Is the length bonus factor, when calculating the accuracy pp of a play, independent from the accuracy percentage?
For SS, the expected probability of hitting each hit correctly tends to 100% (perfection) when increasing the amount of circles
[expected in the sense that the (probability of hitting each circle correctly) makes it so the (probability of getting a rate of correctly hit circles equal or higher than the rate of correctly hit circles in the score calculated) is equal to a predetermined probability],
while for 95% rate of 300s, the probability tends to be only 95% (and the value comes closer to 95% more quickly than in the SS case).
Because of that, accuracy-wise, there isn't much difference between getting 96%acc in 200 circles and 300 circles, while the difference is more notable between a SS in 200 circles and 300 circles.
As a way to quantify the difference, here is a set of graphs comparing how much the Expected Unstable Rate changes when changing the amount of circles (with a formula based on the expected unstable rate formula I showed months ago, but this time accounting for the probability of getting 50's and MISSES; this new formula shows similar values with high accuracy, but more accurate values with low accuracy,
the downside is that it takes a ridiculous amount of time to calculate: calculating only 140 points for the graphs took about 1 hour Made a new algorithm to calculate it's value much faster). Here is a sample of the 2 formulas in a graph varying accuracy with the other variables constant:
http://i.imgur.com/6LdrKIg.png (The one with the lower values on low accuracy is the new formula).
All the graphs are calculated with OD10 (changing the OD doesn't make much difference in the graphs with high accuracy, since with high accuracy the prevalent hit window is the one for 300s, so changing from OD7 to OD10 just roughly halves the expected unstable rate).
In the X axis is the amount of circles, and in the Y axis is (Expected UR with 100 circles) / (Expected UR with X circles) (that way the influence of accuracy alone and OD is discarded from the graph). That way, a Y value of "2" means half the expected unstable rate compared to 100 circles.
The 2 sets have the following difference:
The first set sets the (probability of getting the accuracy inputted in the formula, or more) with the (expected UR calculated) to 50% (that way, the accuracy corresponds to the median accuracy with only 1 try).
The second set tries to model the amount of expected retries a map would get based on the amount of circles: A map with 100 circles would be retried 108 times, with 200 circles half that amount, etc... (this assumes the player has the patience to play each map only a set amount of time, and that the amount of circles is directly proportional to the time each retry takes) This way maps that are more likely to get "fluke" accuracies caused by a lot of retries would give less (for example, a player who only has the skill needed to get 90% chance of hitting a 300 would eventually get a SS in a map with 30 circles if he retries a lot of times, but that would be practically impossible if the map has over 100 circles). If a map is replayed 50 times all with the same Unstable Rate, then the (probability of getting the accuracy inputted in the formula, or more) with the (expected UR calculated) is equal to 1/(1+50) (this corresponds to the formula of the expected value of the lowest value obtained in a perfect die with infinite sides with values that range from 0 to 1). I know this is not the best way to model the amount of retries, but it is something.
First Set:
http://i.imgur.com/SaWoo9P.png http://i.imgur.com/jPIsZa6.png Second Set:
http://i.imgur.com/hgVcl5O.png http://i.imgur.com/OrmhWu2.png Blue: Graph for SS.
Purple: Graph for 99%acc
Brown: Graph for 95%acc
Green: Graph for 85%acc (95% and 85% graph lines overlap each other since they have very similar values)
As you can see, in the SS case, the amount of circles has a bigger impact compared to the other cases with lower accuracy. Also, the point where increasing the amount of circles doesn't change significantly the expected unstable rate anymore is set farther to the right of the X axis when the accuracy increases. For any accuracy inferior to 100%, there is an horizontal asymptote of the graphs, but for 100%accuracy, the Y value goes to infinity (since, with an infinite amount of circles, the expected UR to get 100%accuracy is 0).
For high accuracy on circles (Acc > 0.7, so the hit window of the 300 judgment is predominant), the value of the asymptote for the expected unstable rate is approximately:
(The asymptote is the same both for the cases with only 1 retry and several expected retries).