Is this probability distribution known in the literature?

In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution?

$\Pr(X\le x)=\exp\left[-\frac{1}{2}\left(\frac{1}{2}x-\sqrt{1+\frac{x^{2}}{4}}\right)^{2}\right]$

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That square root reminds me of the Wigner semicircle distribution. You might try seeing if there is some nice transformation that takes you to it. –  Steve Huntsman Jan 11 '12 at 14:45
The square root in my distribution comes from a hyperbola. The x/2 term flattens the positive-x asymptote against the x-axis, and the exponential maps 0 (well, $-\delta$) to 1 (or rather, $1-\epsilon$). –  Cyan Jan 11 '12 at 15:41
Can I assume you've already looked through the Johnson, Kotz & Balakrishnan volumes? –  cardinal Jan 12 '12 at 23:31
Nope. Thanks for the pointer! –  Cyan Jan 17 '12 at 15:39