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Enhanced Devoted

Description?
4.5% chance per rank on being hit that protection prayers will work at 100% (or 75% in PvP) for 3 seconds. This does not stack with devoted.
Release date	5 September 2016 (Update)
Gizmo type	Armour
Maximum rank	3

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.

Enhanced devoted takes priority over the devoted perk if both are active.

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

${\displaystyle r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum\limits_{i=1}^{n}d^{(0)}_{n,i}\right) + \left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil \right]}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\sum\limits_{i=1}^{n}d^{(0)}_{n,i} + \sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}d^{(1)}_{n,i} \right]}}$

• The average damage reduction is then, after some rearranging,

${\displaystyle 1-r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}\left(d^{(1)}_{n,i} - 1\right)}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\sum\limits_{i=1}^{n}d^{(0)}_{n,i} + \sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}d^{(1)}_{n,i} \right]}}$

For PvP :

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Enhanced Devoted to that of without Enhanced Devoted is

${\displaystyle r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\sum\limits_{i=1}^{n}d^{(0)}_{n,i} + \sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil} \left\lfloor{.5 \times d^{(1)}_{n,i}}\right\rfloor \right]}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\sum\limits_{i=1}^{n}d^{(0)}_{n,i} + \sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}d^{(1)}_{n,i} \right]}}$

• The average damage reduction is then, after some rearranging,

${\displaystyle 1-r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}\left(d^{(1)}_{n,i} - \left\lfloor{.5 \times d^{(1)}_{n,i}}\right\rfloor \right)}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\sum\limits_{i=1}^{n}d^{(0)}_{n,i} + \sum\limits_{i=1}^{\left\lceil{\frac{t_{ed}}{t_{AS}}}\right\rceil}d^{(1)}_{n,i} \right]}}$

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.
• ${\displaystyle d^{(j)}_{n,i}}$ 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 ${\displaystyle n}$.
  • ${\displaystyle d^{(0)}_{n,i}}$ 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.
  • ${\displaystyle d^{(1)}_{n,i}}$ 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 ${\displaystyle d^{(j)}_{n}}$ 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 ${\displaystyle d^{(j)}_{n,i}}$ ${\displaystyle \forall}$ 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

Material		Rarity	Perk ranks with X materials
			1	2	3	4	5
	Crystal parts	Common	0	1	1	1	1
	Strong components	Uncommon	1	1	1	1–2	1–2
	Faceted components	Rare	1	1	1–2	2	2–3
	Zaros components	Rare	1	1	1–2	2	2–3

Suggested gizmos

Gizmo layout	Possible perks
	Enhanced Devoted 2–3 Other possible perks: Crystal Shield

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