Button Machine

Button Machine is one of the simplest features in the game, in which you click a large circular button and gain rewards. The closer you click to the center of the button, the higher rewards you gain. The button will occasionally break, but it can be repaired using Coins. Grandpas can be unlocked and upgraded, who can click the Button Machine automatically even when another feature is used.
== Basics ==
Button Machine is a single-screen feature which fits the standard feature window. The machine's button sits on a track in the center of the screen with several useful numbers below it. At the bottom are other buttons labeled Repair, Shop, and Grandpas.
=== Clicking ===
Left-clicking the button will grant Coins and EXP, increase Hit Combo, reduce the button's durability, and cause the button to move along the track. Left-clicking on the track, but not on the button, gives no rewards and resets Hit Combo to 0. Left-clicking anywhere else in Button Machine gives no rewards, but does not reset Hit Combo.
There is a limit to how quickly the button can be pressed. When the Button Machine is loaded, there is a 0.05-second delay before the first click can be made, and at least 0.075 seconds must elapse between consecutive manual clicks after the first click, limiting the maximum possible click rate to 40 clicks per 3 seconds (~13.3 clicks per second). 
=== Breaking and Repairing ===
The button begins with durability of 5000. Each time a manual or auto-clicker click is made which hits the button, the button loses (100 - perfection) durability (this loss is multiplied by (1/3) before being applied if Invincibility Stars Card is active). See Accuracy Judgment for the meaning of "perfection".
When the button's durability is less than 4% of maximum (that is, less than 200), the track will change color from grey to red to indicate that the button is about to break. When the button's durability reaches zero, the button breaks, and depending on how well the player clicked before the button broke, one or more Mystery Boxes may be rewarded.
When the button has broken, clicking the Repair button restores the button to full durability and changes the slant of the track to +(3-6) degrees or -(3-6) degrees from horizontal. The Repair button may also be used while the button is still intact; however, it costs 10x as many Coins to make repairs before the button breaks.
=== Auto-Clicking ===
Button Machine has a built-in auto-clicker which can be activated by holding down Left or Right Shift. However, Multiplier decreases to 60% of its normal value while the auto-clicker is active.
The auto-clicker's click rate varies depending on the number of Clicks Today:
* Clicks Today < 500: 40 clicks per 3 seconds (~13.3 clicks per second)
* 500 <= Clicks Today < 2,500: 10 clicks per second
* 2,500 <= Clicks Today < 10,000: 8 clicks per second
* 10,000 <= Clicks Today < 50,000: 20 clicks per 3 seconds (~6.7 clicks per second)
* Clicks Today >= 50,000: 40 clicks per 7 seconds (~5.7 clicks per second)
== Accuracy Judgment ==
Each click is assigned a grade based on how far away from the center of the button the click was made. Specifically, the game computes the length of the vector going from the center of the button to the click's location in units of pixels, subtracts this length from 100, then rounds up the result to obtain the click's perfection value. This perfection value is then compared to the accuracy criteria to determine the click's grade.
From worst to best, the possible grades are as follows:
* Miss: Clicks on the track, but not on the button, are assigned this accuracy rating. For practical purposes, perfection values for missed clicks can be considered to be less than 0.
* Hit: 0 <= perfection < 79 (over 21 pixels away from center)
* Good: 79 <= perfection < 89 (11-21 pixels away from center)
* Nice: 89 <= perfection < 94 (6-11 pixels away from center)
* Great: 94 <= perfection < 96.5 (3.5-6 pixels away from center)
* Excellent: 96.5 <= perfection < 98.5 (1.5-3.5 pixels away from center)
* Perfect: perfection >= 98.5 (0-1.5 pixels away from center)
== Rewards ==
==== Multiplier ====
Multiplier is a modifier to EXP and Coin gained from Button Machine clicks. It starts at 100%, but can be increased to a much higher value with upgrades and extensive use of the Button Machine.
Several factors contribute to Multiplier (in the following sections, sqrt stands in for the square-root symbol):
* Total Clicks: +sqrt(0.5*Total Clicks)%, up to +2,500% at 12,500,000 Total Clicks.
* Clicks Today: +sqrt(Clicks Today)%, up to +500% at 250,000 Clicks Today.
* Hit Combo: +sqrt(10*Hit Combo)%, up to +500% at 25,000 Hit Combo.
* Shop Multiplier: Purchased from Button Machine Shop in 2% increments, up to +1,200%.
* Invincibility Stars Card: +200% while active.
* Auto-Clicker: x0.6 while in use.
* Daily Click Limit: x0.2 while Clicks Today is over 1,000,000.
* Button Blessing: x2 while active (indicated by the button being blue instead of orange).
With all beneficial factors at maximum values and without any detrimental contributors active, Multiplier can reach a maximum value of 10,000%, multiplying EXP and Coin rewards by 100.
==== Reward per click ====
Each click grants EXP and Coins based on several factors. Base reward depends on A:TG level, perfection value of the click, current Multiplier, and rank of the click. Rank is an integer associated with the grade of the click. The rank values for Miss, Hit, Good, Nice, Great, Excellent, and Perfect are 1, 2, 3, 4, 5, 6 and 7, respectively.
