AP
It would be nice to build the AP mainly just to ensure it is on our continent.
A Buddhist AP could prove problematic for our war vs Brennus however. He has more cities than us, so I'm guessing it is also likely he has more Buddhist population than us = he votes himself in as the pope. He could then use this to force the end of our war, which we presumably would have to veto and incur

in all our Buddhist cities.
GE
Our choices for the GE are bulb for 1000+

on Machinery. Since our raw

gets multiplied by libraries and optional prereq bonuses, etc, that 1000

actually is only worth ~700

.
Or rush something like the AP. AP costs something like 400

? But again, we have multipliers of Forge and Industrious giving a raw cost of 236

.
(If this implies 1

= 3

, then a settled GE is worth 4

, or 12

. It would take 59 turns for a settled GE to give an equal return on investment to the options given above)
Axes vs Crosbows
Another way to look at it:
How many

could Machinery save us? Crossbowmen cost 60

compared to 40 for swordsmen, or 35 for axes. Swords are actually better attackers if we're attacking Archers, though Crossbows are clearly better if cities defended by axes or Brennus Gallic Warrior UU.
So, we shouldn't compare crossbows with swords, but rather axes, since we'll want to build swords regardless. And I'll assume the target unit is a defending Gallic Warrior.
I'll also assume our units have 2 promos, CR1/2, and defenders have 1 promo plus the free one, Guerilla1+Combat1. Also assume city defenses have been bombarded away, and units fully fortified.
Axe vs GW on flat land (defender mods -0.7) = 5.0 vs 6.6/1.7 = 5.0 vs 3.53. Odds about 76%.
Axe vs GW on hill (defender mods -0.25) = 5.0 vs 6.6/1.25 = 5.0 vs 5.28. Odds about 34%.
If we attack 20 Gallic Warriors and half the cities are hills, we expect to need ~43 axes. Total cost = 43*35 = 1505

.
Assume our Crossbows have Combat1+Shock promo. They also have a first strike.
CB vs GW on flat (defender mods -0.5) = 6.6 vs 6.6/1.5 = 6.6 vs 4.4. Odds about 93%.
CB vs GW on hill (defender mods -0.05) = 6.6 vs 6.6/1.05 = 6.6 vs 6.29. Odds about 71%.
Versus 20 GWs we expect to need about 25 CBs. Total cost = 25 * 60 = 1500. Wow! That's a pretty crazy coincidence! Machinery saves us a negligible number of
This analysis is actually quite generous toward crossbows, because it assumes we haven't used suicide cats. With suicide cats it doesn't really matter if it is an axe or a CB attacking, both should have very high odds.
Ummm... so... in what situation would crossbows be better than axes? Only if we are attacking other crossbows or axes really.