Recently, I have analyzed on Sharpshooter Demon Hunter's Sharpshooter. And I have gathered some data and produced a few charts as a reference. From the following charts, perhaps, you can easily see the damage gain of Sharpshooter.

At first, the damage gain of Sharpshooter relates to attack speed. You can find out the APS chart which is closest to your attacks number. Then find the corresponding critical hit damage and the cross coordinates of critical hit chance. If your coordinate position locates below the blue line, then it shows that the damage gain of Sharpshooter is more than 5%; if under the Green Line, it is 10%, and so on.

The following is how to calculate:
The damage gain of opening Sharpshooter = (1 + the average critical hit chance * critical hit damage after opening A) / (1 + critical hit chance * critical hit damage).

 
Because critical hit chance and critical hit damage have fixed values, the key is how to calculate the average critical hit chance.

Well ... I work out it...
 
First of all, we get the cumulative duration of the average critical hit chance.  Plainly, it is how long the average critical hit chance will be reset.
 
Based on the concept of expected value, we can list the following formula:
R is the original critical-hit chance; A IS representative of APS, T is the required consuming time when critical hit chance accumulates to 100%.

The cumulative duration of average critical hit chance = 11 (sec) * {the 0 ~ 1 second probability of critical-hit occurrence}
                                                + 2 (sec) * {the 1 ~ 2 seconds probability of critical-hit occurrence } * { the 0~1 second improbability of critical-hit occurrence }
                                                + 3 (seconds) * {the 2 ~ 3 seconds probability of critical-hit occurrence} * {the 0 ~ 2 seconds improbability of critical-hit occurrence}
                                                + ..................
                                                + T (sec)
                                             = 1 * [1 - (1 - R) ^ A]
                                                + 2 * [1 - (1 - (R +3%)) ^ A] * [1-R] ^ A
                                                + 3 * [1 - (1 - (R +6%)) ^ A] * [1 - (R +3%)] ^ A * [1-R] ^ A
                                                + ...
                                                + T * [1 - (1 - (R + T * 3%)) ^ A] * [1 - (R + T * 3%)] ^ A *..... *[1-R] ^

Please note that the approximation will make the average cumulative duration of critical-hit chance to be slightly underestimated. Take the first line for example. You have critical- hit incidents to happen within 0~ 1 second. If critical hit chance occurs at 0 seconds, then critical hit chance will be reset in 1 second. Then accumulate duration of critical hit chance is 1.  If critical hit occurs in 0.9 seconds, then critical hit chance will be reset at 1.9 seconds. Then continued accumulation time is actually 1.9. However, there is not great impact.

 
We can make the average critical hit chance approximately equal to (1 /the average cumulative duration of critical hit). In other simple words, if the average cumulative duration of critical hit is X seconds, it means  critical hit will be produced once in average X seconds,  that is, the average per second will produce 1 / X times critical hit.

 
Then, the above charts are obtained according to those formulas which are put into different APS, critical- hit chance and critical damage.