0.30000000000000004(0.30000000000000004.com) |
0.30000000000000004(0.30000000000000004.com) |
It used to be widespread because floating point processors were rare and any floating point computation was costly.
That's not longer the case and everyone seems to immediately use floating point arithmetic without being fully aware of the limitations and/or without considering the precision needed.
I just use Zarith (bignum library) in OCaml for decimal calculation, and pretty content with performance.
I don't think much domains needs decimal floating point that much, honestly, at least in finance and scientific calculations.
But I could be wrong, and would be interested in cases where decimal floating-point calculations are preferable over these done in decimal fixed-point or IEEE floating-point ones.
1. Constants have arbitrary precision 2. When you assign them, they lose precision (example 2) 3. You can format at as a arbitrary precision in a string (example 3)
In that last example, they are getting 54 significant digits in base 10.
The status quo is that even Excel defaults to floats and wrong calculations with dollars and cents are widespread.
This is why you should never do “does X == 0.1” because it might not evaluate accurately
Take that Rust and C ; )
My mental model of floating-point types is that they are useful for scientific/numeric computations where values are sampled from a probability distribution and there is inherently noise, and not really useful for discrete/exact logic.
https://discourse.julialang.org/t/posits-a-new-approach-coul...
I use integer or fixed-point decimal if at all possible. If the algorithm needs floats, I convert it to work with integer or fixed-point decimal instead. (Or if possible, I see the decimal point as a "rendering concern" and just do the math in integers and leave the view to put the decimal by whatever my selected precision is.)
But again, there are clearly plenty of use cases where it's insufficient, as you can vouch. I still don't think you can call it "disgusting", though.
It’s kind of a hallmark of bad design when you have to go into a long-winded explanation of why even trivial use-case examples have “surprising” results.
Fixed point is perfectly OK, if all your numbers are within a few orders of magnitude (e.g. money)
The way people rely on assumptions.
The word you're looking for is "surprising," which is a far cry from non-deterministic. IEEE 754 is so thoroughly deterministic that there exists compiler flags whose sole purpose is to say "I don't care that my result is going to be off by a couple of bits from IEEE 754."
Why? Fintech uses decimal fixed-point all the way, there are libraries for them for any major language. Apps like GnuCash or ledger use them as well.
But the person you replied to said programming logic, not programming anything.
Honestly I think if you care about the difference between `<` and `<=`, or if you use `==` ever, it's a red flag that floating-point numbers might be the wrong data type.
If you're handling money, or numbers representing some other real, important concern where accuracy matters, most likely any number you intend to show to the user as a number, floats are not what you need.
Back when I started using Groovy, I was very pleased to discover that Groovy's default decimal number literal was translated to a BigDecimal rather than a float. For any sort of website, 9 times out of 10, that's what you need.
I'd really appreciate it if Javascript had a native decimal number type like that.
When handling money, we care about faithfully reproducing the human-centric quirks of decimal numbers, not "being more accurate". There's no reason in principle to regard a system that can't represent 1/3 as being fundamentally more accurate because it happens to be able to represent 1/5.
DaysInYear = 366
InterestRate = 215
DayBalanceSum = 0
for each Day in Year
DayBalanceSum += Day.Balance
InterestRaw = DayBalanceSum * InterestRate
InterestRaw += DaysInYear * 5000
Interest = InterestRaw / (DaysInYear * 10000)
Balance += Interest
The value of floating point is that it can represent extremely huge or extremely infinitesimal values.
If you're working with currency / money, floating point is the wrong thing to use. For the entire history of human civilization, currency has always been an integer type, possibly with a fixed decimal point. Money has always been integers for as long as commerce has existed, and long before computers.
If you're building games, or AI, or navigating to Pluto, then floating point is the tool to use.
True but irrelevant. The problem isn't with the math fundamentals, it's the programmers.
The issue is if you get your integer handling wrong it usually stands out. Maybe that's because integers truncate rather than round, maybe it's because the program has to handle all those fractions of cents manually rather than letting the hardware do it so he has to think about it.
In any case integer code that works in unit tests usually continues to work, but floating point code passing all unit tests will be broken on some floating point implementations and not others. The reason is pretty obvious: floating point is inexact, but the implementations contain a ton of optimisations to hide that inexactness so it rarely raises it's ugly head.
When it does it's in the worst possible way. In a past day job I build cash registers and accounting systems. If you use floating point where exact results are required I can guarantee you your future self will be haunted by a never ending stream of phone calls from auditors telling you code that has worked solidly in thousands of installations over a decade can not add up. And god help you if you ever made the mistake of writing "if a == b" because you forgot a and b are floating point. Compiler writers should do us all a favour and not define == and != for floating point.
