'Are floating point numbers really commutative?

It's said that floating point addition is commutative but not associative.

An example of it being non associative is the following:

(1 + 1e100) + -1e100 = 0, and 1 + (1e100 + -1e100) = 1

But doesn't this also prove that they are not commutative by the following:

1 + 1e100 + -1e100 = 0, and 1e100 + -1e100 + 1 = 1

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