Base reward is calculated as follows:reward = floor((level * perfection * rank) * 0.65 * (button multiplier / 100) * 0.45)This base reward is modified by Boost, Button Basher Career bonus, and Pet bonus (EXP only) to produce the final EXP and Coin rewards for each click. These final values may be modified further in certain situations:
* Clicks Today > 1,000,000: x0.1
* Level [1/2/3] of Perk #24 is Active: x[1.8/2.2/3.0] for EXP only
* Hard Ascension (before level 8999): x0.7
* Impossible Ascension (before level 8999): x0.5
* Perfect Clicks in a row exceeds 14: EXP and Coin gains set to 1, Hit Combo set to 0.
In addition to EXP and Coin rewards, clicks grant other rewards based on their grade:
* Good: +ceil(Multiplier/300) Button Basher Career EXP
* Nice: +ceil(Multiplier/150) Button Basher Career EXP, +1 Green Coin
* Great: +ceil(Multiplier/75) Button Basher Career EXP, +3 Green Coins
* Excellent: +ceil(Multiplier/37.5) Button Basher Career EXP, +10 Green Coins, +1 Blue Coin (50% chance, only if Clicks Today <= 1,000,000), +0.3 Insta-Progress (if Button Basher Career is at level 200), +1 Purple Button (50/75/100% chance if level 1/2/3 of Perk #18 is active)
* Perfect: +ceil(Multiplier/20) Button Basher Career EXP, +30 Green Coins, +2 Blue Coins (only if Clicks Today <= 1,000,000), +1 Insta-Progress (if Button Basher Career is at level 100 or higher), +1 Purple Button, +1% Boost (up to current yellow limit)
==== Reward for breaking the Button Machine ====
When the button breaks, Mystery Boxes may be given out based on the Hit Combo gained since the last time the button was repaired, referred to as the score:
* Explosion Crate: (score/7.5 - 100)% chance. 0% chance if score <= 750, guaranteed if score >= 1500.
* Awesome Crate: (score/15 - 100)% chance. 0% chance if score <= 1500, guaranteed if score >= 3000.
* Chaos Crate: (score/30 - 100)% chance. 0% chance if score <= 3000, guaranteed if score >= 6000.
* Legendary Crate: (score/60 - 100)% chance. 0% chance if score <= 6000, guaranteed if score >= 12000.
If score is sufficiently large, White Coins will also be awarded. One will be awarded if score >= 1125, 2 more will be awarded if score >= 2250, and 7 more will be awarded if score >= 4500.
If a score of 20,000 or greater is obtained, Mystery Boxes and White Coins will not be awarded. Additionally, breaking the button while Clicks Today is over 1,000,000 will not give any awards.
== Purple Buttons ==
Purple Buttons, the currency of Button Machine, are rewarded whenever a Perfect click is made on the button. If Perk #18 is active, Excellent clicks can also award Purple Buttons. They can be spent at the Button Machine Shop or on hiring Grandpas.
== Button Machine Shop ==
The Button Machine Shop has three items for sale:
* Regular Boost Potion: 250 Purple Buttons
* 2x Button Machine reward (2,000 clicks): 500 Purple Buttons [this adds 2,000 clicks of Button Blessing]
* +2% Multiplier (+Shop Multiplier / +1200%): 200 Purple Buttons
== Grandpas ==
Grandpas are helpers which automatically click the button, granting button-pressing rewards without moving the button or reducing its durability. They will click the button even when the Button Machine is not the current feature, but with greatly diminished effectiveness.
Up to 20 Grandpas can be purchased using Purple Buttons.
Grandpas' effectiveness can be upgrading using Coins and Green Coins:
* Speed Upgrade: Increases the clicking speed of all Grandpas | Every second, each Grandpa has a (Level)% chance to click the Button Machine.
* Accuracy Upgrade: Increases the clicking accuracy of all Grandpas | Grandpas have a (Level)% chance to get PERFECT clicks, and a (Level * 3)% chance to get EXCELLENT clicks.
* Discipline Upgrade: Increases the clicking speed and accuracy of Grandpas when you are not viewing the Button Machine screen. | When you are not viewing the Button Machine screen, Grandpas' click rate and perfect rate are reduced to (Level * 2)% of original rates.
== External Upgrades ==
=== Invincibility Stars Card ===
While this card is active, Multiplier is increased by +200%, and durability loss due to clicks is multiplied by (1/3), effectively acting as an x3 multiplier to the button's durability.
=== Button Basher Career ===
While Button Basher Career is active, EXP and Coin gains from clicks are increased by +(Career Level / 2)%.
If Button Basher Career is level 100 or higher, PERFECT clicks begin to give 1 Insta-Progress along with their other rewards.
If Button Basher Career is level 200, EXCELLENT clicks begin to give 0.3 Insta-Progress (30% Progress) along with their other rewards.
=== Ascension Perks ===
The following Ascension Perks specifically augment Button Machine:
* Perk #18: [Button] EXCELLENT clicks have a 50/75/100% chance to give 1 Purple Button.
* Perk #24: [Button] You gain 80/120/200% extra EXP from Button Machine.
=== REST ===
Initially, while REST is active, button clicks which are not PERFECT or MISS have a 10% chance of having their grade upgraded. This chance can be upgraded in 1% increments until it reaches a maximum of 20%. The White Coin cost for each upgrade is 100*(Upgrade Level^2).
=== Progress Bar Modules ===
Purple Button Modules can be equipped to give a chance for Purple Buttons to be granted when progress bar rewards are claimed.
Non-shiny Purple Button Modules have a (2*(T^2))% chance to produce 1 Purple Button where T is (1+.1*(Tier-1)) [for tiers 1-8] or 2 [for tier 9]. Shiny Purple Button Modules have a (10*(T^2))% chance to produce 1 Purple Button.