- Nov 3, 2016
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Hello again everyone. So unrelated to the realm of Sanctuary, I was recently trying to figure out at what AC values my Ranger in D&D 5e would be better off utilizing his Sharpshooter feat. A handful of graphs and calculations later I came to my answer and realized that it's not that difficult to figure these things out, and I wondered where else it could be applicable.
Coincidentally, I've been putting together a Passive Amazon here and there, and one thing I was considering was whether I should be using Inner Sight or Penetrate (outside of the answer of both). I've generally assumed the conventional wisdom is to just use Penetrate, but I was wondering if there had been calculations done for it. I haven't found any solid answers (if they exist, please let me know) so I thought I'd do some fiddling with it myself. Unfortunately Diablo 2 mechanics can be a lot trickier than pen and papers, but I thought I'd take a stab at it regardless.
The first case I wanted to look at was a straightforward one: I have 20 points to either put into Inner Sight or Penetrate, what do I do?
This pulls in Attack Rating (or rather, the To Hit formula) which as follows:
Given what I managed to find about how the formula for Attack Rating and Defense is concerned, there are some things to keep in mind. For one, we know that % Increases to Attack Rating are additive. This unfortunately means I can't just slap on a (Penetrate's Bonus)*Attack Rating and call it a day, I'll have to account for what your character's % Attack Rating bonus is. Second, the details for calculating defense are.. slightly hazy. But since I am mostly concerned about PvM I'm just going to assume that the monster defense values are "final" and there's no cheeky % Increases to Defense that would be adversely affected by me subtracting from their defense value at the end of their calculation.
Let's compare Level 20 Inner Sight with Level 20 Penetrate then. If we let:
Level 20 Inner Sight: -815 Defense
Level 20 Penetrate: +225% Bonus Attack Rating
P = 1 + (Your Character's % Increase to Attack Rating / 100) without Penetrate
A = Your Integer Attack Rating Total
D = Monster's Base Defense
I believe I can make the following formula:
This takes a bit of manipulation, but the end result after some math (although correct me if I'm wrong) is:
Let's say I don't have any % Increase to Attack Rating, that would mean that I set P = 1 and solve for D. Why one and not zero? Because you still need to account for the base "1" in the Percentage Attack Rating formula (1 + Attack Rating Bonus % / 100). If that feels weird to you, you can instead alter the formula to be: P = (2.25)/(815)D - (2.25+1) and set P = 0 (Or exactly what your +% Attack Rating / 100 is). Either way, we get..
D = 1177.22
This would tell us that the two options are equal when the defense of a monster is ~1177. That would also mean that Inner Sight would be considered better for all Defense values less than 1177, whereas Penetrate would be better for all values greater than this number.
Some (Hell) References: Fallen have 933 Defense, Venom Lords have 1879 Defense and Baal has 2847 Defense.
Okay, so it sounds like Penetrate is just better then, right? Hmm, not necessarily. One thing that is sometimes brought up in Inner Sight vs. Penetrate debates that I've read is how Inner Sight scales, while Penetrate remains a flat 10% increase per level. Maybe that effects things? Let's try both skills at 25 and 30. Since we already solved the formula, this is really easy to compute, as we now have the general formula:
P = (Penetrate Bonus)/(Inner Sight Bonus)*Defense - Penetrate Bonus
Inner Sight at 25: -1175 Defense
Penetrate at 25: 275% Bonus Attack Rating
P = (2.75)/(1175)D - 2.75 (Using our first formula where P = 1)
D = 1602.27
Inner Sight at 30: -1615 Defense
Penetrate at 30: 325% Bonus Attack Rating
P = (3.25)/(1615)D - 3.25 (Using our first formula where P = 1)
D = 2111.92
This goes to show it does pick up once Skill Levels increase, although how rapidly that is to you depends on how many +Skills you have available.
Finally, there's the question of actually accounting for P. In all of these examples, I've been assuming that your Percentage Increase to AR is either none, or entirely coming from Penetrate. What if that's not the case? Maybe you're some fancypants running around with a Faith Mercenary. If you had a Level 15 Fanaticism Aura (110%) on you, how would that alter things?
