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.
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