Crystal Shield (perk)/Proof of the simplification

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Regarding Crystal Shield

We start from the main equation:

The second summation within the brackets in the numerator can be split up.

Split up the damage reduction component.

The first term of the numerator is equal to the denominator.

If we want to know the damage reduction over a large number of hits, we could use the statistical average of the received damage, this means becomes a constant value . We also drop the floor function in the numerator and add a value to correct for the error.

The second summation in the numerator has no terms that change on increasing , the summation can thus be simplified to a product of the range of the index and the summands.

Place terms not dependent on outside of the summations.

Drop common term in numerator and denominator.

To simplify even further we need to find the infinite sum equivalents of the remaining summations. Since (p = perk proc chance), the following is true[math 1]

Take the derivative of (2):

The second term in (3) is equal to (2):

Combine (2) and (3) with diffent constants:

If we take and in (2) we get:

For , and in (4), we get:

The summations (5) and (6) occur in (1) and can be substituted:

Where . This represents the rounding error introduced by dropping the floor function. , so for we have <4% error for ignoring and can therefor safely be ignored for most practical situations.

References