Hello, There were some recent discussions on the floating point support in Haskell and some not-so-pleasant "surprises" people encountered.There is an Eq instance defined for these types! So I tried this: *Main> sqrt (10.0) ==3.1622776601683795 True *Main> sqrt (10.0) ==3.16227766016837956 True *Main> sqrt (10.0) ==3.1622776601683795643 True *Main> sqrt (10.0) ==3.16227766016837956435443343 True It seems strange. So my doubts are:
1. I wonder how the Eq instance is defined in case of floating point types in Haskell? 2. Can the Eq instance for floating point types be considered "meaningful"? If yes, how? In general, programmers are **advised** not to base conditional branching on tests for **equality** of two floating point values. 3. Is this particular behaviour GHC specific? (I am using GHC 6.12.1) If there are references on this please share. Thanks and regards,
-Damodar Kulkarni _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On ghc 7.6.3:
Prelude> 3.16227766016837956 3.1622776601683795 So if you specify a number with greater-than-available precision, it will be truncated. This isn't an issue with (==), but with the necessary precision limitations of Double. On Fri, 20 Sep 2013, damodar kulkarni wrote: > Hello, > There were some recent discussions on the floating point support in Haskell > and some not-so-pleasant "surprises" people encountered. > > There is an Eq instance defined for these types! > > So I tried this: > *Main> sqrt (10.0) ==3.1622776601683795 > True > *Main> sqrt (10.0) ==3.16227766016837956 > True > *Main> sqrt (10.0) ==3.1622776601683795643 > True > *Main> sqrt (10.0) ==3.16227766016837956435443343 > True > > It seems strange. > > So my doubts are: > 1. I wonder how the Eq instance is defined in case of floating point types > in Haskell? > 2. Can the Eq instance for floating point types be considered "meaningful"? > If yes, how? > In general, programmers are **advised** not to base conditional branching > on tests for **equality** of two floating point values. > 3. Is this particular behaviour GHC specific? (I am using GHC 6.12.1) > > If there are references on this please share. > > Thanks and regards, > -Damodar Kulkarni > -- Scott Lawrence _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Ok, let's say it is the effect of truncation. But then how do you explain this? Prelude> sqrt 10.0 == 3.1622776601683795 True Prelude> sqrt 10.0 == 3.1622776601683796 True Here, the last digit **within the same precision range** in the fractional part is different in the two cases (5 in the first case and 6 in the second case) and still I am getting **True** in both cases. So the truncation rules seem to be elusive, to __me__. And also observe the following:Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.0 False Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000002 True Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000003 False Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000001 True Prelude> (with GHC version 7.4.2)
But more importantly, if one is advised NOT to test equality of two floating point values, what is the point in defining an Eq instance? So I am still confused as to how can one make a *meaningful sense* of the Eq instance? Is the Eq instance there just to make __the floating point types__ members of the Num class? Thanks and regards,
-Damodar Kulkarni On Fri, Sep 20, 2013 at 5:22 PM, Scott Lawrence <[hidden email]> wrote: On ghc 7.6.3: _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Fri, Sep 20, 2013 at 09:47:24PM +0530, damodar kulkarni wrote:
> Ok, let's say it is the effect of truncation. But then how do you explain > this? > > Prelude> sqrt 10.0 == 3.1622776601683795 > True > Prelude> sqrt 10.0 == 3.1622776601683796 > True > > Here, the last digit **within the same precision range** in the fractional > part is different in the two cases (5 in the first case and 6 in the second > case) and still I am getting **True** in both cases. What do you mean the "same precision range"? Notice: Prelude> 3.1622776601683795 == 3.1622776601683796 True Prelude> 3.1622776601683795 == 3.1622776601683797 True Prelude> 3.1622776601683795 == 3.1622776601683798 False The truncation happens base 2, not base 10. Is that what's confusing you? Tom _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by damodar kulkarni
On Fri, Sep 20, 2013 at 12:17 PM, damodar kulkarni <[hidden email]> wrote:
Because there's no reliable difference there. The truncation is in bits (machine's binary representation) NOT decimal digits. A difference of 1 in the final digit is probably within a bit that gets truncated.
