Precession is the secular and long-periodic component of the motion of the Earth's spin axis in the celestial reference frame, approximately exhibiting a motion of about $50''$ per year around the pole of the ecliptic. The presently adopted precession model, IAU2006, approximates this motion by polynomial expansions of time that are valid, with very high accuracy, in the immediate vicinity (a few centuries) of the reference epoch J2000.0. For more distant epochs, this approximation however quickly deviates from reality. As a reaction to this problem, a new model, comprising very long-period terms fitted to a numerical integration of the motion of solar system bodies on scales of several thousand centuries, was recently published by the present author with co-authors from France and United Kingdom in Astronomy & Astrophysics. Here a shorter description of the new model, including a new assessment of its accuracy and comparisons with other models, is given.