MacroHard
09-26-2011, 06:38 PM
The Basics
Passively reduces the damage Martyr takes by 0.2 / 0.3 / 0.4 / 0.5 times the percentage of Health he is missing.
This simply means that the lower Martyr’s health, the less damage he takes. Think of it like Armadon’s ability, but with the damage reduction scaling from 0% to 20/30/40/50% depending on Martyr’s current health.
Damage Reduction and Damage Factor Formulae
Damage Reduction, DR =
L1: 0.2*(1 - x)
L2: 0.3*(1 - x)
L3: 0.4*(1 - x)
L4: 0.5*(1 - x)
Damage Factor, DF =
L1: 0.2x + 0.8
L1: 0.3x + 0.7
L1: 0.4x + 0.6
L1: 0.5x + 0.5
Damage reduction is post-mitigation
All incoming damage is first reduced by any shield (Accursed, MoA, Chipper), armor, magic armor, and blocking. Any damage that is left over is then reduced by Guardian Angel.
The damage reduction works for damage received from casting Guardian Angel on an ally.
The damage reduction does not work for direct HP removal (such as activating Sacrificial Lamb or Blood Chalice).
Effective Health Impact (Ideal Case)
Effective Health Points (EHP) is how much unmitigated damage a hero can withstand. For most cases, EHP is an effective measurement of how well a hero can survive physical damage. EHP is determined by multiplying health by a mitigation factor calculated from armor, blocking, and evasion. For more on EHP see this thread: http://forums.heroesofnewerth.com/showthread.php?t=337020
Since Martyr’s passive ability reduces damage after mitigation, his EHP gains an additional factor that stacks with all other factors.
This additional factor is =
L1: 1.116 (equivalent to 10.3% damage reduction)
L2: 1.189 (equivalent to 15.9% damage reduction)
L3: 1.277 (equivalent to 21.7% damage reduction)
L4: 1.386 (equivalent to 27.9% damage reduction)
...which means Martyr with level 4 Guardian Angel will survive approximately 38% longer than another hero with identical health, armor, magic armor, blocking, and evasion. Please be aware that this factor is the ideal case where he receives an infinite number of attacks of very small incidents of damage.
These factors are calculated by integrating EHP from X=0 to X=1:
EHP = 1 / DF
Level 1:
http://i52.tinypic.com/5kehie.png
EHP = 1 / (0.2*(x + 4))dx, from x = 0 to x = 1
= 5*ln(x + 4)|
= 5*ln(5) - 5*ln(4)
= 1.116
Level 2:
http://i55.tinypic.com/33atahy.png
EHP = 1 / (0.3*(x +7/3))dx, from x = 0 to x = 1
= 10/3*ln(x + 7/3)|
= 10/3*ln(10/3) - 10/3*ln(7/3)
= 1.189
Level 3:
http://i51.tinypic.com/345y360.png
EHP = 1 / (0.4*(x + 1.5))dx, from x = 0 to x = 1
= 2.5*ln(x + 1.5)|
= 2.5*ln(2.5) - 2.5*ln(1.5)
= 1.277
Level 4:
http://i53.tinypic.com/9gztl1.png
EHP = 1 / (0.5*(x +1))dx, from x = 0 to x = 1
= 2*ln(x + 1)|
= 2*ln(2) - 2*ln(1)
= 1.386
Effective Health Impact (Specific Case)
Now, effective EHP for infinitely small attacks is all well and good, but how useful is Guardian Angel against real attacks? Obviously the magnitude of the attacks has a huge effect; if you get one shotted by a level 5 codex hellflower spellshards from full health your damage reduction was zero.
Given that each attack does X% of your total health, what is the effective EHP that Guardian Angel provides? What is the equivalent Damage Reduction?
