Pathfinder Power Attack Calculator?

[Thrawn183]

So, I've checked around on google and I haven't found a calculator for the new Pathfinder version of Power Attack. If anyone here has, could you provide a link please?

[Yora]

Why would you want to use a calculator for it?

[Rixx]

You don't need a calculator for Pathfinder Power Attack. (If you needed a calculator in 3.5, that's probably why they changed it...)

[Starbuck_II]

So, I've checked around on google and I haven't found a calculator for the new Pathfinder version of Power Attack. If anyone here has, could you provide a link please?

Isn't it a straight formula based on BAB?

You can't choose.
Assuming 2 handed:
BAB 0: -1, +3
BAB 4: -2, +5
BAB 8: -3, +7
BAB 12: -4, +9
BAB 16: -5, +11
BAB 20: -6, +13.

You have no choice. If you use Power attack you must use above formula.

[Thrawn183]

Just to see whether or not you should actually be power attacking.

It would also really help me in my comparisons between the average damage output of a 3.5 fighter vs. a pathfinder fighter to see how the new system stacks up.

[Rixx]

Isn't it a straight formula based on level?

It's based on your BAB and wether or not you're using a two-handed weapon.

You can choose to take a –1 penalty on all melee attack rolls and combat maneuver checks to gain a +2 bonus on all melee damage rolls. This bonus to damage is increased by half (+50%) if you are making an attack with a two-handed weapon, a one handed weapon using two hands, or a primary natural weapon that adds 1-1/2 times your Strength modifier on damage rolls. This bonus to damage is halved (–50%) if you are making an attack with an off-hand weapon or secondary natural weapon. When your base attack bonus reaches +4, and every 4 points thereafter, the penalty increases by –1 and the bonus to damage increases by +2. You must choose to use this feat before making an attack roll, and its effects last until your next turn. The bonus damage does not apply to touch attacks or effects that do not deal hit point damage.

Considering you're only losing 1 to-hit and gaining up to 3 damage, Power Attacking is still worth it.

[Epinephrine]

Just to see whether or not you should actually be power attacking.

It would also really help me in my comparisons between the average damage output of a 3.5 fighter vs. a pathfinder fighter to see how the new system stacks up.

I did the whole thing in spreadsheets. Depends on what the roll is that is needed to hit with your attack, as well as the number of iteratives you are swinging, as well as your base damage (since huge base damage means that losing attack roll hurts more).

I could throw it together as a google document and share it to you if you'd like (and if you have a gmail account). I was tinkering with adding critical hits, etc, but that seemed like overkill.

[Thrawn183]

Critical hits are kind of important if someone is say, wielding a falchion with improved critical. That's a 30% chance to threaten, and if you happen to have a class ability that automatically confirms...

[Epinephrine]

Critical hits are kind of important if someone is say, wielding a falchion with improved critical. That's a 30% chance to threaten, and if you happen to have a class ability that automatically confirms...

They are important to total damage, but not much for what happens with power attack. A critical hit multiplies the base damage, most damage bonuses a power attacker uses, and the power attack damage - it seemed unecessary for the relative gains of power attacking.

If you are power attacking with characters with damage sources that don't multiply (for example, a sneak attack), then the critical hit begins to matter.

That was the logic. Not super hard to add critical hits into the calculation, but why bother if 99% of the time it won't make a difference. The people who power attack (or who care much about critical hits) typically don't have substantial amounts of non-multiplying damage. You wouldn't bother with a keen scimitar or improved critical (Dire Pick) if you are dealing all your damage from a sneak attack.

[Morquard]

I don't read anything about light weapons in there anymore. I know it mentiones off-hand weapons, which are usually light, but they don't have to be.

So can a light weapon user theoretically use Power Attack?

Would it make sense for a dual wielding rogue?

[Epinephrine]

I don't read anything about light weapons in there anymore. I know it mentiones off-hand weapons, which are usually light, but they don't have to be.

So can a light weapon user theoretically use Power Attack?

