Re: Floating-Point Numbers and Internal Representation

["Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de>]

On Sun, 4 Dec 2005 14:11:49 +0000 (UTC), Martin Dowie wrote:

> Martin Krischik wrote:
>> Independent of CPU/Language used I would allways suggest to use
>> 
>> abs (X - Y) < epsilon
>> 
>> with a sufficient but not to small epsilon instead.
> 
> Suggested reading:
> 
> http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm
>
> There is more than one answer to this problem...

Epsilon above can well be relative that depends solely on the algorithm.
Usually epsilon is estimated in the course of calculations together with X
and Y, as a part of the algorithm. So the answer is still same. (:-))

BTW, I wouldn't use division to evaluate relative errors as the paper
suggest. Rather:

Half_Epsilon * (abs X + abs Y)

A really different answer would be interval arithmetic. If X and Y were
intervals they would carry the accuracy estimation with them. So one could
directly compare them:

case X < Y or X > Y is -- The result is not Boolean!
   when False | Uncertain =>
       -- The difference cannot be distinguished from accumulated
       -- inaccuracy
   when True =>
       -- They are sufficiently different
end case;

-- 
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de