On 12 Feb 2008, at 4:35 am, Andrew Butterfield wrote: [floating point addition is not associative] And this is an excellent example of why violating expected laws is BAD. The failure of floating point addition to be associative means that there are umpteen ways of computing polynomials, for example, and doing it different ways will give you different answers. This is *not* a good way to write reliable software. I did enough Numerical Analysis papers in my pre- PhD years to get quite scared sometimes. Oh, here's a good one: dot1 [] [] = 0 dot1 (x:xs) (y:ys) = x*y + dots1 xs ys Obvious naive code for dot product. Switch over to tail recursion dot2 xs ys = aux xs ys 0 where aux [] [] s = s aux (x:xs) (y:ys) s = aux xs ys (s + x*y) The problem is that (a) in floating point arithmetic these two functions give DIFFERENT answers, and (b) NEITHER of them is wrong (arguably, neither of them is right either). For integers, of course, they must agree (if I haven't made any silly mistakes). This kind of thing makes it incredibly hard to think about numerical calculations. Basically, laws are stated so that implementors will make stuff that clients don't have to think about until their brains melt.