From: broeker@acp3bf.knirsch.de (Hans-Bernhard Broeker)
Subject: Re: Comparison of 2 floating point values
Date: 21 Dec 1999 00:00:00 GMT
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Paul Hsieh (DONT.qed.SPAM.ME@pobox.com) wrote:
> Usually to compare two numbers assuming a certain amount accuracy you do
> it like this:

> if( fabs(a/b-1) < MachEps )

Caution with this; you may get divide by zero problems. Avoid
division, like this:

 if ( fabs(a-b) <= Machine_Epsilon * fabs(a))

Which is what the comp.lang.c FAQ recommends. And even _that_ doesn't
work if a is zero... you'ld need a special case, like:

#define EQ(a,b,precision) \
  (fabs(a-b) <= precision * (a==0? (Machine_Epsilon*fabs(a)) \
                                 : (Machine_Smallest)))

Where 'precision' is a factor that describes how much precision has
already been lost, relative to what the machine floating point format
can describe. This has to be a factor bigger than 1, to make sense.
It's this 'precision' factor you have to hand-tune to for every single
application of such a comparison, on every platform.

The latter is because comparing to a relative difference of
'Machine_Epsilon' is not a useful thing. By definition of Epsilon and
the structure of floating point numbers, it is roughly the smallest
*relative* difference between any two FP numbers. I.e. two numbers
with a relative difference of less than Epsilon would be exactly
equal, already, not just approximately, and the whole point of using a
test other than simple '==' would be moot.

As I hope you can see, dealing with floating point numbers can be
surprisingly complicated, as soon as you get down to the gory details.
--
Hans-Bernhard Broeker (broeker@physik.rwth-aachen.de)
Even if all the snow were burnt, ashes would remain.