Newbie: Haskell Sine Oddities
The Sine function in the prelude is not behaving as I expected. In the following Hugs session I tested sin on 0, 90,180 & 360 degrees. Prelude> sin 0 0.0 --correct Prelude> sin (pi/2) 1.0 --correct Prelude> sin pi 1.22460635382238e-16 --WRONG! Prelude> sin (2*pi) -2.44921270764475e-16 --WRONG! Is this normal behaviour? Or am I using the trig functions in an unexpected way? Thanks... Deech
1.22460635382238e-16 --WRONG! Thats just another way of writing: 0.000000000000000122460635382238
Which you can consider as 0 +- error Floating point numbers are not exact, the value of pi is not exact either, and I guess that between them they are giving you errors. Thanks Neil
Aditya Siram wrote:
Prelude> sin pi 1.22460635382238e-16 --WRONG!
Neil Mitchell wrote:
Floating point numbers are not exact, the value of pi is not exact either, and I guess that between them they are giving you errors.
Yes. Actually, this particular inexactness is entirely due to the value of pi. The calculation of sin pi is being performed using the Double data type, which cannot represent pi exactly. Since Double uses binary fractions, doing Hugs.Base> pi 3.14159265358979 shows a decimal approximation to the binary approximation. To investigate the representation of pi, subtract from it a number which _can_ be represented easily and exactly as a binary fraction, as follows: Hugs.Base> pi-3.140625 0.000967653589793116 This shows that pi is represented using an approximation that is close to 3.141592653589793116 This value, the computer's pi, differs from true pi by 0.000000000000000122... so the sin function is working perfectly.
G'day all. Aditya Siram wrote:
Prelude> sin pi 1.22460635382238e-16 --WRONG!
Quoting Scott Turner
This value, the computer's pi, differs from true pi by 0.000000000000000122...
Exercise for those doing numeric analysis 101: Explain why these two numbers are approximately the same. And if you found that one too easy, show how quickly following iteration converges to pi. iterate (\x -> x + sin x) 1 Cheers, Andrew Bromage
As others have pointed out, floating point representations of numbers
are not exact. You don't even have to use fancy functions like sine to
see all kinds of nice algebraic properties break down.
let x = 1e8; y = 1e-8 in (y + x) - x == y + (x - x)
evaluates to False.
So from this you can see that addition is not even associative,
neither is multiplication. Distributivity also fails in general.
Floating point computations are always approximate and have some level
of error associated with them. If you want proper real numbers, things
like equality testing become impossible in general. If you look
around, I think there are a couple of libraries in Haskell which let
you work with arbitrary precision reals though.
Try: http://www.haskell.org/haskellwiki/Exact_real_arithmetic
- Cale
On 29/04/06, Aditya Siram
The Sine function in the prelude is not behaving as I expected. In the following Hugs session I tested sin on 0, 90,180 & 360 degrees.
Prelude> sin 0 0.0 --correct Prelude> sin (pi/2) 1.0 --correct Prelude> sin pi 1.22460635382238e-16 --WRONG! Prelude> sin (2*pi) -2.44921270764475e-16 --WRONG!
Is this normal behaviour? Or am I using the trig functions in an unexpected way?
Thanks... Deech
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On Sat, Apr 29, 2006 at 04:51:40PM -0400, Cale Gibbard wrote:
Floating point computations are always approximate and have some level of error associated with them. If you want proper real numbers, things like equality testing become impossible in general. If you look around, I think there are a couple of libraries in Haskell which let you work with arbitrary precision reals though.
That's not really true. The exact cases of floating point arithmetic can be important, and it's really annoying when compilers break them. For small integers, floating point arithmetic *is* exact, for example, and also for arithmetic (not division) involving integers divided by powers of two, provided there's no overflow or underflow. These exact properties allow the moderately careful programmer to do exact calculations that could have done using clever integer arithmetic while reusing code that works with floating point numbers. It can be handy, for example, when computing the symmetries of a basis set, since you don't need a separate integer 3-vector class (in C++, for example). This isn't a big deal, and it's much less of a deal in Haskell, where you can profitably use typeclasses to make the integer 3-vectors relatively easy to work with, but on the other hand, why bother with an integer class that will behave identically to the floating-point one whenever it's used? (Yes, the answer is the safety of *knowing* that you made no approximation, but for such a small piece of easily audited code, that's not likely to be worth the effort.) -- David Roundy http://www.darcs.net
participants (6)
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Aditya Siram -
ajb＠spamcop.net -
Cale Gibbard -
David Roundy -
Neil Mitchell -
Scott Turner