Is there a faster way to find distances on a sphere than Haversine?

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My task is to find the distance between the arbitrary point and many others on the sphere. Unfortunately, Haversine, even in vectorized form, does not satisfy in terms of speed, and I would like to find an option that will be faster for realtime calculation of the large data-set. I noticed that for a point lying on the pole, the distances to all others will be equal to their latitude. Maybe there are methods to transform the view of the coordinate system to center the point on the pole? Also I can adapt dataset before starting calculations, maybe I can somehow use the matrix of the precalculated distances to find distances to a new point? I will appreciate any ideas!