Level 20 Skills: D = 1575.67
Level 25 Skills: D = 2072.27
Level 30 Skills: D = 2658.54
As we could see from the formula, the more percentage increases to attack rating we already have, the more beneficial the use of Inner Sight is becoming.
This raises an interesting notion. Surely, it'd be entirely possible I'd wind up with some amount of points in each Skill, rather than this all or nothing hypothetical, right? What if I had say, 1 Point in Inner Sight and 20 in Penetrate, or vice versa? That seems reasonable, since both are prerequisites for potentially useful Skills depending on the build. How does this change things? The example of maxing Inner Sight and using a base level of Penetrate can be covered fairly easily by our earlier formulas. Just increase P by the level of Penetrate (0.35 in the case of first level) and subtract the difference from the penetrate bonus, so:
[20 Inner Sight, 1 Penetrate] (1.9)/(815)D - 1.9 = 1.35 --> D = 1394.08
[25 Inner Sight, 6 Penetrate] (1.9)/(1175)D - 1.9 = 1.85 --> D = 2319.08
[30 Inner Sight, 11 Penetrate] (1.9)/(1615)D - 1.9 = 2.35 --> D = 3612.5
Much more competitive values. Remember, this is comparing 20 Inner Sight and 1 Penetrate versus just 20 Penetrate by itself. Since they'd both scale by +Skills, this is why the gap doesn't keep increasing (the value remains at 1.9 or 190% extra from the 19 hard points put in).
For kicks, let's just compare 20 Points of Inner Sight versus 1 Point of Inner Sight and 20 Penetrate. I'm going to breeze through the formula explanation just because I'm assuming this will be a fairly minimal change.
[1 Inner Sight, 20 Penetrate] (2.25)/(775)D - 2.25(815/775) = 1 --> D = 1159.44
[6 Inner Sight, 25 Penetrate] (2.75)/(1010)D - 2.75(1175/1010) = 1 --> D = 1542.27
[11 Inner Sight, 30 Penetrate] (3.25)/(1265)D - 3.25(1615/1265) = 1 --> D = 2004.23
We would want to compare these to the very first values we calculated, namely 1177, 1602, and 2112 respectively. This is showing us how much value a maxed our Inner Sight loses compared to someone who maxed Penetrate and is using a base level version of Inner Sight. This shows us the value difference is fairly minimal, which is what we'd expect-- you don't expect huge differences casting Level 1 Inner Sight on Hell enemies.
Before we wrap up, I want to use a few real examples just to try and put this all into perspective.
Let's say we're trying to shoot at Hell Baal and we're a Level 90 Amazon. Before accounting for anything related to Attack and Defense, we'll do the Level portion.
To Hit % = 200 * (90/189) * Attack Rating Formula = (18000/189) * Attack Rating Formula
Let's say we're trying to go for an 75% Chance to Hit Baal because we're modest. So let's solve for what our Attack Rating would need to be. Baal has 2847 Defense.
Let's do two of the more likely options. We'll say we have +5 To Skills, and our first case will be 20 in Penetrate, no Inner Sight. The other option will be 20 in Inner Sight, 1 in Penetrate. We won't have any extra +% Bonuses to Attack Rating besides Penetrate.
Penetrate Only = (3.75A)/(3.75A+2847)
75 = (18000/189) * (3.75A)/(3.75A+2847)
A = 2813
Inner Sight = (1.85A)/(1.85A+2847)
75 = (18000/189) * (1.85A)/(1.85A+(2847-1175))
A = 3349
Here's one of the most drastic examples in the entire game-- the highest level enemy with some of the highest defense around. What about something more mild, at say, a 90% chance To Hit?
A Venom Lord in Chaos Sanctuary? (Level: 84, Defense: 1879)
Penetrate Only:
90 = (18000/174) * (3.75A)/(3.75A+1879)
A = 3353
Inner Sight:
90 = (18000/174) * (1.85A)/(1.85A+(1879-1175))
A = 2547
A Bone Warrior in the Pit? (Level: 83, Defense: 1616)
Penetrate Only:
90 = (18000/173) * (3.75A)/(3.75A+1616)
A = 2761
Inner Sight:
90 = (18000/173) * (1.85A)/(1.85A+(1616-1175))
A = 1527
Remember, we are assuming we have no other ways of getting +% Bonus Attack Rating besides the given Penetrate multipliers, so these values would be pretty difficult to actually achieve on integer attack rating alone. Perhaps a better way to go about this would be to input your attack rating into these formulas and see what chance to hit you wind up with. Either way, this was just to put things out of the realm of theoretical.
