if you know what you are doing, but the point of the warning against it is that you have to know what you are doing, and chances are you don't, or if you do, then you know where you can ignore it anyway.
(That is, unless you mean nothing more than that you can encode the information contained in an n-digit decimal in a float/double--which of course is true, but not particular to floating point numbers, as any state with a certain number of bits can, of course, encode any information of no more than that many bits, somehow.)
In a previous discussion, someone was worrying about using floats to represent price in JS. I think this is a consequence on the fear mongering on using floats to store currencies.
Floating Points are hard. There is a study done with academics that shows that even researches that works with float point everyday forget about the format intricacies. And the study didn't even look into the compiler mess.
But I agree with you, some (a lot?) of caution is needed when working with float point is good.
If you're certain that your software will never handle national-economy-scale or hyperinflationary use cases, then sure, you may be able to get away with 64-bit floats, but I think "no need to worry" is overstating your case. Please do worry about precision until you've proven you don't need to.
You probably want some smaller unit than a dollar for currency as well, in which case it becomes of a problem with even smaller amounts.
I really see no reason to use any other representation for currency than decimal fixed point. Store the amount as mils or whatever unit suits your use case.
For the vast majority of person, there is no need to worry so much about using fp to represent currencies. There are other issues with float that will bite you in your back before precision became one of them.
Depending on context, you can assume that precision will bite you.
The problem is that rounding is kind of a big deal in certain financial contexts, and the process of rounding can greatly magnify floating point's decimal precision problems when you're dealing with numbers that are close to the .5's.
When I said up above that there are some contexts where IEEE floats are fine, those contexts are largely ones where you never have to round, or where you can guarantee that an accountant is never going to see or care how you rounded. So, to an approximation: Go ahead and fearlessly implement the Black-Scholes formula using doubles, but never, ever use them to do something simple like calculating an invoice.
I worked with a CSV containing, among other things, phone numbers. A coworker called and complained that the phone numbers were all wrong. He'd edited the thing in MS Excel, which promptly converted the phone numbers to floating point with a loss in precision. When he saved it, those new numbers were happily written back to the disk.
I agree with your overall point: it most likely does not matter when the values are close enough. However :)
There can be two companies with 100M market cap.
Corp A has issued 10M shares @ 10 each, Corp B has 10B shares priced at 0.01
A +/-0.001 change in Corp A share price is just 0.01% and moves the market cap by +/-10k, so probably nothing significant. The same nominal change in Corp B amounts to 10%, or +/- 10M in the company value, which is quite a big deal.
Also I think there may be some money to be made in changes at the 7th decimal place with large enough volume of high frequency transactions.
Because of the way floating point numbers work, you'd get an accurate amount for both cases, as it's really the number of significant figures, not decimals.
And that roughly captures the spot where I was seeing doubles used.
Yes, they could have used fixed point. I am guessing that what happened is that someone who had thought way more deeply about this than I ever needed to (I worked on the accounting side, where, yep, we always used decimals) either determined that, where the modeling was concerned, floating point errors were not worth worrying about, or estimated that the expected cost to the company stemming from bugs due to to fixed point math being easier to goof up on would have been smaller than the expected cost to the company due to floating point error.
To see 0.1 error using _double_ you have to do at least 2*10^17 operations (assuming the worst case scenario and no subnormals).
If you are working with such huge numbers, 0.1 cents is probably a cost you are willing to pay to avoid expending thousands in a software solution. The saving with power saving using a floating point is likely greater than power your computers will have to expend to get a precise solution.
When adding numbers with large magnitudes differences (around 10^17 I think) it might exceed the format precision. I should have taken that in account when defining the error boundaries.
In dollars, you start having issues with cents when working with a tens of trillions.
For the vast majority of people this won't be an issue.
I can give you 1.0 error. Take a handful of numbers that add up to 1.5, sum them, and then round that result to the nearest unit.
I'm too lazy to figure out a specific example, but sets of numbers where doubles round up and decimals round down (or vice versa) aren't terribly uncommon.
My day job is high performance financial model implementation. Floats storing dollar amounts are the norm for predictions. Operating on values that are linear combinations of integer fractions multiplied by irrational constants (such as Euler’s number) is perfectly possible, but it’s much more performant to be aware of floating point epsilon when writing modeling code.
