Bitwise and Precision in Python

a = (1 << 52)

print((a + 0.5) == a)

Code Explanation:

a = (1 << 52)
print((a + 0.5) == a)

1. 1 << 52:
   • The << operator is a bitwise left shift.
   • 1 << 52 shifts the binary representation of 1 to the left by 52 places, resulting in $2^{52}$.
   • So, a = 1 << 52 sets a to $2^{52}$, which is 4,503,599,627,370,496.
2. a + 0.5:
   • Adds 0.5 to the value of a. In this case, $a + 0.5 = 4,503,599,627,370,496.5$.

3. Equality Check (==):

• The expression (a + 0.5) == a compares whether $a + 0.5$ is equal to $a$.

Why does the result evaluate to True?

This happens because of the limitations of floating-point precision in Python:

• Python uses 64-bit floating-point numbers (IEEE 754 standard).
• A 64-bit floating-point number can precisely represent integers up to $2^{53}$ (inclusive), but not fractional values beyond this precision.
• $2^{52} = 4,503,599,627,370,496$ is close to the upper limit of this precision. When adding 0.5 to $2^{52}$, the fractional part (0.5) is effectively rounded off due to the lack of precision.
• As a result, $a + 0.5$ is rounded back to $a$, making (a + 0.5) == a evaluate to True.