# Enhanced Devoted

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Enhanced Devoted is an improved version of the Devoted Invention perk that has a chance on each hit of replicating the Devotion ability for 3 seconds, which causes protection and deflection prayers to reduce damage from the appropriate style to 1. The perk does not put the ability itself on cooldown. This perk takes up two slots in a gizmo, meaning that it cannot be paired with any other perk in the same gizmo. It is created in armour gizmos.

Any type of damage, including typeless damage and self-inflicted damage can activate the perk. The opponent's hit that activates Enhanced Devoted is also affected by the perk. Enhanced Devoted will play the same animation as Devotion whenever it activates, but has an icon of its own in the buffs/debuffs interface.

Having Devoted and Enhanced Devoted on one's gear has a benefit[1]. Enhanced Devoted takes priority over Devoted. However, as Enhanced Devoted is unable to proc while it is currently active, this allows for a small window of time for Devoted to proc. Both Devoted and Enhanced Devoted, on their own, last 5 game ticks (3 seconds). This also means that it is possible (though unlikely) to have continuous procs of Enhanced Devoted and Devoted. The chance of continuous procs occurring is highly dependent on the attack speed of the enemy. Further information about the combination can be found here.

An example of Enhanced Devoted and Devoted proccing together to increase damage reduction.
Activation chance
Rank Chance
1 4.5% (4.95%)
2 9.0% (9.90%)
3 13.5% (14.85%)
PvM average damage reduction
Rank ${\displaystyle t_{AS}=1}$ ${\displaystyle t_{AS}=2}$ ${\displaystyle t_{AS}=3}$ or ${\displaystyle 4}$ ${\displaystyle t_{AS}=5+}$
1 19.07% (20.66%) 12.39% (13.51%) 8.61% (9.43%) 4.50% (4.95%)
2 33.09% (35.46%) 22.88% (24.79%) 16.51% (18.02%) 9.00% (9.90%)
3 43.83% (46.58%) 31.89% (34.35%) 23.79% (25.86%) 13.50% (14.85%)
PvP average damage reduction
Rank ${\displaystyle t_{AS}=1}$ ${\displaystyle t_{AS}=2}$ ${\displaystyle t_{AS}=3}$ or ${\displaystyle 4}$ ${\displaystyle t_{AS}=5+}$
1 9.53% (10.33%) 6.19% (6.76%) 4.31% (4.72%) 2.25% (2.48%)
2 16.54% (17.73%) 11.44% (12.40%) 8.26% (9.01%) 4.50% (4.95%)
3 21.92% (23.29%) 15.94% (17.17%) 11.89% (12.93%) 6.75% (7.43%)
• ${\displaystyle t_{AS}}$ is the attack speed of the enemy in game ticks.
• All numbers in parentheses refer to level 20 gear.

## Calculations

For PvM :

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Enhanced Devoted to that of without Enhanced Devoted is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil \right]}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }\left(d_{n,i}^{(1)}-1\right)}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$

For PvP :

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Enhanced Devoted to that of without Enhanced Devoted is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }\left\lfloor {.5\times d_{n,i}^{(1)}}\right\rfloor \right]}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }\left(d_{n,i}^{(1)}-\left\lfloor {.5\times d_{n,i}^{(1)}}\right\rfloor \right)}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
Notes
• A calculator for this is at the top of this page.
• Assumptions :
• The enemy is always attacking with the same style that matches the protect prayer (deflect curse) used.
• The enemy is always attacking at the same attack speed.
• ${\displaystyle R}$ is the rank of the Enhanced Devoted perk.
• ${\displaystyle p_{R}}$ is the proc chance of Enhanced Devoted to activate (if the perk is on level 20 gear, then this value is multiplied by 1.1).
• ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left(1-p\right)^{n}p=1}$ when ${\displaystyle 0\leq p<1}$ or ${\displaystyle \left|1-p\right|<1}$. This represents summing over all possibilities where integer ${\displaystyle n\in [0,\infty )}$ represents the amount of hits prior to the proc of Enhanced Devoted. The probability of Enhanced Devoted proccing on hit ${\displaystyle {n+1}}$ is therefore ${\displaystyle \left(1-p\right)^{n}p}$.
• ${\displaystyle t_{ed}}$ is the time that Enhanced Devoted is active. This is taken to be 5 game ticks.
• ${\displaystyle t_{AS}}$ is the attack speed of the enemy in game ticks.
• ${\displaystyle d_{n,i}^{(j)}}$ is the ${\displaystyle i^{th}}$ random value uniformly sampled between the enemy's minimum hit and maximum hit. The ${\displaystyle j}$ describes different sets of hits. The sets are regenerated for every new value of ${\displaystyle n}$.
• ${\displaystyle d_{n,i}^{(0)}}$ has ${\displaystyle n}$ elements with integer ${\displaystyle i\in [1,n]}$. This is the damage taken before the perk procs.
• Clarification : This does mean that for ${\displaystyle n=0}$ that there are no elements in this set.
• ${\displaystyle d_{n,i}^{(1)}}$ has ${\displaystyle \left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil }$ elements with integer ${\displaystyle i\in \left[1,\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil \right]}$. This is the damage taken during the time Enhanced Devoted procced.
• The set of values in ${\displaystyle d_{n}^{(j)}}$ for integer ${\displaystyle j\in [0,1]}$ is the same in both the numerator and denominator of the ratios for any given ${\displaystyle n}$.

Simplifications
• This can be simplified if the assumption is that every value of ${\displaystyle d_{n,i}^{(j)}}$ ${\displaystyle \forall }$ integer ${\displaystyle i\in [1,n]}$, integer ${\displaystyle n\in [0,\infty )}$, integer ${\displaystyle j\in [0,1]}$, is taken to be the same ${\displaystyle \left(d_{n,i}^{(j)}\rightarrow d\right)}$. In this scenario, there is no random element and the above ${\displaystyle r_{avg}}$ is then only dependent on the proc chance and the attack speed of the enemy. This simplification is increasingly accurate in the limit of large ${\displaystyle d}$. All examples are using large enough values of ${\displaystyle d}$ such that the values should not change to the first two decimal places in percent. Two tables are provided at the top of the page. The first covers all values of ${\displaystyle t_{AS}}$ for PvM situations. The second table is for PvP for all values of ${\displaystyle t_{AS}}$ (normal attack speed of player is ${\displaystyle t_{AS}=3}$ in non-legacy mode , but all values of ${\displaystyle t_{AS}}$ have been provided to account for proccing on damage-over-time abilities, 4 tick auto attack, etc., as well as the varied attack speeds of weapons if using legacy combat).

Using these simplifications, the damage reduction reduces to

For PvM :

${\displaystyle 1-r_{avg}={\frac {\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil p_{R}}{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil p_{R}+\left(1-p_{R}\right)}}}$

For PvP :

${\displaystyle 1-r_{avg}={\frac {.5\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil p_{R}}{\left\lceil {\frac {t_{ed}}{t_{AS}}}\right\rceil p_{R}+\left(1-p_{R}\right)}}}$

## Sources

MaterialRarityPerk ranks with X materials
12345
Crystal partsCommon01111
Strong componentsUncommon1111–21–2
Faceted componentsRare111–222–3
Zaros componentsRare111–222–3

### Suggested gizmos

Gizmo layout Possible perks
• Other possible perks:

## Update history

The update history project is a work-in-progress – not all updates to this topic may be covered below. See here for how to help out!

## References

1. ^ Mod Pi. Devoted and enhanced devoted stack. 20 November 2018.*