Round Off Error Of One Digit Integer

Question

Find the relative error of the function $$f=\frac{a+2b}{c}$$ given the numbers $$a=1,b=1,c=2$$ that were round up to a one digit integer

So intuitively if we take for example $4.2$ and round it to $4$ we change the $10^{-1}$ digit and if it was between $0.0$ to $0.5$ we will round it to $4$ else if it was between $0.5$ to $0.9$ we will round it to $5$ so it is "half the way" or $\frac{1}{2}\cdot 10^{-1}$

1. I do not know if this intuitively explanation is correct.

2. There is a formula $\frac{1}{2}\beta^{1-p}$ where $\beta$ is the base and $p$ is the precision or the significant digits, why in this case it is $2$?

Answer 1

You can just right down a first order approximation to the relative error:

$$ \varepsilon_f \approx \frac{a}{a+2b} \varepsilon_a + \frac{2b}{a+2b}\varepsilon_b - \varepsilon_c $$

So, for $a=b=1$ and $c=2$, the relative error is approximately given by

$$ \frac 13 \varepsilon_a + \frac 23 \varepsilon_b - \varepsilon_c. $$