Question

Calculate the product of$\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{0546875}\mathbf{\times }{\mathbf{10}}^{\mathbf{0}}$and$\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{79931640625}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$by hand, assuming A and B are stored in the 16-bit half precision format described in Exercise 3.27. Assume 1 guard, 1 round bit, and 1 sticky bit and round to the nearest even. Show all the steps; however, as is done in the example in the text, you can do the multiplication in human readable format instead of using the techniques described in Exercises 3.12 through 3.14. Indicate if there is overflow or underflow. Write your answer in both 16-bit floating point format described in Exercise 3.27 and also as a decimal number. How accurate is your result? How does it compare to the number you get if you do multiplication on a calculator?

Short answer

The result of the multiplication is $1.0111001100$. In decimal form, the value is 1.44921875. The overflow occurs and the result is off by 0.0000743865966796875.

Step-by-step solution

Convert -8.0546875×100 and -1.79931640625×10-1 to 16-bit half precision format:

The 16-bit half precision format is 16-bit wide. The most significant bit is sign bit, next 5 bits are for exponent and the size of mantissa is 10-bits. The binary representation of $-8.0546875\times {10}^0$ is $-1000.0000111\times 2^0$. It can also be written as$-1.0000000111\times 2^3$. The value $-1.79931640625\times {10}^{-1}$is equal to $-0.179931640625\times {10}^0$. The binary representation of $-0.179931640625\times {10}^0$is $-0.001011100001\times 2^0$. This value is equal to$-1.011100001\times 2^{-3}$.

Identify the exponent and the sign of the result

The value obtained by adding the biased exponents of the two numbers is as follows:

$-3+3=0$

The exponent is, thus,

$16+0=16$

The binary representation of 16 is 10000.

The values to be multiplied have negative signs. The multiplication of two negative numbers results in positive value.

Multiply the significands

The multiplication of significands is performed as below:

Identify if there is overflow or underflow

The result of the multiplication is $1.0111001100$. The binary equivalent of the result is 1.44921875. The multiplication of $-8.0546875\times {10}^0$and$-0.179931640625\times {10}^0$ is 1.4492931365966796875. Because this number cannot be represented in the 10-bit field, the overflow occurs. The result obtained is off by 0.0000743865966796875.