There are other similarly mathy questions on this site such as "How much damage does Great Weapon Fighting add on average?" and "Are features that allow −5 to attack to get +10 to damage mathematically sound?" but I was wondering how the math pans out with the Durable Feat.
The feat states:
When you roll a Hit Die to regain hit points, the minimum number of hit points you regain from the roll equals twice your Constitution modifier (minimum of 2).
There are two interpretations for how the Durable feat could work "How does the Durable feat work?", but for the purposes of this question I would like you to assume that "the roll" refers to the total of the die's result and the Constitution modifier.
As the answers there explain, this means the feat is useless to those with a Constitution modifier of +1 but it prevents the alternative interpretation's unusual case of a d6 Hit Die class with a +5 Constitution modifier gaining 15 Hit Points from spending a Hit Die whereas their normal maximum is 11.
This is also shown to be the intended way for the feat to work (thank to user @Rykara for finding this) as Jeremy Crawford has made this tweet:
If you have the Durable feat and spend a Hit Die to regain hit points, the minimum number of hit points you regain is equal to twice your Constitution modifier (minimum of 2 hit points). For example, if your modifier is +1, you regain a minimum of 2 hit points.
How much does this feat increase the average Hit Point gained when spending Hit Dice?
Best Answer
The feat always benefits smaller hit dice more, but by how much depends on your Constitution modifier. The average increase is 0.60454545 Hit Points.
I'll first present the math/formulas, then the raw numerical results and then the more natural English generalizations with a graph. (you can scroll down to go straight to the results).
The section on "Short Rest" states:
And the Durable Feat states:
This increases our minimum to the higher of double our Constitution modifier (unless our modifier is zero or less in which case the minimum is 2).
Because of the negative modifiers there are actually two separate formulas we will have to use:
If we had a d6 Hit Die and a -3 Constitution modifier our possible Hit Point gains would be:
0, 0, 0, 1, 2, 3.
There will always be a number of zeroes equal to the negative of our Constitution modifier. Here we have zeroes until we roll a 4 on the die as 4-3 (the die result plus our Con modifier) is 1.
Using the Durable Feat our new possible Hit Point gains would be:
2, 2, 2, 2, 2, 3.
The new minimum is two, so we have increased every result that was less than 2. In particular, all of the zeroes are replaced by twos and the single result of one that we had is also replaced by a two.
Thus we have increased the total by 2 times our negative Constitution modifier plus 1: -2MOD+1.
Dividing this by the number of results (as they are all equally likely) will get us our average increase from the feat, (HD represent the size of our Hit Die):
(-2MOD+1)/HD.
If we had a d6 Hit Die and a +3 Constitution modifier our possible Hit Point gains would be:
4, 5, 6, 7, 8, 9.
The lowest we can roll will always be 1 plus our Constitution modifier.
Using the Durable Feat our new possible Hit Point gains would be:
6, 6, 6, 7, 8, 9.
The new minimum is double our Constitution modifier, so we have increased every result that was less than that number.
The first result will increase by our Constitution modifier minus 1 (double our con mod minus the sum of our con mod and 1), the second result wil increase by our Constitution modifier minus 2, and so forth.
So will get every number from 0 to out constitution modifier minus 1. Summing these numbers we get this: (MOD-1)(MOD)/2.
Dividing this by the number of results (as they are all equally likely) will get us our average increase from the feat, (HD represent the size of our Hit Die):
(MOD-1)(MOD)/(2HD).
The raw numerical results
Your average increases by (-2MOD+1)/HD for modifiers less than +1.
Your average increases by (MOD-1)(MOD)/(2HD) for modifiers +1 and above.
I had a python script run these formulas over the range of hit die sizes (from d6 to d12) and constitution modifier values (from -5 to +5) and got the following results:
You always gain a greater amount of Hit Points from the roll if your Hit Die size is smaller.
The x-axis if your Hit Die size. The y-axis is your Constitution modifier. The values are the gain that you get from this feat.
There is also the notable case of a level 20 Barbarian as their Constitution modifier can reach +7 on their d12 Hit Die. For them the feat increases the average by 1.75 Hit Points.
The generalizations
The feat benefits the a negative modifier more than its positive counterpart, but this is because the feat provides no increase for any dice if you have a +1 modifier:
If two modifiers are the same distance from +1 (like +3 and -1), the positive modifier will benefit from the feat equally, unless the modifier is +4 or above, in which case it will benefit from the feat more than the negative one.
The graphical representation
The x-axis is your modifier, the y-axis is the gain from the feat. The smaller hit dice have the higher benefits.
Purple is the d6, Green is the d8, Blue is the d10, and Red is the d12.