I kind of assumed "a" was the width of the street (or half the width?) and "y" was the vertical distance from the cross street to the turn-around point. And you are solving for "y" for known values of "a" and "d." But just wanted to confirm that's the case.
I think the problem becomes trivial if you assume the path turns abruptly at a single point back at the cross street rather than following a turning radius. And the change in distance will also be trivial between the those two paths. If you use the solution to that simpler path as the starting point for iteratively solving the equation for the more complex path, you would probably be able to converge pretty quickly.