Can I recover lost data after formatting Mac HD?

Unfortunately, it is currently impossible (rather: computationally unfeasible) to recreate the private keys from the public keys (otherwise they wouldn't be "private", and public keys wouldn't be, as they are, safely discloseable to all and sundry). The passphrase is just an encryption of the private keys, and while required for their recovery, it is not per se sufficient - you still need the encrypted private keys.

There is, perhaps, some small hope of recovering said encrypted private keys from the formatted disk, since formatting is not the same thing as erasing. The private keys are small and their format is distinctive, so if they have not been overwritten by the OS reinstalled files, it is possible they're not only still lying somewhere in the free space of the disk, but they're in a single, recognizable and thus recoverable fragment.

The disk must be set off line as soon as possible to prevent further writes to increase the risk of overwriting the data, and the free area examined with forensic tools by an expert or at least someone familiar with the mechanics of this undertaking. The operation can take several hours depending on the hard disk size.

If the disk has been already filled with data, or worse, a "secure erase", "secure format", "diagnostic rewrite", or similar operation has been performed prior to the reinstallation, then recovery chances are nil - that's literally the point of a "secure erase". Sorry.