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What is the decimal value of the floating-point number $C1D00000$ (hexadecimal notation)? (Assume $32$-bit, single precision floating point $\text{IEEE}$ representation)

  1. $28$
  2. $-15$
  3. $-26$
  4. $-28$
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C1D00000 = 1100 0001 1101 0000……0000

In IEEE representation for single precision floating point numbers, first bit is for sign, next 8 bits for exponent and last 23 bits for mantissa which excludes the implicit "1" before ".", unless the number is not normalized (happens only when exponent is 0 and mantissa is non zero).

First bit = sign bit = 1, so number is negative

exponent $= 131 - 127 = 4$ (127 is the bias in IEEE representation). 

significant $= 1. 101\underbrace{000\dots 0}_{\text{20 0's}} \\=(1)2^0 + (1)2^{-1} + (0)2^{-2 }+ (1)2^{-3} \\= 1 + 1/2 + 0 + 1/8 = 13/8$

hence we finally get decimal value

$= \text{sign} \times  2^{\text{exponent}} \times \text{significant} = -26$.

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