Question

Consider the IEEE-754 single precision floating point numbers $P=0xC1800000$ and $Q=0x3F5C2EF4$. Which one of the following corresponds to the product $P \times Q$, represented in IEEE-754 single precision format?

• A. 0x404C2EF4
• B. 0x405C2EF4
• C. 0xC15C2EF4
• D. 0xC14C2EF4

Hint:

To quickly verify IEEE-754 values: check the sign bit for positivity/negativity, compute exponent–bias for scaling, then approximate mantissa. Multiplication/division is faster by decoding exponents and mantissas.

Answer 1

The Correct Option is C

Let's solve the problem by converting the IEEE-754 single precision floating point numbers from hexadecimal to decimal to identify the product:

1. First, we will decode each number from hexadecimal to binary, then to floating point.

Step 1: Decode P = 0xC1800000

• Convert hex \(C1800000_{16}\) to binary: \(1100\ 0001\ 1000\ 0000\ 0000\ 0000\ 0000\ 0000_2\)
• The first bit is the sign bit: \(1\) (indicating a negative number).
• The next 8 bits represent the exponent: \(1000\ 0011_2 = 131_{10}\).
• The exponent bias is 127, so actual exponent: \(131 - 127 = 4\).
• The last 23 bits represent the mantissa: \(1.100\ 0000\ 0000\ 0000\ 0000\ 0000_2\) (because the leading bit is implied for normalized numbers).
• The value of \(P\) is \(-1 \times 1.1 \times 2^4 = -1.5 \times 16 = -24\).

Step 2: Decode Q = 0x3F5C2EF4

• Convert hex \(3F5C2EF4_{16}\) to binary: \(0011\ 1111\ 0101\ 1100\ 0010\ 1110\ 1111\ 0100_2\)
• The first bit is the sign bit: \(0\) (indicating a positive number).
• The next 8 bits represent the exponent: \(0111\ 1110_2 = 126_{10}\).
• The exponent bias is 127, so actual exponent: \(126 - 127 = -1\).
• The last 23 bits represent the mantissa: \(1.0101\ 1100\ 0010\ 1110\ 1111\ 0100_2\).
• The value of \(Q\) is approximately \(1.36279296875 \times 2^{-1} \approx 0.681396484375\).

Step 3: Calculate P × Q

• Calculate: \(-24 \times 0.681396484375 = -16.353515625\). 
• Convert -16.353515625 to IEEE-754 format:
  • Sign bit: \(1\).
  • Binary representation: Approximately \(1.00000101101 \times 2^4\) for normalized form.
  • Exponent: \(4 + 127 = 131\_2 = 10000011_2\).
  • Mantissa: \(00000101101\_2\) after the decimal point.
• The IEEE-754 representation is \(1100\ 0001\ 0110\ 0000\ 0101\ 1010\ 1101\ 0000_2\), which is 0xC15C2EF4 in hex.

Therefore, the product of P and Q, represented in IEEE-754 single precision format, is 0xC15C2EF4.