Yes. Not just theoretically, either. They really can ;)

Would it make sense for a dual wielding rogue?

It could. You'd need the 13 Str, and you'd have to work out whether the increase in damage were worth it. I doubt it is. Since your base damage is very high (say, 1d6 base + many d6 SA) it is very valuable to actually score a hit.

Math stuff:

Spoiler

Show

Assume you have 6d6 sneak attack (level 11?), a 14 strength (so you can get Power Attack if you want), TWF and Improved TWF, and a slightly magical pair of +1 shortswords. You're looking at 7d6+3(+2 on off hand) or so damage, or 27.5/26.5 damage per hit.

If you have any real chance of missing with your attacks (say you hit on a 4 with your main attacks, and with a 9 on your iteratives) you may lose more by taking a penalty to hit than you are gaining. You can take -3 to hit to gain +6/+3 damage. So, you compare your 85% chance and 60% chance to deal 24/23 damage with your 70% and 45% chances of dealing 30/26 damage.

0.85*(27.5+26.5)+0.6*(27.5+26.5) = 78.3 expected damage
0.7*(33.5+29.5)+0.45*(33.5+29.5) = 72.45 expected damage

So you lose out on your power attack in this case. You need to be pretty certain to land your attacks to make the small bonus to damage worth the penalty to your iteratives - if you get into the range where your primary attacks are 95% likely to hit you are looking at a bonus from PA.

I've left criticals out, but checking my spreadsheet the difference is minor - the shortsword expected damages jump to ~80 and ~75, still an edge for not power attacking

[Mando Knight]

Isn't it a straight formula based on BAB?

You can't choose.
Assuming 2 handed:
BAB 0: -1, +3
BAB 4: -2, +5
BAB 8: -3, +7
BAB 12: -4, +9
BAB 16: -5, +11
BAB 20: -6, +13.

You have no choice. If you use Power attack you must use above formula.

The Pathfinder Power Attack damage boost actually follows this table:
{table=head]Penalty|1Hand|2Hand
-1|+2|+3
-2|+4|+6
-3|+6|+9
-4|+8|+12
-5|+10|+15
-6|+12|+18[/table]

[Epinephrine]

So Thrawn (or others) - if you want a google docs spreadsheet, let me know by PM, and I'll share a copy with you - I don't even think you need a gmail account, though I suspect I do need a valid email address to be able to share it.

I'll add the ability to handle sneak attack, criticals, etc.

Edit: Handling of non-multiplying damage and crits added.

[Starbuck_II]

The Pathfinder Power Attack damage boost actually follows this table:
{table=head]Penalty|1Hand|2Hand
-1|+2|+3
-2|+4|+6
-3|+6|+9
-4|+8|+12
-5|+10|+15
-6|+12|+18[/table]

Wait, how did you get +6 at 4th?
Oh, I see you apply +50% after the boost from BAB not before.

[ericgrau]

I just came up with some tables. See spoiler below.

Spoiler

Show

Effective Increase to Damage Per Hit
Average Base Damage: 5
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 2.58 | 4.84 | 6.79 | 8.42 | 9.74 | 10.74
3 | 2.56 | 4.78 | 6.67 | 8.22 | 9.44 | 10.33
4 | 2.53 | 4.71 | 6.53 | 8.00 | 9.12 | 9.88
5 | 2.50 | 4.63 | 6.38 | 7.75 | 8.75 | 9.38
6 | 2.47 | 4.53 | 6.20 | 7.47 | 8.33 | 8.80
7 | 2.43 | 4.43 | 6.00 | 7.14 | 7.86 | 8.14
8 | 2.38 | 4.31 | 5.77 | 6.77 | 7.31 | 7.38
9 | 2.33 | 4.17 | 5.50 | 6.33 | 6.67 | 6.50
10 | 2.27 | 4.00 | 5.18 | 5.82 | 5.91 | 5.45
11 | 2.20 | 3.80 | 4.80 | 5.20 | 5.00 | 4.20
12 | 2.11 | 3.56 | 4.33 | 4.44 | 3.89 | 2.67
13 | 2.00 | 3.25 | 3.75 | 3.50 | 2.50 | 0.75
14 | 1.86 | 2.86 | 3.00 | 2.29 | 0.71 | -1.71
15 | 1.67 | 2.33 | 2.00 | 0.67 | -1.67 | -1.17
16 | 1.40 | 1.60 | 0.60 | -1.60 | -1.00 | -0.40
17 | 1.00 | 0.50 | -1.50 | -0.75 | 0.00 | 0.75
18 | 0.33 | -1.33 | -0.33 | 0.67 | 1.67 | 2.67
19 | -1.00 | 0.50 | 2.00 | 3.50 | 5.00 | 6.50 [/TABLE]

