numerics - Arbitrary-Precision Arithmetic behaves unexpectedly

I want to understand how small numerical errors propagate through a computation and may drastically change the final result. To this end, I consider a toy world in which every number is limited to exactly 5 significant digits.

The output of

$MaxPrecision = 5;
$MinPrecision = 5;

x = SetPrecision[10^(-10), 5];

x // FullForm
y = 1 + x;
y // FullForm
y - 1 // FullForm

1.`5.*^-10

1.`5.

1.`5.*^-10

The last line puzzles me. The FullForm of y shows that the Precision of 5 significant digits is not sufficient to resolve the small deviation x from 1. However, y - 1 still yields the exact result, instead of y - 1 = 0.

If one writes explicitly

1.`5. - 1 // FullForm

this yields, of course,

0

as it should.

Does this mean that FullForm[y] hides something?

Comments

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