The most accurate computation I've come across for a great circle arc is that contained in Thaddeus Vincenty's 1975 paper in Survey Review - Directorate of Overseas Surveys of the Ministry of Overseas Development (Surrey, UK). Copy available here,
http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
I expect Mr. E. Williams' website (linked via post above) has employed Vincenty's formulæ [
wikipedia entry here] as the paper is referenced on his GC Calc page. Mr. Chris Veness has also implemented a javascript calculation using Vincenty, at
Geodesic distance between two Latitude/Longitude points using Vincenty ellipsoid formula in JavaScript (useful as he has provided the complete code).
Accuracy for the specified ellipsoid is claimed to be within half a millimetre.