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Expected Long Term Cookie Production

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These calculations build on the numbers from Golden Cookie Probabilities, by working out the average effect of each cookie combination, to find how many cookies you can expect to produce over the long term.

Version 2.0 Edit

The following variables are used in the calculations:

Variable Meaning Business Christmas
B Base cookies per second
C Maximum Chain payout 7/9*10^E
E Exponent for max Chain payout Int(Log10(194400*B))
G Average seconds between Golden Cookies 104.903046 108.0010206
K Number of cookies per click With all upgrades, 0.11979*B
N Number of clicks per second
The following assumptions are made in the calculations:
  • Over 1 billion cookies in bank, so Chain cookies can have the maximum number of golden cookies in the chain.
  • At least 3.025*10^E cookies in bank, where the max Chain payout is 7/9*10^E, so Chain cookies can always max out. This will also guarantee that Lucky cookies max out.
  • All Golden Cookies and Reindeer are clicked as soon as possible (essentially the auto-clicker scenario).
  • The upgrades Lucky Day, Serendipity, Get lucky, Heavenly luck, Lasting fortune, Golden goose egg, and Startrade have been bought.
  • When factoring in Reindeer, the upgrades Reindeer baking grounds, Ho ho ho-flavored frosting, and Snarsnow have been bought.
  • The framerate is 30 FPS.

The conditions used are:

  • (a) Base cookies per second between 3.600823*10^n and 5.144033*10^n, for some n. (Max chain payout is the same during Frenzy.)
  • (b) Otherwise. (Max chain payout is higher during Frenzy.)
  • (1) Cookies in bank between 3.025*10^E and 10^(E+1).
  • (2) Cookies in bank between 10^(E+1) and 10^(E+2).
  • (3) Cookies in bank between 10^(E+2) and 10^(E+3).
  • (4) Cookies in bank between 10^(E+3) and 10^(E+4).

If you have more than 10^(E+4) cookies in your bank, you should probably reset.

The first round of calculations will assume that the optimal configuration is Appeased Grandmatriarchs with the Radiant Appetite and Dragonflight auras. Future calculations will try to improve on this configuration.

Cookie Effects
Extra:
Cookie Conditions Cookies Seconds
Blab
Frenzy Before non-Frenzy 170*6*(B+N*K)
Before Frenzy G*6*(B+N*K)
Lucky After non-Frenzy B*60*15
After Frenzy B*60*15*7
Click Frenzy After non-Frenzy 29*776*N*K
After Frenzy 29*776*N*K*7
Dragonflight After non-Frenzy 22*1110*N*K
After Frenzy 22*1110*N*K*7
Chain (a)

or

(b) After Frenzy

(1) 1.026424*C 22.944826
(2) 1.036792*C 20.176592
(3) 1.047264*C 17.380396
(4) 1.057843*C 14.555955
(b) After non-Frenzy (1) 0.103680*C 20.176592
(2) 0.104726*C 17.380396
(3) 0.105784*C 14.555955
(4) 0.106853*C 11.702985
Multiplying by the respective probabilities, dividing by the average cookie spawn time, and adding the base B+NK per second, the formulas for Business Day come out as follows:
Cookies per second
Conditions Business Day Christmas
(a) (1) 26.711*B + 94.785*NK + 9.434*10^(E-5) 25.982*B + 92.109*NK + 9.164*10^(E-5)
(2) 26.719*B + 94.815*NK + 9.532*10^(E-5) 25.990*B + 92.138*NK + 9.259*10^(E-5)
(3) 26.728*B + 94.846*NK + 9.632*10^(E-5) 25.998*B + 92.167*NK + 9.356*10^(E-5)
(4) 26.737*B + 94.878*NK + 9.732*10^(E-5) 26.006*B + 92.196*NK + 9.453*10^(E-5)
(b) (1) 26.716*B + 94.801*NK + 4.844*10^(E-5) 25.987*B + 92.124*NK + 4.705*10^(E-5)
(2) 26.724*B + 94.832*NK + 4.894*10^(E-5) 25.995*B + 92.153*NK + 4.754*10^(E-5)
(3) 26.733*B + 94.863*NK + 4.945*10^(E-5) 26.003*B + 92.183*NK + 4.807*10^(E-5)
(4) 26.741*B + 94.895*NK + 4.997*10^(E-5) 26.011*B + 92.213*NK + 4.854*10^(E-5)

