A quote from that old article:
In XSI, we need a way of allowing the chain to gradually fall behind the position of the effector.
This is the original assumption that ends up necessitating a bit of stretch. Because if the effector “falls behind”, it eventually has to catch up, and one way is by making it catch up further than the straight length of the IK chain.
So… I don’t have a math formula, but here is how you could imagine reversing this assumption. Instead of the effector falling behind, it could race ahead slightly. And then they could meet exactly at the final length of the IK chain. (And thus be completely separate from the stretchy limb.) Beyond the length of the straight IK, the position of the effector would match 1:1 linearly.
This is what the formula could look like expressed as a function curve.
Which would animate like this. Black is the soft effect. Faint green is original snappy IK.