I suggest you study IEEE floating point a bit. Also, study why computers do not generally store anything like full precision for real numbers. (Hint: you *cannot* store random real numbers in finite space. Only rationals are guaranteed to be storable in their full precision; irrationals require infinite space, unless you have a very clever representation that can store in terms of some operation like sin(x) or ln(x).)
brandon s allbery kf8nh sine nomine associates
unix, openafs, kerberos, infrastructure, xmonad http://sinenomine.net _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by damodar kulkarni
On 13-09-20 07:47 AM, damodar kulkarni wrote:
> *Main> sqrt (10.0) ==3.1622776601683795 > True [...] > *Main> sqrt (10.0) ==3.16227766016837956435443343 > True This is not even specific to Haskell. Every language that provides floating point and floating point equality does this. (To date, P(provides floating point equality | provides floating point) seems to be still 1.) In the case of Haskell, where you may have a choice: Do you want floating point > < ? If you say yes, then you have two problems. 1. At present, Haskell puts > < under Ord, and Ord under Eq. You must accept Eq to get Ord. If you reject this, you're asking the whole community to re-arrange that class hierarchy just for a few types. 2. With or without your approval, one can still defy you and define: eq x y = not_corner_case x && not_corner_case y && not (x<y) && not (x>y) See, == can be derived from > < . _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by damodar kulkarni
On Fri, Sep 20, 2013 at 6:17 PM, damodar kulkarni <[hidden email]> wrote:
Well, that's easy: λ: decodeFloat 3.1622776601683795 (7120816245988179,-51) λ: decodeFloat 3.1622776601683796
(7120816245988179,-51) On my machine, they are equal. Note that ...4 and ...7 are also equal, after they are truncated to fit in 53 (which is what `floatDigits 42.0` tells me) bits (`floatRadix 42.0 == 2`).
It seems to me that you're not familiar with the intricacies of floating-point arithmetic. You're not alone, it's one of the top questions on StackOverflow. Please find yourself a copy of "What Every Computer Scientist Should Know About Floating-Point Arithmetic" by David Goldberg, and read it. It should be very enlightening. It explains a bit about how IEEE754, pretty much the golden standard for floating point math, defines these precision rules.
Although equality is defined in IEEE754, it's not extremely useful after arithmetic (except perhaps for zero tests). Eq is a superclass of Ord, however, which is vital to using floating point numbers.
That was also a reason before GHC 7.4 (Eq is no longer a superclass of Num). _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Fri, Sep 20, 2013 at 06:34:04PM +0200, Stijn van Drongelen wrote:
> Please find yourself a copy of "What Every Computer Scientist Should Know > About Floating-Point Arithmetic" by David Goldberg, and read it. It should > be very enlightening. It explains a bit about how IEEE754, pretty much the > golden standard for floating point math, defines these precision rules. Ah, this is definitely the best advice in the thread. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by damodar kulkarni
On Fri, Sep 20, 2013 at 11:17 AM, damodar kulkarni <[hidden email]> wrote: > Ok, let's say it is the effect of truncation. But then how do you explain this?
Oh, it's a trunaction error all right. > Prelude> sqrt 10.0 == 3.1622776601683795 > True > Prelude> sqrt 10.0 == 3.1622776601683796
> True > > Here, the last digit **within the same precision range** in the fractional part is different in the two cases (5 in the first case and 6 in the second case) and still I am getting **True** in both cases.
Because you're using the wrong precisision range. IEEE floats are stored in a binary format, not a decimal one. So values that differ by 2 in the last decimal digit can actually be different values even though
values that differ by one in the last decimal digit aren't. > And also observe the following: > > Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.0 > False
> Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000002 > True > Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000003 > False > Prelude> (sqrt 10.0) * (sqrt 10.0) == 10.000000000000001
> True > Prelude> > > Ok, again something like truncation or rounding seems at work but the precision rules the GHC is using seem to be elusive, to me. > (with GHC version 7.4.2)
Here's a quick-and-dirty C program to look at the values. I purposely print decimal digits beyond the precision range to illustrate that, even though we started with different representations, the actual
values are the same even if you use decimal representations longer than the ones you started with. In particular, note that 0.1 when translated into binary is a repeating fraction. Why the last hex digit
is a instead of 9 is left as an exercise for the reader. That this happens also means the number actually stored when you enter 0.1 is *not* 0.1, but as close to it as you can get in the given
representation. #include <stdio.h> union get_int { unsigned long intVal ; double floatVal ; } ;
doubleCheck(double in) { union get_int out ; out.floatVal = in ; printf("%.20f is %lx\n", in, out.intVal) ; } main() {
doubleCheck(3.1622776601683795) ; doubleCheck(3.1622776601683796) ; doubleCheck(10.0) ; doubleCheck(10.000000000000001) ; doubleCheck(10.000000000000002) ; doubleCheck(10.000000000000003) ;
doubleCheck(0.1) ; } > But more importantly, if one is advised NOT to test equality of two floating point values, what is the point in defining an Eq instance? > So I am still confused as to how can one make a *meaningful sense* of the Eq instance?
