Recall that the reciprocal square root of a number x can be approximated using Newton-Raphson method, which iteratively improves an initial guess.

Investigate how the original fast inverse square root algorithm (for 32-bit floats) leverages bit manipulation and floating-point representation tricks—can these be adapted for 64-bit doubles?

Examine the IEEE 754 double-precision format to understand how the exponent and mantissa fields differ from 32-bit floats, and how this affects the bit-level operations needed for the reciprocal sqrt approximation.