Date: Tue, 14 Jul 1998 10:21:11 +0300 (IDT)
From: Eli Zaretskii <eliz AT is DOT elta DOT co DOT il>
To: djgpp AT delorie DOT com
Subject: Re: Use of FLT_EPSILON
In-Reply-To: <6odqk1$p23$1@nnrp1.dejanews.com>
Message-ID: <Pine.SUN.3.91.980714102032.6831H-100000@is>
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On Mon, 13 Jul 1998 plipson AT my-dejanews DOT com wrote:

> > 		if (fabs (a - b) > min (fabs(a), fabs(b))*DBL_EPSILON)
> > 		  printf ("Not equal\n");
> > 		else
> > 		  printf ("Equal\n");
> >
> >   (Use FLT_EPSILON for floats, LDBL_EPSILON for long doubles; all of them
> >   are defined on <float.h>.)
> >
> > In other words, don't EVER compare FP numbers for exact equality, since
> > floating-point computations have inherent inaccuracy, unlike integer
> > numbers.
> 
>   This looks pretty ugly (the multiply and even more the min() ) - isn't
> there a better way?

None that I know of.  Floating-point computations *are* a complex
issue if you need to get them right.

Also, I think you are mistaken in your perception of ugliness: by far
the most inefficient part of the above computation is the call to
`fabs' library function, all the rest is usually very fast on modern
machines.

> And what about comparing an expression to zero?

If the zero is a constant, not a variable, then you can compare
directly, but I would suggest to use FLT_MIN (or DBL_MIN) instead of a
literal zero, since somebody might argue that 1.e-100 is zero at least
in some applications.

>   I tried several; since FLT_EPSILON is defined in terms of 1 -
>    if( fabs(a-b) < FLT_EPSILON....
>    if( (1 + fabs(a-b)) < (1+FLT_EPSILON).....
>   and a few others.
>     Does anyone have an explanation of what & why to do?

These attempts suggest that you are missing a crucial aspect of this
problem: FLT_EPSILON is a *relative* accuracy, not an *absolute* one.
In other words two float variables that differ by less than
FLT_EPSILON times their absolute value, are indistinguishable.  Since
it is a relative accuracy, you MUST multiply it by the absolute value
of the numbers you compare, to get an absolute accuracy.  (Be sure to
use DBL_EPSILON for double variables.)

The reason for this complexity is that floating-point numbers have a
certain number of bits in the mantissa, and the value of the
least-significant bit is proportional to the absolute value of the
number.  (If this doesn't make sense, I suggest to read any of the
available books about floating-point computations; they should explain
these issues on page 2 or thereabouts.)