Transcript Sample-Exam1

UNIVERSITY OF MASSACHUSETTS
Dept. of Electrical & Computer Engineering
Digital Computer Arithmetic
ECE 666
Mid-Term I
Sample Exam
Israel Koren
ECE666/Koren Sample Mid-term 1.1
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1. (a) Multiply the following two SD numbers 010\bar{1}1 and
00\bar{1}01. Perform all intermediate steps in SD arithmetic.
(b) (10 points) Find the minimal representation of the following SD number:
01110\bar{1}\bar{1}\bar{1}.
ECE666/Koren Sample Mid-term 1.2
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2. (a) Show the result (in hexadecimal) of the following multiplication in the
IEEE short format (the operands are given in hexadecimal notation), in all
four rounding schemes (round-to-nearest-even, round toward zero, round
toward  and -  ): 4400 2000 x C300 0200.
(b) Show the result (in hexadecimal) of the following addition in the IEEE
short format in all four rounding schemes. The operands are given in the
hexadecimal notation: 6380 0000 + D7C0 0007
ECE666/Koren Sample Mid-term 1.3
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3. In an attempt to reduce the expected error in the computation A1 
(A2-A3) it has been suggested to calculate instead A1  A2- A1  A3.
Compare the relative error accumulated in these two calculations assuming
that each of the three operands is a result of a previous calculation and
has a relative error of _i (i=1,2,3), i.e., Ai=Ai^c (1+ _i), where Ai^c is
the correct value of Ai. Also assume that the multiply and subtract
operations introduce relative errors of _m and _s, respectively (e.g.,
Fl(x  y)=(x  y)(1+ _m)). (a) Write expressions for the relative errors
of the results of A1  (A2-A3) and A1  A2 - A1  A3.
ECE666/Koren Sample Mid-term 1.4
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(b) Compare the two relative errors for the case A1=1000A3, A2=1.01A3
and _1= _2= _3. Which calculation will result in a higher accumulated
error if _s = 2 _m? Explain.
ECE666/Koren Sample Mid-term 1.5
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4. Convert the number (011011)_{-2} to radix 2 and the number
(001011)_2 to radix -2. Describe a procedure for converting numbers
from radix r to -r and vice versa. Illustrate your procedure by converting
(321)_10 to radix -10.
ECE666/Koren Sample Mid-term 1.6
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ECE666/Koren Sample Mid-term 1.7
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1. Show the exact steps in the non-restoring division with the
(negative) dividend X=1011001 in two's complement representation and
the divisor D=0110.
ECE666/Koren Sample Mid-term 1.8
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2. (a) Show the representation of the following operands in the IEEE short
format (use hexadecimal notation), perform the multiplication and show the
final result in all four rounding schemes (nearest-even, toward zero
(truncate), toward +, and toward -). (1+2^{-23})  (1+2^{-22}).
Note: (1+2^{-23}) is the number 1.00000000000000000000001.
ECE666/Koren Sample Mid-term 1.9
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(b) Show the result of the following subtraction of numbers in the IEEE
short format in all four rounding schemes. The operands are already given
in the hexadecimal notation. 3F80 0000 - 3EFF FFFF
ECE666/Koren Sample Mid-term 1.10
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3. (4.10) Write down the post-normalization steps that might be needed
when performing addition, subtraction, multiplication, and division with two
floating-point operands in the IEEE short format. Indicate how many
guard digits are needed in each case.
ECE666/Koren Sample Mid-term 1.11
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4. Prove that the optimal way to implement a two-level combinatorial
shifter for k bits, where k=m^2, is for the first level to shift by multiples
of m, and the second level to shift from 0 to m. Assume that the speed is
proportional to the number of destinations for each line in the two levels.
Can you generalize this result for any value of k?
ECE666/Koren Sample Mid-term 1.12
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5. (4.12) Two normalized floating-point numbers A and B in the short IEEE
format were added, and the result was equal to A. Does this imply that
B=0?
ECE666/Koren Sample Mid-term 1.13
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(b) (4.13) Given a floating-point number A with an exponent E_A (in any
format), its successor has either the same exponent or the exponent
E_A+1. Is the distance between A and its successor the same in both
cases?
ECE666/Koren Sample Mid-term 1.14
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6. (4.14) (a) Compare the error involved in the serial evaluation of the
product of four numbers, performed as (((A_1  A_2)  A_3)  A_4) to
that of its parallel evaluation performed as ((A_1  A_2)  (A_3  A_4)).
Decide whether one of these methods has a smaller upper bound for the
error when forming the product of n numbers.
(b) Repeat (a) for the sum of four numbers, then n numbers. Can we get
lower error bounds if we know that the numbers are in some order; e.g.,
ascending order?
ECE666/Koren Sample Mid-term 1.15
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