Unless the XML file containing the racial traits doesn't include all the modifiers, then Productive only gives the manufacturing bonus.

Furthermore, checking against a homeworld with 10 population and 50% approval, there is no evidence for a +1 production from Productive. The expected raw production of such a world is 20.02 if there isn't a production bonus from Productive or 21.02 if there is, and the raw manufacturing/research/wealth production at 100% slider position is 20.0 (give or take; sometimes you get 19.9, sometimes you get 20, most likely due to a slider position which isn't exactly 100% but which rounds to 100%). The sum of the raw outputs with the slider in default position is 20.1 (6.7*3), which is within the margin of error for rounding to the tenths place in the manufacturing/research/wealth tool tips; besides which, at 21 base production each of those raw output values should be 7 rather than 6.7.

Therefore, I would rule out the possibility that Productive gives bonuses to raw production. You may, however, have the tech that grants +1 raw production (Enhanced Production in the Colonization tree, same specialization as Soil Enhancement and Enhanced Growth, if I recall correctly; if there are other such techs, I cannot think of them off the top of my head). There are also a few technologies that grant percentage bonuses to production, at least in the Terran tree; they're mostly in the governments section of the Influence tree, if I recall correctly.

As for how the production bonus from a starbase stacks with the production bonus from approval, I believe it is additive. I checked this on a homeworld with 6.7 population, 100% approval (+25% production), 1 economic starbase (+10% production), and Enhanced Production (+1 production). The homeworld's raw manufacturing at 100% manufacturing was 25.0; if we assume that rounding in game always rounds up on the slider values and rounds down on the raw manufacturing values, then the maximum possible production which would give us "25.0" raw manufacturing at "100%" manufacturing is 25.1 / 0.99 = 25.35. The expected value of the planet's production with the starbase bonus stacking multiplicatively with the approval bonus is (11 + 2*6.7^0.7)*1.25*1.1 = 25.54, while the expected value of the planet's production with the starbase bonus stacking additively with the approval bonus is (11 + 2*6.7^0.7)*(1 + 0.25 + 0.1) = 25.07. Setting the sliders to 33%/33%/33% gave raw production values of 8.4/8.4/8.3, which sums to 25.1, while setting the sliders to 50%/50%/0% gave raw output values of 12.5/12.5/0.0; if the production had been 25.5 these should have been roughly 8.5/8.5/8.5 and 12.8/12.8/0 respectively.

I think there's enough there to conclude that the starbase production bonus stacks additively with the approval production bonus, and since this is the case I would further presume that the starbase production bonuses stack additively with one another. This is what I'd expect regardless, as this is how all other multipliers in the game behave.

As far as getting the numbers in Bamdorf's example to work:

Bamdorf probably has +1 production globally from Enhanced Production, accounting for the +1 that Bamdorf misattributed to the Productive trait. The planet has +5 production from its capital and +1 production from the Durantium Refinery. Together, this gives +7 flat production. We see evidence of two economic starbases for +20% production, and of 100% approval for +25% production, giving a known multiplier of 1.45 (which accounts for the -1.5 production from removing the Durantium Refinery, with a little rounding). I suspect that Bamdorf has Interstellar Governance (or its equivalent, if not using the Terran tech tree), which gives a further +10% production, though this wasn't mentioned in the details of the example.

(5 + 1 + 1 + 2 * 4^0.7) * (1 + 0.25 + 2 * 0.1 + 0.1) = 12.28 * 1.55 = 19.03

Bamdorf, does this solve your bookkeeping issues? Have you researched a tech in the Influence tree that gives a +10% production bonus, or do you have a third economic starbase that you forgot about? Also be aware that there's a bit of rounding that goes on in the game; if you round the base production to 12.3, then you'll get a base production of 19.07, which would round to 19.1 when displayed in game.

Well, I am not a math person, but I have learned a lot reading the forum and I use the hints that I understand, if they don't work me to death.

I am mostly an intuitive player, and it is surprising how close intuition comes to math sometimes. I have never built farms on 10 and under planets simply because there isn't room and I don't build more than 1-2 on my best colonies. Nice to know that math supports me.

By late game there are not many 10 and under colonies. They have all been upgraded to 15 or more so I will go back and build some farms on the new space that is available.

 First, you need to keep in mind proper order of operations.  You show 2*10^.7 as being = 8.1, which can only happen if you did (2 * 10) ^ .7

Exponents always happen before multiplication, so 2*10^.7 is supposed to be calculated as (10 ^ .7) * 2 = 10.02

Now that that is covered, your production is:

((2*P^.7) + Flat bonuses) * Multiplicative bonuses

Flat bonuses could include tech, durantium factories, etc.

Multiplicative bonuses would be things like your factories, racial bonuses, starbases, approval mods, etc.  These are generally additive multipliers.

In your example

((2 * (10^.7) + 5 Colony Capital + 5 Civilization Capital) * 1.00 = 20.024

Then you have to multiply in your approval modifier, which joeball mentions in the next post.  At 48% approval you'd have a 1% penalty to production, so 20.024 * .99 = 19.82

If you split it evenly you'd see 6.6 of each.

You're right joeball, my mistake.

