Why Python cannot perform bit operations on floats?

Python can represent floats in binary but cannot use bit operations (&, |, ^, ~, >>, and <<) on them. Why?

Let’s take bitwise conjunction operation:

>>100.0 & 101.0 # raises a TypeError

>>100 & 101 # works fine outputting 100

Bitwise operators perform at the level of individual bits, treating each bit independently. To illustrate, if we assume a typical integer variable occupies 64 bits, you can imagine the bitwise operation as a 64-fold evaluation of the logical operator for each pair of bits of the arguments. This analogy is obviously imperfect, as in the real world all these 64 operations are performed simultaneously.

So, bitwise operators in Python can deal with integers but not floats.

Binary float representation in Python

Floating-point numbers in Python are represented using the IEEE 754 standard, which defines a binary representation for real numbers. The representation consists of three main components: the sign bit, the exponent, and the fraction (also known as the mantissa or significand). Here’s how it works:

1. Sign Bit: The leftmost bit (the most significant bit) represents the sign of the number. It is 0 for positive numbers and 1 for negative numbers.
2. Exponent: Following the sign bit, there is a group of bits that represent the exponent. This exponent indicates how many positions to shift the binary point (analogous to the decimal point) to the left or right to get the actual value. The exponent is biased, meaning that it is adjusted by a constant value to allow both positive and negative exponents to be represented. In the IEEE 754 standard, the exponent is usually in “excess notation.”
3. Fraction: After the exponent, there is the fraction part. This part represents the actual binary digits of the number. It is used to represent the significant digits of the number.

The format of the representation can vary depending on whether it’s single-precision (32 bits) or double-precision (64 bits). Double-precision is the most commonly used format for representing floating-point numbers in modern computers, and it provides greater precision compared to single-precision.

Why bitwise operators doesn’t work on floats

When you use bitwise operators like & (bitwise AND) on floating-point numbers, you are performing bitwise operations on the internal binary representations of these numbers. However, this typically doesn’t yield meaningful or expected results for several reasons:

1. Complex Representation: The internal representation of floating-point numbers is quite complex, including sign bits, exponents, and fractions. Applying bitwise operations directly to these representations doesn’t correspond to arithmetic operations on the actual numeric values.
2. Loss of Significance: Floating-point numbers are approximations of real numbers, and small rounding errors can occur during arithmetic operations. Bitwise operations don’t take these rounding errors into account, leading to unexpected results.
3. NaN and Infinity: Floating-point numbers can represent special values like NaN (Not-a-Number) and positive/negative infinity. Bitwise operations on these values can result in undefined or unexpected behavior.
4. Bit Patterns: Even if you could apply bitwise operations, the bit patterns of floating-point representations are not designed for direct manipulation like integers. Changing bits directly could lead to invalid or nonsensical values.

While bitwise operators like & are useful for integers, they are not suitable for performing meaningful operations on floating-point numbers due to the complex nature of floating-point representation, potential rounding errors, and the fact that the bit patterns aren’t intended for direct manipulation as with integers.

Break down of the Python’s evaluation of "100 & 101":

1. Binary Representation of 100 and 101: First, let’s convert the decimal numbers 100 and 101 into their binary representations:
   • 100 in binary is 1100100.
   • 101 in binary is 1100101.
2. Bitwise AND Operation: The & operator performs a bitwise AND operation on the individual bits of the binary representations of the operands. In this case, we are performing on 1100100 & 1100101. Here’s how the bitwise AND operation works for each pair of corresponding bits:
   • 1 & 1 results in 1.
   • 1 & 1 results in 1.
   • 0 & 0 results in 0.
   • 0 & 0 results in 0.
   • 1 & 1 results in 1.
   • 0 & 1 results in 0.
   • 0 & 0 results in 0.
3. Result: After performing the bitwise AND operation on each pair of corresponding bits, we get the binary result 1100100, which is the binary representation of the decimal number 100.