IEEE Standard for Floating Point Numbers

Introduction

• Discussion about IEEE standard for floating point numbers
• Previous video: normalization of binary numbers, floating point representation, memory storage.
• Floating point numbers have:
  • 1 bit for sign
  • Few bits for exponent
  • Remaining bits for mantissa
• IEEE 754 standard: defines storage of floating point numbers.

IEEE 754 Formats

• Half Precision: 16 bits
• Single Precision: 32 bits
• Double Precision: 64 bits
• Quad Precision: 128 bits
• Octuple Precision: 256 bits
• Commonly used formats: Single and Double Precision

Single Precision Format

• 32-bit Structure:
  • 1 bit: Sign
  • 8 bits: Exponent
  • 23 bits: Mantissa
• Normalization:
  • Binary number is normalized to store only the fractional part in mantissa.

Exponent Storage

• 8-bit Exponent: Represents unsigned integers 0-255
• Exponent can be positive or negative:
  • Stored using bias representation
  • Bias = 2^(n-1) - 1 (for n = number of bits)
  • Example: 8 bits, bias = 127
• Range after subtracting bias: -127 to +128
• Bias representation ensures continuity in numbers from negative to positive.

Special Exponent Values

• All zeros and all ones reserved for special purposes.
• Available range: -126 to +127
• Continuity and ease of comparison via bias representation.

Comparing Floating Point Numbers

• Steps:
  1. Compare sign bits
  2. Compare exponents
  3. Compare mantissas
• Bias representation aids comparison due to continuity in number ordering.

Example Calculations

• Converting IEEE 754 format to decimal:
  • Use sign bit, exponent value (after bias subtraction), and mantissa
  • Normalized binary numbers converted to true binary for final decimal value
• Example conversions illustrate process.

Decimal to IEEE 754 Format

• Conversion of decimal to binary, normalize, adjust exponent, and store.
• Example: Converting 12.625 into IEEE format.

Range and Precision

• Largest and smallest numbers in single precision format:
  • Max exponent: 127
  • Min exponent: -126
• Precision limited by stored mantissa bits.
• Fixed vs floating point representation: floating point covers greater range but less precision.

Double Precision Format

• 64-bit Structure:
  • 1 bit: Sign
  • 11 bits: Exponent
  • 52 bits: Mantissa
• Bias: 1023
• Enhanced range and precision compared to single precision.

Conclusion

• IEEE 754 standard helps store floating point numbers with defined structure.
• Double precision offers greater precision than single.
• Special cases for all-zero and all-one exponents to be explored in future content.