That's not true. The axiom is correct.
Say you're using double (i.e., you have 16 decimal digits of accuracy, roughly) and you are doing ((1.0 + 1e-20) - 1.0).
That becomes (1.0 - 1.0) = 0.0 because 1e-20 was dropped during the addition.
The substraction was perfectly accurate given its inputs (1.0 - 1.0 is exactly 0.0).
The issue is that float(1.0 + 1e-20) = 1.0. But that's also very accurate ! (All the 16 decimal digits of 1.0 are correct). But yeah the 20th (decimal) digit got dropped.
You see how each individual computation is maximally accurate. So the axiom is correct.
What it boils down to is that you are manipulating (adding/subtracting) numbers of very different magnitudes. So cancellations, which would be negligible if all numbers where of similar magnitude, becomes catastrophic. Hence the name catastrophic cancellations.
That becomes (1.0 - 1.0) = 0.0 because 1e-20 was dropped during the addition. The substraction was perfectly accurate given its inputs (1.0 - 1.0 is exactly 0.0).
The issue is that float(1.0 + 1e-20) = 1.0. But that's also very accurate ! (All the 16 decimal digits of 1.0 are correct). But yeah the 20th (decimal) digit got dropped.
You see how each individual computation is maximally accurate. So the axiom is correct. What it boils down to is that you are manipulating (adding/subtracting) numbers of very different magnitudes. So cancellations, which would be negligible if all numbers where of similar magnitude, becomes catastrophic. Hence the name catastrophic cancellations.