matlab - Why in MATLAB is realmax minus something still equal to realmax?

Question

In MATLAB, if I do realmax - 1000000 == realmax I get a logical 1 (true) as an answer. Why is that?

Answer 1

You get a true result because the value of 1000000 (i.e. 1e6) is much smaller than the floating point relative accuracy of a double precision variable for values at or near the maximum limits. For example:

>> realmax-1e6==realmax  % Subtract 1 million

ans =
  logical

   1       % Still equal; not big enough to register a change

>> realmax-eps(realmax)==realmax  % Subtract the distance to the next largest value

ans =
  logical

   0       % Unequal; yeah, that's big enough to matter

In short, the distance between representable numbers at the maximum limit (i.e. eps(realmax)) is on the order of 10^292. Subtracting a much smaller value of 1e6 gives a result that just gets rounded back to what it was before.

You can find more comprehensive explanations of dealing with floating point numbers here and here.