> Is the Eq instance there just to make __the floating point types__ members of the Num class? You can do equality comparisons on floats. You just have to know what you're doing. You also have to be aware of how NaN's (NaN's are float
values that aren't numbers, and are even odder than regular floats) behave in your implementation, and how that affects your application. But the same is true of doing simple arithmetic with
them. Note that you don't have to play with square roots to see these issues. The classic example you see near the start of any numerical analysis class is: Prelude> sum $ take 10 (repeat 0.1) 0.9999999999999999 Prelude> 10.0 * 0.1 1.0 This is not GHC specific, it's inherent in floating point number
representations. Read the Wikipedia section on accuracy problems
more information. Various languages have done funky things to deal with these issues, like rounding things up, or providing "fuzzy" equality. These things generally just keep people from realizing when they've done something
wrong, so the approach taken by ghc is arguably a good one. <mike _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Stijn van Drongelen
Me: "My Lord, I just used the (==) on floats and it gave me some unpleasant surprises." Lord Haskell: "You fool, why did you tested floats for equality? Don't you know a bit about floating points?" Me: "My Lord, I thought it'd be safe as it came with the typeclass guarantee you give us." Lord Haskell: "Look, you fool you scum you unenlightened filthy soul, yes I know I gave you that Eq instance for the floating point BUT nonetheless you should NOT have used it; NOW go enlighten yourself." Me: "My Lord, thank you for the enlightenment." I don't know how many people out there are being enlightened by His Excellency, the Lord Haskell, on floating point equality and other things. Yes, many a good old junkies, like the filthier kinkier C, were keen on enlightening people on such issues. But, see, C is meant to be used for such enlightenment. Although I am not an expert on floating point numbers, the paper is not surprising as I have learnt, at least some things given in the paper, the hard way by burning myself a couple of times because of the floating point thing while programming some things in the good old C. But even the Haskell tempted to define an Eq instance for that scary thing __that__ was a new enlightenment for me. Life is full of opportunities to enlighten yourself. That was also a reason before GHC 7.4 (Eq is no longer a superclass of Num). This seems a good step forward, removing the Eq instance altogether on floating point types would be much better; (unless as pointed out by Brandon, "you have a very clever
representation that can store (floats) in terms of some operation like sin(x) or
ln(x) (with infinite precision)") I know I might be wrong in expecting this change as it might break a lot of existing code. But why not daydream? [1] Please read His/Her Thanks and regards,
-Damodar Kulkarni On Fri, Sep 20, 2013 at 10:04 PM, Stijn van Drongelen <[hidden email]> wrote:
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On Fri, Sep 20, 2013 at 7:35 PM, damodar kulkarni <[hidden email]> wrote: > This seems a good step forward, removing the Eq instance altogether on
> floating point types would be much better; (unless as pointed out by > Brandon, "you have a very clever representation that can store > (floats) in terms of some operation like sin(x) or ln(x) (with
> infinite precision)") Please don't. The problem isn't with the Eq instance. It does exactly what it should - it tells you whether or not two floating point
objects are equal. The problem is with floating point arithmetic in general. It doesn't obey the laws of arithmetic as we learned them, so they don't behave the way we expect. The single biggest gotcha is that two calculations
we expect to be equal often aren't. As a result of this, we warn people not to do equality comparison on floats. So people who don't understand that wind up asking "Why doesn't this
behave the way I expect?" Making floats not be an instance of Eq will just cause those people to ask "Why can't I compare floats for equality?". This will lead to pretty much the same explanation. It
will also mean that people who know what they're doing who want to do so will have to write their own code to do it. It also won't solve the *other* problems you run into with floating
point numbers, like unexpected zero values from the hole around zero. Given that we have both Data.Ratio and Data.Decimal, I would argue that removing floating point types would be better than making them
not be an instance of Eq. It might be interesting to try and create a floating-point Numeric type that included error information. But I'm not sure there's a good
value for the expression 1.0±0.1 < 0.9±0.1. Note that Brandon was talking about representing irrationals exactly, which floats don't do. Those clever representations he talks about
will do that - for some finite set of irrationals. They still won't represent all irrationals or all rationals - like 0.1 - exactly, so the problems will still exist. I've done microcode implementations of
floating point representations that didn't have a hole around 0. They still don't work "right". <mike _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Making floats not be an instance of Eq will
Yes, and then all the torrent of explanation I got here about the intricacies of floating point operations would seem more appropriate. Then you can tell such a person "how is the demand for general notion of equality for floats tantamount to a demand for an oxymoron? because depending on various factors the notion of equality for float itself floats (sorry for the pun)." In this case, such people might implement their __own__ notion
of equality for floating points. And if they intend to do such a thing,
then it would not be much of an issue to expect from them the detailed
knowledge of all the intricacies of handling equality for floating
points... as anyway they themselves are asking for it and they are NOT
relying on the Haskell's Num typeclass for it.But in the given situation, such an explanation seems uncalled for as it goes like: "we have given you the Eq instance on the floating point types BUT still you are expected NOT to use it because the floating point thingy is very blah blah blah..." etc. It not much of a problem with that as then it would be more like people who do unsafePerformIO, where Haskell clearly tells you that you are on your own. You might provide them `unsafePerformEqOnFloats` for instance. And then if someone complains that the `unsafePerformEqOnFloats` doesn't test for equality as in equality, by all means flood them with "you asked for it, you got it" type messages and the above mentioned explanations about the intricacies of floating point operations.
