Chapter 1, Problem 39CRP

Explanation of Solution

The best approximation to the square root of $2$ that can be expressed in the 8-bit floating-point format described in figure $1.24$. If the approximation is squared by a machine, then what is the value obtained using floating-point format.

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right, start with 1.
• Copy the bit pattern into the mantissa field from left to right, start with 1.
• The exponential field is determined by assuming the pattern of mantissa field with a radix at its left and it obtains the original binary number by determining the number of bits and direction of radix that must be moved.

The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	Value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive, the bit assigned on high order bit is 0 and when the given value is negative, the bit assigned on higher bit is 1.
• The approximation $\sqrt{2}$ is $1\frac{3}{8}$.
• Convert the number $1$  into binary equivalent.

$\begin{array}{ccc}2& 1& \\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom ${\left(1\right)}_{10}={\left(1\right)}_2$
• Convert the number $\frac{3}{8}$ into binary representation