29 Jan, 2025
Floating-Point Representation
Notes

Floating-Point Representation

Floating-point representation is a method used in computing to represent real numbers in a way that can accommodate a wide range of values, from very small to very large, with a finite number of digits. This representation is crucial for scientific computations, engineering applications, and any domain where precision and range are important.

Key Components of Floating-Point Representation

  1. Sign Bit (S):
    • The sign bit determines whether the number is positive or negative.
    • Typically, 0 represents a positive number, and 1 represents a negative number.
  2. Exponent (E):
    • The exponent is a biased integer that represents the power to which the base (usually 2) is raised.
    • The bias is used to allow the representation of both positive and negative exponents.
    • For example, in the IEEE 754 standard for 32-bit floating-point numbers, the exponent is 8 bits long with a bias of 127.
  3. Mantissa (M) or Significand:
    • The mantissa represents the significant digits of the number.
    • It is a fractional part that, when combined with the exponent, gives the precise value of the number.
    • In normalized form, the mantissa is typically represented with an implicit leading 1 (i.e., 1.xxxx...), which is not stored explicitly.

IEEE 754 Standard

The IEEE 754 standard is the most widely used standard for floating-point representation. It defines several formats, including:

  1. Single-Precision (32-bit):
    • 1 bit for the sign.
    • 8 bits for the exponent.
    • 23 bits for the mantissa.
    • The bias for the exponent is 127.
  2. Double-Precision (64-bit):
    • 1 bit for the sign.
    • 11 bits for the exponent.
    • 52 bits for the mantissa.
    • The bias for the exponent is 1023.

Representation of a Floating-Point Number

A floating-point number NN can be represented as:

N=(−1)S×M×2(E−bias)

Where:

  • S is the sign bit.
  • M is the mantissa (including the implicit leading 1 in normalized form).
  • E is the exponent.
  • The bias is a constant that depends on the precision (e.g., 127 for single-precision).

Special Values

The IEEE 754 standard also defines special values:

  1. Zero:
    • Represented by an exponent and mantissa of all zeros.
    • Can be positive or negative depending on the sign bit.
  2. Infinity:
    • Represented by an exponent of all ones and a mantissa of all zeros.
    • Can be positive or negative depending on the sign bit.
  3. NaN (Not a Number):
    • Represented by an exponent of all ones and a non-zero mantissa.
    • Used to represent undefined or unrepresentable results, such as the result of dividing zero by zero.