Financial models are predictive, they don't have to be accurate to a penny, right? Unlike processing actual money people own.
(I do some work with predictive simulations about money, but outside finance, and there we care that the result has accurate order of magnitude. Floats were used extensively in the project; I actually upgraded them to doubles for the sake of handling larger order of magnitude spans.)
That’s right. The trading desk also uses floats for analysis and regulatory reporting. Actual account balances come through an API that gives us floats, but rumor has it that it’s backed by Hollerith a punch card library maintained by cybernetic undead, encoded in 1215-EBCDIC-BLACKTONGUE.
>“They found out that if you priced your gas 1/10 of a cent below a break point, let’s say 40 cents a gallon, ‘.399’ just looked to the public like 39 cents…”
Tarsnap goes as low as counting attodollars. Yes, that's 10^-18 dollars, judging by the precision with which individual line items and total account funds are reported. Storage price is 250 picodollars per byte-month.
Storing money in floating point is fine. Just round to the nearest atomic unit when displaying. Sometimes this is a necessity when working with money in e.g. existing JSON APIs. You lose a few bits of range relative to fixed point storage but it's almost never a practical issue.
Performing arithmetic operations against money in floating point is the dangerous part, as error can accumulate beyond an atomic unit.
> Performing arithmetic operations against money in floating point is the dangerous part, as error can accumulate beyond an atomic unit.
A good example of this is trying to compute the sales tax on $21.15 given a tax rate of 10%. The exact answer would be $2.115, which should round to $2.12.
IEEE 64-bit floating point gives 2.1149999999999998, which is hard to get to round to 2.12 without breaking a bunch of other cases.
Here are three functions that try to compute tax in cents given an amount and a rate, in ways that seem quite plausible:
I did some exhaustive testing and determined that storing a money amount in floating point is fine. Just convert to integer cents for computation. Even though the floating point representation in dollars is not exact, it is always close enough that multiplying by 100 and rounding works.
Similar for tax rates. Storing in floating point is fine, but convert to an integer by multiplying by an appropriate power of 10 first. In all the jurisdictions I have to deal with, tax rate x 10000 will always be an integer so I use that.
Give amt and rate, where amt is the integer cents and rate is the underlying rate x 10000, this works to get the tax in cents:
> IEEE 64-bit floating point gives 2.1149999999999998, which is hard to get to round to 2.12 without breaking a bunch of other cases.
Your issue is on how to print the float, not with the precision of fp. For instance, `21.15 * 0.1` can be print both as 2.115 or 1.12 depending on how many decimal digits of precision you set your print function. I manage to get those results with printf using `%.3f` and `%.2f`, respectively.
To produce one cent (0.0x) error with the default FP rounding, it takes more than 1 Quadrillion of operation. Each operation can only introduce 1*10^17/2 error.
The "you shouldn't be using float to do monetary computation" is likely one the most spread float point misinformation.
The issues with your others examples is that you are rounding the data (therefore, discarding information). If you don't do any manual round, the result should be correct (I haven't test thought).
> Your issue is on how to print the float, not with the precision of fp. For instance, `21.15 * 0.1` can be print both as 2.115 or 1.12 depending on how many decimal digits of precision you set your print function. I manage to get those results with printf using `%.3f` and `%.2f`, respectively.
I get 2.115 with %.3f and 2.11 with %.2f. Here's my test program. Same result on my Mac with clang and my Debian 8 server with gcc.
The thing is that if 2.115 represents a calculated dollar figure, such as the value of some transaction or the cost of something or whatever, then we should round it to 2.12. (Unless we are working in a financial domain that deals with fractions of a cent.) Now in floating-point, we don't exactly have the exact value 2.12, but we have something that is extremely close. So close that if we happen to print it to %.3f, we better get 2.120, and if we print it to %.4f, we better see 2.1200.
That some monetary calculation works out to $2.115 (and is left that way) instead of being correctly rounded $2.12 doesn't add up to a valid argument against using floating-point for money.
I think piadodjanho does have a point there in the grandparent comment; "don't use floating-point for money" may just be a repeated mantra that doesn't entirely hold water. If extremely accurate engineering and scientific calculations can be done with floating-point, surely we can get floating-point values to measure stacks of pennies with the proper care in the programming.
> If extremely accurate engineering and scientific calculations can be done with floating-point, surely we can get floating-point values to measure stacks of pennies with the proper care in the programming.