Average Base Damage: 10
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 2.32 | 4.32 | 6.00 | 7.37 | 8.42 | 9.16
3 | 2.28 | 4.22 | 5.83 | 7.11 | 8.06 | 8.67
4 | 2.24 | 4.12 | 5.65 | 6.82 | 7.65 | 8.12
5 | 2.19 | 4.00 | 5.44 | 6.50 | 7.19 | 7.50
6 | 2.13 | 3.87 | 5.20 | 6.13 | 6.67 | 6.80
7 | 2.07 | 3.71 | 4.93 | 5.71 | 6.07 | 6.00
8 | 2.00 | 3.54 | 4.62 | 5.23 | 5.38 | 5.08
9 | 1.92 | 3.33 | 4.25 | 4.67 | 4.58 | 4.00
10 | 1.82 | 3.09 | 3.82 | 4.00 | 3.64 | 2.73
11 | 1.70 | 2.80 | 3.30 | 3.20 | 2.50 | 1.20
12 | 1.56 | 2.44 | 2.67 | 2.22 | 1.11 | -0.67
13 | 1.38 | 2.00 | 1.88 | 1.00 | -0.63 | -3.00
14 | 1.14 | 1.43 | 0.86 | -0.57 | -2.86 | -6.00
15 | 0.83 | 0.67 | -0.50 | -2.67 | -5.83 | -5.33
16 | 0.40 | -0.40 | -2.40 | -5.60 | -5.00 | -4.40
17 | -0.25 | -2.00 | -5.25 | -4.50 | -3.75 | -3.00
18 | -1.33 | -4.67 | -3.67 | -2.67 | -1.67 | -0.67
19 | -3.50 | -2.00 | -0.50 | 1.00 | 2.50 | 4.00 [/TABLE]

Average Base Damage: 15
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 2.05 | 3.79 | 5.21 | 6.32 | 7.11 | 7.58
3 | 2.00 | 3.67 | 5.00 | 6.00 | 6.67 | 7.00
4 | 1.94 | 3.53 | 4.76 | 5.65 | 6.18 | 6.35
5 | 1.88 | 3.38 | 4.50 | 5.25 | 5.63 | 5.63
6 | 1.80 | 3.20 | 4.20 | 4.80 | 5.00 | 4.80
7 | 1.71 | 3.00 | 3.86 | 4.29 | 4.29 | 3.86
8 | 1.62 | 2.77 | 3.46 | 3.69 | 3.46 | 2.77
9 | 1.50 | 2.50 | 3.00 | 3.00 | 2.50 | 1.50
10 | 1.36 | 2.18 | 2.45 | 2.18 | 1.36 | 0.00
11 | 1.20 | 1.80 | 1.80 | 1.20 | 0.00 | -1.80
12 | 1.00 | 1.33 | 1.00 | 0.00 | -1.67 | -4.00
13 | 0.75 | 0.75 | 0.00 | -1.50 | -3.75 | -6.75
14 | 0.43 | 0.00 | -1.29 | -3.43 | -6.43 | -10.29
15 | 0.00 | -1.00 | -3.00 | -6.00 | -10.00 | -9.50
16 | -0.60 | -2.40 | -5.40 | -9.60 | -9.00 | -8.40
17 | -1.50 | -4.50 | -9.00 | -8.25 | -7.50 | -6.75
18 | -3.00 | -8.00 | -7.00 | -6.00 | -5.00 | -4.00
19 | -6.00 | -4.50 | -3.00 | -1.50 | 0.00 | 1.50 [/TABLE]