Back when I was doing this no complier implemented anything beyond 32 bit integer arithmetic, in fact there was no open source either. So you had to write a multi precision library and all expression evaluation had to be done using function calls. Despite floating point giving you hardware 56 bit arithmetic (which was enough), you were still better off using those clunky integers.
As others have said here: if you need exact results (and, yes currency is the most common use case), for the love of god do it using integers.
Um... that really depends. If you have an algorithm that is numerically unstable, these errors will quickly lead to a completely wrong result. Using a different type is not going to fix that, of course, and you need to fix the algorithm.
Now, when you’re talking about central bank accruals (or similar sized deposits) that’s a bit different. In these cases, you have a very specific accrual multiple, multiplied by a balance in the multiple hundreds of billions or trillions. In these cases, precision with regards to the interest accrual calculation is quite significant, as rounding can short the payor/payee by several millions of dollars.
Hence the reason bond traders have historically traded in fractions of 32.
A sample bond trade:
‘Twenty sticks at a buck two and five eights bid’ ‘Offer At 103 full’ ‘Don’t break my balls with this, I got last round at delmonicos last night’ ‘Offer 103 firm, what are we doing’ ‘102-7 for 50 sticks’ ‘Should have called me earlier and pulled the trigger, 50 sticks offer 103-2’ ‘Fuck you, I’m your daughter’s godfather’ ‘In that case, 40 sticks, 103-7 offer’ ‘Fuck you, 10 sticks, 102-7, and you buy me a steak, and my daughter a new dress’ ‘5 sticks at 104, 45 at 102-3 off tape, and you pick up bar tab and green fees’ ‘Done’ ‘You own it’
That’s kinda how bonds are traded.
Ref: Stick: million Bond pricing: dollar price + number divided by 32 Delmonicos: money bonfire with meals served
https://developer.mozilla.org/en-US/docs/Web/JavaScript/Refe...
Floats are a digital approximation of real numbers, because computers were originally designed for solving math problems - trigonometry and calculus, that is.
For money you want rational numbers, not reals. Unfortunately, computers never got a native rational number type, so you'll have to roll your own.
Prior to the IBM/360 (1964), mainframes sold for business purposes generally had no support for floating point arithmetic. They used fixed-point arithmetic. At the hardware level I think this is just integer math (I think?), but at a compiler level you can have different data types which are seen to be fractions with fixed accuracy. I believe I've read that COBOL had this feature since I-don't-know-how-far-back.
This sort of software fixed-point is still standard in SQL and many other places. Some languages, and many application-specific frameworks, have pre-existing fixed-point support. So it's also not accurate to say that you necessarily need to roll your own, though certainly in some contexts you'll need to.
And for money, you very much do not want arbitrary rational numbers. The important thing with money is that results are predictable and not fudgable. The problem with .1 + .2 != .3 is not that anyone cares about 4E-17 dollars, it's that they freak out when the math isn't predictable. Using rationals might be more predictable than using floats, but fixed-point is better still. And that's fixed-point base-10, because it's what your customers use when they check your work.
The type of any numeric literal is any type of the `Num` class. That means that they can be floating point, fractional, or integers "for free" depending on where you use them in your programs.
`0.75 + pi` is of type `Floating a => a`, but `0.75 + 1%4` is of type `Rational`.
Is there a way to exploit the difference between numeric precision underlying the neural network and the precision used to represent the financial transactions?
Was proposed in the late 90's Mike Cowlishaw but the rest of the standards committee would have none of it.
Proposal: https://github.com/littledan/proposal-bigdecimal
Slides: https://docs.google.com/presentation/d/1qceGOynkiypIgvv0Ju8u...
Give it =0.1+0.2-0.3 and it will see what you are trying to do and return 0.
Give it anything slightly more complicated such as =(0.1+0.2-0.3) and this won't trip, in this example displaying 5.55112E-17 or similar.
https://people.eecs.berkeley.edu/~wkahan/Mind1ess.pdf
(and plenty of other rants...:
2015.000000000000: https://news.ycombinator.com/item?id=10558871
What I’m trying to show here is that both integers and floating point are not suitable for doing ‘simple’ financial math. But we get used to this Bresenhamming in integers and do not perceive it as solving an error correction problem.
I have written a test framework and I am quite familiar with these problems, and comparing floating point numbers is a PITA. I had users complaining that 0.3 is not 0.3.
The code managing these comparisons turned out to be more complex than expected. The idea is that values are represented as ranges, so, for example, the IEEE-754 "0.3" is represented as ]0.299~, 0.300~[ which makes it equal to a true 0.3, because 0.3 is within that range.
Maybe the creator's theory is that people will search for 0.30000000000000004 when they run into it after running their code.
FWIW - the only way I can ever find my own website is by searching for it in my github repositories. So I definitely agree, it's not a terribly memorable domain.
