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> 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) 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 wrote:

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

On Fri, Sep 20, 2013 at 12:17 PM, 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

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 allbery.b@gmail.com ballbery@sinenomine.net unix, openafs, kerberos, infrastructure, xmonad http://sinenomine.net

On Sat, Sep 21, 2013 at 12:35 PM, Bardur Arantsson wrote:

On 2013-09-20 18:31, Brandon Allbery wrote: [--snip--]

...

unless you have a very clever representation that can store in terms of some operation like sin(x) or ln(x).)

I may just be hallucinating, but I think this is called "describable numbers", i.e. numbers which can described by some (finite) formula.

Not sure how useful they would be in practice, though :).

I was actually reaching toward a more symbolic representation, like what Mathematica uses. -- brandon s allbery kf8nh sine nomine associates allbery.b@gmail.com ballbery@sinenomine.net unix, openafs, kerberos, infrastructure, xmonad http://sinenomine.net

On Sat, Sep 21, 2013 at 12:43 PM, David Thomas wrote:

Sure. An interesting, if not terribly relevant, fact is that there are more irrational numbers that we *can't* represent the above way than that we can (IIRC).

I think that kinda follows from diagonalization... it does handle more cases than only using rationals, but pretty much by the Cantor diagonal argument there's an infinite (indeed uncountably) number of reals that cannot be captured by any such trick. -- brandon s allbery kf8nh sine nomine associates allbery.b@gmail.com ballbery@sinenomine.net unix, openafs, kerberos, infrastructure, xmonad http://sinenomine.net

On Fri, Sep 20, 2013 at 6:17 PM, 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

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`). Ok, again something like truncation or rounding seems at work but the

precision rules the GHC is using seem to be elusive, to me.

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. 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?

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. Is the Eq instance there just to make __the floating point types__ members

of the Num class?

That was also a reason before GHC 7.4 (Eq is no longer a superclass of Num).

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.

I can imagine the following dialogue happening between His[1] Excellency, the Lord Haskell (if I am allowed to anthropomorphize it) and me: 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 wrote:

On Fri, Sep 20, 2013 at 6:17 PM, 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

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`).

Ok, again something like truncation or rounding seems at work but the

...

precision rules the GHC is using seem to be elusive, to me.

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.

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?

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.

Is the Eq instance there just to make __the floating point types__ members

...

of the Num class?

That was also a reason before GHC 7.4 (Eq is no longer a superclass of Num).

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.

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)." 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

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.

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. 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.

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. 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. Thanks and regards, -Damodar Kulkarni On Sat, Sep 21, 2013 at 9:46 AM, Mike Meyer wrote:

On Fri, Sep 20, 2013 at 7:35 PM, damodar kulkarni 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".

On 21 September 2013 08:34, Stijn van Drongelen wrote:

* 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).

how do you get a total order when nan compares false with everything including itself?

Let me quibble.

* Float and Double are imprecise types by their very nature. That's exactly what people are forgetting, and exactly what's causing misunderstandings.

Float and Double are precise types. What is imprecise is the correspondence between finite precision floating point types (which are common to all programming languages) and the mathematical real numbers. This imprecision is manifest in failures of the intended homomorphism from the reals to the floating point types.

On Sat, Sep 21, 2013 at 2:21 AM, Bardur Arantsson 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.

On 2013-09-21, at 4:46 AM, Stijn van Drongelen wrote:

I do have to agree with Damodar Kulkarni that different laws imply different classes. However, this will break **a lot** of existing software.

You could argue that the existing software is already broken.

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

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: http://lambda-the-ultimate.org/node/1301 Cheers, Bob

On Sep 21, 2013 9:17 AM, "Bob Hutchison" wrote:

On 2013-09-21, at 4:46 AM, Stijn van Drongelen wrote:

...

I do have to agree with Damodar Kulkarni that different laws imply different classes. However, this will break **a lot** of existing software. You could argue that the existing software is already broken.

I'd argue that it's not all broken, and you're breaking it all.

