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The result of sha256(s) that miners attempt is a 256 bits long number (64-digits hexadecimal number).

I understand that this 256bits number should be lower than the target threshold, the target threshold being itself a 256 bits long number with a number of leading zeros.

Target threshold is encoded with 4 bytes in the block's headed, as 0xEEMMMMMM in hex notation. EE being the 1byte exponent and MMMMMM being the 3bytes mantissa.

How exactly are these exponent and mantissa translated to the actual 256-bits long target threshold?

1I like the explaination in the edit. – croraf – 2017-11-07T09:05:31.837