Also the "field" of floating point numbers is not commutative†, (can run on JS console:)
x=0;for (let i=0; i<10000; i++) { x+=0.0000000000000000001; }; x+=1
--> 1.000000000000001
x=1;for (let i=0; i<10000; i++) { x+=0.0000000000000000001; };
--> 1
Although most of the time a+b===b+a can be relied on. And for most of the stuff we do on the web it's fine!††
† edit: Please s/commutative/associative/, thanks for the comments below.
†† edit: that's wrong! Replace with (a+b)+c === a+(b+c)
0.1 = 1 × 10^-1, but there is no integer significand s and integer exponent e such that 0.1 = s × 2^e.
When this issue comes up, people seem to often talk about fixing it by using decimal floats or fixed-point numbers (using some 10^x divisor). If you change the base, you solve the problem of representing 0.1, but whatever base you choose, you're going to have unrepresentable rationals. Base 2 fails to represent 1/10 just as base 10 fails to represent 1/3. All you're doing by using something based around the number 10 is supporting numbers that we expect to be able to write on paper, not solving some fundamental issue of number representation.
Also, binary-coded decimal is irrelevant. The thing you're wanting to change is which base is used, not how any integers are represented in memory.
Yes, there are some exceptions where you can reliably compare equality or get exact decimal values or whatever, but those are kind of hacks that you can only take advantage of by breaking the abstraction.
printf("%.4g", 1.125e10) --> 1.125e+10
printf("%.4f", 1.125e10) --> 11250000000.0000Edit: Oh wait, it's listed in the main article under Raku. Forgot about the name change.
The other (and more important) matter, — that is not even mentioned, — is comparison. E. g. in “rational by default in this specific case” languages (Perl 6),
> 0.1+0.2==0.3
True
0.1+0.2
0.3
⎕PP←20 ⋄ 0.1+0.2
0.30000000000000004
(0.1+0.2) ≡ 0.3
1In Raku, the comparison operator is basically a subroutine that uses multiple dispatch to select the correct candidate for handling comparisons between Rat's and other numeric objects.
And the length (but not value) winner is GO with: 0.299999999999999988897769753748434595763683319091796875
The explanation then goes on to be very complex. e.g. "it can only express fractions that use a prime factor of the base".
Please don't say things like this when explaining things to people, it makes them feel stupid if it doesn't click with the first explanation.
I suggest instead "It's actually rather interesting".
Most languages have classes for that, some had them for decades in fact. Hardware floating point numbers target performance and most likely beat any of those classes by orders of magnitude.
> 0.1 + 0.2;
< 0.30000000000000004
> (0.1 + 0.2).toPrecision(15);
< "0.300000000000000"
I never heard anyone complain that it would be simple to fix. But complaining? Yes - and rightfully so. Not every webprogrammer need to know the hw details and don't want to, so it is understandable that this causes irritation.
?- A is rationalize(0.1 + 0.2), format('~50f~n', [A]).
0.30000000000000000000000000000000000000000000000000
A = 3 rdiv 10.Fun times.
https://tools.ietf.org/html/rfc1123#page-13
One aspect of host name syntax is hereby changed: the restriction on the first character is relaxed to allow either a letter or a digit. Host software MUST support this more liberal syntax.
The mandate of a starting letter was for backwards compatibility, and mentions it in light of keeping names compatible with email servers and HOSTS files it was replacing.
Taking a numeric label risks incompatibility with antiquated systems, but I doubt it will effect any modern browser.
That said, it's one of my favorite trivia gotchas.
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.
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.
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).
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.
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.
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.
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 still remember that warning well.
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.
Which specific places have you seen it used in?
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.
https://en.m.wikipedia.org/wiki/Arbitrary-precision_arithmet...
and gets harder when you want exact irrationals too https://www.google.com/search?q=exact+real+arithmetic
But maybe that just results in all the floating point weirdness again, just not for small rationals.
Let's say we need to do a comparison. Set
a = 34241432415/344425151233
b = 45034983295/453218433828
Or even more feindish, Set
a = 14488683657616/14488641242046
b = 10733594563328/10733563140768
By what algorithm would you do the computation, and could you guarantee me the same compute time as comparing 2/3 and 4/5?
(Shameless Common Lisp plug: http://clhs.lisp.se/Body/t_ration.htm)
That's why we need regular expressions support in every search box, browser history, bookmarks and Google included.
"simple" discussion: https://floating-point-gui.de/errors/comparison/
more advanced: https://randomascii.wordpress.com/2012/02/25/comparing-float...
