Inherent imprecision of floating point variables

[Tim Writer]

In article <b3f.2688bfce at ibmpcug.co.uk> dylan at ibmpcug.co.uk (Matthew Farwell) writes:

>main()
>{
>	float f;

>	f = 0.0;
>	while (1) {
>		if (f == 10.0) break;
>		printf("%f\n", f);
>		f += 0.1;
>	}
>	printf("Stopped\n");
>}

>If its all to do with conversion routines ...

As has been mentioned in several earlier postings, it is *not* all to do with
conversion routines.  It *is* to do with the fact that no floating point
system can represent any real number exactly.  Your documentation will tell
you that a float can represent real numbers within in a specific interval.
However, since we know there are an infinite number of reals in any given
interval, we can deduce that your floating point system can only represent
a relative few of these numbers exactly.  Thus, when you write `0.1' and
`10.0' in your program, your computer may not store exactly `0.1' or `10.0'.
Then, when you continuously add `0.1', you are not actually adding `0.1' but
some value close to `0.1'.  The result is you never get `10.0' because you
have accumulated some error in `f'.

Try this program.

main()
{
        float  f,
               stop;

        f = 0.0;
        stop = 10.0;

        while (1) {
                if (f == stop)
                break;
		f += 0.1;
		(void) printf("f=%.20f\tstop=%.20f\n",f,stop);
	}
}

Now try this program.

main()
{
        float   eps,
		tst;

	eps=1.0;

	do {
		eps=0.5*eps;
		tst=eps+1.0;
	} while (tst>1.0);

	(void) printf("%.10e\n",eps);
}

This gives an approximation of machine epsilon which differs from its real
value by at most a factor of 2.  Machine epsilon is defined to be the
smallest number which when added to 1 produces a result which is greater than
1.  It is a measure of the relative error in the floating point
representation of real numbers.  On my system, this program gives a value
of roughly 5.9e-8.  In other words, I can represent floats with about 8,
yes *only* 8 digits accuracy.  Or, to rephrase, any digits following the
eighth digit of the mantissa are garbage.

Tim