I guess I missed that you were changing the food surplus in the middle, to get a 7-food surplus on the first and last turns and zero surplus on the other turns. I was assuming you'd have the same food surplus each turn. Are you assuming that you can switch back and forth between high-food and high-production tiles?
Slavery: overrated?
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VoiceOfUnreason
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Not zero - a little bit less than 2, actually ( 1 + P/9)
Not quite - closer to say that I'm observing that you can tune the loop if you can switch back and forth between fungible tiles.
Example: consider a 6->4 whip with dry rice. Dry rice and three farms is +7, so we can hit the beginning and end targets. In the middle, rice + specialist + 3 cottages is a little bit too fast (3 food to many), but we can tune that with a farm and a second specialist. Easy.
Now, replace the dry rice with wet corn, hitting the 7 got easier, and in the middle we've got two specialists and two cottages. That's still a little hot, but we can tune it with a farm and a specialist... whoops - so we need a third specialist slot to make that work.
Now try wet corn and a brown cow. Hitting the 7 is easier still, but it's getting harder to slow down in the middle .
With wet corn and flat pigs, it's basically impossible. You have to resort to other measures - training a worker or a settler, not working a good tile (horrors! but maybe another city can use it for a time) etc.
On the other hand, we have situations where the tiles aren't fungible.
Consider dry rice, with four green mines. The sixth pop is working a cottage.
If we whip from here, we end up switching from mines to farms on the growing turns, and we also don't get to work all the mines while trying to grow. Assuming I've done the math right, you get 29 mine turns instead of 44. 15 lost turns on the mine is 45 hammers, so all this work earned +15
(ignoring setup cost).If we had a fifth mine - well, it's useless in the whipping case, but parking at size six you can alternate the mine with a farm, so you are effectively working another half mine. In this example, the whip is actually a little bit behind ( either behind by 1F, or behind by 3H-1F, per loop).
That result shouldn't be a surprise - CW has long held that you shouldn't be expecting extra production when whipping away mines.
I'd say it's been proven that slavery is one of the best techs in its civics bracket. However, I'd say it is even better than the numbers show.
In most of my games, slavery is the difference between settling a good city or not. A whip's value is not only 30P - you can't really quantify the difference. Same for rushes - some of them would be hopelessly late on higher difficulties without slavery, making the difference between a quick successful war or a failed war with lots of lost production on units.
I think the greatest boon of slavery is that it lets you time everything a lot better than without it at a time in the game where doing well is particularly important. Later on it will get less good but with its civics bracket being so terrible I often keep it until the end.
I guess caste system is worthy too, however it is strange that a tech which amplifies workshops comes with code of laws - they are pretty bad at the time. Serfdom is so bad it seems like nobody even bothers defending it and emancipation is rather lame compared to both CS and slavery, especially lame when you consider how the game shoehorns you into taking emancipation.
In most of my games, slavery is the difference between settling a good city or not. A whip's value is not only 30P - you can't really quantify the difference. Same for rushes - some of them would be hopelessly late on higher difficulties without slavery, making the difference between a quick successful war or a failed war with lots of lost production on units.
I think the greatest boon of slavery is that it lets you time everything a lot better than without it at a time in the game where doing well is particularly important. Later on it will get less good but with its civics bracket being so terrible I often keep it until the end.
I guess caste system is worthy too, however it is strange that a tech which amplifies workshops comes with code of laws - they are pretty bad at the time. Serfdom is so bad it seems like nobody even bothers defending it and emancipation is rather lame compared to both CS and slavery, especially lame when you consider how the game shoehorns you into taking emancipation.
Larwin
Prince
A side note: I'm playing a game as Montezuma. He has a UB that allows you to whip more often. 
Peregrine
The Swift
On Marathon, slavery's good and bad aspects are, imo, amplified. Infrastructure can take soooooo long to emplace, that the whip is often the best way to go. I regularly whip the classical infra into place; courthouses, markets, temples, and sometimes barracks, though very rarely stables, monasteries, granaries, or monuments. Granaries and monuments usually get chopped simply because at that point, so early in the game, there just isn't the population available to whip. I hesitate whipping 1 out of 2 pop points. Is there an argument for that? The "bad" is that unhappiness due to whipping takes 3 times as long to go away. I don't even do the math. I just whip when it seems necessary/advisable; it'll take 30+ turns for that courthouse? WHIP. Sometimes a city will pile up so much unhappy that it takes a small army (HR) to calm things down. On the plus side, a small army in the city tends to deter, somewhat, the AIs from perceiving moi as a target. Costs a bit to maintain the force, but then I'm after long-term goals. Usually.
Celebithil
Warlord
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The way I see whipping is that it gives an opportunity to trade food for hammers once at a (generally great) rate of (10+pop)F/30H, at the cost of some tile-yield turns (10 plus how many you lose because you are below the (possible depreciated) happy cap). So we can see that the efficiency increases the more population you whip at once, as the cost remains the same no matter the size of the transaction. Examples given in this thread show that even with 2pop whips it is possible to lose only 11 tile yield turns (thus being almost as expensive as a 1 pop whip, while giving a much larger amount of trade advantage). When you whip 3 pop, you will end up with losing a lot more tile yield turns (as you average population tends to be lower), so while in principle this could be more efficient, in practice it hardly ever is.