What is failing is associativity, i.e. (a+b)+c==a+(b+c)
For example
(.0000000000000001 + .0000000000000001 ) + 1.0
--> 1.0000000000000002
.0000000000000001 + (.0000000000000001 + 1.0)
--> 1.0
In your example, you are mixing both properties,
(.0000000000000001 + .0000000000000001) + 1.0
--> 1.0000000000000002
(1.0 + .0000000000000001) + .0000000000000001
--> 1.0
but the difference is caused by the lack of associativity, not by the lack of commutativity.
[1] Perhaps you must exclude -0.0. I think it is commutative even with -0.0, but I'm never 100% sure.
(Well, it's a big document. I searched for the string "addition", which occurs just 41 times.)
I failed, but I believe I can show that the standard requires addition to be commutative in all cases:
1. "Clause 5 of this standard specifies the result of a single arithmetic operation." (§10.1)
2. "All conforming implementations of this standard shall provide the operations listed in this clause for all supported arithmetic formats, except as stated below. Unless otherwise specified, each of the computational operations specified by this standard that returns a numeric result shall be performed as if it first produced an intermediate result correct to infinite precision and with unbounded range, and then rounded that intermediate result, if necessary, to fit in the destination’s format" (§5.1)
Obviously, addition of real numbers is commutative, so the intermediate result produced for addition(a,b) must be equal to that produced for addition(b,a). I hope, but cannot guarantee, that the rounding applied to that intermediate result would not then depend on the order of operands provided to the addition operator.
3. "The operation addition(x, y) computes x+y. The preferred exponent is min(Q(x), Q(y))." (§5.4.1). This is the entire definition of addition, as far as I could find. (It's also defined, just above this statement, as being a general-computational operation. According to §5.1, a general-computational operation is one which produces floating-point or integer results, rounds all results according to §4, and might signal floating-point exceptions according to §7.)
4. The standard encourages programming language implementations to treat IEEE 754 addition as commutative (§10.4):
> A language implementation preserves the literal meaning of the source code by, for example:
> - Applying the properties of real numbers to floating-point expressions only when they preserve numerical results and flags raised:
> -- Applying the commutative law only to operations, such as addition and multiplication, for which neither the numerical values of the results, nor the representations of the results, depend on the order of the operands.
> -- Applying the associative or distributive laws only when they preserve numerical results and flags raised.
> -- Applying the identity laws (0 + x and 1 × x) only when they preserve numerical results and flags raised.
This looks like a guarantee that, in IEEE 754 addition, "the representation of the result" (i.e. the sign/exponent/significand triple, or a special infinite or NaN value - §3.2) does not "depend on the order of the operands". §3.2 specifically allows an implementation to map multiple bitstrings ("encodings") to a single "representation", so it's possible that the bit pattern of the result of an addition may differ depending on the order of the addends.
5. "Except for the quantize operation, the value of a floating-point result (and hence its cohort) is determined by the operation and the operands’ values; it is never dependent on the representation or encoding of an operand."
"The selection of a particular representation for a floating-point result is dependent on the operands’ representations, as described below, but is not affected by their encoding." (both from §5.2)
HOWEVER...
6. §6, dealing with infinite and NaN values, implicitly contemplates that there might be a distinction between addition(a,b) and addition(b,a):
> Operations on infinite operands are usually exact and therefore signal no exceptions, including, among others,
> - addition(∞, x), addition(x, ∞), subtraction(∞, x), or subtraction(x, ∞), for finite x (§6.1)
OK.
>> x = 0;
0
>> for (let i=0; i<10000; i++) { x+=0.0000000000000000001; };
1.0000000000000924e-15
>> x + 1
1.000000000000001
>> 1 + x
1.000000000000001
1.0 + 1e-16 == 1e-16 + 1.0 == 1.0 as well as 1.0 + 1e-15 == 1e-15 + 1.0 == 1.000000000000001
however (1.0 + (1e-16 + 1e-16)) == 1.0 + 2e-16 == 1.0000000000000002, whereas ((1.0 + 1e-16) + 1e-16) == 1.0 + 1e-16 == 1.0
But this isn't a sales pitch. Some people are just bad at things. The explanation on that page require grade school levels of math. I think math that's taught in grade school can be objectively called simple. Some people suck at math. That's ok.
I'm very geeky. I get geeky things. Many times geeky things can be very simple to me.
I went to a dance lesson. I'm terribly uncoordinated physically. They taught me a very 'simple' dance step. The class got it right away. The more physically able got it in 3 minutes. It took me a long time to get, having to repeat the beginner class many times.
Instead of being self absorbed and expect the rest of the world to anticipate every one of my possible ego-dystonic sensibilities, I simply accepted I'm not good at that. It makes it easier for me and for the rest of the world.
The reality is, just like the explanation and the dance step, they are simple because they are relatively simple for the field.
