This should be clearly explained in the tooltip.
Raw Manufacturing: +11
* Colony Capital: +5
* Population: +3.8
* Approval Bonus: +25%
The approval bonus is a bonus to base production, not to any of the three output types.
I.e. the output equations are:
Production = ((flat production) + 2 *(population)^0.7)*(approval multiplier)
Manufacturing = (manufacturing fraction)*(Production)*(manufacturing multiplier)
Research = (research fraction)*(Production)*(research multiplier)
Money = (money fraction)*(Production)*(money multiplier)
This means that the approval multiplier stacks multiplicatively with the output bonuses rather than additively; since all the bonuses in the tool tips stack additively, including the approval bonus in the tool tip would be deceptive or require a change to the formulas.
It would be better to have a breakdown of production showing the base production and any bonuses given to it somewhere, such as on the Govern Planet screen, if the formulas remain the same, rather than having the approval bonus included in the tool tips listing all the bonuses to each output type when the approval bonus doesn't stack in the same way that all the other bonuses in those tool tips do.
Perhaps total production should be calculated by:
(raw production bonuses + pop^(approval rating)) * (%bonuses)
I think not. 0% approval gives a 25% penalty to base production, which is bad; your formula would more or less remove population from the equation at low levels of approval, which is generally far worse. I could go for pop^F(approval) for some function F that gives a reasonable range of exponents, but not for F(approval) = approval.
Additionally, this formulation hides the effect of approval from the player even more than the current form does (where it's a percentage penalty to the whole thing and simply isn't listed in the tool tip for the output types but can be determined from looking at the influence, population growth, or resistance tool tips). Consider - if the exponent on population is equal to the approval rate expressed as a decimal, what is the difference between 50% approval and 70% approval? How much of a penalty is it? ((base production) + (population)^0.5) / ((base production) + (population)^0.7) is not something that I can do in my head, nor is it something where the effects of approval can be easily quantified. ((base production) + H(population))*F(approval) is a function where the approval effects can easily be quantified, assuming you know what F(approval) is.
For the record, F(approval) is a piecewise linear function defined as follows:
F =
1 + ((approval) - 0.95) + 0.2 if approval > 0.95
1 + ((approval - 0.85)*0.5 + 0.15 if 0.95 >= approval > 0.85
1 + ((approval) - 0.7)*2/3 + 0.05 if 0.85 >= approval > 0.7
1 + ((approval) - 0.3)*0.25 - 0.05 if 0.7 >= approval > 0.3
1 + ((approval) - 0.15)*2/3 - 0.15 if 0.3 >= approval > 0.15
1 + ((approval) - 0.05)*0.5 - 0.2 if 0.15 >= approval > 0.05
1 + (approval) - 0.25 if 0.05 >= approval
A rough approximation of this is G(approval) = 1 + ((approval) - 0.5)^3 + 0.25*((approval) - 0.5).