Average Base Damage: 20
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 1.79 | 3.26 | 4.42 | 5.26 | 5.79 | 6.00
3 | 1.72 | 3.11 | 4.17 | 4.89 | 5.28 | 5.33
4 | 1.65 | 2.94 | 3.88 | 4.47 | 4.71 | 4.59
5 | 1.56 | 2.75 | 3.56 | 4.00 | 4.06 | 3.75
6 | 1.47 | 2.53 | 3.20 | 3.47 | 3.33 | 2.80
7 | 1.36 | 2.29 | 2.79 | 2.86 | 2.50 | 1.71
8 | 1.23 | 2.00 | 2.31 | 2.15 | 1.54 | 0.46
9 | 1.08 | 1.67 | 1.75 | 1.33 | 0.42 | -1.00
10 | 0.91 | 1.27 | 1.09 | 0.36 | -0.91 | -2.73
11 | 0.70 | 0.80 | 0.30 | -0.80 | -2.50 | -4.80
12 | 0.44 | 0.22 | -0.67 | -2.22 | -4.44 | -7.33
13 | 0.13 | -0.50 | -1.88 | -4.00 | -6.88 | -10.50
14 | -0.29 | -1.43 | -3.43 | -6.29 | -10.00 | -14.57
15 | -0.83 | -2.67 | -5.50 | -9.33 | -14.17 | -13.67
16 | -1.60 | -4.40 | -8.40 | -13.60 | -13.00 | -12.40
17 | -2.75 | -7.00 | -12.75 | -12.00 | -11.25 | -10.50
18 | -4.67 | -11.33 | -10.33 | -9.33 | -8.33 | -7.33
19 | -8.50 | -7.00 | -5.50 | -4.00 | -2.50 | -1.00 [/TABLE]

Average Base Damage: 25
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 1.53 | 2.74 | 3.63 | 4.21 | 4.47 | 4.42
3 | 1.44 | 2.56 | 3.33 | 3.78 | 3.89 | 3.67
4 | 1.35 | 2.35 | 3.00 | 3.29 | 3.24 | 2.82
5 | 1.25 | 2.13 | 2.63 | 2.75 | 2.50 | 1.88
6 | 1.13 | 1.87 | 2.20 | 2.13 | 1.67 | 0.80
7 | 1.00 | 1.57 | 1.71 | 1.43 | 0.71 | -0.43
8 | 0.85 | 1.23 | 1.15 | 0.62 | -0.38 | -1.85
9 | 0.67 | 0.83 | 0.50 | -0.33 | -1.67 | -3.50
10 | 0.45 | 0.36 | -0.27 | -1.45 | -3.18 | -5.45
11 | 0.20 | -0.20 | -1.20 | -2.80 | -5.00 | -7.80
12 | -0.11 | -0.89 | -2.33 | -4.44 | -7.22 | -10.67
13 | -0.50 | -1.75 | -3.75 | -6.50 | -10.00 | -14.25
14 | -1.00 | -2.86 | -5.57 | -9.14 | -13.57 | -18.86
15 | -1.67 | -4.33 | -8.00 | -12.67 | -18.33 | -17.83
16 | -2.60 | -6.40 | -11.40 | -17.60 | -17.00 | -16.40
17 | -4.00 | -9.50 | -16.50 | -15.75 | -15.00 | -14.25
18 | -6.33 | -14.67 | -13.67 | -12.67 | -11.67 | -10.67
19 | -11.00 | -9.50 | -8.00 | -6.50 | -5.00 | -3.50 [/TABLE]

