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:
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 .
- So, a will hold the value .
- 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 , the smallest representable change (called the machine epsilon) in floating-point arithmetic is . This means any value smaller than 1.0 added to is effectively ignored because it cannot be represented precisely.
What happens with (a + 0.5)?:
- Since is less than the floating-point precision at (which is ), adding to does not change the value of a in floating-point arithmetic.
- Therefore, (a + 0.5) is rounded back to a.
Result:
- The expression (a + 0.5) == a evaluates to True.
Key Insight:
- Floating-point arithmetic loses precision for very large numbers. At , is too small to make a difference in the floating-point representation.