How do I indicate approximation by rounding when manipulating equations?

Question

I manipulate a set of inequalities and arrive at

$$\Leftrightarrow \frac{(||\nu|| - \lambda^{-1})^2}{2\tau^2} \geq 53\ln(2)$$

Note the $\Leftrightarrow$ sign, indicating that this inequality is equivalent to the previous one (not shown here). I want the next line to be

$$||\nu|| \leq \lambda^{-1} - 8.5728\tau \;\vee\;||\nu|| \geq \lambda^{-1} + 8.5728\tau$$

but I don't feel comfortable preceding it with the $\Leftrightarrow$, because my rounding of $53\ln(2)\sqrt{2}$ makes it not equivalent anymore. What is the correct notation in such a case?

Answer 1

Anyone with the mathematical maturity to be reading this will certainly understand that, for instance, "8.5728" denotes not the precise rational number 5358/625 but rather some real number in the interval [8.57275, 8.57285]. So I don't think it's necessary to do anything special here.

That said, if it really bothers you you could change the $\Leftrightarrow$ to a $\Rightarrow$ and change the $\leq$ to an $\lesssim$ (by analogy with $\approx$ for approximate equality.)