So, can we draw any sort of conclusions? It's tough. There's a lot of variables and moving parts, but what I'd want to encourage people is to look at their bonuses to attack rating that they have on their Amazons, and try fiddling with some of these formulas. Assuming my math is correct (which it might not be) here are some observations.
Things this comparison doesn't account for:
Things to keep in mind / things I learned while researching this:
I think that's all! A lot of rambling and a lot of numbers, but I hope it was at least somewhat comprehensible. I'd be curious to hear your thoughts, as well as the inevitable moment where someone points out where the error in my math/logic is, as this was a little off the cuff.
Coincidentally, I've been putting together a Passive Amazon here and there, and one thing I was considering was whether I should be using Inner Sight or Penetrate (outside of the answer of both). I've generally assumed the conventional wisdom is to just use Penetrate, but I was wondering if there had been calculations done for it. I haven't found any solid answers (if they exist, please let me know) so I thought I'd do some fiddling with it myself. Unfortunately Diablo 2 mechanics can be a lot trickier than pen and papers, but I thought I'd take a stab at it regardless.
The first case I wanted to look at was a straightforward one: I have 20 points to either put into Inner Sight or Penetrate, what do I do?
This pulls in Attack Rating (or rather, the To Hit formula) which as follows:
To Hit (%) = 200 * (Attack Rating / (Attack Rating + Defense)) * (Attacker Level / (Attacker Level + Defender Level))
Which admittedly, can look a little daunting. But for the sake of what Inner Sight and Penetrate are concerned about, we really only need this part right here:
(Attack Rating / (Attack Rating + Defense))
Because everything else will remain the same regardless of our skill selection. We also are not particularly worried about the exact numbers. We know that this equation will always give us a result between a very small positive result up to 1, so for our sake "bigger is better" works just fine, we want the maximal value here.
Given what I managed to find about how the formula for Attack Rating and Defense is concerned, there are some things to keep in mind. For one, we know that % Increases to Attack Rating are additive. This unfortunately means I can't just slap on a (Penetrate's Bonus)*Attack Rating and call it a day, I'll have to account for what your character's % Attack Rating bonus is. Second, the details for calculating defense are.. slightly hazy. But since I am mostly concerned about PvM I'm just going to assume that the monster defense values are "final" and there's no cheeky % Increases to Defense that would be adversely affected by me subtracting from their defense value at the end of their calculation.
Let's compare Level 20 Inner Sight with Level 20 Penetrate then. If we let:
Level 20 Inner Sight: -815 Defense
Level 20 Penetrate: +225% Bonus Attack Rating
P = 1 + (Your Character's % Increase to Attack Rating / 100) without Penetrate
A = Your Integer Attack Rating Total
D = Monster's Base Defense
I believe I can make the following formula:
(AP) / (AP + (D-815)) = (A(P+2.25)) / (A(P+2.25) + (D))
This takes a bit of manipulation, but the end result after some math (although correct me if I'm wrong) is:
P = (2.25)/(815)D - 2.25
Okay, so what does this tell us then? Well it's quite straightforward. You need to either know the Base Defense of the Monster you're fighting or more easily, you need to know what the Percentage increase to Attack Rating your character has without Penetrate. Given that Bow Skills are bugged to not give their AR, this is almost entirely going to come from Equipment and/or Auras, so it should be easy to figure out.
Let's say I don't have any % Increase to Attack Rating, that would mean that I set P = 1 and solve for D. Why one and not zero? Because you still need to account for the base "1" in the Percentage Attack Rating formula (1 + Attack Rating Bonus % / 100). If that feels weird to you, you can instead alter the formula to be: P = (2.25)/(815)D - (2.25+1) and set P = 0 (Or exactly what your +% Attack Rating / 100 is). Either way, we get..
