Instructions

Introduction
Welcome to Exponential Idle, a math-inspired incremental game. Your goal is to stack up money by taking advantage of exponential growth. To do so, you have to step through time by tapping the equation or simply let the time follow its course. You can perform change of variables to accelerate the process, buy upgrades, and unlock achievements while earning virtual money.

Ticks
Expect changes regarding variable tick rate.

A tick is automatically generated every tenth of a second, which increases $$t$$ by $$\frac{dt}{10}$$ and update the function value using the equation. A tick can also be generated with a tap, which increases $$t$$ by 1.

Variables & Upgrades
You can improve the growth rate by performing changes of variables. The first variable is free. You can improve other variables once you gain enough money. You can also purchase new variables once you have enough star currency. You can buy in bulk by clicking on the Cost label or buy all available variables using the 'Buy All' button, once available, which iteratively buy the cheapest variable while it can afford it. Similarly, you can buy upgrades to boost your progress. Once an upgrade is available, the two small arrows above the variable list will be highlighted. Auto buyer bonuses will automatically buy affordable variables and upgrades using the same logic.

Cost
Buying a variable or an regular upgrade will DIVIDE the current function value by the cost instead of subtracting it. This cost model makes more sense in the context of exponential growth using multiplicative updates. All other costs ($$\mu$$, $$\psi$$, and stars) follow the usual subtractive cost model.

The cost of these purchases are driven by a double exponential model, which follows the formula $$2^{2^{a(x-1)+b}}$$, where x is the level you are buying (starts at one and increments by one after each purchase).

For variable purchase, the following table of a and b can be used:

For regular upgrade purchase, the following table of $$a$$ and $$b$$ can be used:

All variables have a $$2^n$$component that increases as the level of the variable increases. Except for x, which is capped at n = 3, all variables follow these rules for assigning n:


 * 1) It starts at 0 and increases by one at level 10 and 25 (after buying 10 and 25 times).
 * 2) Then, it increases every 25 levels until level 1500
 * 3) Then, it increases every 50 levels until level 6000
 * 4) Then, it increases every 100 levels until level 10000
 * 5) Then, it increases every 200 levels until level 24000
 * 6) Finally, it increases every 400 levels forever.

Equation
The equation shows how things are calculated at every tick. The right side of the arrow is calculated using the values shown in the summary bar above the equation. For example, if $$f(t)=10, b=1,x=2,dt=3$$, then the right hand side will be equal to:

$$\begin{align} f(t) \times e^{b x dt} &= 10 \times e^{1 \times 2 \times 3} \\ &= 4034.29 \end{align}$$

The left hand side represents the new updated value of $$f(t)$$, and the left arrow represents the update. So after the tick, the new value of $$f(t)$$ will be 4034.29. At the next tick, this calculation is repeated using the new values of $$f(t), b, x$$, and $$dt$$, which can change if you buy variables or upgrades. The higher these values, the faster you'll progress. So the trick is to buy variables and upgrades, and perform Prestiges to increase $$b$$ (see below).

Summary Bar
The summary bar shows the current state of the game. The parameter $$t$$is the current time, and $$dt$$is how much $$t$$will increase in one second. $$f(t)$$ is the currency that you can spend for variables and upgrades. $$b$$ is a parameter that will increase after specific milestones (Prestige), and $$db$$ shows how much $$b$$ will change after a Prestige. $$x$$is the result of all the variables in the list below. Just remember: The higher these numbers are, the faster your progression will be.

Currency Bar
The bar below the equation shows the Prestige ($$\mu$$, mu) and Supremacy ($$\psi$$, psi) currency that you can spend. These currencies are used to buy special upgrades. Prestige upgrades gives bonuses to regular upgrades and Supremacy upgrades improve the equation itself.

For prestige and supremacy upgrades, the cost of purchases are driven by an exponential model, which follows the formula $$b\times 2^{a(x-1)}$$, where x is the level you are buying (starts at one and increments by one after each purchase). The following values of $$a$$ and $$b$$ are used:

Note: y exponent and function layers share the same level, despite having different values for a and b. For example, buying y exponent 2.2 will use x = 6.

