[plt-dev] Inexact integers

On Tue, May 25, 2010 at 12:34 PM, Joe Marshall <jmarshall at alum.mit.edu> wrote:
> On Tue, May 25, 2010 at 7:23 AM, Michael Sperber
> <sperber at deinprogramm.de> wrote:
>> However, the IEEE operations aren't defined in terms of
>> those intervals: they are defined (simplifying somewhat) as operations
>> on "exact" numbers followed by rounding.
>
> Followed by rounding *if necessary*.  And it often isn't necessary.
> Many so-called rounding errors come from the translation from base 10
> input to base 2, or from base 2 to base 10 on printing.  The computation
> itself can often proceed without rounding.  For example, integer arithmetic
> for add, subtract, and multiply are *exact* for floating-point integer values
> in the range -2^52 to 2^52.  There will be *no* rounding whatsoever.
>
> Floating point isn't `magic' or a `black art', it's just a little trickier than
> rationals, and maybe on par with complex numbers.

If the best we could say about IEEE floating point were that it's a
valid alternative for 53-bit signed integers, then it would be a
failure, wouldn't it?  Fortunately we can say much better, but then we
have to start admitting to ourselves that it can't precisely represent
things like 64-bit MAXINT or the simple fraction 1/3.  Floating point
is really useful, but it will always be a black art.  And much
trickier than complex numbers, which are just real numbers that
brought along a(n imaginary) friend.

I hope I'm not too far off topic here, though.  I think the original
topic had more to do with the semantics of the `integer?` predicate
than with the actual representation of inexact numbers.

--Carl