...

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 As soon as you start doing computations with fp numbers things get much worse.

Exactly. The Eq and Ord instances aren't what's broken, at least when you're dealing with numbers (NaNs are another story). That there are pairs of non-zero numbers that when added result in one of the two numbers is broken. That addition isn't associative is broken. That expressions don't obey the substitution principle is broken. But you can't tell these things are broken until you start comparing values. Eq and Ord are just the messengers.

On 2013-09-21 23:08, Mike Meyer wrote:

...

Exactly. The Eq and Ord instances aren't what's broken, at least when you're dealing with numbers (NaNs are another story). That there are

On Sat, Sep 21, 2013 at 5:28 PM, Bardur Arantsson wrote: pairs

According to Haskell NaN *is* a number.

Trying to make something whose name is "Not A Number" act like a number sounds broken from the start.

...

Eq and Ord are just the messengers. No. When we declare something an instance of Monad or Applicative (for example), we expect(*) that thing to obey certain laws. Eq and Ord instances for Float/Double do *not* obey the expected laws.

I just went back through the thread, and the only examples I could find where that happened (as opposed to where floating point calculations or literals resulted in unexpected values) was with NaNs. Just out of curiosity, do you know of any that don't involve NaNs? Float violates the expected behavior of instances of - well, pretty much everything it's an instance of. Even if you restrict yourself to working with integer values that can be represented as floats. If we're going to start removing it as an instance for violating instance expectations, we might as well take it out of the numeric stack (or the language) completely.

2013/9/22 Mike Meyer :

On Sat, Sep 21, 2013 at 5:28 PM, Bardur Arantsson wrote: Trying to make something whose name is "Not A Number" act like a number sounds broken from the start.

The point here is that IEEE floats are actually more something like a "Maybe Float", with various "Nothing"s, i.e. the infinities and NaNs, which all propagate in a well-defined way. Basically a monad built into your CPU's FP unit. ;-)

I just went back through the thread, and the only examples I could find where that happened (as opposed to where floating point calculations or literals resulted in unexpected values) was with NaNs. Just out of curiosity, do you know of any that don't involve NaNs?

Well, with IEEE arithmetic almost nothing you learned in school about math holds anymore. Apart from rounding errors, NaNs and infinities, -0 is another "fun" part: x * (-1) is not the same as 0 - x (Hint: Try with x == 0 and use recip on the result.)

Float violates the expected behavior of instances of - well, pretty much everything it's an instance of. Even if you restrict yourself to working with integer values that can be represented as floats. If we're going to start removing it as an instance for violating instance expectations, we might as well take it out of the numeric stack (or the language) completely.

Exactly, and I am sure 99.999% of all people wouldn't like that removal. Learn IEEE arithmetic, hate it, and deal with it. Or use something different, which is probably several magnitudes slower. :-/

On Tue, Sep 24, 2013 at 11:36 AM, Stijn van Drongelen wrote:

On Tue, Sep 24, 2013 at 5:39 PM, Sven Panne wrote:

...

2013/9/22 Mike Meyer :

...

On Sat, Sep 21, 2013 at 5:28 PM, Bardur Arantsson

wrote: Trying to make something whose name is "Not A Number" act like a number sounds broken from the start.

The point here is that IEEE floats are actually more something like a "Maybe Float", with various "Nothing"s, i.e. the infinities and NaNs, which all propagate in a well-defined way.

So, `Either IeeeFault Float`? ;)

Sort of, but IeeeFault isn't really a zero. Sometimes they can get back to a normal Float value: Prelude> let x = 1.0/0 Prelude> x Infinity Prelude> 1/x 0.0 Also, IEEE float support doesn't make sense as a library, it needs to be built into the compiler (ignoring extensible compiler support via the FFI). The whole point of IEEE floats is that they're very fast, but in order to take advantage of that the compiler needs to know about them in order to use the proper CPU instructions. Certainly you could emulate them in software, but then they'd no longer be fast, so there'd be no point to it.