Question

7. Perform the sum of the following two decimal numbers: [10 Marks]
a. -1203.232 and -253.1398
b. -2037.4235 and 653.25
All the operation should be performed in IEEE 754 single precision.

Answer 1

Solution:

Given,

=>All the operations should be performed in IEEE 754 single precision.

(a)

Given,

=>First number = -1203.232

=>Second number = -253.1398

Explanation:

Single precision frame format:

Sign(S)

Exponent(E)

Mantissa(M)

1 bit 8 bits 23 bits

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

Converting given decimal numbers to binary:

=>First number = -1203.232 in decimal

=>First number in binary = -10010110011.0011101101100100011

=>First number in normalized form = -1.00101100110011101101100*2^10

=>Second number = -253.1398 in decimal

=>Second number in bianry = -11111101.00100011110010011111

=>Second number in normalized form = -1.11111010010001111001001*2^7

Now performing addition:

=>First number in normalized form = -1.00101100110011101101100*2^10

=>Second number in normalized form = -1.11111010010001111001001*2^7

Step 1:

=>Rewrite the smaller larger such that exponent of smaller number matches with higher number

=>Second number in normalized form = -1.11111010010001111001001*2^7

=>Second number in normalized form = -0.00111111010010001111001*2^10

Step 2:

=>Add mantissa part of both the numbers

=>-1.11111010010001111001001*2^10 -0.00111111010010001111001*2^10 = -10.0011100110010000100001*2^10

Step 3:

=>Now normalizing the sum

=>-10.0011100110010000100001*2^10 = -1.00011100110010000100001*2^11

Step 4:

=>Checking overflow

=>As -126 < 11 < 127 so there is no overflow

Step 5:

=>Now converting into decimal

=>-1.00011100110010000100001*2^11 = -100011100110.010000100001

=>-100011100110.010000100001 = 2278.258056640625

=>Hence sum of -1203.232 and -253.1398 = -2278.258056640625

(b)

Given,

=>First number = -2037.4235

=>Second number = 653.25

Explanation:

=>First number = -2037.4235 in decimal

=>First number in binary = -11111110101.01101100011010101

=>First number in nomarlized form = -1.11111101010110110001101*2^10

=>Second number = 653.25 in decimal

=>Second number in binary = 1010001101.01

=>Second number in normalized form = 1.01000110101000000000000*2^9

Now performing additon:

=>First number in nomarlized form = -1.11111101010110110001101*2^10

=>Second number in normalized form = 1.01000110101000000000000*2^9

Step 1:

=>Make the exponent part of smaller number same as higher number's exponent.

=>Second number in normalized form = 1.01000110101000000000000*2^9

=>Second number = 0.10100011010100000000000*2^10

Step 2:

=>Now adding mantissa part.

=>-1.11111101010110110001101*2^10 + 0.10100011010100000000000*2^10 = -1.01011010000010110001101*2^10

Step 3:

=>Checking overflow

=>-126 < 10 < 127 hence no overflow

Step 4:

=>Normalize the summation. As summation is already normalized so no need to do that.

Step 5:

=>Convert into decimal

=>-1.01011010000010110001101*2^10 = -10101101000.0010110001101

=>10101101000.0010110001101 = -1384.1734619140625

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