That was for a long time my position. I definitely have commented before either here or in /r/programming to the effect that floating point is fine for money as long as you are aware that it is not exact and not associative, and take that into account when doing your calculations.
Any intermediate result in a calculation chain might be off a tiny amount from the exact value, but if you just rounded to the nearest 0.01 before you accumulated enough error to not < 0.005 off, you'd be fine.
I think that's probably true for addition of money amounts. If you have a large number of costs to add up, for example, you should be able to add thousands of them, round to nearest 0.01, and get the right result.
But for tax calculations, such as 10% of $21.15, 0.1 x 21.15 = 2.1149999999999998 in 64-bit IEEE floating point, and rounding the nearest 0.01 gives 2.11, not the 2.12 that we want. A call to fesetround(FE_UPWARD) makes that come out 2.115, and then rounding to the nearest 0.01 gives the desired 2.12.
Will FE_UPWARD make this work for all amounts and tax rates, or are there amounts and rates where we need FE_TONEAREST or FE_DOWNWARD? If so, how do we tell which one we need? Like I said earlier:
> I'm not fully convinced that you cannot do all the calculations in floating point, but I am convinced that I can't figure it out.
PS: calculating tax in cents given double amt, rate, using this method:
tax = amt * rate;
cents_tax = round(100 * tax);
almost works if the rounding mode is FE_UPWARD. For all amounts from 0.01 through 99.99, and all tax rates from 0.01% through 10.99% in increments of 0.01% it works except for 3.75% of $67.60 and 7.5% of $33.80.
> but if you just rounded to the nearest 0.01 before you accumulated enough error to not < 0.005 off, you'd be fine.
And in run-of-the-mill, everyday finance, there simply isn't enough calculation stuffed in between the concrete monetary points that are recorded in the ledger.
> If you have a large number of costs to add up, for example, you should be able to add thousands of them, round to nearest 0.01, and get the right result.
Exactly.
> But for tax calculations, such as 10% of $21.15, 0.1 x 21.15 = 2.1149999999999998 in 64-bit IEEE floating point, and rounding the nearest 0.01 gives 2.11, not the 2.12 that we want.
This problem will be there even if we use integers for the currency amounts, but floating-point only for these fractional calculations.
Luckily for us Canadians, I'm pretty sure the Canada Customs and Revenue Agency won't care which way you call this rounding. They also don't collect or refund overall discrepancies of less than around two dollars in a single tax return.
I think I've been mostly rounding taxes down over the years, and tax credits up. E.g. if a tax credit is $235.981..., I make it 235.99.
The myth that has been foisted on programmers is that if you use floating-point for numbers, the actual ledgers won't balance, and sum totals of columns of figures will appear incorrect if verified by pencil-and-paper arithmetic. That will certainly be true if the math is done very carelessly; and it's true that it's easier to get it right with less care using integers.
A percentage calculation whose rounding is called the wrong direction will, in and of itself, not cause such a problem. E.g. if we split some sum of money into two complementary percentages, we can do it such that the two add up to the original.
You have to be careful not to do this as two independent percentages. Like, dont take 10% of 21.15 and then 90% of 21.15, individually round them to a penny, and then expect them to add up to 21.15. It has to be centround(21.15 - centround(.1 * 21.15)) to get the 90% residue.
The trick is that by default rounding happens using banker's rounding. Programming languages use this because this is what CPUs use. When you want to round your way, you need an extra digit and round manually:
You're right. My remainder calculation in my code snippet is incorrect. It should've been a floating point remainder instead.
import math
def tax_f5(amt, rate):
t = amt * rate * 1000
return round(t) // 10 + ((math.fmod(t, 10.0) - 5.0) > -1e-7)
But then since there's now an epsilon, it raises the question of how many digits of precision the tax rates typically need. This is indeed a difficult problem.
Some exhaustive testing on all amounts from $0.01 through $999.99 in $0.01 increments and all taxes from 0.01% through 99.99% in increments of 0.01% show that this is the minimum that does the trick (switching to C from Python for speed):
(Yes, I see that I goofed in translation your code to C and typed -1e-5 instead of -1e-7. It looks like the results are the same with -1e-7).
I also tested that up through $9999.99 with taxes up to 12%, and no problems.