This seems better. Let people have the support for floating point types in some other libraries IF at all they want to have them but then it would bear no burden on the Num typeclass and more importantly on the users of the Num class. Thanks and regards,
-Damodar Kulkarni On Sat, Sep 21, 2013 at 9:46 AM, Mike Meyer <[hidden email]> wrote:
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On 2013-09-21 06:16, Mike Meyer wrote:
> The single biggest gotcha is that two calculations > we expect to be equal often aren't. As a result of this, we warn > people not to do equality comparison on floats. The Eq instance for Float violates at least one expected law of Eq: Prelude> let nan = 0/0 Prelude> nan == nan False There was a proposal to change this, but it didn't really go anywhere. See: http://permalink.gmane.org/gmane.comp.lang.haskell.libraries/16218 (FWIW, even if the instances cannot be changed/removed, I'd love to see some sort of explicit opt-in before these dangerous/suprising instances become available.) Regards, _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by damodar kulkarni
I think you are trying to solve a problem that doesn't exist. * Float and Double are imprecise types by their very nature. That's exactly what people are forgetting, and exactly what's causing misunderstandings. Perhaps(!) it would be better to remove the option to use rational literals as floats, and require people to convert rationals using approx :: (Approximates b a) => a -> b when they want to use FP math (instance Approximates Float Rational, etc). * Pure equality tests make perfect sense in a few situations, so Eq is required. In fact, it's required to have an IEEE754-compliant implementation. * As mentioned, there is a total order (Ord) on floats (which is what you should be using when checking whether two approximations are approximately equal), which implies that there is also an equivalence relation (Eq). _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On 21 September 2013 08:34, Stijn van Drongelen <[hidden email]> wrote:
how do you get a total order when nan compares false with everything including itself? _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Good point. It should be a partial order. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by Bardur Arantsson-2
On Sat, Sep 21, 2013 at 2:21 AM, Bardur Arantsson <[hidden email]> wrote: > On 2013-09-21 06:16, Mike Meyer wrote:
> > The single biggest gotcha is that two calculations > > we expect to be equal often aren't. As a result of this, we warn > > people not to do equality comparison on floats.
> The Eq instance for Float violates at least one expected law of Eq: > > Prelude> let nan = 0/0 > Prelude> nan == nan > False
Yeah, Nan's are a whole 'nother bucket of strange. But if violating an expected law of a class is a reason to drop it as an instance, consider: Prelude> e > 0
True Prelude> 1 + e > 1 False Of course, values "not equal when you expect them to be" breaking equality means that they also don't order the way you expect:
Prelude> e + e + 1 > 1 + e + e True So, should Float's also not be an instance of Ord? I don't think you can turn IEEE 754 floats into a well-behaved numeric
type. A wrapper around a hardware type for people who want that performance and can deal with its quirks should provide access to as much of the types behavior as possible, and equality comparison
is part of IEEE 754 floats. <mike _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Sat, Sep 21, 2013 at 10:26 AM, Mike Meyer <[hidden email]> wrote:
I do have to agree with Damodar Kulkarni that different laws imply different classes. However, this will break **a lot** of existing software.
If we would do this, only Eq and Ord need to be duplicated, as they cause most of the problems. Qualified imports should suffice to differentiate between the two.
import qualified Data.Eq.Approximate as A import qualified Data.Ord.Approximate as A main = print $ 3.16227766016837956 A.== 3.16227766016837955
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On 2013-09-21, at 4:46 AM, Stijn van Drongelen <[hidden email]> wrote:
As soon as you start doing computations with fp numbers things get much worse. Something like Edward Kmett's Numeric.Interval package would likely be helpful, a start at least (and the comments in the Numeric.Interval documentation are amusing) In the distant past when I was worried about maintaining accuracy in a solids modeller we went with an interval arithmetic library that we *carefully* implemented. It worked. Unpleasant in C, but it worked. And this link might be interesting: Cheers, Bob _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
I agree, but that might also be hardly relevant when fixing an existing language. >> If we would do this, only Eq and Ord need to be duplicated, as they cause most of the problems. Qualified imports should suffice to differentiate between the two. Only when you start reasoning about (in)equalities. Really, in (a + b) * c = a * c + b * c, it isn't + or * that's causing problems, but =. I'm going to look at Kmett's work and that ltu link when I'm home ;) _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
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