With all Reindeer upgrades, Reindeer spawn every 114.391 seconds on average, and are worth 120*B cookies, or 120*7*B during a Frenzy. A Frenzy is active on average 67.1% of the time. Working out the numbers, clicking all Reindeer is worth 5.5736*B cookies per second.

With all click upgrades, K = 0.11797*B. Using Benford's Distribution for base rates, 10^(E-5) averages to 0.233586*B for (a) and 0.406743*B for (b), with base rates falling into (a) 15.4902% of the time, and (b) 84.5098% of the time. Taking condition (3) as a reasonable average, the final CpS formulas for Appeased and Dragonflight, where N is clicks per second and B is base cookies per second, are:

Business Day Christmas
Always clicking (28.78 + 11.36*N) *B (33.57 + 11.04*N) *B
Only Click Frenzies (28.78 + 10.87*N) *B (33.57 + 10.56*N) *B
Only Stacked Frenzies (28.78 + 9.38*N) *B (33.57 + 9.12*N) *B

These formulas are equal across seasons when N=14.9, 15.5, and 18.0 respectively, so Business Day season is better when your autoclicker clicks at least 15 times per second all the time, or 16 times per second only during Click Frenzies and Dragonflights, or 18 times per second only during stacked Click Frenzies and Dragonflights.

Version 1.0466 Edit

Variables used will be explained in the next table. The assumptions here are:

  • At least 10 trillion cookies in bank, so chain cookies always have the same chance to end the chain.
  • At least 84000 * base cookies per second in bank, so Lucky cookies can always max out. (This will be the limiting factor if base cps is between 3.600824*10^n and 10.288066*10^n, for some n.)
  • At least 3.025*10^E cookies in bank, where the max Chain payout is 7/9*10^E, so Chain cookies can always max out. (This will be the limiting factor if base cps is between 1.028807*10^n and 3.600824*10^n, for some n.)
  • All Golden Cookies and Reindeer are clicked as soon as possible (essentially the auto-clicker scenario).
  • The upgrades "Lucky Day", "Serendipity", and "Get lucky" have been bought.
  • When factoring in Reindeer, the upgrades "Reindeer baking grounds", "Weighted sleighs", and "Ho ho ho-flavored frosting" have been bought.
  • The framerate is 30 FPS.
Extra:
Conditions Cookie Probability Cookies Clicks Seconds
Blab .00004405198
Before Frenzy Frenzy .04510165421 S*6*(B+N*K)
Before non-Frenzy .42611329519 154*6*(B+N*K)
After Frenzy Lucky .39793016499 B*1200*(7*L+1-L)
After non-Frenzy .07328478441 B*60*20
After Frenzy Click Frenzy .02177728779 776*N*K*(7*D+26-D)
After non-Frenzy .02251232906 776*N*K*26
(a) (1) Chain .01323643238 1.026423872*C 7.648275252 22.94482575
(2) 1.03679178*C 6.725530557 20.17659167
(3) 1.047264323*C 5.793465209 17.38039563
(4) 1.057841741*C 4.85198506 14.55595518
(b) (1) After non-Frenzy .00685167149 .103679178*C 6.725530557 20.17659167
(2) .1047264323*C 5.793465209 17.38039563
(3) .1057841741*C 4.85198506 14.55595518
(4) .106851691*C 3.90099501 11.70298503
Egg3 (1) After Frenzy .00638476088 .998682627*C +

