Real Number Representation in Binary

Fixed Point vs. Floating Point

• Fixed Point Representation:

  • Fixes the binary point's position within a register.
  • Limits the range and precision of real numbers.

• Floating Point Representation:

  • The most common way to encode real numbers in binary.
  • Uses binary scientific notation:
    • Mantissa: The base number.
    • Exponent: Determines the range.
  • Uses two's complement for negative values.
  • Historically, format varied depending on the computer.

IEEE 754 Standard

• Established in 1985 to standardize floating point representation.

• 32-bit Single Precision:

  • Structure: 1 sign bit, 8-bit exponent, 23-bit mantissa.
  • Effective precision of 24 bits.

• 64-bit Double Precision:

  • Structure: 1 sign bit, 11-bit exponent, 52-bit mantissa.

Conversion Example: Converting to IEEE 754

• Example: Convert 19.59375 to IEEE 754 Single Precision:
  1. Determine Sign Bit: Positive number, so sign bit = 0.
  2. Convert to Binary:
     • Use repeated division for whole numbers.
     • Use repeated multiplication for fractional part.
  3. Normalize Binary: Move the binary point to get the exponent.
  4. Calculate Exponent Bias: Add 127.
  5. Drop Leading 1 from Mantissa.
  6. Add Zero Fillers at the Back of Mantissa.

Biased Exponent Encoding

• Allows for more positive or negative values.
• Example: A 4-bit exponent can represent 16 values.
• Bias for 8-bit exponent (single precision): 127.
• Bias for 11-bit exponent (double precision): 1023.

Floating Point Arithmetic

• Rounding:
  • Mantissa fit into storage.
  • Computers may round to the nearest value.

Special Cases in IEEE 754

• Infinity: Exponent all ones, mantissa all zeros.
• NaN (Not a Number): Exponent all ones, mantissa not all zeros.
• Zero: Exponent all zeros, mantissa all zeros.
• Subnormal Numbers: Exponent all zeros, mantissa not all zeros.

Summary

• Floating point representation enhances range and precision compared to fixed point.
• IEEE 754 standardizes the format, simplifying calculations and comparisons.