I think such over-sensitivity is based on a combination of expecting never to encounter ego-dystonic events/words, which is unrealistic and removes many/most growth opportunities in life, and the idea that things we don't know can be simple (basically, reality is complicated). I think we've gotten so used to catering to the lowest common denominator, we've forgotten that it's ok for people to feel stupid/ugly/silly/embarrassed/etc. Those bad feelings are normal, feeling them is ok, and they should help guide us in life, not be something to run from or get upset if someone didn't anticipate your ego-dystonic reaction to objectively correct usage of words.
The idea that you care about the growth of people you are actively excluding doesn't make a whole lot of sense. In this example we're talking about word choice. The over-sensitivity from my point of view is in the person who takes offense that someone criticized their language and refuses to adapt out of some feigned interest for the disadvantaged party. The parent succinctly critiqued the word choice of the author and offered an alternative that doesn't detract from the message in the slightest.
The lowest common denominator is the person who throws their arms up when offered valid criticism.
On the other hand, people say "it's actually pretty simple" to encourage someone to listen to the explanation rather than to give up before they even heard anything, as we often do.
Yep, I've thrown 10,000 round house kicks and can teach you to do one. It's so easy.
In reality, it will be super awkward, possibly hurt, and you'll fall on your ass one or more times trying to do it.
I read the rest of your reply but I also haven’t let go of the possibility that we’re both (or precisely 100.000000001% of us collectively) are as thick as a stump.
We could all learn a lot more from each other if everything wasn't a contest all the time.
My take is that this sentence is badly worded. How do these fractions specifically use those prime factors?
Apparently the idea is that a fraction 1/N, where N is a prime factor of the base, is rational in that base.
So for base 10, at least 1/2 and 1/5 have to be rational.
And given that a product of rational numbers is rational, no matter what combination of those two you multiply, you'll get a number rational in base 10, so 1/2 * 1/2 = 1/4 is rational, (1/2)^3 = 1/8 is rational etc.
Same thing goes for the sum of course.
So apparently those fractions use those prime factors by being a product of their reciprocals, which isn't mentioned here but should have been.
Did the text change in the last 15 minutes?
And in case anyone's wondering about handling it by representing the repeating digits instead, here's the decimal representation of 1/12345 using repeating digits:
0.0[0008100445524503847711624139327663021466180639935196435803969218307006885378
69582827055488051842851356824625354394491697043337383556095585257189145402997164
84406642365330093155123531794248683677602268124746861077359254759011745646010530
57918185500202511138112596192790603483191575536654515998379910895099230457675172
13446739570676387201296071283920615633859862292426083434588902389631429728635074
92912110166059133252328878088294856217091940056703118671526933981368975293641150
26326447954637505062778452814904819765087079789388416362899959497772377480761441
87930336168489266909680032401782098015390846496557310652085864722559740785743215
87687322802754151478331308221952207371405427298501417577966788173349534224382341
02875658161198865937626569461320372620494127176994734710409072498987444309437019
03604698258404212231672742]See also binary coded decimals.
That is true, but most humans in this world expect 0.1 to be represented exactly but would not require 1/3 to be represented exactly. Because they are used to the quirks of the decimal point (and not of the binary point).
This is a social problem, not a technical one.
https://en.wikipedia.org/wiki/Decimal_floating_point#IEEE_75...
But it's still not much used. E.g. for C++ it was proposed in 2012 for the first time
http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n340...
then revised in 2014:
http://open-std.org/jtc1/sc22/wg21/docs/papers/2014/n3871.ht...
...and... silence?
https://www.reddit.com/r/cpp/comments/8d59mr/any_update_on_t...
Seeing the occasional 0.300000000000004 is a good reminder that your 0.3858372895939229 isn't accurate either.
Not really. It would be really cool if fixed point number storage were an option... but I'm not aware of any popular language that provides it as a built-in primitive along with int and float, just as easy to use and choose as floats themselves.
Yes probably every language has libraries somewhere that let you do it where you have to learn a lot of function call names.
But it would be pretty cool to have a language with it built-in, e.g. for base-10 seven digits followed by two decimals:
fixed(7,2) i;
i = 395.25;
i += 0.01;
With many operations this trade off makes sense, however its critical to understand the limitations of the model.
Pretty much all languages have some sort of decimal number. Few or none have made it the default because they're ignominiously slower than binary floating-point. To the extent that even languages which have made arbitrary precision integers their default firmly keep to binary floating-point.
You can strike the "none". Perl 6 uses rationals (Rat) by default, someone once told me Haskell does the same, and Groovy uses BigDecimal.
The opposite.
Decimal floating points have been available in COBOL from the 1960s, but seem to have fallen out of favor in recent days. This might be a reason why bankers / financial data remains on ancient COBOL systems.
Fun fact: PowerPC systems still support decimal-floats natively (even the most recent POWER9). I presume IBM is selling many systems that natively need that decimal-float functionality.
it's horrifying!