To determine whether whipping is smart (and ignoring issues where the main benefit of the whip is getting stuff NOW, which should always be recognized as an important benefit of the whip), there remain two questions: 1. What is the value of an extra tile yield? and 2. What is the normal exchange rate of food to hammers?
The answer to question 1 is that one of the best marginal tiles you usually have available is a grass farm, in which case one tile yield is worth 1F. The answer to the second question seems to vary more, let us assume the best food square is a grass farm. Then if the best production square is a grass mine, you can trade 2F for 3H, so the exchange rate is 1F=3/2 H, if the best production square is a plains mine, you can trade 3F for 4H, so the exchange rate is 1F=4/3 H, while if the best production square is a plains forest (or another 1F2H tile) then you can exchange 2F for 2H so the exchange rate is 1F=1H.
So let us now consider the case of 2pop whipping at population 6. Then the growth we consider is 4->5 and 5->6, so the pop in the formula above is the average of 4 and 5, so 4.5. Thus we get to exchange at a great rate of about 1F=2.07H. However we lose 5.5 yield turns per transaction (two transactions at total cost 11, in the optimal case), each turnyield is (as determined above) 1F, so the net exchange rate is (14.5+5.5)F/30H, or 1F=1.5H. So this is exactly as good as working grass mines, but better than working worse production tiles (plains hills, 1F2H tiles etc.).
Increasing the population makes whipping less good, so this is why whipping away grass mines is generally not efficient above size 6. A similar calculation yields that plains hills are as good as whipping until population 8.5 (i.e. at pop 8 it is better to use the whip, at 9 it is better to work the plains hill mines), but working 1F2H tiles is less efficient until size 16.
If you have more than one food resource and not sufficiently many grass hills to lose all the food on, food becomes cheaper in respect to hammers so whipping becomes better (Yes, usually it is better to use the extra food for the whip than just leave it lying unused), but beware that the lost tileyields become more expensive as well. For example if your best marginal food tile is 4F, then a tileyield is worth 2F, if the best production tile is 1F3H (grass hill), the exchange rate becomes 1F=1H. So in that case it is better to whip until size 10.5 (solution of (10+ (pop-1.5) + 2*5.5)F/30H=1F/1H).
One final remark on the calculation by VOU. It seems he wants to not only minimize the number of tileyields lost (to 11 in a 2-pop whip), but also enforces whipping every 10 turns. The latter ensures that he has to have a huge possible food surplus he can use for two turns (as you have to grow in two subsequent turns, once from happy cap -1 to happy cap, at the end of the cycle, and the next turn from happy cap-2 to happy cap -1, after the new whip. However, as Iranon pointed out, you can
grow at any rate you like at the happy cap after the unhappiness of the whip has worn off and before whipping again, without damaging the efficiency of the whip. This means that being able to whip efficient is not contingent on having the ability to conjure up a huge food surplus (something like 2 or 3 is enough). This is not to say that it is always best to use an efficient cycle longer than 10 turns as Iranon proposes, using the whip more often at a less efficient rate might still be advantageous, but one does not need to satisfy the narrow conditions VOU identifies in which his "whip as often as possible" cycle is also most efficient, in order to succesfully whip. (Then again, VOU's analysis does show that having the ability to quickly change from food to hammer tiles does benefit the whip, and that you want to try to grow as fast as possible at the happy cap after the whip unhappiness wore off, to whip again as quickly as possible, while there is usually not such a rush to grow from cap-1 to cap as you have to wait for the unhappiness to wear off anyway).
To determine whether whipping is smart (and ignoring issues where the main benefit of the whip is getting stuff NOW, which should always be recognized as an important benefit of the whip), there remain two questions: 1. What is the value of an extra tile yield? and 2. What is the normal exchange rate of food to hammers?
The answer to question 1 is that one of the best marginal tiles you usually have available is a grass farm, in which case one tile yield is worth 1F. The answer to the second question seems to vary more, let us assume the best food square is a grass farm. Then if the best production square is a grass mine, you can trade 2F for 3H, so the exchange rate is 1F=3/2 H, if the best production square is a plains mine, you can trade 3F for 4H, so the exchange rate is 1F=4/3 H, while if the best production square is a plains forest (or another 1F2H tile) then you can exchange 2F for 2H so the exchange rate is 1F=1H.
So let us now consider the case of 2pop whipping at population 6. Then the growth we consider is 4->5 and 5->6, so the pop in the formula above is the average of 4 and 5, so 4.5. Thus we get to exchange at a great rate of about 1F=2.07H. However we lose 5.5 yield turns per transaction (two transactions at total cost 11, in the optimal case), each turnyield is (as determined above) 1F, so the net exchange rate is (14.5+5.5)F/30H, or 1F=1.5H. So this is exactly as good as working grass mines, but better than working worse production tiles (plains hills, 1F2H tiles etc.).
