# Floating point arithmetic on division

I am trying to figure out how $$(x/y)$$ in floating point arithmetic

$$fl(fl(x) / fl(y))$$ where $$fl(x) = x(1-\delta_1)$$, $$fl(y) = y(1-\delta_2)$$, $$fl = (1-\delta_3)$$

I have:

$$= x/y \cdot ((1-\delta_1)/(1-\delta_2))(1-\delta_3)$$

after arithmetic

$$= x/y \cdot (\delta_3 \delta_1 - \delta_1 - \delta_3 + 1)/(1-\delta_2)$$

Not sure how to fully write the rest I believe i have to shorten them using the info that $$\delta < \epsilon$$ and $$\delta^2 \leq \delta$$

$$\frac{1}{1 - \delta_2} = 1 + \delta_2 + \delta_2^2 + \delta_2^3 + \cdots$$
Then, for example, if $$\epsilon$$ is the machine epsilon and $$\delta_2 > 0$$:
$$1 + \delta_2 \le \frac{1}{1 - \delta_2} \le 1 + \delta_2 + \epsilon$$
That is quite a tight bound, when you think about it. (Exercise: Prove that this is true if $$\delta_2 \le 0$$, too.)