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.
However, even though algorithm A and B are "correct" they can behave differently when rounding errors are introduced.
For example – if algorithm A uses
https://en.wikipedia.org/wiki/Kahan_summation_algorithm
and B uses naive summation then you can expect the end result of A to be more precise than the end result of B – even though both algorithms are correct.
Formally speaking, no. The problem can be defined precisely. At least one of the algorithms fails to solve the problem.
In practice of course, some amount of error may be acceptable.
One can argue that nothing is important to most people.
The correct calculations involving money, up to the last cent, are in fact important for people who do them or who are supposed to use them. I've implemented them in the software I've made to preform some financial stuff even in eighties, in spite of all the software which used binary floating point routines. And, of course, from the computer manufacturers, at least IBM cares too:
https://www.ibm.com/support/pages/decimal-floating-point-use...
Apparently, there are even processors which supports these formats in hardware. It's just still not mainstream.
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.
See e.g. https://observablehq.com/@jrus/qang
Yes there is something lost. I included it in my post but I'll repeat it: People who aren't good at math are 'shielded from the truth' (they objectively suck at math because they can't grasp something that is objectively simple in the domain of math). Again, feeling bad about not grasping something simple is the necessary element for a humbling experience. Humbling experiences aren't meant to feel great. For me, I've learned the most with humbling experiences. I honestly believe most people in the first world need more of them.
The suggested language is more inclusive, that's an advantage to sales, but less clear, that's a disadvantage to communication/learning. Personally, I like learning and want to see things optimized for that.
BTW; I loved the sly way of you implying that: A- I took offense (I am not, nor do I see anything in my comment that says I'm offended) and B- That I'm the lowest common denominator because of A. It's a subtle way of attacking me and not my point. It says a lot about both the person doing the attack and the strength of their argument that they have to resort to ad-hominems. Though I will credit you with using a smartly disguised one.
Also, you are speaking to me as if I was the website author. I'm not the OP of the article, which if you read TFA you would see the actual author changed in favor of the suggestion.
And, at the end of the day, even if there's a filter bubble and it's the reason I see it first, then so what? The people looking for this site are likely going to fit into the same set of targeted demographics as you and me and most people on this site. So unless you also want to cater to 65-year old retirees that don't care about computer science and what floating numbers are, then why does the filter bubble even matter?
It would be if you both used DuckDuckGo, though :)
I realized something I didn't ever notice or appreciate in 20+ years: oh yeah, they can't just round off the penny in their favour every time. And the code that handles tracking when to charge someone an extra penny must be irritating to have developed and manage. All of a sudden you've got state.
Every programming language that has come since has been designed to be a general purpose programming language, therefore they don't include fixed point as a first class data type.
Therefore the financial industry continues to use COBOL.
Every time someone some tries to rewrite some crusty COBOL thing in the language de jure, they'll inevitably fuck up the rounding somewhere. The financial industry has complicated rounding rules. Or better yet, the reference implementation is buggy and the new version is right, but since the answers are different it's not accepted.
DOOM's implementation of fixed-point is a good example which should work on any platform as long as sizeof(int) >= 4.
https://github.com/id-Software/DOOM/blob/master/linuxdoom-1.... https://github.com/id-Software/DOOM/blob/master/linuxdoom-1....
Addition and subtraction will work normally. Multiplication also works normally except you need to right-shift by FRAC_BITS afterwards (and probably also cast to a larger integer type beforehand to protect against overflow).
Division is somewhat difficult since integer division is not what you want to do. DOOM's solution was to cast to double, perform the division, and then convert that back to fixed-point by multiplying by 1.0 before casting back to integer. This seems like cheating since it's using floating-point as an intermediate type, but it is safe to do because 64-bit floating point can represent all 32-bit integers. As long as you're on a platform with an FPU it's probably also faster than trying to roll your own division implementation.
Fixed point classes have been proposed many times as addition to the standard library, but have yet to be voted in.
[1] see my comment elsethread about trivial problems.
This is how positional number systems work at all.
x = 2^e * (1 + m)
But you could have a fully exponential number format:
x = 2^(m + o)
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.
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.
IIRC some JS engines are capable of detecting many circumstances where floating-point is not needed, particularly for simple cases like loop counters, and their JiT compilers will produce code that uses integer values instead of floats for those purposes - but how reliable that is for cases any more complex than that I don't know.
There are several BigInt libraries out there that you could use, though obviously this is not as convenient and even if they wrap BigInt when available will be less efficient.
Safari (WebKit) actually has a fully working implementation, they just haven't shipped it yet. Search the release notes for "BigInt": https://developer.apple.com/safari/technology-preview/releas...
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):
https://github.com/asik/FixedMath.Net
It is a little terrifying that e.g. normalizing a vector involves a while loop, but all things considered the whole thing runs surprisingly well.
