Question

Calculate $1.666015625\times {10}^0\times \left(1.9760\times {10}^4+-1.9744\times {10}^4\right)$by hand, assuming each of the values is 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 value of $1.666015625\times {10}^0\times \left(1.9760\times {10}^4+-1.9744\times {10}^4\right)$is as follows:

$1.1010101010\times 2^4$

And the decimal value is 26.65625

Step-by-step solution

Determine the 16-bit half-precision format.

The 16-bit half-precision format will have the 1 sign bit, 5 exponent bit, and 10 fraction bits.

If the exponent is large to represent in the exponent field, then it is called overflow.

Guard bit is the first two extra bits in the right of the intermediate calculations that help in rounding.

Round is the nearest number that can be fit into the floating format.

A sticky bit is used to round in addition to guard and to set the round when there are bits present right to the round bit.

Determine the value of1.666015625×100×(1.9760×104+-1.9744×104)

Convert each value to the half-precision format as follows:

Consider $1.666015625\times {10}^0$as A , The exponent is zero, so simply convert the number to binary.

$\begin{array}{c}A=1.666015625\times {10}^0\\ =1.1010101010\times 2^0\end{array}$

Consider as $1.9760\times {10}^4$B. The exponent is 4, so place the decimal point four digits right as 19760.0 . Now convert to binary.

$\begin{array}{c}B=1.9760\times {10}^4\\ =19760.0\\ =1.0011010011\times 2^{14}\end{array}$

The floating point is adjusted to the power of 2.

Consider $-1.9744\times {10}^4$as C. The exponent is 4, so place the decimal point four digits right as -19744.0, Now convert to binary.

$\begin{array}{c}C=-1.9744\times {10}^4\\ =19744.0\\ =-1.0011010010\times 2^{14}\end{array}$

The floating point is adjusted to the power of 2.

The exponents of B and C are same, so subtract them.

$\begin{array}{c}B-\left(-C\right)=B+C=1.0011010011+\left(1.0011010010\right)\\ =0.0000000001\times 2^{14}\\ =1.0000000000\times 2^4\end{array}$

The 2¹⁴ is reduced to 2⁴ in the above result.

The fraction part is calculated as follows:

$\begin{array}{c}A\times \left(B+C\right)=1.1010101010\times 1.0000000000\\ =11010101010\\ =1.10101010100000000000\end{array}$

From the above result, the guard bit is 0, Round is 0, and sticky is 0.

So, the value of $A\times \left(B+C\right)$is $1.1010101010\times 2^4$and the decimal value is 26.65625