Average Base Damage: 30
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 1.26 | 2.21 | 2.84 | 3.16 | 3.16 | 2.84
3 | 1.17 | 2.00 | 2.50 | 2.67 | 2.50 | 2.00
4 | 1.06 | 1.76 | 2.12 | 2.12 | 1.76 | 1.06
5 | 0.94 | 1.50 | 1.69 | 1.50 | 0.94 | 0.00
6 | 0.80 | 1.20 | 1.20 | 0.80 | 0.00 | -1.20
7 | 0.64 | 0.86 | 0.64 | 0.00 | -1.07 | -2.57
8 | 0.46 | 0.46 | 0.00 | -0.92 | -2.31 | -4.15
9 | 0.25 | 0.00 | -0.75 | -2.00 | -3.75 | -6.00
10 | 0.00 | -0.55 | -1.64 | -3.27 | -5.45 | -8.18
11 | -0.30 | -1.20 | -2.70 | -4.80 | -7.50 | -10.80
12 | -0.67 | -2.00 | -4.00 | -6.67 | -10.00 | -14.00
13 | -1.13 | -3.00 | -5.63 | -9.00 | -13.13 | -18.00
14 | -1.71 | -4.29 | -7.71 | -12.00 | -17.14 | -23.14
15 | -2.50 | -6.00 | -10.50 | -16.00 | -22.50 | -22.00
16 | -3.60 | -8.40 | -14.40 | -21.60 | -21.00 | -20.40
17 | -5.25 | -12.00 | -20.25 | -19.50 | -18.75 | -18.00
18 | -8.00 | -18.00 | -17.00 | -16.00 | -15.00 | -14.00
19 | -13.50 | -12.00 | -10.50 | -9.00 | -7.50 | -6.00 [/TABLE]

Average Base Damage: 35
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 1.00 | 1.68 | 2.05 | 2.11 | 1.84 | 1.26
3 | 0.89 | 1.44 | 1.67 | 1.56 | 1.11 | 0.33
4 | 0.76 | 1.18 | 1.24 | 0.94 | 0.29 | -0.71
5 | 0.63 | 0.88 | 0.75 | 0.25 | -0.63 | -1.88
6 | 0.47 | 0.53 | 0.20 | -0.53 | -1.67 | -3.20
7 | 0.29 | 0.14 | -0.43 | -1.43 | -2.86 | -4.71
8 | 0.08 | -0.31 | -1.15 | -2.46 | -4.23 | -6.46
9 | -0.17 | -0.83 | -2.00 | -3.67 | -5.83 | -8.50
10 | -0.45 | -1.45 | -3.00 | -5.09 | -7.73 | -10.91
11 | -0.80 | -2.20 | -4.20 | -6.80 | -10.00 | -13.80
12 | -1.22 | -3.11 | -5.67 | -8.89 | -12.78 | -17.33
13 | -1.75 | -4.25 | -7.50 | -11.50 | -16.25 | -21.75
14 | -2.43 | -5.71 | -9.86 | -14.86 | -20.71 | -27.43
15 | -3.33 | -7.67 | -13.00 | -19.33 | -26.67 | -26.17
16 | -4.60 | -10.40 | -17.40 | -25.60 | -25.00 | -24.40
17 | -6.50 | -14.50 | -24.00 | -23.25 | -22.50 | -21.75
18 | -9.67 | -21.33 | -20.33 | -19.33 | -18.33 | -17.33
19 | -16.00 | -14.50 | -13.00 | -11.50 | -10.00 | -8.50 [/TABLE]