Increasing the population makes whipping less good, so this is why whipping away grass mines is generally not efficient above size 6. A similar calculation yields that plains hills are as good as whipping until population 8.5 (i.e. at pop 8 it is better to use the whip, at 9 it is better to work the plains hill mines), but working 1F2H tiles is less efficient until size 16.
If you have more than one food resource and not sufficiently many grass hills to lose all the food on, food becomes cheaper in respect to hammers so whipping becomes better (Yes, usually it is better to use the extra food for the whip than just leave it lying unused), but beware that the lost tileyields become more expensive as well. For example if your best marginal food tile is 4F, then a tileyield is worth 2F, if the best production tile is 1F3H (grass hill), the exchange rate becomes 1F=1H. So in that case it is better to whip until size 10.5 (solution of (10+ (pop-1.5) + 2*5.5)F/30H=1F/1H).
One final remark on the calculation by VOU. It seems he wants to not only minimize the number of tileyields lost (to 11 in a 2-pop whip), but also enforces whipping every 10 turns. The latter ensures that he has to have a huge possible food surplus he can use for two turns (as you have to grow in two subsequent turns, once from happy cap -1 to happy cap, at the end of the cycle, and the next turn from happy cap-2 to happy cap -1, after the new whip. However, as Iranon pointed out, you can
grow at any rate you like at the happy cap after the unhappiness of the whip has worn off and before whipping again, without damaging the efficiency of the whip. This means that being able to whip efficient is not contingent on having the ability to conjure up a huge food surplus (something like 2 or 3 is enough). This is not to say that it is always best to use an efficient cycle longer than 10 turns as Iranon proposes, using the whip more often at a less efficient rate might still be advantageous, but one does not need to satisfy the narrow conditions VOU identifies in which his "whip as often as possible" cycle is also most efficient, in order to succesfully whip. (Then again, VOU's analysis does show that having the ability to quickly change from food to hammer tiles does benefit the whip, and that you want to try to grow as fast as possible at the happy cap after the whip unhappiness wore off, to whip again as quickly as possible, while there is usually not such a rush to grow from cap-1 to cap as you have to wait for the unhappiness to wear off anyway).
VoiceOfUnreason
Deity
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If we whip from here, we end up switching from mines to farms on the growing turns, and we also don't get to work all the mines while trying to grow. Assuming I've done the math right, you get 29 mine turns instead of 44. 15 lost turns on the mine is 45 hammers, so all this work earned +15
I didn't do the math right, because somewhere I switched to an 11 turn cycle. For 10 turns, it should be 25 mines vs 40, which still leads to a +15 gap. If you have a 5th mine to toggle with a farm, then the non whip case works the mine for 5 turns, exactly covering the gap.
One thing you may want to watch out for is food parity: switching between grassland mines and grassland farms nets 2 food. You've got 8 turns in the middle, so in any stable configuration you're going to end up with an even amount of food. So if you are sitting at 14/30, expecting to land on 29/30 in eight turns, you've got a problem
There are some possible solutions, including
switching between a green mine and a brown mine
switching between dry rice and a green mine (which is OK when you are looking at rice, but not the sort of thing you want to do when your +4 tile is cows)
switching between a grassland farm and a grassland forest
planning your second growth from 28/30 instead of 29/30.
allow yourself one food of overflow on the first growth
Growing into unhappiness makes your regrowths more expensive, so we're behind by the size of your whip in food over the cycle. We only regain ground if this enables us to stay at our pre-whip cap for a reasonable time. If we actually need a reasonable volume of hammers whipped, this requires a bigger whip which may or may not be practical.
My original example was meant to beat a short 1-pop whip cycle, the equivalent alternative to short-cycle 2-pop whips may look like this:
Whip P+1 to P-2
Grow to P-1 (1 turn, large surplus)
Grow to P (9 turns)
Grow to P+1 (5 turns, large surplus on the last from an almost-full bin)
We get the same amount of whip hammers as with VOU's cycle over 30 turns.
We need to spend 3 more food on regrowing, but are 11 citizenturns ahead.
Note that the duration of the last line is arbitrary (chosen not for efficiency but to be directly comparable to 10-turn, 2-pop cycles) and that this version would still be ahead if we lost some citizenturns to slower regrowth.
My original example was meant to beat a short 1-pop whip cycle, the equivalent alternative to short-cycle 2-pop whips may look like this:
Whip P+1 to P-2
Grow to P-1 (1 turn, large surplus)
Grow to P (9 turns)
Grow to P+1 (5 turns, large surplus on the last from an almost-full bin)
We get the same amount of whip hammers as with VOU's cycle over 30 turns.
We need to spend 3 more food on regrowing, but are 11 citizenturns ahead.
Note that the duration of the last line is arbitrary (chosen not for efficiency but to be directly comparable to 10-turn, 2-pop cycles) and that this version would still be ahead if we lost some citizenturns to slower regrowth.
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