Formulas

This page is a collection of formulas used for calculating various in-game numbers.

Formulas

Hero Cost Formula (cost of upgrading from "Level")

The cost to level up Cid, the Helpful Adventurer from level 1 to 15 is:

floor((5 + Level) × 1.07^{Level - 1})

The cost to level up Cid, the Helpful Adventurer from level 16 is:

floor(20 × 1.07^{Level - 1})

The cost to level up other hero by one is:

floor(BaseCost × 1.07^{Level - 1})

Hero DPS

DPS of each hero, the number displayed on the right side of the hero's levelling buttons, is a product of

• Hero Base Damage
• Hero Level
• ×4 Multiplier for each 25 Hero Levels, starting from level 200; at Hero Level 1000, 2000, ..., 8000 the multiplier is ×10. Heroes after  Frostleaf have a ×5 multiplier instead of ×4 for each 25 Hero levels from 525 to 725;
• Personal DPS Upgrade bonuses
• ×2 Damage Bonus, which costs 50 rubies
• Dark Ritual Stacks (×1.05 with each normal DR, ×1.1 with each Energized DR)
• Global DPS Upgrades:
  • ×1.25 from The Wandering Fisherman;
  • ×2.0736 from Betty Clicker;
  • ×1.25 from Leon;
  • ×1.25 from Broyle Lindeoven, Fire Mage;
  • ×1.44 from Amenhotep;
  • ×1.1 from Beastlord;
  • ×1.1 from Shinatobe, Wind Deity;
  • ×1.5625 from Grant, The General;
  • ×1.25 from Frostleaf;
  • ×1.5 from Gog;
  • ×1.5 from Moeru;
  • ×2.34375 from Zilar;
• Bonus from unused Hero Souls and Morgulis, which equals to (1 + Hero Souls × 0.1 + Level of Morgulis × 0.11)
• Gild bonus, which is (1 + (0.5 + Level of Argaiv × 0.02) × NumberOfGilds)

Total DPS, the number displayed on the upper left side of Hero tab, is a product of:

• Sum of all Heroes DPS
• Bonus from Siyalatas (idle mode)
• Bonus from Juggernaut Combo (active mode)
• ×2 if Powersurge is activated (×3 with Energized Powersurge)

Hero DPS per Gold

All character level costs use the same multiplier (Base×1.07^Level). They also have the same scaling WRT DPS / level. The only difference is their base cost to base DPS ratio, and personal modifiers to damage. For example, Treebeast cost 50 gold for level 1, but gives you 5 base DPS (10:1 ratio), while Frostleaf costs 2.1e27 and gives 7.5e22 base DPS (28,113:1 ratio).

Below level 200, the scaling is:

DPS = BaseDPS × StaticModifier × Level

Cost = BaseCost × 1.07Level

StaticModifier includes the global adjustment, personal modifiers and gilding, but does not change based on the level of the character.

DPS/Cost = (BaseDPS*StaticModifier/BaseCost) × (Level/1.07Level) = Static × Level × 1.07-Level

(At level 199, this is Static × 199×(1.07^-199) = Static × 2.828e-4, or 0.03% the DPS/gold you got at level 1).

After level 200, the 4^Level term starts to dominate, so we can ignore the other terms. You get

DPS = Base×Static×Level×4Level/25~Constant×4Level/25=Constant × eln(4)/(25×L)=C×e^(0.0555 L)

DPS/Cost = Static×e^(0.0555 L)/(1.07^L) = Static×e^(0.0555 L)×e-0.0677L=Static×e-0.01222L=Static×0.988L=Static(1-0.012)L

So each level is 1.2% less efficient of a DPS increase vs the cost to do the increase.

Each Hero is a little less efficient than the previous one. Ivan, the Drunken Brawler being 14% less base efficient than Treebeast, down to a low of Broyle Lindeoven, Fire Mage being 189% less base efficient than Bobby, Bounty Hunter. After Broyle Lindeoven, Fire Mage, each one, up to Frostleaf, is a static 36% less efficient than the previous one. At a 25 level upgrade, this is 26.3% less efficient. Which is fairly close to the average difference between heroes. So ignoring personal modifiers, it is close to optimally efficient to level each hero 25 levels higher than the next hero.

Hero Souls awarded for killing Primal Bosses

Hero Souls awarded by defeating Primal Bosses:

floor(((Level - 80) / 25)^1.3 × (1 + (Bonus from Solomon)))

Transcendent Power Hero Souls awarded by defeating Primal Bosses (only for transcendent player):

20 × (1 + (Bonus from Solomon)) × (1 + TP%)^{(Level / 5) - 20}

Note: Omeet (Centurion Boss at Level 100) always gives 1 Hero Souls and 0 Transcendent Power Hero Souls, regardless of the formulas above.

Monster HP for levels

From Level 1 to Level 140:

$\left \lceil 10\times(Level-1+1.55^{Level-1})\times [isBoss\times 10] \right \rceil$

From Level 141 to Level 500:

$\left \lceil 10\times(139+1.55^{139}\times 1.145^{Level-140})\times [isBoss\times 10] \right \rceil$

From Level 501 to Level 200000:

$\left\lceil 10\times (139+1.55^{139}\times 1.145^{360}\times \prod _{i=501}^{Level}(1.145+0.001\times \left\lceil {\frac {i}{500}}\right\rceil ))\times [isBoss\times 10]\right\rceil$

From Level 200001 onwards:

$\left\lceil (1.545^{Level-200001}\times 1.240\times 10^{25409}+((Level-1)\times 10)\right\rceil$

$\prod$ is the product symbol and $\left \lceil \right \rceil$ are ceiling brackets.