(I agree with everything in your post, just thought I could add a real world field report)
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.
1: https://guardiansofatlas.com/
2: It was 2012 when we started.
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.
It was fun discovering all the corner cases while debugging drivers.
> Programmable calculators manufactured by Texas Instruments, Hewlett-Packard, and others typically employ a floating-point BCD format, typically with two or three digits for the (decimal) exponent.
https://en.wikipedia.org/wiki/Fixed-point_arithmetic
> In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point.
> A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type.
But that practically holds only for a reasonable amount of simple arithmetics. Fractional components tend to grow exponential for many numerical methods repeated multiple times. This can happen if you're describing money and want to apply a complex numerical method from an economics article for whatever purpose. Might be worth it but be careful not to carry ever expanding fractions in your system.
If you are running an economic simulation you generally don't have to worry about rounding, the whole thing is only approximate anyway.
I wish I could up vote you more than once. You are bang on.
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.
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 !
a/b > x/y is the same as ay > xb
Assuming you don’t overflow your integer type.
There's your answer :)
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.
This is just not true. If you add 1.5 + 4.25 with IEEE754, there is nothing inexact or rounded. That you cannot exactly represent 0.1 in base2 FP is a problem of base2, not FP.
You get inexact results with FP math for underflows, overflows, or if you don't have enough precision for the result (or an intermediate result). But the same is true for normal integer types.
Well, this is true. But integer math is also not an accurate model of rational-number arithmetic, yet nobody would claim that integer math is inexact.
Compare that to a 32 bit integer, which can have 4 billion unique values, and supports numbers from 0 to 4 billion. It's a 1:1 mapping.
No, they don't must represent all number in the range. I don't know where you get from that they must. An integer also can't represent all real numbers in its range.
That you cannot represent 1/3 as a non-periodic decimal number is a problem of base 10.
That you cannot represent 1/10 as a non-periodic binary number is a problem of base 2.
These are just mathematic facts. Maybe you don't like the world "problem", but it does not change that this is where we are.
The problem that you cannot represent 0.1 in base 2 FP, is a problem of base 2. You can represent it exactly in base 10 FP.
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
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.
https://ruby-doc.org/stdlib-2.5.1/libdoc/bigdecimal/rdoc/Big...
https://www.postgresql.org/docs/9.6/datatype-money.html
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?
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.
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.
Only if you never use anything but addition and subtraction. So, no currency coversions, interest rates, complex taxes or rebate schemes or the like.
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... )
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, ...
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`).
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.
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.
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.
(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.)
More typically, mills[1] (tenth of a cent).
https://aws.amazon.com/emr/pricing/
Azure has some hourly prices with ten-thousandths of a cent ($0.0102/hour):
https://azure.microsoft.com/en-ca/pricing/details/virtual-ma...
Microsoft should use gas station 9/10 pricing conventions to just barely undercut Amazon's lowest price $0.011 with $0.0109.
https://www.marketplace.org/2018/10/11/why-do-gas-prices-end...
>“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…”
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:
def tax_f1(amt, rate):
tax = round(amt * rate,2)
return round(tax * 100)
def tax_f2(amt, rate):
return round(amt*rate*100)
def tax_f3(amt, rate):
return round(amt*rate*100+.5)
1% of $21.50
3% of $21.50
6% of $21.50
10% of $21.15
tax_f1: 21 65 129 211
tax_f2: 22 64 129 211
tax_f3: 22 65 130 212
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:
def tax(amt, rate):
tax = (amt * rate + 5000)//10000
return tax
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”.
And even this is only true if you retrict yourself to unsigned integers. For signed integers you have quirks (-0x8000.. = 0x8000..) or minefields (undefined overflow semantics in C, which can yield non-associativity, tests deleted by the compiler, etc.).
And I'd argue that whoever understands the ring of integers modulo 2^64, will also understand the IEEE754 semantics (which are, I agree, sometimes unfortunate. But not inexact).
Fair point. I've edited my comment to include the word "unsigned".
> I'd argue that whoever understands the ring of integers modulo 2^64, will also understand the IEEE754 semantics
I'm an existence proof that that is not true :). Although I'm sure I could learn the IEEE754 semantics if I put enough effort into reading the spec.
But even if they don't know the word "ring", I think most programmers do understand how modulo arithmetic works, and they have algebraic intuitions about it that turn out to be true: both operations are commutative and associative, multiplication distributes over addition, equality of a forumla involving * and + is true if it's true in the actual integers, and so on.
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).
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.
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).
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.
Poor souls that use FP for accounting are scourge of the industry and source of jokes.
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.
> 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.
But they still can't precisely represent quantities like 1/3 or pi.