Adding another 0 to the 1000000, the two 10000's, and the 5000 works. And another, and another. Past that it starts to fail, but not the simple off-by-one failures you get when you don't use enough digits. These are way way off, so I'm guessing its running into some new class of problem. I haven't looked to see what that is yet.
> Storing money in floating point is fine. Just round to the nearest atomic unit when displaying.
Well, it's not just a display issue. In accounting, associativity and commutativity are important. People do care that `a + b + c - a == c + b` should evaluate to “true”.
It appears you did not see the critical point in the above comment. "Performing arithmetic operations against money in floating point is the dangerous part, as error can accumulate beyond an atomic unit."
There's very little point in storing money in floats if you're not going to do arithmetic in floats; about the only use case I can think of is JavaScript and JSON APIs.
Even decimal floating point is a bad idea (for dealing with money) since you still can't represent a subset of rational numbers without approximation and without introducing rounding error during some calculations. It's just a different subset than what binary floating point can represent without approximation.
Well, this is one of those things where context matters.
In trading, it's super common to use floating point arithmetic for decision logic since it's very fast and straightforward to write. The actual trade execution, however, almost always relies on integer arithmetic because then money is actually being used (and hence must be tracked properly).
It's not therefore inherently incorrect to do currency conversions with floats in some situations provided that the actual transaction execution relies on fixed precision or decimal arithmetic.
When I was in college the professor of my software engineering class explicitly warned us to never use floating point numbers for money. He went on at length of the dangers of floating points for dealing with money and warned us that people can get really upset if they feel like they've been screwed out of money.
He had decades of experience in the software development industry and I got the feeling that he'd seen the effect of this issue personally.
I haven't worked in fintech but I've read that money is often represented (at least in storage) as plain integers, since for example US currency only ever goes to two decimal places. But I guess once you start operating on it you run into potential truncation unless you use rationals.
In finance, US dollars are generally stored to four decimal places, because you need to deal with stuff like compounding interest or stock splits.
COBOL has a built in fixed point integer type, which makes defining a 4 digit decimal and doing math on it easy. (IBM designed it from the ground up to cater to people with a lot of money, who spend a lot of money, to work with lots of money, ie banks) Java has the BigDecimal type, which is a class in the class library, which means you need to import it. And because Java lacks operator overloading, doing calculations is tedious.
In the 90s, there was a huge push to replace COBOL with <something else>, and Java was the Rust of its day, so that's what everyone got behind. However, 4 digit COBOL decimals apparently round differently than 4 digit Java BigDecimals, so all the tests failed. And all the stuff like a\x+b had to be written like BigDecimal.add(BigDecimal.multiply(a,x),b) so development was taking forever.
Eventually they said "fuck it" and 20 years later we're still stuck with COBOL and everyone who remembers the original death march says "never again".
I have a feeling a lot of the problems came down to computer science people thinking money has two decimal digits but domain knowledge people knowing it has four. We programmers, as a group, make a lot of assumptions about other peoples' domains and we're wrong a lot*.
I've had the thought that programmers should note assumptions in flagged comments, and those comments should be automatically collected, and then reviewed occasionally. Assumptions might be sustainable, so to speak, but they can also create one kind of technical debt.
For a brief time in 2008, 1 Zimbabwe dollar was very roughly equivalent to one TRILLIONTH of a United State penny. So technically a value of “1” did exist, but it was meaningless. I have some of the 100 Trillion Dollar notes from Zimbabwe from that time period.
While it isn't physical US currency, my brokerage account represents the value of the account in units of USD- therefore any rules about how US currency works should apply.
Additionally, fractional cents are often presented to the consumer when purchasing gas/fuel.
Money is not as wierd as anyone might guess. I work on a financial application, and money is almost always just a BigDecimal with the scale set to 2 (and stored in a database as a bigint type or equivalent). When its not, its just a higher scale (for say, compound daily interest on small amounts for a significant period of time).
Yes, the parent approach is the same thing but with a decimal point added for convenience. The main thing is that the scale is fixed; it is effectively an integer count of 1s of the least significant index. It is impossible to truncate values or round upward and create money that didn't exist. This makes it perfect for representing actual money.
No it can't. There are systems that track things worth less than a penny for later billing, but at the end of the month when they bill someone, they do some sort of rounding.
If you are earning interest at a bank, and you've earned a fraction of a penny, they will eventually pay it to you once you've earned enough for a whole penny.
i.e. they track your account balance to more than 2 digits, they just only show you 2 digits.
