IEEE 754 Computer Arithmetic: Single Precision and Exceptions, Slides of Computer Aided Design (CAD)

Download IEEE 754 Computer Arithmetic: Single Precision and Exceptions and more Slides Computer Aided Design (CAD) in PDF only on Docsity!

Computer Arithmetic

Integer and Fixed Point

Single Precision IEEE 754 Example (1)

Ex : 3.8125 10 = +1.11101 * 2^1

• Sign is 0 (bit position 31  MSB)
• Exponent is 1 0 0 0 0 0 0 0 (underlined above)
  • 0111 1111 + 0000 0001 = 1000 0000
• Mantissa is 1.1110100…0 (the leading 1 is assumed in the representation above)
• That’s +1.11101 * 2(128-127)^ = 1.90625 10 * 2^1 = 3.8125 10

Exceptions to IEEE 754

• These rules apply when
  • Exponent  00
  • Exponent  FF (= 255 10 )
• Special rules apply for these situations
• These special rules provide for
  • NaN (Not a Number)
  • +/- Inf (infinity)
  • +/- 0 (yes, two zeros!)
  • “unnormalized” numbers  allows very very very small values including “machine epsilon” (the smallest positive number allowed)

Exceptions to IEEE 754

• Special cases: (E is Exponent, F is Mantissa)
• If E=255 and F is nonzero, then Value=NaN ("Not a Number")
• If E=255 and F is zero and S is 1, then Value=-Infinity
• If E=255 and F is zero and S is 0, then Value=Infinity
• If E=0 and F is nonzero, then Value=(-1)^S * 2^(-126) * (0.F) These are "unnormalized" values.
• If E=0 and F is zero and S is 1, then Value=-
• If E=0 and F is zero and S is 0, then Value=