Error bound on square root of relative error bounds

Question

Can you correct me if I am wrong.

If $|d^2 - \tilde d^2| \leq \epsilon d^2$, what is the error on $|d - \tilde d|$?

I think is $\sqrt \epsilon d$.

Answer 1

It whoudl be $\sqrt \epsilon d$.

$$-\epsilon d^2 \leq d^2 - \tilde d^2 \leq \epsilon d^2$$

$$-\epsilon d^2 + d^2\leq \tilde d^2 \leq \epsilon d^2 + d^2 $$

$$\sqrt{-\epsilon d^2 + d^2} \leq \tilde d \leq \sqrt{\epsilon d^2 + d^2} $$

$$-\sqrt{\epsilon} d + d \leq \tilde d \leq \sqrt{\epsilon} d + d $$

$$-\sqrt{\epsilon}d \leq |\tilde d - d| \leq \sqrt{\epsilon} d $$

Thank you.