Solving a general case for this thru analytic means is extremely difficult and time consuming. The below graphs were generated by computation methods only (quite literally subtracting damage in finite increments until death). As such there are no overall formulas to look at. Sorry. =X
EHP
http://i42.tinypic.com/6fsfn6.png
DR
http://i39.tinypic.com/bdv2f4.png
Example:
You have 1500 health and Level 4 Guardian Angel. You get hit by attacks that average 120 damage (8% of total health) after other forms of damage reduction (armor, etc).
Guardian Angel will provide an EHP boost of 36% (26.5% damage reduction) against such attacks.
It will take 17 attacks of 120 damage to kill you (1500 * 1.36 = 2040 EHP; 2040 / 120 = 17).
Commutativity of Guardian Angel
All sources of damage mitigated by Guardian Angel are commutative. In other words, the order in which damage received has no effect on Martyr’s overall survivability.
For example, a 800 damage nuke with spellshards followed by a 150 damage attack will leave Martyr with the exact same health as it would had he been hit by the smaller attack first. Furthermore, three different attacks of damage A B and C will do the same damage whether the order done is ABC, ACB, BAC, BCA, CAB, or CBA. This can be extrapolated to any number of attacks of any amount of damage (provided that Martyr can survive all attacks).
When playing against Martyr, do not worry about throwing big nukes first or saving them until the end. It really makes no difference.
Specific Example
2200 current HP. 2500 max HP. 800 damage attack followed by 150 damage attack.
Attack 1: 2200 - 800*[0.5*(2200/2500)+0.5] = 1448 HP remaining
Attack 2: 1448 - 150*[0.5*(1448/2500)+0.5] = 1329.56 HP remaining
2200 HP. 150 damage attack followed by 800 damage attack.
Attack 1: 2200 - 150*[0.5*(2200/2500)+0.5] = 2059 HP remaining
Attack 2: 2059 - 800*[0.5*(2059/2500)+0.5] = 1329.56 HP remaining
General Proof (Two Attacks)
”X” current HP, in percent of max. Max HP = 1 (100%). “A” damage attack followed by “B” damage attack.
Attack 1: X - A*(0.5*X+0.5) = (X - 0.5AX - 0.5A) HP remaining
Attack 2: (X - 0.5AX - 0.5A) - B*[0.5*( X - 0.5AX - 0.5A) +0.5]
= X - 0.5AX - 0.5A - 0.5BX + 0.5ABX + 0.5AB - 0.5B
= X - 0.5A - 0.5B - 0.5AX – 0.5BX + 0.5AB + 0.5ABX
”X” current HP, in percent of max. Max HP = 1 (100%). “B” damage attack followed by “A” damage attack.
Attack 1: X - B*(0.5*X+0.5) = (X - 0.5BX - 0.5B) HP remaining
Attack 2: (X - 0.5BX - 0.5B) - A*[0.5*( X - 0.5BX - 0.5B) +0.5]
= X - 0.5BX - 0.5B - 0.5AX + 0.5BAX + 0.5BB - 0.5A
= X - 0.5A - 0.5B - 0.5AX – 0.5BX + 0.5AB + 0.5ABX
The same method can be applied to the other damage reduction factors of 0.2, 0.3, and 0.4 with the same results
General Proof (Arbitrary Number of Attacks)
Since A>B is the same as B>A…
A>B>C>D>E is the same as B>A>C>D>E
Since A>C is the same as C>A…
B>A>C>D>E is the same as B>C>A>D>E
This swap can be completed an infinite number of times with adjacent sets to get any pre-prescribed order of attacks.
Without loss of generality,
The order of objects for any permutation of infinite objects, when the order of any two objects is arbitrary, is itself arbitrary.
TL;DR Summary
Under ideal circumstances, Guardian Angel provides...
an average damage reduction of 10.3% / 15.9 % / 21.7% / 27.9%
the equivalent of increasing your health by 11.6% / 18.9% / 27.7% / 38.6%
Furthermore, Guardian Angel has the following property...
the order of attacks you receive does not matter, at all, as long as you survive the attacks
FAQ
Specific question or clarifications that are brought up will be addressed here.
