[Haskell-cafe] Sum and Product do not respect the Monoid laws

[Carter Schonwald]

well said and thanks for taking the time to clarify some of the imprecision
in what I was saying!

On Tue, Sep 23, 2014 at 3:06 AM, Richard A. O'Keefe <ok at cs.otago.ac.nz>
wrote:

>
> On 20/09/2014, at 5:26 AM, Carter Schonwald wrote:
>
> > Indeed,
> >
> > Also Floating point numbers are NOT real numbers, they are approximate
> points on the real line that we pretend are exact reals but really are a
> very different geometry all together! :)
>
> Floating point numbers are *PRECISE* numbers with
> approximate *OPERATIONS*.  This is the way they are
> defined in the IEEE standard and its successors;
> this is the way they are defined in LIA-1 and its
> successors.  If you do not understand that it is
> the OPERATIONS that are approximate, not the numbers,
> you have not yet begun to understand floating point
> arithmetic.
>
> > Floats and Doubles are not exact numbers, dont use them when you expect
> things to behave "exact". NB that even if you have *exact* numbers, the
> exact same precision issues will still apply to pretty much any computation
> thats interesting (ignoring what the definition of interesting is).  Try
> defining things like \ x -> SquareRoot x or \x-> e^x  on the rational
> numbers! Questions of precision still creep in!
>
> You're not talking about precision here but about
> approximation.  And you can simply work with finite representations
> of algebraic numbers.  I have a Smalltalk class that implements
> QuadraticSurds so that I can represent (1 + sqrt 5)/2 *exactly*.
> You can even compare QuadraticSurds with the same surd exactly.
> (This all works so much better in Haskell, where you can make
> the "5" part a type-level parameter.)
>
> Since e is not a rational number, it's not terribly interesting
> that e^x (usually) isn't when x is rational.
>
> If we want to compute with a richer range of numbers than
> the rationals, we can do that.  We could, for example, compute
> with continued fractions.  Haskell makes that a small matter
> of programming, and I expect someone has already done it.
>
> > So I guess phrased another way, a lot of the confusion / challenge in
> writing floating point programs lies in understanding the representation,
> its limits, and the ways in which it will implicitly manage that precision
> tracking book keeping for you.
>
> Exactly so.  There are even people who think the representation
> is approximate instead of the operations!  For things like
> doubled-double, it really matters that the numbers are precise
> and combine in predictable ways.
>
> Interestingly, while floating point addition and multiplication
> are not associative, they are not wildly or erratically
> non-associative, and it is possible to reason about the results
> of operations.
>
>
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