I know that it's simple enough to map the integers, $\mathbb{Z}$, to pairs of integers, $\mathbb{Z}^2$, in a bijective way (i.e. a one-to-one mapping). You can wrap the integers around the origin of the 2D Cartesian grid like a spiral, or you can use some space filling curve, or you can map the integers individually to naturals by "folding up" the real line and then use a common bijection between naturals, like the Cantor pairing function (which is what the answer to this question does).
But so far, I haven't been able to come up with a bijection that has a simple closed form. I know you can express that "folding up" operation using $\mathrm{sgn}$ and $\lfloor\cdot\rfloor$, but those seem inelegant, and I'm looking for something that feels more "natural" (no pun intended) to the domain of $\mathbb{Z}$ than mapping them to naturals first, applying a bijection there, and mapping the result back to an integer.
For the positive naturals, there's this very elegant map:
$$ f(a,b) = 2^{a-1}(2b-1) $$
Extending $b$ to the integers is not a problem, but then a) we can't cover $0$ with this, b) we've already covered all other integers, so there's no room left if we want to extend $a$ to the integers as well (which would require changes anyway, since we can't just work with negative powers of $2$ here).
I also can't find anything like a list of common bijections between integers and pairs thereof, I suspect because most mathematicians stop caring after any one bijection has been found.
So my question is, are there any bijections between $\mathbb{Z}$ and $\mathbb{Z}^2$ which can be expressed in a simple closed form, akin to $f$ above, without reducing the problem to the naturals? I'd take a bijection where only one direction of the mapping has a closed form, but if both directions can be expressed in closed form that would be preferable. Bijections that are formulated in terms of the Gaussian integers, $\mathbf{Z}[i]$, instead of $\mathbb{Z}^2$ would also be acceptable.