D = 1177.22
This would tell us that the two options are equal when the defense of a monster is ~1177. That would also mean that Inner Sight would be considered better for all Defense values less than 1177, whereas Penetrate would be better for all values greater than this number.
Some (Hell) References: Fallen have 933 Defense, Venom Lords have 1879 Defense and Baal has 2847 Defense.
Okay, so it sounds like Penetrate is just better then, right? Hmm, not necessarily. One thing that is sometimes brought up in Inner Sight vs. Penetrate debates that I've read is how Inner Sight scales, while Penetrate remains a flat 10% increase per level. Maybe that effects things? Let's try both skills at 25 and 30. Since we already solved the formula, this is really easy to compute, as we now have the general formula:
P = (Penetrate Bonus)/(Inner Sight Bonus)*Defense - Penetrate Bonus
Inner Sight at 25: -1175 Defense
Penetrate at 25: 275% Bonus Attack Rating
P = (2.75)/(1175)D - 2.75 (Using our first formula where P = 1)
D = 1602.27
Inner Sight at 30: -1615 Defense
Penetrate at 30: 325% Bonus Attack Rating
P = (3.25)/(1615)D - 3.25 (Using our first formula where P = 1)
D = 2111.92
This goes to show it does pick up once Skill Levels increase, although how rapidly that is to you depends on how many +Skills you have available.
Finally, there's the question of actually accounting for P. In all of these examples, I've been assuming that your Percentage Increase to AR is either none, or entirely coming from Penetrate. What if that's not the case? Maybe you're some fancypants running around with a Faith Mercenary. If you had a Level 15 Fanaticism Aura (110%) on you, how would that alter things?
Level 20 Skills: D = 1575.67
Level 25 Skills: D = 2072.27
Level 30 Skills: D = 2658.54
As we could see from the formula, the more percentage increases to attack rating we already have, the more beneficial the use of Inner Sight is becoming.
This raises an interesting notion. Surely, it'd be entirely possible I'd wind up with some amount of points in each Skill, rather than this all or nothing hypothetical, right? What if I had say, 1 Point in Inner Sight and 20 in Penetrate, or vice versa? That seems reasonable, since both are prerequisites for potentially useful Skills depending on the build. How does this change things? The example of maxing Inner Sight and using a base level of Penetrate can be covered fairly easily by our earlier formulas. Just increase P by the level of Penetrate (0.35 in the case of first level) and subtract the difference from the penetrate bonus, so:
[20 Inner Sight, 1 Penetrate] (1.9)/(815)D - 1.9 = 1.35 --> D = 1394.08
[25 Inner Sight, 6 Penetrate] (1.9)/(1175)D - 1.9 = 1.85 --> D = 2319.08
[30 Inner Sight, 11 Penetrate] (1.9)/(1615)D - 1.9 = 2.35 --> D = 3612.5
Much more competitive values. Remember, this is comparing 20 Inner Sight and 1 Penetrate versus just 20 Penetrate by itself. Since they'd both scale by +Skills, this is why the gap doesn't keep increasing (the value remains at 1.9 or 190% extra from the 19 hard points put in).
For kicks, let's just compare 20 Points of Inner Sight versus 1 Point of Inner Sight and 20 Penetrate. I'm going to breeze through the formula explanation just because I'm assuming this will be a fairly minimal change.
[1 Inner Sight, 20 Penetrate] (2.25)/(775)D - 2.25(815/775) = 1 --> D = 1159.44
[6 Inner Sight, 25 Penetrate] (2.75)/(1010)D - 2.75(1175/1010) = 1 --> D = 1542.27
[11 Inner Sight, 30 Penetrate] (3.25)/(1265)D - 3.25(1615/1265) = 1 --> D = 2004.23
We would want to compare these to the very first values we calculated, namely 1177, 1602, and 2112 respectively. This is showing us how much value a maxed our Inner Sight loses compared to someone who maxed Penetrate and is using a base level version of Inner Sight. This shows us the value difference is fairly minimal, which is what we'd expect-- you don't expect huge differences casting Level 1 Inner Sight on Hell enemies.