Someone should tell that to everyone who ever used a ½¢ coin in the US. Also, US law explicitly states (31 USC §5101) that the unit of 1/1000th of a dollar is a mill.
Go to a bank and ask for a $500 or $1000 note too. They won’t give you one as they aren’t in circulation anymore, but most (all?) will let you deposit it for face value.
Take two half penny coins or notes to the bank and they’ll credit your account a penny. Of course, just like with the $500 or $1000 notes, you’ll be losing money on the deal.
> A mill ... is perhaps a tad outdated.
Yet you use mills every time you pay for gas. Pointless in that case? Probably. Still used all the time? Certainly.
You can still get reasonable enough approximations with more than two decimals if you do something like `int64 myWorkingMoneyVal = currentMoney * 100000`, do your work, then divide the final result by 100000. You still risk some potential truncation if your work involves division, but the larger your multiplier that you're working with, the larger divisor at the end, which will help minimize how much of an error this ends up being. The 64 bit integer space is pretty darn big, so you typically don't risk an overflow, and you will typically get better performance than using a regular "decimal" type, since on-chip integer operations are usually very fast.
EDIT:
Just a note, there's nothing special about the number 100000; pick the largest exponent of 10 that you can get away with a reasonable assurance that no overflow is possible. For a vast majority of money applications, I seriously doubt you're going to be hitting the limits of int64, so you could probably even get away with something like 1000000000.
I didn't know that, but it doesn't surprise me (I suspected I wasn't the first person to come to the realization that there's no reason not to choose a giant number :) ).
I have developed a payment plan calculator for asset based finance and you would be amazed how many different rounding schemes and day counters (for fractional periods) exist and are actively used.
It isn't really a price though, it is a rate for an infinitely divisible good, i. e. $/L. You get the price when you multiply with the quantity purchased.
The basic representation is usually good enough at 2 decimals (so a plain int), but it is often needed to have a transient representation during calculations.
For instance if one needs to apply discounts, add taxes, split in equal parts, all of the above one after the other, there will be a more precise intermediary representation before rounding everything in a way that keeps the total amount consistent with the original amount.
> It's actually in use in many places, for things like handling currency and money
Hm, are you sure? I don't believe "rational" types which encode numbers as a numerator and denominator are typically used for currency/money.
If they were, would the denominator always be 100 or 1000? I guess you could use a rational type that way, although it'd be a small subset of what rational data types are intended for. But I guess it'd be "safe"? Not totally sure actually, one question would be if rounding works how you want when you do things like apply an interest percentage to a monetary amount. (I am not very familiar with rational data types, and am not sure how rounding works -- or even if they do rounding at all, or just keep increasing the magnitude of the denominator for exact precision, which is probably _not_ what you'd want with currency, for reasons not only of performance but of desired semantics).
You are correct an IEEE-754 floating point type is inappropriate for currency. I believe for currency you would generally use a fixed-point type (rather than floating point type), or non-IEEE "arbitrary precision floating point" type like ruby's BigDecimal (ruby also offers a Rational type. https://ruby-doc.org/core-2.5.0/Rational.html . This is a different thing than the arbitrary-precision BigDecimal. I have never used Rational or seen it used. It is not generally used for money/currency.) Or maybe even a binary-coded decimal value? (Not sure if that's the same thing as "arbitrary-precision floating point" of ruby's BigDecimal or not).
There are several possible correct and appropriate data encodings/types for currency, that will have the desired precision and calculation semantics... I am not sure if rational data type is one of them, and I don't believe it is common (and it would probably be much less performant than the options taht are common). Postgres, for instance, does not have a "rational" type built in, although there appears to be a third-party extension for it. Yet postgres is obviously frequently used to store currency values! I believe many other popular rdbms have no rational data type support at all.
I'm not actually sure what domains rational data types are good for. Probably not really anything scientific measurement based either (the IEEE-754 floating point types ARE usually good for that, that is their use case!) The wikipedia page sort of hand-wavily says "algebraic computation", which I'm not enough about math to really know what that means. I have never myself used rational data types, I don't think! Although I was aware of them; they are neat.
Good catch! I'm thinking of fixed-point number types. Ruby's Rational was/is cool, but looks like an inherently difficult number-type to work with and keep sanity high.
