Whip 'til your hands bleed
Pop rushing is way overpowered in this game, and should be abused to the max if you intend to master this game. Part of the reason why it's so overpowered is because of a bad calculation that they haven't even bothered to fix in patch 1.61.
With pop rushing at normal speed, you're supposed to get 30 hammers for every pop spent. That's already pretty good, considering that with a granary, a city only needs about 11 (at level 1) to 31 (at level 21) food to grow back the pop you've spent. Spending 11 food to get 30 hammers is already mighty good, but because of the exploit, it gets even better.
Assuming the building or unit is already started (so you don't get a penalty), what the game does is it checks to find how many base hammers you need to complete the build, and charges you an amount of pop based on this. Let's say there is a forge in your city, or you're using organized religion. This means you have a 25% bonus on production. With 30 base hammers, you then get 37 total hammers. If what you need to complete the build is 37 hammers, the game will only charge you one pop.
However, the game always gives you an amount of hammers that's the smallest multiple of 30 needed to complete the build. This is total hammers, not base hammers. So in the case above, you'll get 60 hammers because 30 wouldn't be enough to complete it. Congratulations, you've just received 60 hammers by expending only 11 food (assuming a size 2 city going down to size 1). That's a grand total of 5.45 hammers per food spent! Tell me that isn't overpowered. Even taking into account that normal hammers would receive a normal 25% bonus, you still need 4.36 normal hammers to equal every food used. No improvement anywhere in the game can provide 4.36 hammers to compete with the 1 food a farm provides.
Pop rushing is so overpowered in this game that i find slavery to be by far the best civic, all branches included. I abuse it to such an extent that in a typical game, until gold rushing comes along, about 70% to 80% of all my production is obtained through this mean, and only about 20% to 30% from actual hammers obtained from tiles. It's so overpowered that a city full of farms becomes a better producer than one full of mines!
The best ways to abuse the system are using 1 pop for 60 hammers when needing 31 to 37 with a 25% bonus (getting 30 hammers for free because of the bug), or using 5 pop to get 210 hammers when needing 181 to 187 with a 25% bonus (getting 60 hammers for free). However, it's good at all levels, as long as you make sure to be at a point that will provide more hammers than you spend pop for.
Because of the unhappiness penalty involved, you don't want to whip much more than once every 10 turns (so the unhappiness has time to wear off). However, because it's so overpowered, you definitely want to use it as soon as the unhappiness penalty wares off, if not earlier. You don't want to waste a single turn that's not being spent on making unhappiness disappear. Yes, this means that you want to be in a constant state of unhappiness due to whipping!
For cities with low food, it will take them time to grow back the population, but little time to get rid of the unhappiness. For those, you ideally want to whip a single pop every 10 turns, and no more. This increases the bonus obtained from the bug, while minimizing your pop spent. Time your production and use your queue to make sure that you have a building that will have just the right amount of production done by the time unhappiness falls down to zero. With a 25% bonus, this means you should have something that needs 31 to 37 hammers to finish. With a 50% bonus, that's 31 to 45 hammers.
For cities that have high food outputs, growing is not a problem, and unhappiness quickly becomes huge. In this case, you want to whip as many pop points as possible everytime you whip. This means you should have buildings barely started waiting to be whipped. Those buildings should have somewhere between 1 and 30 hammers done on them, depending on the total hammers required, to ensure the biggest gain possible as described above. For example, a bank requires 200 hammers. Therefore with a 25% bonus you want to whip 5 pop to gain 210 hammers. This can be done by having 181 to 187 hammers left to the build. Therefore you should let normal production of the bank reach 13 to 19 hammers, then put it in the queue until you can whip the rest. Actually, due to a rounding error this will work at 12 hammers as well.
Another thing you want to keep in mind is when to rush. The ideal time is usually not on the very turn where unhappiness disappears, but before that. Ideally, you want to be as close to that point as possible, to keep unhappiness at a minimum and be able to use as many productive tiles as possible for as long as possible. However, there is another factor to consider, one that becomes more significant the more pop you whip in one swoop.
You see, the cost of whipping depends on your city size. Assuming a granary, whipping one pop costs 1 more food for every extra pop your city has before the whip, because it will cost one more food, for example, to go from size 4 to size 5 than it would cost to go from size 3 to size 4. As you whip more pop at a time, however, the effect gets bigger. Whipping 2 pop will cost 2 more food per extra initial level, because each of the two levels you need to grow back will cost 1 extra food. Similarly, whipping 5 pop will cost 5 extra food per initial level, i.e. it will cost 5 more food at level 11 than at level 10, and 10 more food at level 12 than at level 10. This really adds up.
So you want to be at high levels as long as possible to use more tiles, but you want to whip at the lower level possible. Solution : whip at the end of a level. Use you level to the max, wait until you're right about to jump to the next level, and whip right before this happens, rather than right after it does. Ideally, what you want to have is the following. The last turn before whipping, you want to make just enough food to reach the next level, and not one more. Set you governor to keep the city from increasing in size. Then the next turn your city will be at 30/30 food or something like that. Set the governor back to normal, witness your city now at 30/26 or something like that (which is nice because now you know that first two increases will come quickly), and let it grow again.
When to do this and when to let your city grow instead? Well, the gist of it is that if in the time it takes for your whipping related unhapiness to drop down to zero, you could grow an extra size and fill that granary again (or at least come close to filling it), then you should do that; otherwise you should whip at this level and not wait for unhappiness to subside. By going down in pop, happiness won't be a problem for a while anyway. Another thing to keep in mind is that it's sometimes preferable to wait until you're at max happiness before whipping; this maximizes usage of tiles at the cost of less whipping (i.e. less abuse). There are also other factors to consider, like whether or not you've got extra 3-food or 2-food tiles you could use, but i'll let you find that out through your own experimentation.
One last thing. Whenever you've got a big overflow from whipping (around 30 hammers at normal speed), it's often a good idea to use that overflow on a worker or settler. This way, you're not wasting your time making them the normal way and getting only 1 hammer per food. Also, workers and settlers make good candidates for the whip for the same reason.
