Why 564 bits? That’s 70.5 bytes.

Wait, what? I'm fairly certain that you can do a 128-bit by 128-bit division using a x64's 128-bit by 64-bit division instruction that gives you only 64-bit quotient and remainder. The trick is to pre-multiply both dividend and divisor by a large enough power of 2 so that the "partial" quotient and remainders that the hardware instruction would need to produce will fit into 64 bits. On the whole, IIRC you need either 1 or 2 division instructions, depending on how large the divisor is (if it's too small, you need two divisions).