Question

Convert 1101.11011101 x 2²³ to IEEE Standard 754 for single-precision floating-point binary format.

Convert the IEEE Standard 754 number

11001010100011010101000000000000 to its decimal equivalent.

Answer 1

1)
1101.11011101 * 2^23
= 1.10111011101 * 2^26

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+26=153) => 10011001
   Divide 153 successively by 2 until the quotient is 0
      > 153/2 = 76, remainder is 1
      > 76/2 = 38, remainder is 0
      > 38/2 = 19, remainder is 0
      > 19/2 = 9, remainder is 1
      > 9/2 = 4, remainder is 1
      > 4/2 = 2, remainder is 0
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10011001
   So, 153 of decimal is 10011001 in binary
frac/significant bits are 10111011101000000000000

Answer: 0 10011001 10111011101000000000000

2)
1 10010101 00011010101000000000000
sign bit is 1(-ve)
exp bits are 10010101
Converting 10010101 to decimal
   10010101
   => 1x2^7+0x2^6+0x2^5+1x2^4+0x2^3+1x2^2+0x2^1+1x2^0
   => 1x128+0x64+0x32+1x16+0x8+1x4+0x2+1x1
   => 128+0+0+16+0+4+0+1
   => 149
in decimal it is 149
so, exponent/bias is 149-127 = 22
frac bits are 00011010101

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.00011010101 * 2^22
1.00011010101 in decimal is 1.10400390625
   => 1.00011010101
   => 1x2^0+0x2^-1+0x2^-2+0x2^-3+1x2^-4+1x2^-5+0x2^-6+1x2^-7+0x2^-8+1x2^-9+0x2^-10+1x2^-11
   => 1x1+0x0.5+0x0.25+0x0.125+1x0.0625+1x0.03125+0x0.015625+1x0.0078125+0x0.00390625+1x0.001953125+0x0.0009765625+1x0.00048828125
   => 1+0.0+0.0+0.0+0.0625+0.03125+0.0+0.0078125+0.0+0.001953125+0.0+0.00048828125
   => 1.10400390625
so, 1.10400390625 * 2^22 in decimal is 4630528.0
so, 11001010100011010101000000000000 in IEEE-754 single precision format is -4630528.0
Answer: -4630528.0