It seems like there might be a way to speed up mining by a tiny bit. The first 8 0's of the final block hash can be determined by looking at a value calculated in round 61 of the final SHA256 hash (instead of waiting for all 64 rounds and the final step to complete). The second set of 8 0's can be determined by looking at a value calculated in round 62, and the third set from round 63. Therefore, the algorithm should be able to short-circuit and skip some rounds of calculations by checking these values at their respective rounds. Given that there are 128 rounds (http://www.righto.com/2014/09/mining-bitcoin-with-pencil-and-paper.html), there should be ~2.3% reduction in calculations per hash (at most).

Also, when the mining difficulty reaches 33 0's, another value from round 61 can be examined.

I'm assuming the gain is negligible when considering the cost/difficulty in implementing it. Either way I'm curious to hear some feedback.

(this is helpful for understanding: https://bitcointalk.org/index.php?topic=809430.0)