Using the back-and-forth method we can construct an increasing bijection from the set of rational numbers to the set of of rational numbers except zero.

http://en.wikipedia.org/wiki/Back-and-forth_method

I would like to have a "natural" bijection. The algorithm resulting from the back-and-forth method behaves rather chaotically.

It would be nice for example to have an uniform bound on the number of steps needed to evaluate the image of any given rational number $a=\frac{p}{q}$. I'm note sure what should count as a "step" here, maybe adding or multiplying integers.