Re: [Haskell-cafe] Mystery of an Eq instance

Ok, let's say it is the effect of truncation. But then how do you explain

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

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

On Fri, Sep 20, 2013 at 11:17 AM, damodar kulkarni wrote: this? Oh, it's a trunaction error all right. 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. 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 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 (http://en.wikipedia.org/wiki/Floating_point#Accuracy_problems) for 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.