5. Floating Point

5.9. Denormal Numbers

• Exponent = 000…0 → hidden bit is 0

  $$ X = (-1)^S*(0+Fraction)*2^{-Bias} $$

• Smaller than normal numbers

  • allow for gradual underflow, with diminishing precision

• Denormal with fraction = 000…0

  • 0도 $\pm$가 존재

  $$ X = (-1)^S*(0+0)*2^{-Bias} = \pm0.0 $$

5.10. Infinities and NaNs

• Exponent = 111…1, Fraction = 000..0
  • $\pm\infty$
  • Can be used in subsequent calculations, avoiding need for overflow check
• Exponent - 111…1, Fraction ≠ 000…0
  • Not-a-Number(NaN)
  • Indicates illegal or undefined result
    • e.g., 0.0/0.0 or $\infty / \infty$ or $\infty - \infty$
  • Can be used in subsequent calculations

5.11. Floating-Point Addition(1)

Consider a 4-digit decimal example

$9.99910^1 + 1.61010^{-1}$

1. Align decimal point

   • Shift number with smaller exponent
   • $9.99910^1+0.01610^1$

2. Add significands

   $9.99910^1 + 0.01610^1 = 10.015*10^1$

3. Normalize result & check for over/underflow

   $1.0015*10^2$

4. Round and renormalize if necessary

   $1.002*10^2$

5.12. Floating-Point Addition(2)

Now consider a 4-digit binary example

$1.000_22^{-1}+-1.110_22^{-2}$

1. Align binary points

   • Shift number with smaller exponent

   $1.000_22^{-1}+-0.111_2-2^{-1}$$1.002_22^{-1}+-0.111_2-2^{-1}$

2. Add significands

   $1.000_22^{-1}+-0.111_2-2^{-1} = 0.001_2*2^{-1}$

3. Normalize result & check for over/underflow

   $1.000_2*2^{-4}$, with no over/underflow