Why is realmin > eps(0)?

Question

realmin "returns the smallest positive normalized floating point number in IEEE double precision". eps(X) "is the positive distance from ABS(X) to the next larger in mangitude floating point number of the same precision as X".

If I am interpreting the above documentation correctly, then realmin -- the smallest positive number that can be represented -- must be smaller than eps (0). But:

>> realmin; % 2.2251e-308
>> eps(0);  % 4.9407e-324

Obviously, eps(0), which is even smaller, can be represented too. Could someone explain this to me?

Answer 1

This is a floating point issue. You should go read up on denormal numbers.

Briefly, realmin returns the smallest positive normalized floating point number. But it's possible to have denormal numbers that are smaller than this and still representable in floating point, which is what eps(0) returns.

Quick explanation of denormal numbers

A binary floating point number looks like this:

1.abcdef * 2^M

where abcdefg are each either 0 or 1, and M is a number in the range -1022 <= M <= 1023. These are called normalized floating point numbers. The smallest possible normalized floating point number is 1 * 2^(-1022).

The denormal numbers looks like this

0.abcdef * 2^(-1022)

so they can take values that are smaller than the smallest possible normalized floating point number. The denormal numbers linearly interpolate between -realmin and realmin.

Answer 2

Perhaps it is a matter of definition, this is what I see in the documentation of eps:

For all X of class double such that abs(X) <= realmin, eps(X) = 2^(-1074)