Monster Gold Drop

This is how much Gold you will get (on average) from killing a monster. Gold dropped is the product of:

• Monster gold worth (see below),
• 1 + 9 × ( Fortuna Chance),
• Bonus from Libertas (idle mode).
• Bonus from Nogardnit × number of unassigned Auto Clickers (idle mode).

Monster Gold Worth

This is how much killing a monster is "worth" - it is used in determining Gold dropped (see above) and for the Golden Clicks skill. Monster worth is the ceiling (round up) of the product of:

• Monster's HP / 15,
• Min(3, 1.025^{level - 75}) if level is greater than 75,
• Gold Found Upgrades:
  • × 2.9296875 from King Midas,
  • × 1.5 from Bomber Max,
  • × 3 from Tsuchi,
• Bonus from Mammon,
• × 10 × (Bonus from Mimzee), if it is a Treasure Chest,
• × 2 if Metal Detector is activated (×3 with Energized Metal Detector).

Formula for sum of n^1.5

The approximation below can be used to estimate the total cost of raising an Ancient with upgrade cost at level n equals to n^1.5, from level 1 to level n:

$C(n)\approx\frac{2}{5}n^{\frac{5}{2}}+\frac{1}{2}n^{\frac{3}{2}}+\frac{1}{8}n^{\frac{1}{2}}+\frac{1}{1920}n^{-\frac{3}{2}}+O(n^{-\frac{5}{2}})+R$

where R≈-0.0254852. This result can be obtained with the Euler-Maclaurin formula. In practice, the terms after n^1/2 can be ignored, leaving the following:

$C(n)\approx\frac{2}{5}n^{\frac{5}{2}}+\frac{1}{2}n^{\frac{3}{2}}+\frac{1}{8}n^{\frac{1}{2}}$

The cost to level such Ancient, from level x to level y, can be estimated using $C(y)-C(x)$.

When is Siyalatas worth an upgrade?

The Ancient Siyalatas increases your DPS by 25% per upgrade, but depending on how many Hero Souls you have, you may end up on the losing side of the transaction if you're not careful. Here a formula is derived which tells you how many Hero Souls you need to have before a Siyalatas upgrade benefits you.

Let D be your current DPS, including bonuses from Gilded heroes, Dark Rituals, but not including Hero Souls or Siyalatas. Let H be your current number of Hero Souls. Let S be Siyalatas’s current level, including 0 if you haven't bought him yet.

So your total damage is as follows:

TD₁ = D × (1 + 0.1H) × (1 + 0.25S)

Now you want to spend N Hero Souls to increase Siyalatas’ level by 1. This includes the situation where you buy Siyalatas for N Hero Souls. Is it worth it? Well, this is your total damage after spending the Hero Souls:

TD₂ = D × (1 + 0.1 × (H – N))(1 + 0.25 × (S + 1))

You want the second expression to be better than the first one:

TD₂ ≥ TD₁

D × (1 + 0.1 × (H – N))(1 + 0.25 × (S + 1)) ≥ D × (1 + 0.1H) × (1 + 0.25S)

(1 + 0.1 × (H – N))(1 + 0.25 × (S + 1)) ≥ (1 + 0.1H) × (1 + 0.25S)

(1 + 0.1H - 0.1N)(1.25 + 0.25S) ≥ (1 + 0.1H) × (1 + 0.25S)

1.25 + 0.25S + 0.125H + 0.025HS - 0.125N - 0.025NS ≥ 1 + 0.25S + 0.1H + 0.025HS

0.25 + 0.025H - 0.125N - 0.025NS ≥ 0

Solve for H:

H ≥ (0.125N + 0.025NS – 0.25)/0.025

Or:

H ≥ 5N + NS – 10

Example 1: It costs 4 Hero Souls to get Siyalatas from lvl 3 to lvl 4. Here N=4 and S=3. Doing so will benefit you as long as you currently have at least 5 × 4 + 4 × 3 – 10 = 22 Hero Souls. In fact, you would break even if you had exactly 22 Hero Souls before you upgraded Siyalatas (and you would lose DPS if you had fewer than 22).

Example 2: Siyalatas is going to be your fourth Ancient and thus costs 8 Hero Souls. Here N=8 and S=0 since you haven't bought him yet. Is it worth it? Yes, as long as you have 5 × 8 + 8 × 0 – 10 = 30 Hero Souls. Any less and you will lose DPS even with Siyalatas's lvl 1 25% bonus.

Update

Eventually, as players level up Siyalatas, the bonus given drops by 1% every 10 levels, going from from 25% to 15% before finally settling at 15% increases forever. Once this occurs, the formula above becomes too inaccurate to use strategically. I have redone the math for a high-level Siyalatas where the gain from 1 Level is 15%. This is made slightly more difficult by the fact that you have to take all previous upgrades of Siyalatas into consideration, yielding a damage formula like this:

D × (1 + 0.10H)(1 + (0.25 x 9) + (0.24 x 10) + (0.23 x 10) + ..... + (0.16 x 10) + (0.15 x (S – 99)))

Which must be compared to its post-upgrade equivalent, as above, in order to arrive (assuming all my math was performed correctly) at:

H ≥ NS + 43.67N – 10

Since at this point, N is always S+1, the following alternative may be used:

H ≥ S² + 44.67S + 33.67

Thus, if you have Siyalatas lvl 400, an upgrade to lvl 401 is only beneficial if you have at least 400² + 44.67 × 400 + 33.67 = 177 900 Hero Souls (before upgrading).

So yeah, prepare to farm.