GATE CSE 2021 Set 1 | Question: 24

Question

Consider the following representation of a number in $\text{IEEE 754}$ single-precision floating point format with a bias of $127$.$$S: 1\quad\quad  E:\; 10000001\quad\quad F:\;11110000000000000000000$$ Here $S, \;E$ and $F$ denote the sign, exponent, and fraction components of the floating point representation.

The decimal value corresponding to the above representation (rounded to $2$ decimal places) is ____________.

Answer 1

-7.75 is the correct answer.

Here, Sign bit = 1 → number is negative.

Exponent bits $= 10000001 = 129_{10} → E = 129-127 = 2$ as IEEE-754 single precision format uses $127$ as the exponent bias.

Mantissa bits $= 11110000000000000000000$

Number $= – 1.111100\ldots 00 \times 2^{2} = (-111.11)_2$ 

$\therefore$ Number $= (-7.75)_{10}$.

Reference: https://steve.hollasch.net/cgindex/coding/ieeefloat.html

Comments

Helpful

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Those who not getting how $(-111.11)_2=(-7.75)_{10}$
$111.11=1\times 2^2+1\times 2+1\times 2^0+1\times \frac{1}{2}+1\times \frac{1}{4}=4+2+1+0.5+0.25=7.75$

Answer 2

ANS IS -7.75. 

given S = 1 , E = 10000001 M =11110000000000000000000 , BY CALCULATING AS BELOW ANS WILL BE -7.75

Answer 3

 

S (1)

E(8)

F (23)

 

$(-1)^s * (1.M) * 2^{E-127}$ is used to convert IEEE 754 single precision floating point when $1 \leq E \leq 254$  and M any number 

 

Hear,

E= 10000001   Decimal equivalent 129

F = 111100000000000000000000

 

$(-1)^1 * (1.111100000000000000000000) * 2^{129-127} = -(1.1111 * 2^2)   $

$-(111.11)= -7.75$

Ans : -7.75

 

 

Comments

E is 129

E = 129-127 = 2

(1.1111)^2

111.11 =7.75

Sing bit 1 (NEG) so final ans - 7.75

Answer 4

Ans: $-7.75$

S(Sign) is $1$. So number is negative.

E(Biased exponent) in decimal is $129$. Bias is $127$. So real exponent is $129 - 127 = 2.$

F(Fraction) is normalized and has an implicit $1$ before decimal point.

$\therefore$ Number is $-1.11110000000000000000000 * 2^2 = -111.11 = -7.75$