At what point is a Lifedrinker Warlock's expected damage reduced by Sharpshooter/GWM?

Question

At Level 12, a Pact of the Blade Warlock can take the Lifedrinker Eldritch Invocation, which states the following:

When you hit a creature with your pact weapon, the creature takes extra necrotic damage equal to your Charisma modifier (minimum 1).

Assuming a Hexblade Warlock with a 20 in Charisma, this translates to 10 bonus damage per attack. This enormous bonus means that every missed attack results in a significant reduction in damage.

The Sharpshooter and Great Weapon Master feats grant options of the following pattern:

Before you make [an attack] with [a weapon] with which you are proficient, you can choose to take a -5 penalty to the attack roll. If you do so and the attack hits, it deals +10 damage.

At what point is the Warlock's expected damage per attack reduced by using the optional attack modifier of either the Sharpshooter or Great Weapon Master feat?

Assume a Hexblade, Pact of the Blade Warlock with the following characteristics:

• At least 12th level
• Lifedrinker Eldritch Invocation
• 20 in Charisma
• Using a pact weapon, either:
  • A longbow
  • A greatsword
• ... with the respective weapon feat:
  • Sharpshooter
  • Great Weapon Master

With the following variables:

• Enemy AC
• Attack roll and damage roll bonuses (due to spells, proficiency bonus, magic weapons, etc.)

Answer 1

The break even formula for GWM/SS requires a bit of algebra. First, our variables:

• \$DAM\$ is average damage per hit before GWM/SS.
• \$HIT\$ is bonus to hit before GWM/SS.
• \$AC\$ is target armor class.

This is all we need to get our formula. We will set expected damage without GWM/SS equal to expected damage with it, and then solve for \$AC\$.

In this initial equation, the left is expected damage without GWM/SS, right is with.

$$DAM\left(\frac{21+HIT-AC}{20}\right)=(DAM+10)\left(\frac{16+HIT-AC}{20}\right)$$

The steps for solving for \$AC\$, left as an exercise to the reader, yield a break even AC of:

$$AC=16+HIT-\frac{DAM}{2}$$

Note, expected crit damage would be added as a constant on each side of our initial equation, so it is ignored.

Further, when the target's AC is so high that the only way to hit it is with a natural 20, which is when \$AC-HIT\geq20\$, it becomes best practice again to take GWM/SS on every attack since the penalty is meaningless.

The numbers.

With a mundane longbow, assuming a proficiency bonus of +4, \$DAM=1d8+10=14.5\$ and \$HIT=9\$. Then our break even AC is 17.75. This means on average, we should use SS for target ACs 17 and below.

With a mundane greatsword, assuming proficiency bonus of +4, \$DAM=2d6+10=17\$ and \$HIT=9\$. Then our break even AC is 16.5. This means on average, we should use GWM for target ACs 16 and below.

These are just examples, the formula is simple enough to substitute your own numbers with little issue.

A user named Bacon Bits works this result in greater detail in this answer, go give them some love.