Principle_of_Floating-point_Arithmatic

example of principle

x = 0x1.0p1; y = 0x1.4p2

  x + y == (0x1.0p(1 - 2)) + (0x1.4p2)
  x - y == (0x1.0p(1 - 2)) - (0x1.4p2)
  x * y == (0x1.0 * 0x1.4)p(1 + 2)
  x / y == (0x1.0 / 0x1.4)p(1 - 2)

steps description

addition:

• 1 check and process zero
• 2 equalize exponents
• 3 add signed significands
• 4 normalize and round result

subtraction:

• 1 check and process zero
• 2 flip sign of minuend
• 3 equalize exponents
• 4 add signed significands
• 5 normalize and round result

multiplication

• 1 check and process zero
• 2 get sum of exponents
• 3 multiply signed significands
• 4 normalize and round result

division

• 1 check and process zero
• 2 get difference of exponents
• 3 divide signed dignificands
• 4 normalize and round result

special numbers

  Not_a_Number == 0.0 / 0.0
  Positive_Infinity == 1.0 / 0.0
  Negative_Inginity == -1.0 / 0.0

rounding policy

1. round toward nearest
2. round toward positive_infinity
3. round toward negative_infinity
4. round toward zero