For currency, business side should decide the rules (* 100 or * 1000000), and where to funnel the pennies ;) Fixed-point has it's own sort of gotchas, ie. multiplication, power, division, sqrt, etc. So there are fancy techniques to work with the numbers, like https://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm
When you start looking into all of this, it's interesting to see how many ways there are to represent numbers in a computer. It isn't actually obvious or trivial at all, there is no one "true" or "accurate" representation, and they all had to be invented by past computer scientists!
If you're not a bank or similar but just dealing with currency to buy and sell things like for ecommerce, the default semantics of a fixed point type (like postgres 'money' type) or "arbitrary precision floating point" type like ruby's BigDecimal or are probably good enough, and just fine in a way that IEEE-754 floating point definitely are NOT. And probably don't require any additional business side decisions or involve any significant gotchas. Just using them instead of IEEE-754 floating point and not thinking too hard after that is probably just fine.
If you ARE a bank or something similar -- I wouldn't know, I haven't done that! A relevant question: Am I concerned with specifying exactly how fractional pennies get rounded?
Ask the COBOL guys and gals for the true answer ;)
There are accounting, balancing, laws, regulations and reconciliation issues where for the really serious stuff, you use whatever fit spec and requirements, not the other way around. Ruby's BigDecimal will be fine, if you implement the detailed specification about how to calculate each operation every step of the way, with designated precision at various steps, together with truncations along the way that may not make much sense to the developer (or anyone else, but are required to get correct numbers).
Point is, sometimes other parties need to be able to replicate the exact numbers, unrelated to any internal library or coding standards. Code using just plain integers could be easier to certify than a library dependency.
In such cases, you don't just round to make numbers prettier, but may even keep the truncated part of the equation. It's then nice to use simple stuff that are proven to work and not change over time.
Have you worked on ecommerce where you've been given such detailed specs? I have worked on ecommerce, I never have been.
I don't even understand exactly what you mean by "implementing detailed specifications about how to calculate each operation every step of the way" with ruby BigDecimal. Can you provide an example? Ordinarily, you just use ruby `+` and `*` etc operators with BigDecimal values. (Or SQL/postgres arithmetic operators with postgres money type). I don't even know what a "specification about how to calculate each operation every step of the way" would look like. This is something you've had to do with basic ecommerce apps?
I'm not sure what solution you are suggesting could be characterized as "simple stuff that are proven to work and not change over time"
Seriously, most everyone just uses something like BigDecimal or postgres Money type, and it's fine. (IEEE-754 float is NOT though. Neither, probably, is the rational type that you initially suggested... )
>If they were, would the denominator always be 100 or 1000?
The numerator and denominator get automatically reduced to lowest terms (just like you learned in elementary school, so 15/100 becomes 3/20) internally by every implementation I know of. This comes at a performance cost for every operation, but it helps keeping the numerator and denominator from blowing up.
>not sure how rounding works -- or even if they do rounding at all
They do not. The point of a Rational type is to keep precise values, so it's up to the programmer to decide when and how values are rounded.
>"arbitrary precision floating point" type like ruby's BigDecimal
Not sure how Ruby implements BigDecimal, but Java internally represents it as an BigInteger of digits, and a second integer that represents where in the number the decimal point should go. This means that BigDecimal still can't truly represent a value such as 1/3, since you can't have an infinite amount of 3's, but a Rational can.
>I'm not actually sure what domains rational data types are good for.
I'll be honest and say I've never had to use them either, but it's nice to know they exist. The intended use case is when you need to perform calculations and maintain as much precision and accuracy as possible in the intermediate values and such accuracy is more important than speed.
Currency in banking is handled with bigints. Not rationals, just bigints of the smallest unit (i.e., 1 cent). This forces you to order operations so that divisions are done last or not at all.
There is a slight difference where it forces decision-making about rounding at every step. Most importantly, it makes errors that print money (or burn money) impossible states. I'm aware of BigRational libraries or more clever currency abstractions, but this is the least magical way of doing it, kind of like representing datetimes as 64-bit UNIX epochs.
Well, you could look at it as x/100 rationals representing a dollar value. But you could also look at it as an integer amount of a smaller unit (cents). The difference is insignificant; computers support it natively.
