I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b = 10$, the fractional difference is also 10. My quantity, which is only defined if $x>0$, is defined as

$$ \begin{cases} x, & \mbox{if } x \ge 1 \\ \frac{1}{x}, & \mbox{if } x < 1\end{cases} $$

which is equal to (as $x>0$),

$$ e^{|\ln(x)|}. $$

Do mathematicians have a usual word and symbol for this quantity?

  • $\begingroup$ I have been curious about this quantity as well $\endgroup$ – frogeyedpeas Jul 9 '13 at 13:38
  • $\begingroup$ Are you looking for x if $|x| >= 1$ and $1/x$ if $|x| <= 1$ $\endgroup$ – frogeyedpeas Jul 9 '13 at 13:44
  • $\begingroup$ @frogeyedpeas Right. I didn't think of that because in my practical application, I already know that $x > 0$. The issue becomes more involved if $x$ can be negative, because what is the fractional error if $a/b$ is negative? $\endgroup$ – gerrit Jul 9 '13 at 13:47
  • $\begingroup$ I don't know this of the top of my head, but I would look for this in the theory of continued fractions and around the name of Gauss. $\endgroup$ – OR. Jul 9 '13 at 14:01

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