Sins of a Solar Empire Math

Here's a look at how Sins of a Solar Empire processes and displays numbers, in case it helps you understand what you're seeing, or helps you if you're testing or confirming calculations.
== Rounding ==
Sins calculates to a high degree of precision under the hood, and uses the usual rule of rounding when displaying numbers: round up if it's over .5, round down if it's under .5.
If it's exactly .5 past what it's displaying, then it rounds up. Examples:
*8.04999 (exactly) would display as 8.0
*8.05 (exactly) would display as 8.1
== Game Rates ==
Version 1.03 allows you to change certain game rates when loading a game (Income Rate, Build Speed, Ship Speed, Research Speed, and Culture Speed). These all default to Normal, if you've never touched them before.
Just so you know:
*When you save a game, it does save the Rate you chose, but...
*If you load a game while playing a game (i.e., you don't load it from the Main Menu), the Rate changes to Normal.
*You have to load a game from the Main Menu for it to use the Rate setting you saved it with, in which case...
*You also get the chance to view or change the Rate, when you load from the Main Menu.
So, you can't view or change the Rate if you load a game while playing a game; it gets fixed at Normal. Load from the Main Menu if you want something else. If you're curious, there are ways to check your Rate indirectly from inside a game; see the sections below.
=== Research Speed ===
Changing the Research Speed has a dramatic effect on how quickly it is performed.
*If Fast is considered the base speed,
*Normal research takes twice as long and
*Slow research takes five times longer than the base speed.
Viewed from Normal's point of view:
*Fast is twice as fast and
*Slow is 2.5x slower (only 40% the speed).
To see a game's speed from within a game, look at any unresearched 8-Lab topic (Military or Civilian). If it takes 138 seconds, you're on Fast Research Speed. 276 seconds is Normal, and 690 is Slow. But make sure you haven't started researching the topic; the numbers for its next level are different. Also, if you're Advent, don't forget about Knowledge Aggregation... if you're at KA +33%, your numbers will be: Fast 104, Normal 207, Slow 518.
This is a good place to point out that Advent's Knowledge Aggregation description might (or might not) confuse some. It says it reduces the time to research, at a +33% rate. In the game, this is actually Time/1.333, which is equivalent to Time / (4/3) = Time * 3/4. In other words, it reduces it by 25%, to 75%. Something that needed 100 seconds will need 75 seconds. I guess it's a matter of semantics. We've seen this way of handling math before with the Penal Colony planetary bonus (see Bonuses Notes).
For the record, if you look in the game files (RESEARCHUPGRADE.ENTITY files), the game actually uses 1.15 times the game file speed as the Fast speed, and goes from there. So if it says 100 seconds in the .entity, the in-game research speed will be 115 seconds on Fast, 230 on Normal, and 575 on Slow.
=== Income Rate ===
The game Income Rate uses the following modifiers:
*Slow is the base rate.
*Normal is 15% faster than the base rate.
*Fast is 30% faster than the base.
These are global factors that modify all income (credits, metal, and crystal) from all sources, period.
 
This wiki always uses the base rate, unless otherwise indicated. This must be kept in mind if comparing your game results to the wiki.
Note that, from the point of view of the Normal rate, Slow is ~13% less than Normal, and Fast is ~13% more. This is a simple function of the fact that 15 is ~13% of 115; ±15 divided by 115 equals ±13.04%.
If you're in a game and want to check or verify what the Income Rate is, it's easy to do with extractor income, if you take allegiance and research into account:
Game 
Income Rate	No Extractor Research	No Extractor Research	40% Extractor Research	40% Extractor Research
Game 
Income Rate	100% All. 	Neutral 	100% All. 	Neutral
Slow 	0.40 	0.53 	0.56 	0.75
Normal	0.46 	0.61 	0.64 	0.86
Fast 	0.52 	0.69 	0.73 	0.97
If you're at a different extractor research level or not at 100% Allegiance, adjust accordingly. The equation for extractor income is simply Income_Rate X Allegiance X Extractor_Research X 0.40; for example, a Normal, 35% Allegiance, 20% Research extractor's income is 0.19 (1.15 x .35 x 1.20 x 0.40). Neutral extractors (at UCGWs) are always 4/3rds of the base rate or if you like, for the equation, they are "stuck" at 133% allegiance. (And yes, they are subject to extractor research.)
== Resource Equivalents ==
Sometimes you might want to compare the cost of items, without the clutter of three types of resources (credits, metal, and crystal). A popular is to consider the metal and crystal to be worth three times the credits. For example, 100 metal is worth 300 credits. This lets you say that e.g. a TEC refinery (1500$, 125m, 175c) is simply worth 2400 credits. This is sometimes called the 3x equivalency method.
In version 1.02, crystal was scarcer than metal. But in 1.03, there's about as much metal as crystal at Planets. Also, the black market generally has either resource available in the 300-500 credit range. 
If you think other equivalencies might work better, please add them here!