Average Base Damage: 40
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 0.74 | 1.16 | 1.26 | 1.05 | 0.53 | -0.32
3 | 0.61 | 0.89 | 0.83 | 0.44 | -0.28 | -1.33
4 | 0.47 | 0.59 | 0.35 | -0.24 | -1.18 | -2.47
5 | 0.31 | 0.25 | -0.19 | -1.00 | -2.19 | -3.75
6 | 0.13 | -0.13 | -0.80 | -1.87 | -3.33 | -5.20
7 | -0.07 | -0.57 | -1.50 | -2.86 | -4.64 | -6.86
8 | -0.31 | -1.08 | -2.31 | -4.00 | -6.15 | -8.77
9 | -0.58 | -1.67 | -3.25 | -5.33 | -7.92 | -11.00
10 | -0.91 | -2.36 | -4.36 | -6.91 | -10.00 | -13.64
11 | -1.30 | -3.20 | -5.70 | -8.80 | -12.50 | -16.80
12 | -1.78 | -4.22 | -7.33 | -11.11 | -15.56 | -20.67
13 | -2.38 | -5.50 | -9.38 | -14.00 | -19.38 | -25.50
14 | -3.14 | -7.14 | -12.00 | -17.71 | -24.29 | -31.71
15 | -4.17 | -9.33 | -15.50 | -22.67 | -30.83 | -30.33
16 | -5.60 | -12.40 | -20.40 | -29.60 | -29.00 | -28.40
17 | -7.75 | -17.00 | -27.75 | -27.00 | -26.25 | -25.50
18 | -11.33 | -24.67 | -23.67 | -22.67 | -21.67 | -20.67
19 | -18.50 | -17.00 | -15.50 | -14.00 | -12.50 | -11.00 [/TABLE]

Average Base Damage: 45
(Column Headers = Penalty to Hit)
{TABLE] Roll Needed to Hit | -1 | -2 | -3 | -4 | -5 | -6
2 | 0.47 | 0.63 | 0.47 | 0.00 | -0.79 | -1.89
3 | 0.33 | 0.33 | 0.00 | -0.67 | -1.67 | -3.00
4 | 0.18 | 0.00 | -0.53 | -1.41 | -2.65 | -4.24
5 | 0.00 | -0.38 | -1.13 | -2.25 | -3.75 | -5.63
6 | -0.20 | -0.80 | -1.80 | -3.20 | -5.00 | -7.20
7 | -0.43 | -1.29 | -2.57 | -4.29 | -6.43 | -9.00
8 | -0.69 | -1.85 | -3.46 | -5.54 | -8.08 | -11.08
9 | -1.00 | -2.50 | -4.50 | -7.00 | -10.00 | -13.50
10 | -1.36 | -3.27 | -5.73 | -8.73 | -12.27 | -16.36
11 | -1.80 | -4.20 | -7.20 | -10.80 | -15.00 | -19.80
12 | -2.33 | -5.33 | -9.00 | -13.33 | -18.33 | -24.00
13 | -3.00 | -6.75 | -11.25 | -16.50 | -22.50 | -29.25
14 | -3.86 | -8.57 | -14.14 | -20.57 | -27.86 | -36.00
15 | -5.00 | -11.00 | -18.00 | -26.00 | -35.00 | -34.50
16 | -6.60 | -14.40 | -23.40 | -33.60 | -33.00 | -32.40
17 | -9.00 | -19.50 | -31.50 | -30.75 | -30.00 | -29.25
18 | -13.00 | -28.00 | -27.00 | -26.00 | -25.00 | -24.00
19 | -21.00 | -19.50 | -18.00 | -16.50 | -15.00 | -13.50 [/TABLE]

If your AB is equal or greater to the enemy's AC - i.e., free damage at no cost - then simply add the free damage to your base damage so that you still barely hit on a 2, and then consult the tables using the remainder of the PA penalty. If making a full attack, use your average AB when finding "roll needed to hit". So if your AB is 17/12/7, use 12. If it's 17/17/12/7, use (17+17+12+7)/4~=13. Probably something you want to figure out ahead of time if you get hasted a lot.

[Epinephrine]

@ericgrau; neat; I suppose I could have just done multiple tables, gets the main point across.

This is my spreadsheet :) I'll link it rather than put it in the post.
http://img22.imageshack.us/img22/3336/statsam.png

[ericgrau]

Hmm, I could make something in Excel spreadsheet format where you just punch in your stats and it tells you the maximum AC you should target. Then you just write down that number and you're good to go. Does anyone want me to make and e-mail them such a thing?

[Navigator]

If anyone is good at calculus, there is a single clever formula that can tell you everything you need to know.

