Cost of rounding in a probability computation

Status: half-baked

You’re making some quick computations and you don’t want to fuss around with a calculator or computer. You’ve got some fraction \(\frac{x}{y + \epsilon}\) but you’d much rather round it to \(\frac{x}{y}\). What’s the error, \(\Epsilon\)?

\[ \Epsilon = \frac{x}{y+\epsilon} - \frac{x}{y} = x\left ( \frac{1}{y + \epsilon} - \frac{1}{y} \right ) = x\left ( \frac{y}{y(y + \epsilon)} - \frac{y + \epsilon}{y(y + \epsilon)} \right ) = x\left ( \frac{-\epsilon}{y(y + \epsilon)} \right ). \]

Here I used to have a wrong approximation to the above. I’ll make a good one if I can be bothered.