The exploration cost bonus decreases the amount of Tar that is necessary to fully explore a Crypt.

 

Crypt exploration efficiency, on the other hand, enhances the exploration progress made by each march.

 

Some bonuses increase your chance to "strike" a Crypt twice. The chance to double-explore a Crypt grants a probability that a "strike" will explore twice as much of a Crypt without spending any extra Tar. 

In order to explore a Crypt faster, you can take more Tar on a march than the base amount required to explore the relevant Crypt. This also grants you a 10% bonus to exploration efficiency.

 

Please note that in this case, the game first divides the amount of Tar taken by the base amount required for exploring the Crypt. It then adds the 10% bonus to every "strike" performed. Only after that does the game apply the exploration efficiency bonuses granted by technologies, talents, titles, etc.

 

To understand better how exploration bonuses are calculated for Crypts, check out the following video guide: