CMU Computer Systems: Floating Points

Fractional binary numbers

IEEE Floating Point

• IEEE Standard 754

  • Sign + Exp + Frac

• Driven by numerical concerns

Precision options

• Single precision: 32bits: 1+8+23
• Double precision: 64bits: 1+11+52

Normalized Values

• Exp not 000…00 or 111…11
• Exponent coded as a biased value: E = Exp-Bias
• Significand coded with implied leading 1: 1.xxxx…x

Denormalized Values

• Exp is 000…0
• Exponent value: E = 1-Bias
• Significand coded with implied leading 0: 0.xxx…x

Special Values

• Exp all 1, and frac is all 0, infinity number
• Exp all 1, and frac not 0, NaN, not a number

Special Properties of the IEEE Encoding

• FP Zero Same as Integer Zero
• Can Use Unsigned Integer Comparison

Rounding

• Towards zero
• Round down
• Round up

Nearest Even (default)

• Greater than half round up

  • Less than half round down
  • And half round near the even

• FP Multiplication

  • Exact Result
  • Fixing
  • Implementation

• FP Addition

  • Exact Result
  • Fixing

• Multiplication and Addition are commutative and not associative

Summary

• IEEE Floating Point has clear mathematical properties
• Represents numbers of form $M \times 2^E$
• One can reason about operations independent of implementation
• Not the same as real arithmetic