Notations
Since we are dealing with very large numbers, we need mathematical notation to simplify its representation. At first, numbers are displayed normally. When reaching one million, we use scientific notation, a logarithmic representation based on the number's magnitude. For example, $$2.3\text{e}6 = 2.3\times 10^6 = 2,300,000$$. After reaching $$1\text{e}10000$$, we switch to a double logarithmic representation. For example, $$\text{ee}3.3 = 10^{10^{3.3}}$$.

Infinity
While Mathematics defines infinity as an abstract and unreachable number, infinity in this game changes over time. At first, the infinity is defined as $$f(t) = $ee200 = 10^{10^{200}}$$. Once you reach this number, you'll be offered to perform a Supremacy to increase the infinite value to $$f(t) = $ee2000$$. Once that is reached as well, you can do a Supremacy or a Graduation to break it to $$f(t) = $ee50000$$.

Prestige
Prestige lets you reset your current progress while giving a bonus to the constant $$b$$, giving a big boost in progression. It will also allocate a new currency, named μ, with which you can purchase new upgrades. Performing regular Prestige is the only way to progress further. The increase of $$b$$ after Prestige is denoted $$db$$. To increase $$db$$, you need to cumulate money, i.e. $$f(t)$$. In the beginning, it is recommended to wait until the value of b doubles before Prestige. After some time, you can wait until it multiples by 10 or more. For the first prestige, you can expect to Prestige after 15 minutes, when $$db$$ reaches 100%. After some time, you will be able to Prestige within a couple of minutes, multiplying b by 10 or even 100. For the curious, here's how $$db$$ is calculated:

$$db = \frac{\log_{10}^{0.8}\left(f(t)\right)}{4 \times 10^6}$$

Prestige currency is calculated as:

$$d\mu = \log_{10}({f(t)})$$

Star Bonuses
You can collect stars by unlocking achievements and at every tick (automatically or by tapping). Each achievement rewards stars depending on its difficulty and each tick has a chance of generating a star. The star chance per second is $$\sqrt{dt} \times 0.04%$$. When the game is closed, you will get as many stars as if you were in game. Once you reach a certain level (b = 1), you will be able to spend your stars to get permanent upgrades that do not reset after a Prestige. When you collect a star, the star icon will be highlighted.

List of bonuses available:


 * Variables (see Variables & Upgrades)


 * Buy All Button (Single purchase; 15 stars)
 * Acceleration Button (Single purchase; 50 stars)
 * Variable Autobuyer (Single purchase; 200 stars; requires Buy All Button)
 * Upgrade Autobuyer (Single purchase; 200 stars; requires Buy All Button)
 * x2 Automation / s: Multiplies the rate of automation by 2 (3 purchases; prices 500, 1250, 2000 stars respectively; requires Variable Autobuyer OR Upgrade Autobuyer)
 * Auto Prestige (Single purchase; 500 stars; requires Variable Autobuyer AND Upgrade Autobuyer)
 * Permanent variable levels (infinitely buyable, increases with each level bought, requires variable u)
 * Formula is $$\lfloor a (l-1) e^{a (l-1) /200}+1\rfloor$$, where l is the level being bought
 * a is based on a table for each variable (see below)

In the rewards panel, you can use stars to unlock minigames. These stars are permanently used and are not given back during a supremacy (minigames will stay unlocked forever). Current minigames include:


 * 15-Puzzle (10 stars to unlock)
 * Torus Puzzle (75 stars to unlock)
 * Arrow Puzzle (150 stars to unlock)

Supremacy
After reaching a certain point in the game, you will be offered to Supremacy (a new button will appear in the Prestige menu). A Supremacy restarts the game at $$b = 0.001$$ and resets all variables, regular upgrades, and prestige upgrades. It also allocates a new currency, named $$\psi$$ (psi), with which you can purchase new upgrades. Past infinity, you will be able to purchase an upgrade that modifies the equation, which gives a significant boost in progression.

