Stretching is not a warm-up.
The authors state that stretching does not warm up your muscles, and can actually inhibit performance. Some of the warm-ups in the book do stretch your muscles, but the idea is not to lengthen them before the workout, it is just a side effect.
I agree with this new rule. When I run, I tend to stretch a little before a workout, but only after I’ve already jogged around a bit. I save the majority of my stretching (and I stretch a lot) for after my workouts and before I go to bed. I also stretch quite a bit at random points in the day just for the hell of it.
I very rarely stretch before lifting unless I am feeling particularly tight before a set of something. I almost always stretch exclusively when I am done.
Stretching after a workout of any kind will give you the best gains in both flexibility and in reduced soreness. When you stretch after your workout, you are giving your body an opportunity to lose some of the lactic acid build up that can lead to sore muscles the next day.