Explain IEEE-754 standard for floating point representation. Express (314.175)10 in all the IEEE-754 models.

IEEE 754 is the standard format for real numbers in almost every computer. A number is stored as $(-1)^s \times 1.f \times 2^{e}$ in three fields: the sign bit $s$ (0 for positive, 1 for negative), the biased exponent $E = e + \text{bias}$, and the fraction $f$. The leading 1 is implied and not stored.

The bias keeps the stored exponent positive, so numbers can be compared like integers. The all-0 and all-1 exponent patterns are kept for special values.

Normal numbers use the stored exponents 1 to 254 in single precision and 1 to 2046 in double precision. Rounding to the nearest value is the default way to fit a long fraction into the fraction field, and it keeps the error below half of the last bit.

Representing 314.175

The number is positive, so $s = 0$. The integer part is $314 = 100111010_2$. For the fraction, we double $0.175$ again and again and note the digits:

$$0.175 = 0.001011,0011,0011\ldots_2$$

So the number is $100111010.0010110011\ldots_2$. Move the point 8 places left to normalise:

$$314.175 = 1.001110100010110011\ldots_2 \times 2^{8}$$

The true exponent is $e = 8$. The fraction starts with $00111010$ (from the integer part) and then continues with the digits of $0.175$.

Single precision. The stored exponent is:

$$E = 8 + 127 = 135 = 10000111_2$$

The fraction takes 23 bits. The bits after the 23rd are $0110\ldots$, which are less than half, so we round down. Therefore:

In hexadecimal this is $\text{0x439D1666}$.

Double precision. The stored exponent is:

$$E = 8 + 1023 = 1031 = 10000000111_2$$

The fraction takes 52 bits. The bits after the 52nd are $1100\ldots$, which is more than half, so the last bit is rounded up. Therefore:

In hexadecimal this is $\text{0x4073A2CCCCCCCCCD}$.

Neither answer is exact, because $0.175$ has a repeating binary fraction. The error is about $10^{-5}$ in single precision and about $10^{-14}$ in double precision. To check the single answer, decode it: $E = 135$ gives $e = 8$, and the value is $1.00111010001011001100110_2 \times 2^8 \approx 314.17499$.