Calculation Rounding Error/Examples/Arbitrary Example 1

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Example of Calculation Rounding Error

Consider the equation:

$x = \dfrac 1 {1 - \cos 1 \degrees}$

Evaluating the calculation while rounding to $4$ decimal places gives:

$x = 5000$

but the true value is $6565.8$ to $1$ decimal place.

Hence the calculation rounding error of this calculation is $1.6565.8$, or some $24 \%$ or $31 \%$ relative error, depending on how the latter is calculated.


Proof

We have:

$\cos 1 \degrees = 0.999847695 \ldots$

which is $0.9998$ to $4$ decimal places.


So to $4$ decimal places:

\(\ds \dfrac 1 {1 - \cos 1 \degrees}\) \(=\) \(\ds \dfrac 1 {1 - 0.9998}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {0.0002}\)
\(\ds \) \(=\) \(\ds 5000\)


To $9$ decimal places:

\(\ds \dfrac 1 {1 - \cos 1 \degrees}\) \(=\) \(\ds \dfrac 1 {1 - 0.999847695}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {0.000152305}\)
\(\ds \) \(=\) \(\ds 6565.772 \ldots\)


Hence we calculate the relative error as follows, using the two variants:

\(\ds \dfrac {\size {6565.8 - 5000} } {6565.8} \times 100 \%\) \(=\) \(\ds 23.85 \%\)
\(\ds \dfrac {\size {6565.8 - 5000} } {5000} \times 100 \%\) \(=\) \(\ds 31.32 \%\)

$\blacksquare$


Sources

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