And don't forget, the length of a solar day changes over time, mostly due to redistribution of mass: liquid core, tectonic plate movement, tides (thanks, moon!), weather moving huge quantities of water, filling the Three Gorges Dam, and other stuff. That's why we have occasional leap seconds.

Oh, and the place where you are is also moving around, thanks again to tectonic plate movement. Some places on earth only move millimeters/century. Others, like locations near the San Adreas fault can move at a nippy 5-7 centimeters per year (and presumably much faster when the next big earthquake hits, but in that case equinoxes might be the least of your problems).

So if you need a very precise forecast, you may be out of luck. (If you need the to measure the day cycle to nano-second precission, you should also include air density, local gravity and your own speed to account for most relativistic effects).