Number Representation

How do we represent the Significand and Exponent?
(32-bit IEEE standard)

Exponent

• Have 8 bits available (256 values), need to represent positive and negative exponents
  (ideally   -127 – 128, not   -128 – 127).
• Could use sign & magnitude, one's complement, or two's complement
  BUT we want negative exponents to look smaller than positive exponents
  • So shift   -127 – 128 to 0 – 255 by adding 127.

    Let E be the actual exponent of the normalized binary form, (sig * 2E)
    and exp be the representation we will use,
        exp = E + 127
    

  • Example:
        101101 = 1.01101 * 25
    E = 5; exp = 132 ( =100001002 )
     
    

    31	30       23	22                   0
    0	10000100
    ↑	↑	↑
    sign	exp	significand
    	(8 bits)	(23 bits)

Significand

• Note that in normalized form, the bit to the left of the binary point is always 1!
  (except for 0 — special case).

  Why store a bit we already know when every bit counts? Make it implicit.

• Example:
      101101 = 1.01101 * 25
   
  

  31	30       23	22                        0
  0	10000100	0110100000000000...
  ↑	↑	↑
  sign	exp	significand
  	(8 bits)	(23 bits)

---

0 is a special case

• Cannot be represented as described above (has no normal form).
  Represent as all 0's.
   
  

  31	30       23	22                         0
  0	00000000	0000000000000000...
  ↑	↑	↑
  sign	exp	significand
  	(8 bits)	(23 bits)

• Advantages: Very quick tests for  <0,  0,  >0
  (S == 1,  all zeros,  S == 0 but not all zeros)
• Disadvantage: Lose ability to represent 2^-127 — small loss!

---

Examples

45.5₁₀ = 101101.12 = 1.011011 * 25
 

31	30       23	22                         0
0	10000100	0110110000000000...

---

1/8 = 0.0012 = 1.000 * 2-3
 

31	30       23	22                         0
0	01111100	0000000000000000...

---

Your turn: 34 1/4 = ?
 

31	30       23	22                         0

---

Trick Question: 1/10 = ?
 

31	30       23	22                         0

---

64-bit word

• Same as for 32-bit word, except both exponent and significand have more bits
  and exp = E + 1023.
   
  

  63	62        52	51                                       0
  S	Exp	Significand
  ↑	↑	↑
  sign	exp	significand
  	(11 bits)	(52 bits)