Mixing Integers and Floats in Python

 a = (1 << 52)

print((a + 0.5) == a)

This Python code explores the behavior of floating-point numbers when precision is stretched to the limits of the IEEE 754 double-precision floating-point standard. Let me break it down:

Code Explanation:

1. a = (1 << 52):

   • 1 << 52 is a bitwise left shift operation. It shifts the binary representation of 1 to the left by 52 bits, effectively calculating $2^{52}$.
   • So, a will hold the value $2^{52} = 4,503,599,627,370,496$.
2. print((a + 0.5) == a):
   • This checks whether adding 0.5 to a results in the same value as a when using floating-point arithmetic.
   • Floating-point numbers in Python are represented using the IEEE 754 double-precision format, which has a 52-bit significand (or mantissa) for storing precision.
   • At $2^{52}$, the smallest representable change (called the machine epsilon) in floating-point arithmetic is $1.0$. This means any value smaller than 1.0 added to $2^{52}$ is effectively ignored because it cannot be represented precisely.

3. What happens with (a + 0.5)?:

   • Since $0.5$ is less than the floating-point precision at $2^{52}$ (which is $1.0$), adding $0.5$ to $a$ does not change the value of a in floating-point arithmetic.
   • Therefore, (a + 0.5) is rounded back to a.

4. Result:

   • The expression (a + 0.5) == a evaluates to True.

Key Insight:

• Floating-point arithmetic loses precision for very large numbers. At $2^{52}$, $0.5$ is too small to make a difference in the floating-point representation.