> I'm an existence proof that that is not true :). Although I'm sure I could learn the IEEE754 semantics if I put enough effort into reading the spec.
This was sloppy writing on my side. I wanted to say "whoever understands the ring of integers modulo 2^64, can also understand". And I'm sure you could :)
And you don't even have to read the spec. The core idea (mantissa, exponent, and sign) is super easy and writing a FP emulation for addition and multiplation is a really nice task to understand what is actually going on. The only really unfamiliar idea is binary fractions and I think this is a cool idea to understand on its own.
> But even if they don't know the word "ring", I think most programmers do understand how modulo arithmetic works, and they have algebraic intuitions about it that turn out to be true: both operations are commutative and associative, multiplication distributes over addition, equality is true if it's true in the actual integers, and so on.
Well that is all fine but scrolling back to the grand grand grand parent: That would also be a completely wrong abstraction to model financial stuff. I'm not saying FP is the solution, but for sure modulo arithmetic is also how you not want to do finance :)
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.
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.
fn main() {
let x: f64 = 9007199254740992.0;
assert_eq!(x + 1.0, x);
}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'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.
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.)
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.
$ ruby -e 'pp 87e12 + 0.01'
87000000000000.02
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.
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.
For anything imprecise and scientific, doubles will normally work well.
Accounting rules regarding truncation and rounding as specified, seems unaccounted for by most until they meet such stringent reqs.
Like commonly happens doing financial calculations, especially doing interest calculations.
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.
#include <stdbool.h>
#include <stdio.h>
#include <string.h>
bool compare(int cents, float sum) {
char buf[20], floatbuf[24];
int len;
bool result;
len = sprintf(buf, "%d", cents / 100 ) ;
sprintf(buf + len , ".%02d" , cents % 100 ) ;
sprintf(floatbuf, "%0.2f", sum) ;
result = ! strcmp(buf, floatbuf) ;
if (! result)
printf( "Cents: %d, exact: %s, calculated %s\n", cents, buf, floatbuf) ;
return result;
}
int main() {
float cent = 0.01f, sum = 0.0f;
for (int i=0 ; compare(i, sum) ; i++) {
sum += cent;
}
return 0;
}
Cents: 10959, exact: 109.59, calculated 109.60
Precisely. That's why I specified ~ 10^26 addition operations.
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).
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.
#include <stdio.h>
double tax_on(double amt, double rate);
int main(void)
{
double amt = 21.15;
double rate = 0.1;
double tax = tax_on(amt, rate);
printf("%.3f\n", tax);
printf("%.2f\n", tax);
return 0;
}
double tax_on(double amt, double rate)
{
return amt * rate;
}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.
def tax_f4(amt, rate):
tax = round(amt * rate * 1000)
return tax // 10 + (tax % 10 > 4)However, for 10.14% of $21.15, it gives 215, but it should be 214. Another example is 3.5% of $60.70, for which it gives 213 but correct is 212.
import math
def tax_f5(amt, rate):
t = amt * rate * 1000
return round(t) // 10 + ((math.fmod(t, 10.0) - 5.0) > -1e-7)
Good example of this, in Python 3:
>>> (0.1 + 0.2) + 0.3
0.6000000000000001
>>> 0.1 + (0.2 + 0.3)
0.6A gentleman and a scholar.
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*.
What do you mean this person has no surname? That's unpossible, surname is never null, error error.
http://blogs.reuters.com/ben-walsh/2013/11/18/do-stocks-real...
I guess it's time for someone to write an "Assumptions Programmers make about money" post.
Additionally, fractional cents are often presented to the consumer when purchasing gas/fuel.
i.e. they track your account balance to more than 2 digits, they just only show you 2 digits.
That is not correct. Stock settlement transactions often list four decimal places.
That's not a significant difference compared to two decimal places, so brundolf's point still stands. There's no need for arbitrary precision.
Just store all dollars in PIPs so 5$ will be stored as 50000.
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.
Edit: And they forbid equality comparisons for rationals. For some reason even >= is not allowed.
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.
unsigned long tax = (unsigned long)round(amt * rate * 1000000);
return tax/(10000) + (fmod(tax, (double)(10000)) - (double)(5000) > -1e-5 ? 1 : 0);
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.
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.
See JEP 306.
That's something FP does not provide and it makes it completely unusable for accounting.
That seems like a completely arbitrary requirement. Do accounting laws prohibit sort? Does 1 + 1 have to equal green on Tuesdays?
Wonder if JS does something similar or not.
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);
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.
It's not an optimisation if it changes the result! And if you use non-standard flags that's your problem.
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.
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.
If you use it. Which is not the default. Your original claim remains false.
All 10 of them?
It goes without saying that half pennies are dead. A mill seems to be from the Coinage Act of 1792, which is perhaps a tad outdated.
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.
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.
> 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.