How useful are early gardens?

I often find myself neglecting gardens as there are so many things to build early until the medieval that directly affect production, growth, science, and culture. However, recently I've been wondering if that is hurting me. That's a lot of turns the effect could've been applied earlier and an early GE or GS is amazing.

What would inform my decision is knowing the mechanics a little better. 25% boost to generation is very nice, but usually your only source of people (exception the writer city) early is the one or two wonders you snagged, usually resulting in only a solitary point (in my case 1 point usually on immortal from something like pyramids or Oracle). So, my question is this, does that 25% boost actually benefit the solitary point or is it rounded down so it only affects the accumulation after you get at least 2 points? If it is rounded down I think my strategy is fine. I might build one in the writer city but doesn't matter beyond that.

If it does make those early 1 points faster though I've been missing out. Does anyone know the math or want to defend an early garden in a city with a single GP point like from pyramids? I'm too lazy to play this out in a practice game lol as I've passed that point in my two current games.

The hanging garden of Babylon is the only way I'll get early gardens. Occasionally if I'm having money troubles I may build a garden soon after building a market to speed up getting a great merchant...same could be said for a great writer, but usually I have other higher priorities. One thing that can make gardens more interesting early is the religious option that gives you happiness for gardens but I don't usually get that one. Also not sure about your question but I've only ever noticed a straight 25% reduction in great person turns from the garden, the basic problem is 25% is not really a huge reduction although combined with another 25% reduction it might be worth the hammers depending on numerous other factors.

Gardens for me are done right before or after Education. Unless I haven't bothered with religion(or been beaten to it) then I try to get a garden and National Epic in my cap before Education.

I never find any time to build gardens or national epic...My cities are always busy building other things. There's a rush to get the settlers, trade routes, libraries, national college, then I need markets, and by the time that's done Education is up, so I need universities, but by the time they are up, metal casting is done, so I need workshops, and by the time that is done, I have Scientific Theory, so I need public schools, and they take a looooong time to build, and after that is done, I need factories, and by the time factories are finally done, I have ideologies and it's a bit late for gardens...

Best done if you have a specific need for lots of Great People beyond the usual bulbs; Culture game, Alhambra/Sistine synergy, Persian or Brazilian GA farming, maximising Babylonian or Swedish bonuses. If your GP farm isn't the Capital, it's not going to have as urgent a need to build everything.

just tested my question in my new game, the single points DO NOT round up. So there is absolutely no point in building the garden until you have at least 2-4 points. It might be useful to build them in your artist/writer/musician cities for instance, but it will not affect the single points given for the various early wonders. So my habit is probably best.

Lindsay said it already, but if you're interested in Optimal gains, you only build a Garden if you're a Civ with special bonuses, or after you have a building that you're going to work with specialists, and preferably 2 at that (i.e. capital University).

Build as you like, particularly if you're on Immortal and below, but if you're looking to get benefits from a Garden, you need the math to work in your favor.

I usually play solely immortal, but I'll keep these tips in mind. I rarely try for a cultural VC tbh so the extra rate of cultural GP will probably not be terribly useful to me. I've had a few games where I was unable to get the theming bonuses for a few triple, same-age wonders so I can see how doing the math and timing it right with gardens/etc. could help with that.

On a related subject how does the bonus great person points from Sweden work when you are their friend? Does it take 10 turns to get your extra GPP Point or does it round-up or down?

For example suppose you would normally generate 4 scientists:
100+200+300+400 = 1000. The 25% will provide 250 extra points during the production of those 1000 points. That is half of whats needed to get the 5th.

Suppose you are already getting 8 scientists:
100+200+300+400+500+600+700+800 = 3600
Now the 25% adds 900 points, exactly whats needed to get the 9th for free.

If you are for example getting 2 scientists, 2 musicians, 2 writers:
100+200 = 300. 35% = 75.
75 = 1/4th of the cost of the next one. In this case you get that 3 times though so you get 3/4th of a GP

So basically, the 25% GP points is worth 12.5% extra GPs.

When you are aiming for a science based victory (pretty much any victory except early conquest) a scientist is worth 8 turns of research and thus victory date. (if you keep them till the end when your research is highest and thus they are worth most, if you use them to reach key techs like for example plastics 8 turns earlier its probably worth slightly less)

So if 1 scientist is worth 8 turns. This improvement is worth 1 turn on victory condition per scientist you are already getting. It is of course heavilly influenced by rounding. (meaning half a scientist may just turn out to help you get an extra one, or it may be totally useless)

In practice (about 4 scientists from GPP production is quite normal) it means building the garden in every city gives you a 50% chance to win 8 turns faster. You can also choose to build it not in every city but only in part of your cities, that would reduce the chance. You may however be able to calculate ahead and predict what city can or cannot be producing an extra scientist.

After university and guilds are up and being worked, otherwise it's of no benefit and the hammers would be better off spent elsewhere.

An extra GS and eight turns off a science win isn't bad. But eight GSs is somewhat situational. Anyone care to figure out how sooner the first bulb will come if your first uni city has NE and garden vs no garden?