.021369298*10800*B

7.640252412 22.92075724
(2) 1.028687902*C +

.006884077*10800*B

6.723843088 20.17152926
(3) 1.045559809*C +

.001561079*10800*B

5.79323042 17.37969126
(4) 1.05760458*C +

.0002285*10800*B

4.8519652 14.5558956
No

Egg3

(1) .898288281*C +

.079824985*10800*B

7.573797742 22.72139323
(2) .961561972*C +

.047613139*10800*B

6.694679306 20.08403792
(3) 1.016101444*C +

.022526603*10800*B

5.783510679 17.35053204
(4) 1.04778666*C +

.008080105*10800*B

4.849614078 14.54884223
The conditions are:

(a) Base cookies per second between 7.201646*10^n and 10.288066*10^n, for some n.

(b) Base cookies per second between 1.028807*10^n and 7.201646*10^n, for some n.

(1) Cookies in bank between 3.025*10^E and 10^(E+1).

(2) Cookies in bank between 10^(E+1) and 10^(E+2).

(3) Cookies in bank between 10^(E+2) and 10^(E+3).

(4) Cookies in bank between 10^(E+3) and 10^(E+4).

If you have more than 10^(E+4) cookies in your bank, you should probably reset.

Variable Meaning With Egg3 Without Egg3
B Base cookies per second
C Maximum Chain payout 7/9*10^E
D Average length of double frenzy, in seconds 25.84414583 25.00775674
E Exponent for max Chain payout Int(Log10(97200*B))
K Number of cookies per click With all upgrades, .0847*B
L Probability of Lucky happening inside Frenzy interval 1 .999999908
N Number of clicks per second
R Reindeer (1 during Christmas season, 0 otherwise)
S Average length of short Frenzy (cut off by another Frenzy) 115.4927084 121.157839
The average length of time between cookies is 115.4760418 seconds with Egg3, or 121.1578391 seconds without, not including Chains. There are an average of 70.8303597 seconds of Frenzy per cookie cycle. Reindeer spawn an average of 120.1526946 seconds apart. Adding up the effects in the table above, multiplied by their probabilities from the table before, the following table is obtained:
Conditions Extra Cookies Clicks per cycle Seconds per cycle
Egg3 (a) (1) 3855.537285152*B + 3939.037976552*NK

+ 1.05670368*10^(E-2)

1.101235878 115.7797494
(2) 3855.537285152*B + 3939.037976552*NK

+ 1.06737744*10^(E-2)

1.089022030 115.7431079
(3) 3855.537285152*B + 3939.037976552*NK

+ 1.07815893*10^(E-2)

1.076684810 115.7060962
(4) 3855.537285152*B + 3939.037976552*NK

+ 1.08904839*10^(E-2)

1.064222972 115.6687107
(b) (1) 3857.010814017*B + 3939.037976552*NK

+ 5.51189756*10^(E-3)

1.094862311 115.7606287
(2) 3856.011979555*B + 3939.037976552*NK

+ 5.66648241*10^(E-3)

1.082625051 115.7239170
(3) 3855.644930006*B + 3939.037976552*NK

+ 5.75590383*10^(E-3)

1.070232599 115.6867396
(4) 3855.553041465*B + 3939.037976552*NK

+ 5.82140614*10^(E-3)

1.057706974 115.6491627
No Egg3 (a) (1) 3857.070062131*B + 3855.765301889*NK

+ 1.05670368*10^(E-2)

1.101235878 121.4615467
(2) 3857.070062131*B + 3855.765301889*NK

+ 1.06737744*10^(E-2)

1.089022030 121.4249052
(3) 3857.070062131*B + 3855.765301889*NK

+ 1.07815893*10^(E-2)

1.076684810 121.3878935
(4) 3857.070062131*B + 3855.765301889*NK

+ 1.08904839*10^(E-2)

1.064222972 121.3505080
(b) (1) 3862.574427299*B + 3855.765301889*NK

+ 5.01334676*10^(E-3)

