Formal notations of equivalencence between floating point expressions
One plausible notion of "nearly-equivalent" for floating-point expressions is that the relative error between the two is guaranteed small.

For example, if $x\ne 0$ and $y\ne 0$, define

$$D(x,y) = \max(\left|{x \over y} - 1\right|, \left|{y \over x} - 1\right|).$$

This measures the relative error between two real numbers.  Now, say that $x$ is nearly equivalent to $y$ if $D(x,y) \le \varepsilon$ (for some $\varepsilon$ fixed in advance; e.g., $\varepsilon = 10^{-6}$ or something).

Now, we can define two floating-point expressions e, e' to be nearly-equivalent if, for all possible assignments to the variables of these expressions, we have $D(e,e') \le \varepsilon$.

This implies that replacing e with e' will have only a small effect on the value of the expression (with "small" measured using relative error, rather than absolute error).

With these definitions, if you know that a, b, and c are positive and bounded away from zero, you should be able to prove that (a+b)+c and a+(b+c) are nearly-equivalent, for very small values of $\varepsilon$ (dependent on machine epsilon and the specific floating-point format used).  However, you can't prove them nearly-equivalent when nothing is known about a, b, c, because it is possible that (a+b)+c = 0 but a+(b+c) &gt; 0, and then the relative error is infinite in this case.

Perhaps a more nuanced definition could allow them to be treated as nearly equivalent in the general case; I don't know.  For instance, we could alternatively imagine defining $D(x,y)$ to be the smallest value $\delta&gt;0$ such that
$$\begin{align*}
(1-\delta) x - \delta &amp;\le y \le (1+\delta) x + \delta\\
(1-\delta) y - \delta &amp;\le x \le (1+\delta) y + \delta
\end{align*}$$
(This effectively captures a combination of relative error and absolute error.)  I suspect it might be possible to prove that (a+b)+c and a+(b+c) are nearly-equivalent, under this distance metric.  Is this useful?  I don't know.