NIMCET 2024 — Computer PYQ

In the IEEE 754 double-precision floating-point format (also known as Binary64), a total of 64 bits are allocated to store a number:

• Sign bit: 1 bit

• Biased Exponent ($E_{\text{biased}}$): 11 bits

• Mantissa / Fraction: 52 bits

1. Range of the Biased Exponent

With 11 bits allocated for the exponent, the total number of possible binary values ranges from:

$$\text{Minimum value}=(00000000000{)}_2=0$$

$$\text{Maximum value}=(11111111111{)}_2=2^{11}-1=2047$$

So, the biased exponent range is $0\le E_{\text{biased}}\le 2047$.

2. Reserved Values for Special Cases

In the IEEE 754 standard, the extreme minimum and maximum values of the biased exponent are reserved for special representations:

• When $E_{\text{biased}}=0$: Reserved for representing subnormal numbers (denormalized numbers) and zero.

• When $E_{\text{biased}}=2047$: Reserved for representing infinity ($\infty$) and Not-a-Number ($\text{NaN}$).

Therefore, the valid range for the biased exponent of a normal (normalized) floating-point number is:

$$1\le E_{\text{biased}}\le 2046$$

3. Calculating the Actual Exponent ($E$)

To support both positive and negative exponents, a bias is subtracted from the biased exponent value. For double precision, the fixed exponent bias is calculated using the formula:

$$\text{Bias}=2^{11-1}-1=2^{10}-1=1023$$

The relationship between the actual exponent $E$ and the biased exponent is given by:

$$E=E_{\text{biased}}-\text{Bias}$$

$$E=E_{\text{biased}}-1023$$

Substituting the minimum and maximum valid biased exponent values:

• Minimum Actual Exponent ($E_{\text{min}}$):

  $$E_{\text{min}}=1-1023=-1022$$

• Maximum Actual Exponent ($E_{\text{max}}$):

  $$E_{\text{max}}=2046-1023=1023$$

Thus, the standard actual exponent range for normalized numbers is:

$$-1022\le E\le 1023$$