Solutions8
Computer Architecture
Tutorial 4b – Floating Point Numbers
1) Convert –31.3 to IEEE Single Precision format.
First convert to a binary number -31.3 = -11111.01001 1001 1001
Next Normalise
1.11110 1001 1001 1001 1001 1001 x 2
4
Significand field is 1111 0100 1100 1100 1100 110 (23 bits with 1. omitted)
Exponent field is 4+127 = 131 = 1000 0011
Number is -ve therefore Sign field is 1
Sign Exponent Significand
1 1000 0011 1111 0100 1100 1100 1100 110
2) Convert the IEEE Single Precision format hex value C154 0000 to decimal.
C154 0000 = 1100 0001 0101 0100 0000 0000 0000 00000
Sign Exponent Significand
1 1000 0010 1010 1000 0000 0000 0000 000
Exponent field = 1000 0010 = 130 => Exponent = 130 – 127 = 3
Significand field = 10101 Adding Hidden Bit => 1.10101
Therefore number is 1.10101 x 2
3
= 1101.01 = Decimal 13.25
Sign is 1 therefore number is -13.25
3) Carry out the operation 31.3 + 13.25 in IEEE single precision arithmetic
Number Sign Exponent Significand
31.3 0 1000 0011 1111 0100 1100 1100 1100 110
13.25 0 1000 0010 1010 1000 0000 0000 0000 000
Significand of Larger Number = 1.1111 0100 1100 1100 1100 110
Significand of Smaller Number= 1.1010 1000 0000 0000 0000 000
Exponents differ by 1. Therefore shift binary point of Smaller Number 1 place.
Significand of Larger Number = 1.1111 0100 1100 1100 1100 1100
Significand of Smaller Number= 0.1101 0100 0000 0000 0000 0000
Significand of Sum = 10.1100 1000 1100 1100 1100 1100
Sum = 10.1100 1000 1100 1100 1100 1100 x 2
4
Normalise 1.01100 1000 1100 1100 1100 1100 x 2
5
Sign Exponent Significand
0 1000 0100 0110 0100 0110 0110 0110 011
4)
Bits Binary value
or special value
Decimal value
or special value
0 00 00 0 0
0 00 01 0.01 0.25
0 00 10 0.10 0.50
0 00 11 0.11 0.75
0 01 00 1.00 1
0 01 01 1.01 1.25
0 01 10 1.10 1.5
0 01 11 1.11 1.75
0 10 00 10.0 2
0 10 01 10.1 2.5
0 10 10 11.0 3
0 10 11 11.1 3.5
0 11 00
0 11 01 NaN NaN
0 11 10 NaN NaN
0 11 11 NaN NaN