The exploration cost bonus decreases the amount of tar necessary to fully explore a crypt. Crypt exploration effectiveness, on the other hand, increases the exploration progress within every march. Chance to double strike a crypt grants the probability of exploring a crypt another time within one march without spending any extra tar.
The base maximum allowed march time to a crypt is limited by 20 minutes. Technologies and hero talents affecting this parameter allow you to increase this time. Note that Ayrin avatar takes off all restrictions immediately.
In order to explore faster, you can take more tar to a march than its base quantity. This also grants you a 10% bonus to exploration effectiveness. Note that in this case the game first divides the amount of tar taken by the base one for the crypt being explored and adds the bonus to every “strike” performed, and only then applies the exploration effectiveness bonus granted by technologies, talents, titles etc.!