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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 $2$?

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  • $\begingroup$ Is this a part of the context where, for example, $c=a+b$? Please state so, if that is the case. (I suspect there is more context to this, because there isn't much to learn/say about three numbers that wouldn't already be valid for one number.) $\endgroup$
    – user700480
    Dec 31, 2019 at 10:33
  • $\begingroup$ @StinkingBishop I have written the full question $\endgroup$
    – newhere
    Dec 31, 2019 at 10:36
  • $\begingroup$ But writing like that $f$ is not a function but a number.. $f = 3/2$ $\endgroup$
    – Gabrielek
    Dec 31, 2019 at 10:38

1 Answer 1

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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

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

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