Precision is the player’s ability to consistently and accurately aim circles right to the center. Simply put, maps with more of a risk to aim inaccurately would be weighed more than maps that are very easy to aim consistently. Examples of this would be patterns that are widely spaced and quick, but low cs-maps have an exponential impact on precision difficulty.
t+pazolite & Camellia feat. Nanahira - Boku no Yume, Mecha Kuso Mugen Waki · beatmap info | osu!
Aiming accurately and consistently to the center of a circle is the one of the most important aspects of Precision, and the Circle SIze of a map is the key aspect that forces the player to aim more and more precisely. Circle Size should be weighted greatly because of that aspect.
A map’s difficulty when you increase it’s CS doesn’t increase linearly, despite the circles themselves becoming linearly smaller. It increases very rapidly. The difference between CS9 and CS10 is massive in from a gameplay standpoint, while the difference between CS0 and CS1 may feel like almost nothing.
An exponential equation would represent that difficulty accurately.
$$ C_{sWeight}=40\cdot1.4^{\left(2C_{S}-12\right)} $$
In the equation, CS is multiplied by $2$ in order to increase the speed the curve increases.
That doubled CS is then subtracted by $12$ so that the beginning of the curves starts more slowly to represent the little change in difficulty between CS0 and CS1. This also still allows the end of the curve to rise quickly as the CS starts to approach CS9 and CS10.
The $50$ is a scalar constant for weightage.