Question

Check if (A x B) x C = A x (B x C) using half-precision format.

Let

• A = 3.41796875x10 ^-3 = 1.1100000000 x 2 ^-9
• B = 4.150390625 x10^-3 = 1.0001000000 x 2^-8
• C = 1.05625x10² = 1.1010011010 x2⁶

Calculate (A x B) x C 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.

First, we calculate AxB = 1.M x 2^N  

M =_______ (10-bit binary)

N = _________(a negative (decimal) integer)

The product of A and B is an underflow. That is, the result of A*B cannot be represented in the normal 16-bit half precision format.

However, it could be represented as a denorm number. What is the encoding of the this denorm number? (a 4-digit hexadecimal number with uppercase letter digit.)

Answer 1

Solution:--

To prove the (A x B) x C = A x (B x C)

First see
(AxB) x C =

• A = 3.41796875x10 -3 = 1.1100000000 x 2 -9
• B = 4.150390625 x10-3 = 1.0001000000 x 2-8
• C = 1.05625x102 = 1.1010011010 x26

AxB=  1.1100000000 x 1.0001000000 x 2-17

AxB= 1.1101110000 x 2-17

Here M = 1101110000

N = -17 =11101110 (representing Negative number using 2's complement )

(AxB) x C= 1.1101110000 x1.1010011010 x 2-11

(AxB) x C=11.0000100011 x 2-11

Here P = 0000100100 (rounded to even)

Q=-11= 11110101 (representing Negative number using 2's complement )

Final ans 1= 11.000100011 x 2-11

Now check for

Ax(BxC)

BxC= 1.0001000000 x1.1010011010 x 2-2

BxC=1.1100000010 x 2-2

Here M=1100000010

N= -2 =11111110

Ax(BxC) = 1.1100000000 x1.1100000010 x 2-11

Ax(BxC) = 11.0000100011 x 2-11

P = 0000100100 (rounded to even)

Q=-11= 11110101 (representing Negative number using 2's complement )

Final ans 2 = 11.000100011 x 2-11

So final ans 1 = final ans 2

Hence proved