Passively reduces the damage Martyr takes by 0.2 / 0.3 / 0.4 / 0.5 times the percentage of Health he is missing.
This simply means that the lower Martyr’s health, the less damage he takes. Think of it like Armadon’s ability, but with the damage reduction scaling from 0% to 20/30/40/50% depending on Martyr’s current health.
Damage Reduction and Damage Factor Formulae
Damage Reduction, DR =
L1: 0.2*(1 - x)
L2: 0.3*(1 - x)
L3: 0.4*(1 - x)
L4: 0.5*(1 - x)
Damage Factor, DF =
L1: 0.2x + 0.8
L1: 0.3x + 0.7
L1: 0.4x + 0.6
L1: 0.5x + 0.5
Damage reduction is post-mitigation
All incoming damage is first reduced by any shield (Accursed, MoA, Chipper), armor, magic armor, and blocking. Any damage that is left over is then reduced by Guardian Angel.
The damage reduction works for damage received from casting Guardian Angel on an ally.
The damage reduction does not work for direct HP removal (such as activating Sacrificial Lamb or Blood Chalice).
Effective Health Impact (Ideal Case)
Effective Health Points (EHP) is how much unmitigated damage a hero can withstand. For most cases, EHP is an effective measurement of how well a hero can survive physical damage. EHP is determined by multiplying health by a mitigation factor calculated from armor, blocking, and evasion. For more on EHP see this thread: http://forums.heroesofnewerth.com/showthread.php?t=337020
Since Martyr’s passive ability reduces damage after mitigation, his EHP gains an additional factor that stacks with all other factors.
This additional factor is =
L1: 1.116 (equivalent to 10.3% damage reduction)
L2: 1.189 (equivalent to 15.9% damage reduction)
L3: 1.277 (equivalent to 21.7% damage reduction)
L4: 1.386 (equivalent to 27.9% damage reduction)
...which means Martyr with level 4 Guardian Angel will survive approximately 38% longer than another hero with identical health, armor, magic armor, blocking, and evasion. Please be aware that this factor is the ideal case where he receives an infinite number of attacks of very small incidents of damage.
These factors are calculated by integrating EHP from X=0 to X=1:
EHP = 1 / DF
Level 1:
http://i52.tinypic.com/5kehie.png
EHP = 1 / (0.2*(x + 4))dx, from x = 0 to x = 1
= 5*ln(x + 4)|
= 5*ln(5) - 5*ln(4)
= 1.116
Level 2:
http://i55.tinypic.com/33atahy.png
EHP = 1 / (0.3*(x +7/3))dx, from x = 0 to x = 1
= 10/3*ln(x + 7/3)|
= 10/3*ln(10/3) - 10/3*ln(7/3)
= 1.189
Level 3:
http://i51.tinypic.com/345y360.png
EHP = 1 / (0.4*(x + 1.5))dx, from x = 0 to x = 1
= 2.5*ln(x + 1.5)|
= 2.5*ln(2.5) - 2.5*ln(1.5)
= 1.277
Level 4:
http://i53.tinypic.com/9gztl1.png
EHP = 1 / (0.5*(x +1))dx, from x = 0 to x = 1
= 2*ln(x + 1)|
= 2*ln(2) - 2*ln(1)
= 1.386
Effective Health Impact (Specific Case)
Now, effective EHP for infinitely small attacks is all well and good, but how useful is Guardian Angel against real attacks? Obviously the magnitude of the attacks has a huge effect; if you get one shotted by a level 5 codex hellflower spellshards from full health your damage reduction was zero.
Given that each attack does X% of your total health, what is the effective EHP that Guardian Angel provides? What is the equivalent Damage Reduction?
