Question

IEEE 754 format of 32-bit floating-point is as follows.

1

8 (bits)

23 (bits)

1. What’s stored in each region?
2. What’s the bias value and how to get it?
3. For decimal fraction: – 0.625, please represent it as given format (Note: you must show the specific procedure/stepsin order to get full credits. If you only present final result directly, you will only get half of the credits even if it is correct.).  

Answer 1

Solution:

Given,

=>IEEE 754 format of 32 bit floating point is given.

1 bit

8 bits

23 bits

(a)

Explanation:

IEEE 754 32 bit frame format:

Sign(S)

Exponent(E)

Mantissa(M)

   1 bit                          8 bits                                          23 bits

=>Hence 1 bit is for sign(S), 8 bits for biased exponent and 23 bits for mantissa part.

(b)

Explanation:

Calculating value of bias:

=>Bias(k) = 2^(exponent bits - 1) - 1

=>Bias(k) = 2^(8 - 1) - 1

=>Bias(k) = 2^7 - 1

=>Bias(k) = 127

=>Hence bias = 127

=>Normalized form = (1-)^S*(1.M)*2^(E-127)

(c)

Given,

=>Decimal value = -0.625

Explanation:

Converting decimal value to binary:

=>-0.625 = (-0.101)₂

Converting binary value to normalized form:

=>(-0.101)₂ = (-1)^1*(1.01000000000000000000000)*2^-1

=>Hence S = 1 as number is negative

=>E - 127 = -1

=>E = 126 in decimal

=>E = 11111110 in binary

=>M = 01000000000000000000000

=>Hence number in 32 bits = 1 11111110 01000000000000000000000

I have explained each and every part with the help of statements attached to the answer above.