So... anyone good at calculus?

[Gralamin]

If anyone is good at calculus, there is a single clever formula that can tell you everything you need to know.

So... anyone good at calculus?

Sounds like its just an optimization formula. Write out the attack formula (Have the Power Attack Penalty and damage as a variables. All other values will be constants that are represented by the variable. Solve for Damage), take the derivative, find points at which the derivative is = 0, compare to them to the area around them to find if they are max / mins, and find the maximum.

So...

Math goes in here.
I haven't looked at pathfinder, so I'm going to assume combat works just like 3.5. I'm treating PowerAttackPenalty as a NEGATIVE number.

ChanceToHit = (21 - EnemyAC + PowerAttackPenalty + AttackBonus)/20
So if you need a 15 to hit (30%), this formula would give you (21-15)/20 = 30%
If you need a 10 to hit (55%) this formula would give you (21-10)/20 = 55%

DamageOnHit = AverageDamage+PowerAttackDamageBonus
DamageOnCrit = DamageOnHit*Modifier

Now, we need to express PowerAttackDamageBonus as a factor of PowerAttackPenalty.

Case 1: One-Handed. PowerAttackDamageBonus = -2*PowerAttackPenalty.
Case 2: Two-Handed. PowerAttackDamageBonus = -3*PowerAttackPenalty

So, Make for a general case:
    PowerAttackDamageBonus = constant*PowerAttackPenalty
Where Constant is -2 or -3 depending.
So
DamageOnAHit=AverageDamage+constant*PowerAttackPenalty
DamageOnCrit=(AverageDamage+constant*PowerAttackPenalty)*Modifier

I'm going to ignore a few edge cases (An attack that would score a threat, but does not hit AC, for example, or always hitting on a 20. These are best handled as special cases, which is simple if your using a computer.)

TotalDamage = (ChanceToHit-ChanceToCrit)*DamageOnAHit+ChanceToCrit*ChanceToHit*DamageOnACrit.

Expand it out.
TotalDamage = ((21 - EnemyAC + PowerAttackPenalty + AttackBonus)/20 -ChanceToCrit) * (AverageDamage + constant * PowerAttackPenalty) + ChanceToCrit * ((21 - EnemyAC + PowerAttackPenalty + AttackBonus)/20 * (AverageDamage+constant*PowerAttackPenalty)*Modifier

This is nasty, Lets simplify a Bit.
TotalDamage = ((21 - EnemyAC + PowerAttackPenalty + AttackBonus - 20 * ChanceToCrit) (AverageDamage + constant * PowerAttackPenalty))/20 + ChanceToCrit * ((21 - EnemyAC + PowerAttackPenalty + AttackBonus) * (AverageDamage+constant*PowerAttackPenalty)*Modifier / 20

Multiply both sides by 20. Let C = AttackBonus - EnemyAC

20*TotalDamage = (21 + PowerAttackPenalty + C - 20 * ChanceToCrit) (AverageDamage + constant * PowerAttackPenalty) + ChanceToCrit * (21  + PowerAttackPenalty  + C ) * (AverageDamage + constant * PowerAttackPenalty ) * Modifier

Where Modifier is the Crit Modifier, Constant is -2 or -3 (Depending on handiness), and AC = AttackBonus-EnemyAC.

I simplify the formula to a long abomination for the next step, It won't be shown.
Treat Everything but TotalDamage and PowerAttackPenalty as Constants. Take Derivative of the abomination. You end up with:

20*TotalDamagePrime = AverageDamage + AverageDamage * Modifier * ChanceToCrit + 21 * Constant + 2 * Constant * PowerAttackPenalty + Constant * C - 20 * ChanceToCrit * Constant + 21 * Constant * Modifier * ChanceToCrit + 2 * Constant * Modifier * ChanceToCrit * PowerAttackPenalty + Constant * C * Modifier * ChanceToCrit

Simplify it, find the points at which it equals to 0. Note, P is an Integer in [-6, -1].
PowerAttackPenalty = (20 * ChanceToCrit * Constant - AverageDamage - AverageDamage * ChanceToCrit - 21 * Constant - Constant * AttackBonus + Constant * EnemyAC - 21 * Constant * Modifier * ChanceToCrit - Constant * Modifier * ChanceToCrit * AttackBonus + Constant * Modifier * ChanceToCrit * EnemyAC)/(2*Constant + 2 * Constant * Modifier * ChanceToCrit)

I'll put in the rest of it in a bit.