Solving a general case for this thru analytic means is extremely difficult and time consuming. The below graphs were generated by computation methods only (quite literally subtracting damage in finite increments until death). As such there are no overall formulas to look at. Sorry. =X
EHP
http://i42.tinypic.com/6fsfn6.png
DR
http://i39.tinypic.com/bdv2f4.png
Example:
You have 1500 health and Level 4 Guardian Angel. You get hit by attacks that average 120 damage (8% of total health) after other forms of damage reduction (armor, etc).
Guardian Angel will provide an EHP boost of 36% (26.5% damage reduction) against such attacks.
It will take 17 attacks of 120 damage to kill you (1500 * 1.36 = 2040 EHP; 2040 / 120 = 17).
Commutativity of Guardian Angel
All sources of damage mitigated by Guardian Angel are commutative. In other words, the order in which damage received has no effect on Martyr’s overall survivability.
For example, a 800 damage nuke with spellshards followed by a 150 damage attack will leave Martyr with the exact same health as it would had he been hit by the smaller attack first. Furthermore, three different attacks of damage A B and C will do the same damage whether the order done is ABC, ACB, BAC, BCA, CAB, or CBA. This can be extrapolated to any number of attacks of any amount of damage (provided that Martyr can survive all attacks).
When playing against Martyr, do not worry about throwing big nukes first or saving them until the end. It really makes no difference.
Specific Example
2200 current HP. 2500 max HP. 800 damage attack followed by 150 damage attack.
Attack 1: 2200 - 800*[0.5*(2200/2500)+0.5] = 1448 HP remaining
Attack 2: 1448 - 150*[0.5*(1448/2500)+0.5] = 1329.56 HP remaining
2200 HP. 150 damage attack followed by 800 damage attack.
Attack 1: 2200 - 150*[0.5*(2200/2500)+0.5] = 2059 HP remaining
Attack 2: 2059 - 800*[0.5*(2059/2500)+0.5] = 1329.56 HP remaining
General Proof (Two Attacks)
”X” current HP, in percent of max. Max HP = 1 (100%). “A” damage attack followed by “B” damage attack.
Attack 1: X - A*(0.5*X+0.5) = (X - 0.5AX - 0.5A) HP remaining
Attack 2: (X - 0.5AX - 0.5A) - B*[0.5*( X - 0.5AX - 0.5A) +0.5]
= X - 0.5AX - 0.5A - 0.5BX + 0.5ABX + 0.5AB - 0.5B
= X - 0.5A - 0.5B - 0.5AX – 0.5BX + 0.5AB + 0.5ABX
”X” current HP, in percent of max. Max HP = 1 (100%). “B” damage attack followed by “A” damage attack.
Attack 1: X - B*(0.5*X+0.5) = (X - 0.5BX - 0.5B) HP remaining
Attack 2: (X - 0.5BX - 0.5B) - A*[0.5*( X - 0.5BX - 0.5B) +0.5]
= X - 0.5BX - 0.5B - 0.5AX + 0.5BAX + 0.5BB - 0.5A
= X - 0.5A - 0.5B - 0.5AX – 0.5BX + 0.5AB + 0.5ABX
The same method can be applied to the other damage reduction factors of 0.2, 0.3, and 0.4 with the same results
General Proof (Arbitrary Number of Attacks)
Since A>B is the same as B>A…
A>B>C>D>E is the same as B>A>C>D>E
Since A>C is the same as C>A…
B>A>C>D>E is the same as B>C>A>D>E
This swap can be completed an infinite number of times with adjacent sets to get any pre-prescribed order of attacks.
Without loss of generality,
The order of objects for any permutation of infinite objects, when the order of any two objects is arbitrary, is itself arbitrary.
TL;DR Summary
Under ideal circumstances, Guardian Angel provides...
an average damage reduction of 10.3% / 15.9 % / 21.7% / 27.9%
the equivalent of increasing your health by 11.6% / 18.9% / 27.7% / 38.6%
Furthermore, Guardian Angel has the following property...
the order of attacks you receive does not matter, at all, as long as you survive the attacks
FAQ
Specific question or clarifications that are brought up will be addressed here.