Question

1. Compute the decimal value for the following bit pattern, assuming it is a single-precision floating point number (show major steps):
1100 0011 0001 0010 0100 1001 0010 0100

2. Convert the decimal -2118.75 into single-precision floating point number (show major steps).

3. Assume -75 and -122 are signed decimal integers stored in 8-bit
sign-magnitude binary format. Calculate -75 + -122. Is there overflow, underflow,
or neither?

Answer 1

Answer 1: Convert 1100 0011 0001 0010 0100 1001 0010 0100 to Decimal
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Enter the 32 bit IEEE single precision number:
11000011000100100100100100100100

IEEE 32 bit format is of the form <sign bit> <8 bit biased exponent> <23 bit fraction>
Sign bit: 1, so the number is -ve

Exponent in binary = 10000110 = 134 in decimal
Decode the exponent by subtracting 127 = 134 - 127 = 7

Significand = 00100100100100100100100
Add the implicit 24th bit to significand, so 24 bit significand = 100100100100100100100100

The 24 bits from bit 23 to bit 0 represent a value, starting at 1 and halves for each bit
Add up the corresponding values for 1 bits
bit 23 = 1.0
bit 20 = 0.125
bit 17 = 0.015625
bit 14 = 0.001953125
bit 11 = 2.44140625E-4
bit 8 = 3.0517578125E-5
bit 5 = 3.814697265625E-6
bit 2 = 4.76837158203125E-7
Decoded significand = 1.0 + 0.125 + 0.015625 + 0.001953125 + 2.44140625E-4 + 3.0517578125E-5 + 3.814697265625E-6 + 4.76837158203125E-7 = 1.1428570747375488
Multiply this with base 2 to power of exponent i.e 2^7
= 1.1428570747375488 x 2^7 = 146.28570556640625
The decimal number along with sign is -146.28570556640625

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Answer 2: Convert -2118.75 to IEEE format
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Enter a decimal number:
-2118.75
The given number -2118.75 has Integral part = 2118, Fractional part = 0.75
Step 1:
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Convert integral part using repeated division by 2, accumulating remainders
Divide 2118 by 2, now n = 1059, remainder = 0
Divide 1059 by 2, now n = 529, remainder = 1
Divide 529 by 2, now n = 264, remainder = 1
Divide 264 by 2, now n = 132, remainder = 0
Divide 132 by 2, now n = 66, remainder = 0
Divide 66 by 2, now n = 33, remainder = 0
Divide 33 by 2, now n = 16, remainder = 1
Divide 16 by 2, now n = 8, remainder = 0
Divide 8 by 2, now n = 4, remainder = 0
Divide 4 by 2, now n = 2, remainder = 0
Divide 2 by 2, now n = 1, remainder = 0
Divide 1 by 2, now n = 0, remainder = 1
The binary equivalent of 2118 is 100001000110
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Step 2:
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Convert fractional part using repeated multiplication by 2, accumulating integral parts (until fraction part becomes zero or upto 23 accumulated digits)
Multiply 0.5 * 2 = 1.5, accumulate 1, next use fraction 0.5
Multiply 0.0 * 2 = 1.0, accumulate 1, next use fraction 0.0
The fractional part 0.75 in binary is .11
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The decimal number -2118.75 is equivalent to binary 100001000110.11

Step 3:
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Normalize the binary number by writing in the form 1.XXXXX * 2^exponent
Shift decimal point to left by 11 digits
Normalized form is 1.0000100011011 x 2^11
From the normalized form, we find that exponent = 11 and fraction = 0000100011011
Exponent is 11. Add 127 to exponent to get the biased form i.e 11 + 127 = 138

IEEE 32 bit format is of the form <sign bit> <8 bit biased exponent> <23 bit fraction>
The number is -ve, so sign bit = 1
The biased exponent 138 in 8 bit binary is 10001010
The fraction in 23 bit binary is 00001000110110000000000

The IEEE 32-bit single precision format is 1-10001010-00001000110110000000000