The performance of ray casting is determined by how efficiently ray object intersections can be determined. Let's rethink the series of computations used to determine the ray-sphere intersection while looking for ways to eliminate unnecessary work.

Step 1:

Thus, we can test if (v - r) is greater than the closest intersection thus far, t_best. If it is then this intersection cannot be closer. We can use this test to avoid the rest of the intersection calculation.