Question

1. Assume we are using the simple model for floating-point representation as given in this book (the representation uses a 14-bit format, 5 bits for the exponent with a bias of 15, a normalized mantissa of 8 bits, and a single sign bit for the number): a) Show how the computer would represent the numbers 100.0 and 0.25 using this floating-point format. b) Show how the computer would add the two floating-point numbers in part a by changing one of the numbers so they are both expressed using the same power of 2. c) Show how the computer would represent the sum in part b using the given floating point representation. What decimal value for the sum is the computer actually storing? Explain. 5x3

Answer 1

a)

The representation of 100.0 in the floating-point representation is computed as follows:

First convert the given number 100.0 in the binary form.

100₁₀ = 1100100₂

Therefore, the binary representation of 100.0 is 1100100.

Now, convert the binary number to the power of 2.

1100100.0 = 0.1100100 * 2⁷

Therefore, the value of the exponent is 7.

After the above process, convert the exponent into 15 bits excess. Add 15 in 7, 15 + 7 = 22.

22₂ = 10110₂

Hence, the 14-bit floating point representation of 100.0 is given as follows:

Sign bit = 0

Exponent (5 bits) = 10110

Significant bits of 8 = 11001000

0

1

0

1

1

0

1

1

0

0

1

0

0

0

The representation of 0.25 in the floating-point representation is computed as follows:

First convert the given number 0.25 in the binary form.

0.25₁₀ = 0.01₂

Therefore, the binary representation of 0.25 is 0.01.

Now, convert the binary number to the power of 2.

0.01 = 0.1 * 2^-1

Therefore, the value of the exponent is -1.

After the above process, convert the exponent into 15 bits excess. Add 15 in -1, 15 - 1 = 14.

14₂ = 01110₂

Hence, the 14-bit floating point representation of 0.25 is given as follows:

Sign bit = 0

Exponent (5 bits) = 01110

Significant bits of 8 = 10000000

0

0

1

1

1

0

1

0

0

0

0

0

0

0

b)

The addition of the floating-point numbers is computed as follows:

100.0 = .11001000 * 2⁷

0.25 = .1 * 2^-1

The 0.25 can also be represented as 0.000000001 * 2⁷

.11001000 × 2⁷

+ .000000001 × 2⁷

.110010001 × 2⁷

Thus, the addition of the two floating-point numbers is 110010001 * 2^7.

c)

The decimal representation of the obtained sum is computed as follows:

Since, the sign bit would be 0 because the sum obtained is also positive.

The value of the exponent is 7.

Convert the exponent to the 15-bit excess by adding 15 into 7. So, 15 + 7 = 22.

The binary representation of 22 is 10110.

The significant bits are 11001000.

Therefore, the 14-bit floating point representation is shown below:

0

1

0

1

1

0

1

1

0

0

1

0

0

0

.11000100*2⁷=1100100

Hence, the decimal representation of the sum obtained is 100.

thank you hope you understand easily............

if we like the answer plz vote up