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It was proven that extremely short "rebirths" where you invest into power before rebirthing, over and over is the best way but that even with that new system it's still quite demanding if you want to do it every couple seconds for hours if not days.
The main thing is to spend the exp as often as possible since it increases the exp received.
As far as I know, there is no real optimal timing that works all the time for AFKY.
Relatively short durations tend to give you more but outside of the farming mentionned above the gap between short and medium lengths isn't that big (provided that you continue to invest your exp for the whole duration of course).
How long is too long? A day? A week?
What I found the most helpful for getting that afky multiplier up was to "train" myself to spend that exp frequently every day more than the rebirth length itself since we are talking about months no matter what length you pick.
But yeah, someone a while back (like a few years ago, I think) did some scripting and automated it to where it just killed Hyperion and dumped starter afky xp into power and rebirthed over and over and over. The method racked up... was it in a range of tens of thousands or was it in a range of hundreds of thousands? It was a ****ton of rebirths over the course of a few months, getting nothing else accomplished in that time aside from building afky multi. (Or maybe the powerleveling was only done in shifts for a few hours each day and there was still regular play in between? Still, those are solid chunks of time deliberately missing out on other more influential parts of the game.) I think this person stopped using this approach when the multi got high enough to be able to push for enough power for the pet unlock in a reasonable amount of time.
The key to that method had been the fact that afky's starter xp at the start of every rebirth is... I don't recall the formula, but I know it scales with the number of rebirths and the afky multi. So each next rebirth was starting with an ever so slightly larger amount of starter xp which then got dumped into power which affects the amount of multi gained. It just feeds back into itself from there.
This new change gives afky its own rebirth button now, but I'm pretty sure that afky's starter xp is still scaling with the player's rebirth count. So rapid afky rebirths without actual player rebirths would fail to improve starter xp as rapidly as we saw it happen with that scripter's experiment. If you don't get in some regular rebirths as well your afky rebirths will only be improving each next afky rebirth's starter xp amount based on the amount converted from power at the end of each afky rebirth.
Of course, now that you can rebirth when you like it might be harder to find when halfway is and well, it's a mathematical optimal.
As of late though I've been pretty lazy and just hit max on power and clone count after a few iterations of hp/count/power (exp multi 5e16, I should now be able to max afky pet bonus thanks to the rebirth change) and that leaves me with needing something like 1e24 power to cap which wouldn't get reached (you could also take this as buying the clone count in advance which won't matter wrt power as the cost is negligible). If I buy HP at that point it can increase exp a few % more than buying power would. In the rebirth period I estimate that I still end up spending around 99.9% of EXP on power.
If one wants to visualise how afky god has been going, has a lot of old stat exports and some time on their hand you can give the log sheet in the ITRTG Compiled spreadsheet[docs.google.com] a spin (you'll have to shift rows around and edit date values or the graphs will get weird) though for the afky multi chart I find editing the y-axis to use a logarithmic scale produces more meaningful results. Note that a lack of datapoints might mislead (for example, I have a 5 month gap where long rebirths did take place but I can't measure their impact)
Let's see what my numbers actually look like though...
April 2018 is when my afky multi first started appearing in my data...
2018 datapoints: 311.15, 901.13, 1390, 1961, 4178, 9528, 15789.25, 37597.31, 84863.03
2019 datapoints: 224031.85, 538223.03, 1.029e6, 2.941e6, 7.671e6, 12.519e6, 18.639e6, 29.592e6, 38.958e6, 94.722e6, 197.739e6, 504.543e6
2020 datapoints: 869.436e6, 1.765e9, 4.217e9, 13.550e9, 43.396e9, 136.164e9, 413.218e9, 2.377e12, 11.231e12, 37.590e12, 136.423e12, 480.413e12
2021 datapoints: 2.033e15, 6.404e15, 14.261e15
Huh, got a few months in there where I actually got it 4 or 5 times larger. What else was I doing during one of those bigger surges...? Let's see, from the 7/2020 report to the 8/2020 report... Went from 3281 rebirths to 3292 rebirths. And... no challenges. I was just being really really lazy that month. And I boosted my afky multi to about 5.75 times larger over that lazy month.
To contrast with one of my weaker months for afky multi growth... From the 5/2018 report to the 6/2018 report I went from 1088 rebirths to 1248 rebirths. Much busier month, there. Looks like I blasted through most of my UUCs and BHCs that month. Also did my last few GSCs and a few of my CPCs that month. Only boosted my afky multi to about 1.54 times larger over such an active month.
I stopped doing afky a few months ago (708 E21 is no longer useful) but it was increasing faster and faster to the point where it became very easy to skyrocket it near the end.
Was contemplating submitting "Feedback" requesting a Afky rebirth timer on the page itself, but I imagine it's already in the works.
So to me, I think a couple days is probably the best, mostly because of down time and being asleep. While you're asleep you obviously don't increase in power, which means if you say go to sleep after a 15 minute rebirth it basically means the entire time you were asleep is completely wasted. Whereas going to sleep after a 2 day rebirth is a pretty tiny dent compared to optimal power gain.
I think those periods of time where you're asleep and your power isn't growing are much more impactful than the small 2-3% increase in base power you could get from rebirthing multiple times in a day, or even on a daily basis.
So, while I quite obviously haven't done the math, I think it should be like a once every 2-3 days thing and you should rebirth in the morning (or some time of the day where you will have a lot of time to grow your afky power before you go to sleep).