I am going over the description of Sha-512. It is mentioned that the initial hash value consists of the sequence of 64-bit words that are obtained by taking the fractional part of the first eight primes. I am trying to replicate these values in Python, but I am not getting the same results. To include more digits, I am using the mpmath library.
from mpmath import *
mp.dps = 50
sqrt(2)
# mpf('1.4142135623730950488016887242096980785696718753769468')
mpf(0.4142135623730950488016887242096980785696718753769468 * 2 ** 64)
# mpf('7640891576956012544.0')
hex(7640891576956012544)
# '0x6a09e667f3bcc800'
However, the description indicates this value must be 6a09e667f3bcc908. As it can be seen, the result I get differs in the last three digits from what I should be getting according to the description. I was wondering why that is the case, and what is the correct approach.
I have come across a similar question, but adjusting it for 64-bit word would yield:
import math
hex(int(math.modf(math.sqrt(2))[0]*(1<<64)))
# '0x6a09e667f3bcd000'
which actually differs in the last four digits.
As a comment already explained, you're actually only using 53 bits in your calculations (native CPython float precision).
Here's an easy way to reproduce the result you're apparently after:
>>> import decimal
>>> x = decimal.getcontext().sqrt(2) - 1
>>> x
Decimal('0.414213562373095048801688724')
>>> hex(int(x * 2**64))
'0x6a09e667f3bcc908'
Nothing really magical about decimal. It just happens to use enough precision by default. You could certainly do the same with mpmath.
For example,
>>> import mpmath
>>> mpmath.mp.prec = 80
>>> hex(int(mpmath.frac( mpmath.sqrt(2) ) * 2**64))
'0x6a09e667f3bcc908'