Before we wrap up, I want to use a few real examples just to try and put this all into perspective.
Let's say we're trying to shoot at Hell Baal and we're a Level 90 Amazon. Before accounting for anything related to Attack and Defense, we'll do the Level portion.
To Hit % = 200 * (90/189) * Attack Rating Formula = (18000/189) * Attack Rating Formula
Let's say we're trying to go for an 75% Chance to Hit Baal because we're modest. So let's solve for what our Attack Rating would need to be. Baal has 2847 Defense.
Let's do two of the more likely options. We'll say we have +5 To Skills, and our first case will be 20 in Penetrate, no Inner Sight. The other option will be 20 in Inner Sight, 1 in Penetrate. We won't have any extra +% Bonuses to Attack Rating besides Penetrate.
Penetrate Only = (3.75A)/(3.75A+2847)
75 = (18000/189) * (3.75A)/(3.75A+2847)
A = 2813
Inner Sight = (1.85A)/(1.85A+2847)
75 = (18000/189) * (1.85A)/(1.85A+(2847-1175))
A = 3349
Here's one of the most drastic examples in the entire game-- the highest level enemy with some of the highest defense around. What about something more mild, at say, a 90% chance To Hit?
A Venom Lord in Chaos Sanctuary? (Level: 84, Defense: 1879)
Penetrate Only:
90 = (18000/174) * (3.75A)/(3.75A+1879)
A = 3353
Inner Sight:
90 = (18000/174) * (1.85A)/(1.85A+(1879-1175))
A = 2547
A Bone Warrior in the Pit? (Level: 83, Defense: 1616)
Penetrate Only:
90 = (18000/173) * (3.75A)/(3.75A+1616)
A = 2761
Inner Sight:
90 = (18000/173) * (1.85A)/(1.85A+(1616-1175))
A = 1527
Remember, we are assuming we have no other ways of getting +% Bonus Attack Rating besides the given Penetrate multipliers, so these values would be pretty difficult to actually achieve on integer attack rating alone. Perhaps a better way to go about this would be to input your attack rating into these formulas and see what chance to hit you wind up with. Either way, this was just to put things out of the realm of theoretical.
So, can we draw any sort of conclusions? It's tough. There's a lot of variables and moving parts, but what I'd want to encourage people is to look at their bonuses to attack rating that they have on their Amazons, and try fiddling with some of these formulas. Assuming my math is correct (which it might not be) here are some observations.
- The more +Skills you have, the more valuable Inner Sight becomes compared to Penetrate.
- Having even a base level Penetrate to go along with your Inner Sight can make a big difference, especially considering +Skills.
- Comparatively, you're not getting a huge amount of value from a base level of Inner Sight-- a boon to those who are lazy and don't want to use it.
- The more +% Bonus to Attack Rating you have, the more valuable the Inner Sight can be compared to adding even more from Penetrate. This could be particularly noticeable for non-Bowzons, since they're more likely to get this attribute, or those who are using an aura like Fanaticism.
- The values are relatively close either way, with the values starting to equalize at some very critical junctures (namely, around 2000 Monster Defense).
- This also could infer that the difference between these two options is also fairly minimal, since they're relatively similar around these points, which could add further value to Penetrate, since it requires less setup.
Things this comparison doesn't account for:
- Ways Monsters could get increased Defense / in what order that applies.
- The fact that Penetrate is always active, but Inner Sight requires a cast, and the laziness this may bring.
- The AoE and sight/screen requirements of Inner Sight compared to Penetrate
- The fact that Inner Sight can be used by others, rather than just yourself (and your Valkyrie)
- Any use of ITD, Target Defense X%, -X Defense on Hit.
- Just how marginal of a difference these differences may wind up being.
Things to keep in mind / things I learned while researching this:
- Bow Skills don't apply the +AR% listed on the skills due to a bug.
- Inner Sight does stack with other curses.
- Penetrate is overridden completely by Blessed Aim, regardless of respective levels.
I think that's all! A lot of rambling and a lot of numbers, but I hope it was at least somewhat comprehensible. I'd be curious to hear your thoughts, as well as the inevitable moment where someone points out where the error in my math/logic is, as this was a little off the cuff.
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