Edit: There we go. I'm guessing that can be reduced a bit, But thats essentially the equation that you need to find the maximums. Note that Equation gives results that may be maximums OR are Minimums - This forms a line that damage always increases before or after a certain point.

Edit 2: Lets use this in an example
Say: Using a two-handed weapon (19-20/x2; AverageDamage 15) attacking AC 16, and you have a +5 Attack bonus normally.

PowerAttackPenalty = (20 * 0.1 * -3 - 15 - 15 * 0.1 - 21 * -3 - -3 * 5 + -3 * 16 - 21 * -3 * 2 * 0.1 - -3 * 2 * 0.1 * 5 + -3 * 2 * 0.1 * 16)/(2*-3 + 2 * -3 * 2 * 0.1)

PowerAttackPenalty = (-6 - 15 - 1.5 + 63 + 15 - 48 + 12.6 + 3 - 9.6)/(-6 -1.2)
PowerAttackPenalty = 13.5/-7.2 = -1.875

So, In this case the Maximum / Minimum is in between a -1 penalty and a -2 penalty, leaning towards 2. Check for what it is. Lets Go with -1.7 for a Greater then, and -2 for a less then. Note C = 5 - 16 = -11.

20*TotalDamagePrime = 15 + 15 * 2 * 0.1 + 21 * -3 + 2 * -3 * -1.7 + -3 * -11 - 20 * 0.1 * -3 + 21 * -3 * 2 * 0.1 + 2 * -3 * 2 * 0.1 * -1.7 + -3 * -11 * 2 * 0.1
This works out to be positive. So at values > -1.875 the damage is increasing. You'd want to power attack for 1.

For Checking Purposes:
20*TotalDamagePrime = 15 + 15 * 2 * 0.1 + 21 * -3 + 2 * -3 * -1.7 + -3 * -11 - 20 * 0.1 * -3 + 21 * -3 * 2 * 0.1 + 2 * -3 * 2 * 0.1 * -1.7 + -3 * -11 * 2 * 0.1
Works out to positive as well. So at values < -1.875 the damage is increasing

Which makes me remember: The Derivative is a line. In that case you actually need only determine if the slope is negative or positive (easy way: Take derivative again), and if its positive, a higher number is always better, except for corner cases.

[Epinephrine]

Calculus is a poor tool for this problem:

1) You need several equations anyway, for the various situations (BAB <6, BAB 6-10, BAB 11-15, BAB 16+, for TWF, ITWF, GTWF, 2H, and 1H)
2) the functions are not continuous - they have discontinuities when to-hit exceeds 20 or drops below 2.

[Gralamin]

Calculus is a poor tool for this problem:

1) You need several equations anyway, for the various situations (BAB <6, BAB 6-10, BAB 11-15, BAB 16+, for TWF, ITWF, GTWF, 2H, and 1H)
2) the functions are not continuous - they have discontinuities when to-hit exceeds 20 or drops below 2.

1) I just followed information is given: You have a penalty from 1 to 6. Other then that, all that matters is the multiplication constant, which depends on Handiness.
2) Limited Domain, corner case formulas.

Granted, I'd just program it usually instead of doing calculus

[Doc Roc]

1) I just followed information is given: You have a penalty from 1 to 6. Other then that, all that matters is the multiplication constant, which depends on Handiness.
2) Limited Domain, corner case formulas.

Granted, I'd just program it usually instead of doing calculus

Hello Python! Aren't you a good little lad, eating all my math problems?
:: hugs the interpreter ::

[Raewyn]

Ow.... my brain...

I was never very good at math.