I've got to think that the wheel-measured distance includes a "wobble factor" in that you are not on a perfectly straight line (in fact, it's impossible to ride a bike in a perfectly straight line - its not stable). So the "measured mile" would be a mile plus a fraction of a percent to cover the wobble. A GPS-measured distance will not have the resolution to account for the wobble. Civilian GPS has a horizontal confidence interval of 7.8 meters and worse than that vertically. To get better than that, I think you need a GPS unit that is Defense-Department accurate. But even the US newest and most accurate nuclear weapon (the B61-12) has a Circular Error Probability of 30 meters*.
*I guess that this adds another category, besides horseshoes and handgrenades, where "close" is good enough.