Supremacy currency is calculated, based on cumulative $$f(t)$$, as:

$$d\psi = 2^{\frac{\log_{10}(\log{10}(f(t)))}{25} - 1}-0.5$$

Graduation
Graduation is the third prestige layer of the game. A Graduation resets all variables and normal, prestige, and supremacy upgrades in exchange for students. Stars and bonuses are not reset. After a Graduation, you will get new students that you can assign to different research projects. These projects introduce a new parameter, $$\varphi$$, into the main equation. You can access the students panel using the icon in the navigation bar. Don't be afraid to assign students and experiment, you can reassign them whenever you want. The total amount of students depends on your lifetime $$f(t)$$ using the following formula:

$$d\sigma = \frac{\log_{10}(\log_{10}(f(t)))-1000}{200}$$

Theories
Theories can be seen as smaller idle games. Each theory increases $$\tau$$, which contributes to the progression of the main game. However, theories are not influenced by the parameters of the main game. Only one theory can be active at any time. You can change the current theory whenever you want and you can resume a theory without losing previous progress.

A theory has its own equation that generates its own currency, $$\rho$$, that can be used to buy upgrades. The generated $$\rho$$currency is then converted into a value, e.g., $$\tau_1$$. The $$\tau_i$$values from all theories are then multiplied to give the total $$\tau$$, which is used in the main equation. In other words, maximize $$\tau_i$$to progress in the main game.

Tap the "Upgrade" label to access a second panel of features.

Upgrade cost is driven by an exponential model, same as for prestige and supremacy upgrades. The formula can be expressed as $$b a^x $$ where x is the level preceding the one to be purchased, or equivalently $$b 2^{\alpha x}\!,\, \alpha = \log_2 a$$.

(first) Each listed theory offers the first level of its first upgrade for free, transforming the formula into $$b a^{x-1} $$ or equivalently $$ b 2^{\alpha (x-1)} $$.

Each upgrade displays current value. Values written in scientific notation increase by $$2^{\bigl\lfloor \frac{x}{10} \bigr\rfloor} $$ and are equal to $$(10 + x \bmod 10)2^{\bigl\lfloor \frac{x}{10} \bigr\rfloor} - 10 $$.

(offset) Each listed theory has $$c_1 $$ upgrade start from value 1 instead of 0 unless it is (first), transforming the formula for total value into $$(10 + x \bmod 10)2^{\bigl\lfloor \frac{x}{10} \bigr\rfloor} - 9 $$.

The following tables list $$a$$, $$b$$ values exactly and $$\alpha$$ rounded to 6 significant digits:

Automation
Some automation features are available to simplify the progress, which can be purchased with stars. First, you will get the 'Buy All' button that buys all affordable items. Then, there are the variable and upgrade auto buyers that automatically buy all affordable items every 5 seconds. Then, there is the auto prestige that will perform a prestige as soon at the ratio db/b is above the given ratio in the auto prestige settings. For example, if you want to prestige when $$db $$ is 10x higher than b, you input 10 as the ratio. Accepted format are the same as the one used in the UI. This means that you can use 1e6 or 1.000e6 for 1 million. The check for prestige is done along with the auto buyers. After the 1.4 Upgrade, you can buy an auto Supremacy which can take the same type of input as the auto Prestige.

Rewards
You can increase $$dt $$ by 50% for 2 hours by watching a short advertisement. You can cumulate rewards by watching more than one ad. For example, watching three ads in a row will increase $$dt $$ by 50% for 6 hours. You can also play minigames, unlocked using stars, which rewards you some amount of stars once you succeed, depending on the difficulty.

The formula for stars given by a game is $$\lfloor \text{gameBaseReward} \times \max(1, 0.02 \times dt^{0.5}) \rfloor $$, where gameBaseReward is the number of stars given by a set of game and difficulty initially (updated on 2020-08-05).

$$0.02 \times \sqrt{dt} $$ is the expansion of $$50.0 \times \text{starChance} $$.

Accelerator
An accelerator button is available after you collect enough stars. The button will appear in the equation section. Maintain your finger on the button to multiply the value of dt by some amount, which provides a boost in progression. For more comfort, you can drag your finger around the screen after pressing the button. While you finger remains on the screen, the accelerator will be active and you will be able to continue purchasing variables.

The formula for acceleration is $$(9 s + 1)^{\frac{1}{9}}$$, where s is the time in seconds since you started holding the button. As of 1.4, the acceleration has a limit of 1 hour (3600 seconds, ~x3.17), before releasing the multiplier back to 1.

Leaderboard
After signing in, you can access the leaderboard from the Settings menu. The displayed score corresponds to the current 'ee' income. For example, if you reached $$f(t) = $ee400 $$, then your leaderboard score will be 400.