Question

Write down the binary bit pattern to represent $-1.5625\times {10}^{-1}$assuming a format similar to that employed by the DEC PDP-8 (the left most 12 bits are the exponent stored as a two’s complement number, and the rightmost 24 bits are the fraction stored as a two’s complement number). No hidden 1 is used. Comment on how the range and accuracy of this 36-bit pattern compares to the single and double precision IEEE 754 standards.

Short answer

Representation in binary bit patternis

Exponent	Mantissa
111111111110	10110000 0000 0000 0000

Step-by-step solution

Convert -1.5625×10-1 into binary

We can write$-1.5625\times {10}^{-1}$ as $-0.15625$

Now, we convert it into binary representation

$$\begin{gathered} 0.15625\times 2=0.3125 \\ 0.3125\times 2=0.625 \\ 0.625\times 2=1.25 \\ 0.25\times 2=1.25 \\ 0.25\times 2=0.5 \\ 0.5\times 2=1 \end{gathered}$$

So, we write it

${(-0.15625)}_{10}={(-0.00101)}_2$

Normalize the binary representation

For normalization, we shift the decimal

$-0.00101=-0.101\times 2^{-2}$

Step 3

As per the given instruction in the question, We don't use hidden 1, so in this format, the exponent is negative

$-2=-000000000010$

Find two's complement of exponent (-2)

To find two's compliments first we have to find one's complement

So, one's complement of (-2) is $111111111101$

Now to find two's complements we have to add 1 in one's complement of the number

$$\begin{gathered} 111111111101+000000000001 \\ =111111111110 \end{gathered}$$

Find two's complement of the fraction part

We have to find two’s complement because the number given in the question is negative

So first we find one’s complement of 0101 0000 0000 0000 0000 0000

One’s complement of fraction part is $101011111111111111111111$

Now two’s compliment

$$\begin{gathered} 101011111111111111111111+000000000000000000000001 \\ =101100000000000000000000 \end{gathered}$$

Binary representation of the number -1.5625×10-1

Exponent	Mantissa
111111111110	10110000 0000 0000 0000

36-bit pattern has more accuracy than the single-precision IEEE 754 standard, but In IEEE 754 standard accuracy of the double-precision s greater.