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I'm looking for a real function $f$ such that for all real $x$

$$e^{-e^{f(x)}} + e^{-e^{f(-x)}} = 1$$

$\sinh(x) + \log(\log(2))$ gets close, but not quite.

EDIT: Sorry, my problem is way underspecified... I need to think about the other constraints. Ideally I'm looking for $f(x)$ of the form $$f(x) = \left(\sum_{i=1}^n a_i x^i\right)\left(\sum_{j=1}^m b_j e^{c_j x}\right)$$ with $f(-\infty)=-\infty$ and $f(\infty)=\infty$

but I doubt such a solution exists.

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    :( What am I supposed to do now? I already answered. – 2017-02-07
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    What do you think now, @ArthurB.? Does this satisfy your interests? – 2017-02-07
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    I'll accept the answer if you insist, because I wasn't clear in the first place, but this isn't really what I'm looking for. – 2017-02-07
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    Okay, sorry. But if you don't like the answer on this one, just ask another question, being more specific that your looking for a specific type of function. – 2017-02-07
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    It isn't as if this is your only one chance to ask this question. It is alright to ask a similar one. – 2017-02-07
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    My [question here](http://math.stackexchange.com/questions/1790087/how-to-solve-fxf-left-frac1x-right-ex-frac1x) was not answered the way I wanted it to, but I didn't actually edit the question later. Good luck on finding your answer! – 2017-02-07

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EDIT (Do not edit the question after someone's given the answer!)

We know $e^{e^{f(x)}}=\frac{1}{2}+g(x)$ for some odd function $g(x)$.

This gives us that $f(x)$ is of the form $f(x)=\ln \ln \left(\frac{2}{2g(x)+1} \right)$. The closest example of such a function as you mentioned, which satisfies$$\lim_{x \to \infty} f(x)=\infty, \; \lim_{x \to -\infty}f(x)=-\infty$$ and is nontrivial would be $$f(x)=\ln \ln \frac{2 \pi}{-2 \arctan x+ \pi}$$