I have a question regarding the BSR formula, specifically for RPS RSS. The formula reads as follows:

RPS = SAtk * (SAtk * SSpe + 415) / (SAtk * (1 - SSpe) + 415)

RSS = SSpA * (SSpA * SSpe + 415) / (SSpA * (1 - SSpe) + 415)

My question is: What is the purpose of the +415 in this formula? And why is simply [Attacking Stat]*[Speed Percentile] not sufficient?

The simple answer is that it's 415 because it's 415. It really could be other things. Over the history of BSR, it has not always been 415. You should be able to even find old articles on site where it was other values. Brief research turned up examples where it was 315 and 630.

Now the longer answer...

This is how I understand things, but

DougJustDoug is probably best suited to address what X-Act did. Please correct me if I'm wrong.

Let's look at an example: Consider a pokemon with 120 SpA and 120 Spe.

This gives its a Special Sweepiness of 206.239 under the current formulation (Special Sweepiness is a metric that includes the RSS along the away, RSS itself isn't really a metric we ever talk about). But what if it used a different valuation of 315 instead of 415? Well, without making similar adjustments to later steps in the formula meant to normalize the data, the SS would jump to 228.365. What if we used 630? It would drop to 181.773. Do these necessarily mean anything? Not really. We could adjust our ending scale of what constitutes "Amazing" Special Sweepiness however we wanted in order to attain the same results. So why is one better than the others?

Think about it very basically, if SSpA is 10 and SSpe is .9, you get an equation like (9+x)/(1+x). If x is 1, you get 5. If x is 3, you get 3. Basically, as x increases, your RSS goes down, all else equal, for all positive x. This is true across the board. But it's not true

*equally* across the board. As x moves from 1 to 3, RSS goes down by 40%. But if we move x from 3 to 5, it goes from 3 to 2.33, aka 22%. So the higher x goes, you get a narrower distribution in RSS. Right if we take SSpA of 10 and SSpe of .9 again, but now x is 200... you get 209/201 = 1.04. If we instead used SSpA of 5 and SSpe of .6, we'd get 203/202 = 1.005. This is a difference of only 3% in outcome, despite the Pokemon being much slower and also less powerful.

This is why if SSpA typically looked like 10 or 5, we would never use x of 200. It makes all Pokemon essentially look the same. X that is too high makes all Pokemon be perceived to be basically identical. We could adjust our eventual definitions of SS accordingly so the exact same Pokemon were called Excellent or Above Average etc, but perceptually, they would appear identical. It's good to have a sensible spread in data for easy understanding.

And this is also why we use 415. It fits what SSpA typically looks like and produces results that are logical. It was obvious that using X that is too high is bad, but what about using X that is too low?

Let's try another example: compare the 120/120 Pokemon to one that is 120 SpA / 110 Spe.

In our current formula we get SS of 206.239 and 199.41. Basically, you get that they are comparable Pokemon. One is 3.3% more sweepy than the other.

But if we had a much lower adjustment X of 100, (again leaving later stuff unchanged), these same two Pokemon would give 414.159 and 373.059. This is a difference of 9.9%.

Which intuitively feels right? Is that Pokemon 10% better or only more marginally better? At this high of a speed tier, moving from 110 to 120 impacts the Pokemon's matchup against fewer and fewer Pokemon. You outspeed slower Pokemon, which is most every Pokemon, whether you compare to 110 or 120. The change only impacts matchups against Pokemon between 110 and 120, which is way less than 10% of matchups, so it feels right to say the Pokemon with 120 Spe is considerably less than 10% better of a special sweeper than the one with 110 speed.

All this being said, 415 could change. But we don't change it. Instead, the only thing that CAP has tinkered with is the ending normalization process that happens converting RSS to SS. This is much easier to tinker with and so we just don't touch X-Act's magic 415.