How do you figured out that double effectiveness and weakness is work in Pokemon GO?

data mined

As for the .64, and 1.56, it's simply 80% applied twice, or 125% applied twice.

In other words, base damage is 1, effective damage adds 25%, leading to a 1.25 damage modifier. And then double effective adds another 25% to THAT. Meaning it's 1.25 * 1.25 = 1.5625 (rounded to 1.56).

This thread for details:
https://www.reddit.com/r/TheSilphRoad/comments/4wzll7/testing_gym_combat_misconceptions/

The way we collected exact damage is stick the Attacking Pokemon into a friendly gym, then train on the gym with a selection of Pokemon. I retreat after the initial double attacks of the defender and record the damage taken, via Pokemon details screen (hp lost/2).

For figuring out double effectiveness, I found a pair of Pokemon whose damage output would be different if the multipliers were x1.25, x1.5, and x1.25^2=1.5625:

Arcanine 1219 CP 94 HP (IV 15/9/3)
Fire Fang / Bulldoze
Paras 447 CP 45 HP (IV 12/15/10)
Scratch / Seed Bomb

Paras is Bug/Grass, both fire weaknesses.

If type modifier is x1.25 = floor{.5 * 245 / 135 CPM(15)/CPM(18) * 10 * 1.25 * (1.25) } + 1 = 13 damage

If type modifier is x1.25 x1.25 = floor{.5 * 245 / 135 CPM(15)/CPM(18) * 10 * 1.25 * (1.25*1.25) } + 1 = 17 damage

If type modifier is x1.5 = floor{ .5 * 245 / 135 CPM(15)/CPM(18) * 10 * 1.25 * (1.5) } + 1 = 16 damage

Since the actual damage was measured as 17, we can determine that the game uses x1.5625 instead of x1.5.

Likewise for double resistence.

Similar methodology with selected Pokemon-pairs allow us to determine that immunity had been converted to simple resistance (x1.25) as well.