Question

Calculate $\left(3.41796875{10}^{-3}\times 6.34765625\times {10}^{-3}\right)\times 1.05625\times {10}^2$by hand, assuming each of the values are stored in the 16-bit half precision format described in Exercise 3.27 ( and also described in the text). Assume 1 guard, 1 round bit, and 1 sticky bit, and round to the nearest even. Show all the steps, and write your answer in both the 16-bit floating point format and in decimal.

Short answer

The answer for the given calculation is 0.00230.

Step-by-step solution

Given Values

Consider the following values:

$A=3.41796875\times {10}^{-3}$

$B=6.34765625\times {10}^{-3}$

$C=1.05625\times {10}^2$

Converting decimal to binary equivalent

$$\begin{gathered} 3.41796875\times {10}^{-3} \\ =0.00341796875 \\ =0.00000000111\times 2^0 \\ =1.11\times 2^{-9} \end{gathered}$$

$$\begin{gathered} 6.34765625\times {10}^{-3} \\ =0.00634765625 \\ =0.00000001101\times 2^0 \\ =1.101\times 2^{-8} \end{gathered}$$

$$\begin{gathered} 1.05625\times {10}^2 \\ =105.625 \\ =1101001.101\times 2^0 \\ =1.101001101\times 2^6 \end{gathered}$$

Multiplying the values

Consider the multiplication of the values:

$$\begin{gathered} \left(3.41796875\times {10}^{-3}\times 6.34765625\times {10}^{-3}\right)\times 1.05625\times {10}^2 \\ =\left(1.11\times 2^{-9}\times 1.101\times 2^{-8}\right)\times 1.101001101\times 2^6 \\ =\left(1.011011\times 2^{17}\right)\times 1.101001101\times 2^6 \\ =1.0010110010\times 2^9 \end{gathered}$$

The value in decimal is 0.00230

Result in 16-bit floating point format.

Consider the following binary number:

$1.0010110010\times 2^{-9}$

S value is 0 since number is positive.

Exponent can be computed as:

$\begin{array}{rcl}Exponent-15& =& -9\\ Exponent& =& 15-9\\ Exponent& =& 6\end{array}$

General representation of half precision number is

$$\begin{gathered} {\left(-1\right)}^s\times \left(1+fraction\right)\times 2^{Exponent-15} \\ {\left(-1\right)}^0\times \left(1+.0010110010\right)\times 2^6 \end{gathered}$$

The floating point representation is as shown below:

01 2 3 4 5 6 7 8 9 10 11 12 13 14 15

00 0 1 1 0 0 0 1 0 1 1 0 0 1 0