[Haskell-cafe] Re: Defaulting to Rational [was: Number overflow]

[Jonathan Cast]

On Friday 13 July 2007, Jon Fairbairn wrote:
> Henning Thielemann <lemming at henning-thielemann.de> writes:
> > On Thu, 12 Jul 2007, Jon Fairbairn wrote:
> > > (ie limited precision, but unbounded magnitude). If we were
> > > to use BigFloat the base would need to be a power of ten to
> > > get the desired results for things like Don's example)
> >
> > People will be confused, that 'sin pi' won't lead to a result since the
> > correct result is zero and it will need forever to normalize that number.
>
> Surely the first few digits can be computed?

That was my first thought, too.

We can't define

data Real = Real{
  wholePart :: Integer,
  fractionPart :: [Int]}

because you can't yield e.g. sin pi as an infinite list of digits, but you can 
define

data Real = Real{
  exponent :: Int,
  mantissa :: Int -> [Int]}

where mantissa rounds the number when it's called.  But unless these can be 
memoized fairly well, I would expect performance to be *quite* surprising to 
new users. . .

> I thought 
> sin pi was a computable numer, anyway. Note that in my
> representation I didn't specify what form the small part
> would take; I'm not sufficiently familiar with computing on
> proper reals to know the best choice for that, but as I
> undestand it, once we reach the point of showing a number to
> a finite precision, we /can/ compute the necessary digits.
>
> > They will be confused, that the result of 'sqrt 2 ^ 2' cannot be shown in
> > usual decimal notation, since the formatting algorithm cannot decide
> > between starting with 2.0000 and 1.9999.
>
> Again, if it's being shown to finite precision, it can look
> at the next digit after the last one to be shown and use
> that to decide what to start with.  There's no reason why
> show should be defined to truncate rather than defined to
> round after the last digit, is there?
>
> > In short, the Real number type will leed to all
> > difficulties known from computable reals.
>
> All the real ones, anyway :-).

Jonathan Cast
http://sourceforge.net/projects/fid-core
http://sourceforge.net/projects/fid-emacs