1.094438014 121.4411531
(2) 3860.353246009*B + 3855.765301889*NK

+ 5.33314007*10^(E-3)

1.082438847 121.4051556
(3) 3858.623393446*B + 3855.765301889*NK

+ 5.60961579*10^(E-3)

1.070170540 121.3683507
(4) 3857.627229145*B + 3855.765301889*NK

+ 5.77265108*10^(E-3)

1.057691963 121.3309150
And the final formulas:
Conditions Cookies per second
Egg3 (a) (1) (34.3006+4.6647*R)*B + 35.0218*NK + 9.1268*10^(E-5)
(2) (34.3112+4.6658*R)*B + 35.0326*NK + 9.2220*10^(E-5)
(3) (34.3218+4.6670*R)*B + 35.0435*NK + 9.3181*10^(E-5)
(4) (34.3326+4.6682*R)*B + 35.0545*NK + 9.4152*10^(E-5)
(b) (1) (34.3188+4.6653*R)*B + 35.0274*NK + 4.7615*10^(E-5)
(2) (34.3208+4.6664*R)*B + 35.0382*NK + 4.8966*10^(E-5)
(3) (34.3283+4.6676*R)*B + 35.0492*NK + 4.9754*10^(E-5)
(4) (34.3384+4.6688*R)*B + 35.0602*NK + 5.0337*10^(E-5)
No Egg3 (a) (1) (32.7555+4.4932*R)*B + 32.7447*NK + 8.6999*10^(E-5)
(2) (32.7651+4.4942*R)*B + 32.7543*NK + 8.7904*10^(E-5)
(3) (32.7748+4.4953*R)*B + 32.7640*NK + 8.8819*10^(E-5)
(4) (32.7845+4.4964*R)*B + 32.7738*NK + 8.9744*10^(E-5)
(b) (1) (32.8061+4.4938*R)*B + 32.7501*NK + 4.1282*10^(E-5)
(2) (32.7973+4.4948*R)*B + 32.7595*NK + 4.3928*10^(E-5)
(3) (32.7927+4.4959*R)*B + 32.7691*NK + 4.6220*10^(E-5)
(4) (32.7943+4.4969*R)*B + 32.7789*NK + 4.7578*10^(E-5)
With all upgrades, K = .0847*B, and using Benford's Distribution for base rates, 10^(E-5) averages to .11679*B for (a) and .42815*B for (b), with base rates falling into (a) 15.49% of the time, and (b) 84.51% of the time. So we can generate approximate weighted averages as follows:
Christmas Not Christmas
Egg3 (40.95+2.968*N)*B (36.28+2.968*N)*B
No Egg3 (39.10+2.775*N)*B (34.60+2.775*N)*B
The Golden Goose Egg is worth about a 5% increase in non-click production (but including golden cookies and reindeer) and a 7% increase in clicking power, and Reindeer are worth about a 13% increase in non-click production and no increase in clicking power. Non-click production is worth about 12 to 14 clicks per second.

Click Frenzies account for the majority of click-based production. If you are not using an autoclicker, and simply clicking all the golden cookies, reindeer, and clicking during Click Frenzies, just multiply the coefficient of N by .8678 with the Golden Goose Egg, or .8633 without it. If you're only clicking during stacked frenzies, then instead multiply by .7556 with the Egg, or .7490 without.

Probably a negligible effect: Clicking golden cookies will decrease your clicks per second by about 1/120 (see tables for exact numbers), and Reindeer will decrease it by about the same amount.

Keep in mind that with the Century egg, base cookies per second changes over time. This also does not account for production achievements being reached, which changes the base rate. These are long-term averages, and are highly variable, mostly being dependent on the frequency of stacked Frenzies. An advanced Monte Carlo simulation could probably determine confidence intervals over defined periods of time.

The same process could be repeated for the other Grandmapocalypse stages, but it would be much more complicated. Since Appeased is the most profitable stage, this should be good enough.

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