I'm just another nerd, trying to make the game(s) I love better

Also, note that increased population makes it harder for an invasion to succeed.  Approval affects the "Resistance" modifier, but the actual invasion success is calculated against the total population in absolute terms.

Also, each "level" bonus from adjacency or tile varies in effect, based on the structure.  The way to see the effect is hover over the structure, and look at the popup stats, and in the middle of it, you will see a section headlined "Level X", and the bonus associated with that level.

For example, most 1- and 2- level buildings give a +5% per level bonus. Level 3 buildings often give a +10% per level bonus, and certain buildings don't do percentages (e.g. Food and Approval buildings give base number bonuses, not percentage bonuses).

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REPLY

Ok, so I see three things.  One is that I have the techs interstellar governance (production 10%) and enhanced production ( 1 flat).    Also, apparently the productive racial trait, whiich is listed in the manufacturing bonuses and says +1 to production AND +15% does not add to base production.   Third, all the percentage production pluses are added beefore being multiplied.    So 25% approval + 2x 10% starbase factories + 10% for interstellar governance are one factor 1 + .25 + .2 + .1 = 1.55.   Thus I get to where you got and it agrees with what the game said.   OK.

So I decided to do one more. I chose my Capital, what the heck.

Input information:  Approval 100% -> 25% bonus.   1 Starbase factory for 10%, and enhanced governance for 10%.    Flat bonuses are 5 for colony capital and 1 for enhanced production.    The total raw production was 28.7.   So I got

(5  + 1 + 2 * 8.3^.7) * (1 +.25 + .1 + .1)  = something not even close.  Aha.  It is the civilization capital which ADDS TO the colony capital (not immediately obvious!) so there is another additive 5 at the beginning.   This gives the right answer.

(5 + 5   + 1 + 2 * 8.3^.7) * (1 +.25 + .1 + .1) = 28.7

Sigh.  Yes, perhaps it was time for me to retire.

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Summary.

1.  Thanks very much for helping me sort this.  The "multiplicative" items being added before being multiplied, and a more careful look at the tech tree, were the keys.

2.  The labeling of various items in the data displays is very confusing, both as to how a bonus applies AND to how it figures into the calculation.  A study of the Defs directory isn't a lot of help.  The productive bonus being listed as production+1 .... 15% in the manufacturing table is one of the worst examples.

3.  I would comment that people who talk about "intuitive" results being pretty good have a point.   A few games, paying a little attention to your choices and varying things will give one a decent idea, or should.  I agree.   

4.  I am old school, retired scientist.   So I don't believe anything until I can calculate it spot on.   And then I can objectively confirm my intuition, and maybe able to improve my choices, or my order of choices,  because I know what their effect will be.

Thanks again.

Fair enough I have no actual knowledge of the formula just that ^ normally means to the power of.

You make a great point. 

For some time I  figured that a durantium refinery isn't all that great a building, despite the nice +3 manufacturing adjacency bonuses.   But that +1 flat addition to base production is more than it looks like - bonuses to raw production are always 100% in use because they build up the raw production of  a colony which can contribute to Manufacturing, Research, or Wealth, as you say, and a flat increase is added in before multiplicative raw production bonuses, and the approval bonus, are applied.  (And before specific Manufacturing, Wealth, or Research multipliers are applied, it goes without saying.)   And that 0.1 multiplicative contribution to total production from the first ring of an economy starbase is similarly very useful.  Four econ Starbases early in the game will give a 0.4 additive multiplier to a colony's Raw production - that's a serious improvement.

Clearly, building stuff or researching tech that increases Raw production should have a high priority.  But it's rather important to clearly understand what you are getting!   I think there needs to be a fix put in;  not a big fix, since it is just relabeling stuff I should think.  Just a wee bit of care applied, por favor.

(None of this discussion has gotten to the business of project contributions....arrgh!)

Im not sure how you arrive at the "building econ starbases sucks" theory. Level 1 is amazing - i try and get that asap. 

But even further levels are really good, provided you can build multiple constructors on a single ship. Sure, the first few upgrades suck, but since you can build 4+ constructors early on in a cargo ship (after a few techs), the total bonus to your planet can be +50% or more off a single turn worth of production from a manu world. Better yet, it gives your manu worlds something productive to do in peacetime - and they can support research/wealth worlds in a really productive and efficient way. 

Finally, getting the bonus manu from starbases means you need to invest in less factories to upgrade your res/wealth worlds - another bonus. 

Then there's the +approval upgrades, and the +research if you play ancient with a few relics can be huge. 3-4 relics can easily be worth 15 base research, amplified by research tech, relics and starbases can be 300+ on a world with no labs!

Improvement upgrades are in a much more dire spot than starbase - since they require an entire turn each and require you to sacrifice research/wealth to get them.

I was able to figure a way for 5 bases near a star, but in general I think 3 is the most practical.

Yes, that's only considering the base range of 5. At the maximum range of 9, you can make at least 12 starbases cover a central region, with 9 spaced evenly around the outer ring and three in the center in the pattern indicated in Arumba's diagram by the pink tiles. There might be a 'better' configuration at a 9 tile range, though I'm not aware of it.