Question

Assignment (show how you calculate these): 1) According to the IEEE 754 protocols, compute the following values: i) The greatest, positive, normal, floating-point number has a binary representation of Mantissa Sign Exponent 0 1 1 1 1 1 1 1 1 Derive its decimal value? ii) The smallest, positive, normal, floating-point number has a binary representation of Sign Exponent O O O Mantissa Jo Jo O 1 O O O O O Derive its decimal value? iii) The greatest, positive, subnormal, floating-point number has a binary representation of Sign Exponent O O O O Mantissa 1 1 1 1 1 1 1 Derive its decimal value? iv) The smallest, positive, subnormal, floating-point number that is greater than 0 has a binary representation of recall that 0,0 = 0 0000 00000002) Sign Exponent O O O Mantissa O O O O O O O O 1 Derive its decimal value?

Answer 1

i)
0 1110 1111111
sign bit is 0(+ve)
exp bits are 1110
   => 1110
   => 1x2^3+1x2^2+1x2^1+0x2^0
   => 1x8+1x4+1x2+0x1
   => 8+4+2+0
   => 14
in decimal it is 14
so, exponent/bias is 14-7 = 7
frac bits are 1111111

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.1111111 * 2^7
1.1111111 in decimal is 1.9921875
   => 1.1111111
   => 1x2^0+1x2^-1+1x2^-2+1x2^-3+1x2^-4+1x2^-5+1x2^-6+1x2^-7
   => 1x1+1x0.5+1x0.25+1x0.125+1x0.0625+1x0.03125+1x0.015625+1x0.0078125
   => 1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125
   => 1.9921875
so, 1.9921875 * 2^7 in decimal is 255.0
so, 011101111111 in IEEE-754 format is 255.0
Answer: 255.0

ii)
0 0001 0000000
sign bit is 0(+ve)
exp bits are 0001
   => 0001
   => 0x2^3+0x2^2+0x2^1+1x2^0
   => 0x8+0x4+0x2+1x1
   => 0+0+0+1
   => 1
in decimal it is 1
so, exponent/bias is 1-7 = -6
frac bits are

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1. * 2^-6
1. in decimal is 1
   => 1.
   => 1x2^0
   => 1x1
   => 1
   => 1
so, 1 * 2^-6 in decimal is 0.015625
so, 000010000000 in IEEE-754 format is 0.015625
Answer: 0.015625

iii)
0 0000 1111111
sign bit is 0(+ve)
exp bits are 0000
   => 0000
   => 0x2^3+0x2^2+0x2^1+0x2^0
   => 0x8+0x4+0x2+0x1
   => 0+0+0+0
   => 0
in decimal it is 0
so, exponent/bias is 0-7 = -7
frac bits are 1111111

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.1111111 * 2^-7
1.1111111 in decimal is 1.9921875
   => 1.1111111
   => 1x2^0+1x2^-1+1x2^-2+1x2^-3+1x2^-4+1x2^-5+1x2^-6+1x2^-7
   => 1x1+1x0.5+1x0.25+1x0.125+1x0.0625+1x0.03125+1x0.015625+1x0.0078125
   => 1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125
   => 1.9921875
so, 1.9921875 * 2^-7 in decimal is 0.01556396484375
so, 000001111111 in IEEE-754 format is 0.01556396484375
Answer: 0.01556396484375

iv)
0 0000 0000001
sign bit is 0(+ve)
exp bits are 0000
   => 0000
   => 0x2^3+0x2^2+0x2^1+0x2^0
   => 0x8+0x4+0x2+0x1
   => 0+0+0+0
   => 0
in decimal it is 0
so, exponent/bias is 0-7 = -7
frac bits are 0000001

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.0000001 * 2^-7
1.0000001 in decimal is 1.0078125
   => 1.0000001
   => 1x2^0+0x2^-1+0x2^-2+0x2^-3+0x2^-4+0x2^-5+0x2^-6+1x2^-7
   => 1x1+0x0.5+0x0.25+0x0.125+0x0.0625+0x0.03125+0x0.015625+1x0.0078125
   => 1+0.0+0.0+0.0+0.0+0.0+0.0+0.0078125
   => 1.0078125
so, 1.0078125 * 2^-7 in decimal is 0.00787353515625
so, 000000000001 in IEEE-754 format is 0.00787353515625
Answer: 0.00787353515625