Properly convert FSRS 4.5 parameters into FSRS 5 parameters

[user1823]

@L-M-Sherlock @Expertium

The initial difficulties (D0) calculated using default FSRS 4.5 parameters with the formulas of FSRS 4.5 and FSRS 5 are quite different.

This implies that if the user doesn't re-optimize their parameters after updating to 24.10+, the scheduling is very likely to be worse than with FSRS 4.5. Obviously, we don't want the Anki update to make the scheduling worse.

Ideally, the user should re-optimize their parameters after updating to 24.10+. But, we can't assume that all users will do that. So, to reduce the damage, we should try to find out how to properly convert FSRS 4.5 parameters into FSRS 5 parameters (apart from just adding two zeroes).

Specifically, we want to find out how to convert w4, w5 and w7.

[Expertium]

w7 is the same as before, though the exact numerical value isn't guaranteed to be the same, of course.
As for w4 and w5, I'm not sure. I made this to help you (and anyone else reading this) to visualize initial D: https://www.desmos.com/calculator/ujpkwjxgrl

[user1823]

Thanks for the link. It was indeed helpful.

I propose the following:

Let the values of w4 and w5 in the FSRS 4.5 parameters be $a$ and $b$ respectively.

Then, we should perform the following calculations:

$$ c = a + 2b $$

$$ d = \frac{\ln\left(3b+1\right)}{3} $$

And these values should be used as the converted values of w4 and w5 in FSRS 5.

In the graph below,

• the black line represents the initial D calculated using the FSRS 4.5 formula and the default FSRS 4.5 parameters
• the purple curve represents the initial D calculated using the FSRS 5 formula and the default FSRS 4.5 parameters
• the blue curve represents the initial D calculated using the FSRS 5 formula and the converted FSRS 4.5 parameters

[user1823]

If you think that the converted parameters should produce exactly the same value of initial D for Again and Good (instead of Again and Easy), just use this formula for d:

$$ d = \frac{\ln\left(2b+1\right)}{2} $$

The downside is that the init D for Easy will become much lesser than in FSRS 4.5. However, this is probably not a serious issue because Easy is used less commonly than Good and it makes sense to optimize for the most commonly used buttons.

[Expertium]

@L-M-Sherlock unless this is terribly impractical for some reason, use user1823's formulas (c and d) to internally convert w4 and w5 in Anki 24.10 beta (without notifying the user).

[L-M-Sherlock]

The converting is executed in FSRS-rs. Please open an issue there to remind me.