"Distance" (or angular magnitude) between two quaternions?

Question

I want to find the "distance" between two quaternions. By "distance" I mean a single float or int, not another quaternion (that would be the difference, i.e. inverse(q1)*q2). I guess you could call what I want "angular magnitude".

I need to apply more torque to a physics object the further it's rotated from its original angle.

I don't understand the maths involved in quaternions, so a code-based example would be most helpful. I've looked at several other questions but I don't believe any answer my question, or at least not in a way I understand it.

Answer 1

Find the difference quaternion qd = inverse(q1)*q2).

Than find the angle between q1 and q2 by angle = 2 * atan2(qd.vec().length(), qd.w()) // NOTE: signed

The "angle" here, is the angle of rotation from q1 to q2 by shortest arc.

Answer 2

Also you can use this lib function from pyquaternion. Quaternion.absolute_distance(q0, q1)