Taking the example of a two promo catapult vs a CGIII, drill IV longbow.
Let's stick that longbow (100hp) in a city, and fortify him as well.
That gives him 125% in bonuses (CG3: 75%, City:25%, Fortify:25%) for a strength of 13.5. He also has 4-7 first strikes
For a baseline, we use an unpromoted catapult, so strength 5.
So in each combat round, if the longbow wins (72.973%), he deals 31hp of damage. If the catapult wins, we deal 12hp of damage.
Crunching lots of numbers, the change of doing some damage to the longbow is about 12.36% ~ roughly one in eight. That seems about right, given my world builder tests, so I think I've got the calculation right.
If the catapult is promoted with Drill II, then the catapult has 1-2 first strikes. Our first strikes cancel out some of his, so he'll have 2-6 first strikes (no longer evenly distributed). Net result: a 28.08% chance of doing some damage.
With City Raider II instead, you cut down the strength of the longbow by 45% to 10.8. In each combat round, the longbow wins 68.354%, dealing 28hp damage; the catapult delivers 13hp of damage. Net result: a 17.92% chance of doing some damage.
If we add walls to the city, we buff the Longbow by another 50%. The new chances of doing some damage are
No promotions: 2.34%
Drill II: 9.95%
CRII : 11.90%
Yeah, the walls changed which promotion is "better". Here's the dope: when the ratio of the strengths is about 3.15, the amount of damage per hit goes from 33hp to 34hp. Which means that the number of rounds required to kill drops from 4 to 3.
In this case, the CR promotions take us from a ratio of 3.3 to a ratio of 2.76, crossing the jump point. The CRII catapult can survive one more hit, which for the purposes of this exercise is essentially the same as giving it a first strike (in addition to the extra damage and extra probability to hit).
That 3.15 ratio roughly translates to 162% defensive bonus. So if the longbow is below 162%, Drill II offers the better chance to do some damage. If the longbow is above 207%, Drill II is probably the right answer again. Between those numbers, the two CR promotions will cross the jump point, and probably give the better result.