I actually ran into a bug recently while implementing my first raytracer, where the point calculated from the sphere-intersect test would just occasionally end up inside the sphere due to floating point imprecision, so the diffuse sample rays would have their origins completely in the dark, leading to randomly black pixels. Solved it by bumping every intersection out by 0.01 in the direction of its normal.
And then of course there have been several other "x.abs() < 0.01" cases for various purposes. So I could definitely see that being an interesting experiment.
That's really interesting - hadn't thought of that before. To fix that, would you be able to do a square of the magnitude comparison with the radius and just bump the borderline cases, or is it more efficient without the extra branching?
I just did it across the board; since the error is in the floating-point noise I don't know if I'd even trust a comparison on that. Plus, the discrepancy between "bumped" and "unbumped" samples might cause some visible artifacts.
The result is that your number system (a) makes many common operations dramatically more computationally expensive, (b) has less predictable rounding which is very tricky to reason about, (c) generally gives worse results for the same bit budget. Instead of e.g. evenly spaced numbers, you get spacing like https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_fu...
One thing you can try is storing a floating point numerator and a floating point denominator, and renormalizing them by bit shifts instead of finding GCDs. This lets you avoid rounding errors for small ratios. For general purposes this advantage isn’t really worth doubling the number of bits and complicating arithmetic for though.
But rationals are more expensive to compute with (compared to floating-point; this is another example of the trade-off between performance and accuracy.)
it's also a range-storage trade-off. if you use two fixed width integers to represent a rational, the minimum and maximum values are the same as that of the integer type. floating point gives a far wider range for the same number of bits.
I'm sure there's some subtlety I'm missing, but isn't it actually the same trade-off? A 64-bit float can only represent 52-bit integers exactly. Anything above that, and you don't even have integer-level precision on the number anymore... This sliding scale of precision is exactly why floats are terrible at the kinds of operations that would cause you to use a rational instead.
> I'm sure there's some subtlety I'm missing, but isn't it actually the same trade-off?
not exactly, unless you consider space efficiency to be an aspect of performance (which is certainly reasonable). a naive implementation of rationals using two int32_t's only covers the range of a single int32_t, despite using as many bits as the double. it's also a trade-off between range and consistent precision, of course.
this certainly isn't some deep insight into number representation, just a quick point for the benefit of people who haven't thought much about rational data types before.
Once you care about that level of performance, you can surely optimise your representation to have a greater range (use more bits for the numerator) or greater precision (more bits for the denominator) or some boutique solution like using three integers to store the number a + b/c.
You can store slightly fewer numbers with rationals, because it's hard to avoid having a representation for both 2/4 and 3/6. But the loss of range or precision due to that is pretty small.
It's far too easy to overflow your integer type by simply adding a bunch of rationals whose denominators happen to be coprime, or just by multiplying rationals. For this reason, the vast majority of rational implementations use arbitrary precision integers, and of course arithmetic on those isn't constant time.
One approach would be to hold on to rationals for as long as possible, to eliminate drift, and then dump them out to the nearest floating-point at the very last moment
IEEE floats are pretty complicated, but today’s CPUs have dedicated support for those and not for rationals, so we use them where we probably shouldn’t.
IEEE floats are absolutely great for many applications where rationals would be overkill or even inappropriate. A videogame doesn't care if the result is 0.3 or 0.30000000000000004. Even some scientific applications can use floats if the coder knows what they're doing.
The problem is devs who don't understand what they're doing and just think that they can use floats in every situation and it'll just work out fine. This is not helped by many popular scripting languages who just default to floats when a result doesn't fit an integer (something more reasonable languages like Common Lisp don't do for instance).
For instance, to speak of videogames again, very tight precision isn't usually an issue but loss of granularity when numbers get very big can cause problems, especially if you have very large levels. That being said rationals wouldn't really help you here, you'd have the same problem except now you have to keep two numbers within bound instead of one. Imagine having a very small offset in a complex operation and ending up with a number like 100000000000000000000000000/100000000000000000000000001 !
Yeah, I assumed as much. I wasn't really thinking about that at the time, but knowing now that it exists in the wild, the only conceivable reason for it not to be used everywhere would be some kind of performance penalty.
When I was reading about this I thought why don't the print functions just by default round to the nearest 10 decimal places or similar so 0.30000000000000004 prints as 0.3 unless you specify you don't want that. And I wrote a function in javascript to round like that though it was surprisingly tricky and messy to do so.
