You just paid $5 for a very old-looking computer at a yard sale. To your surpris
ID: 3885990 • Letter: Y
Question
You just paid $5 for a very old-looking computer at a yard sale. To your surprise, it turns on and, during the boot sequence, mentions that its microprocessor uses the IEEE 754 standard (but it does not specify how many bits). To your even bigger surprise, it has an early version of Matlab installed on it, and you can't resist playing with it to determine exactly which standard the computer uses.
a. To determine the size of the mantissa, you perform the following operations and observe the following results:
>> 1 + exp(-7.6) == 1 ans = 0
>> 1 + exp(-7.7) == 1 ans = 1
From this, determine the number of bits t in the fraction f of the mantissa. Hint: assume that the computer relies on chopping and use the information to bracket the machine epsilon (i.e. write: ... < < ...).
b. To determine the exponent, you perform the following operations and observe the following results:
>> exp(5.5) ans = 244.6919
>> exp(5.6) ans = Inf
From this, determine the number of bits n used for the characteristic c. Hints: use the information to bracket the realmax for this version of Matlab (remember that the characteristic that corresponds to realmax is € c = 2b) and use the fact that for the IEEE standard, the bias is given by: b = 2^(n1) 1
Explanation / Answer
A circuit which performs an AND operation is shown in Fig. 1.2. It has N inputs (N
2) and one output. Digital signals are applied at the input terminals marked A, B, …, N, the
other terminal being ground, which is not shown in the diagram. The output is obtained at the
output terminal marked Y (the other terminal being ground) and it is also a digital signal. The
AND operation is defined as : the output is 1 if and only if all the inputs are 1.
Mathematically, it is written as
Y = A AND B AND C … AND N
= A B C … N
= ABC …N …(1.1)
Fig. 1.2 The standard symbol for an AND gate
where A, B, C, … N are the input variables and Y is the output variable. The variables are
binary, i.e. each variable can assume only one of the two possible values, 0 or 1. The binary
variables are also referred to as logical variables.
Equation (1.1) is known as the Boolean equation or the logical equation of the AND gate.
The term gate is used because of the similarity between the operation of a digital circuit and a
gate. For example, for an AND operation the gate opens (Y = 1) only when all the inputs are
present, i.e. at logic 1 level.
Truth Table Since a logical variable can assume only two possible values (0 and 1),
therefore, any logical operation can also be defined in the form of a table containing all
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