[Haskell-cafe] about integer and float operations

[Richard O'Keefe]

On 5 Feb 2009, at 10:38 am, Manlio Perillo wrote:
> I'm looking for an exact integer division that avoids overflows, if  
> possible.

What this sounds like to me is a request that the Prelude
function 'fromRational' should work well.  Since
"The floating point literal f is equivalent to
  fromRational (n Ratio.% d), where fromRational is a method in class
  Fractional and Ratio.%constructs a rational from two integers,
  as defined in the Ratio library.  The integers n and d are chosen
  so that n/d = f."
If you cannot divide two Integers n, d accurately using
    (fromRational (n Ratio.% d) :: Double)
that casts doubt on the trustworthiness of floating point literals.

Suppose we have a function
     decodeIntegerAsFloat :: RealFloat a => Integer -> (Integer,a)
such that if (s,m) = integer_to_scaled_float x
then either x = 0 and s = 0 and m = 0
          or x = m * 2**s (mathematically) and abs m \in [0.5,1.0).

Then
     integer_ratio_as_float :: Floating a => Integer -> Integer -> a
     integer_ratio_as_float p q = (mp/mq)*(2.0^(sp-sq))
         where (sp,mp) = decodeIntegerAsFloat p
               (sq,mq) = decodeIntegerAsFloat q

You'd actually use scaleFloat; if the difference sp-sq is outside
the range of Int the answer is going to be a signed zero or a signed
infinity anyway.  decodeIntegerAsFloat would sit very well in the
RealFloat class alongside its model, decodeFloat.  It has other uses.
For example, you can use it to compute logarithms of Integers with
much less worry about overflow.