# Cancellation of inequalities in floating point arithmetic

In finite precision floating point arithmetic the associative property of addition is not satisfied. This is, it is not always the case that $$(a+b)+c=a+(b+c)$$ Even $$a=(a+b)-b$$ is not always true.

To prove that $$x+y is equivalent to $$x with real numbers we can add $$-y$$ on both sides of $$x+y to get $$(x+y)-y and then from this $$x=x+(y-y). But I can’t repeat the last step for floating point.

Question: Are the inequalities $$x+y and $$x equivalent in finite precision floating point arithmetic?