Chapter 3

3.5

Question

• What is 4365 - 3412 when these values represent signed 12-bit octal numbers stored in sign-magnitude format? The result should be written in octal. Show your work.

Answer

• http://bit.ly/2MtqyLX
• 7777(-3777)

3.9

Question

• http://bit.ly/3rUoXiH
• Assume 151 and 214 are signed 8-bit decimal integers stored in two’s complement format. Calculate 151 + 214 using saturating arithmetic. The result should be written in decimal. Show your work.

Answer

• -105 -42 = -128 (-147)

3.12

Question

• Using a table similar to that shown in Figure 3.6, calculate the product of the octal unsigned 6-bit integers 62 and 12 using the hardware described in Figure 3.3. You should show the contents of each register on each step.

fig 3.6

fig 3.3

Answer

• $62\times 12$

3.16

Question

• Calculate the time necessary to perform a multiply using the approach given in Figure 3.7 if an integer is 8 bits wide and an adder takes 4 time units.

fig 3.7

Answer

3.19

question

• Using a table similar to that shown in Figure 3.10, calculate 74 divided by 21 using the hardware described in Figure 3.11. You should show the contents of each register on each step. Assume A and B are unsigned 6-bit integers. Th is algorithm requires a slightly different approach than that shown in Figure 3.9. You will want to think hard about this, do an experiment or two, or else go to the web to figure out how to make this work correctly. (Hint: one possible solution involves using the fact that Figure 3.11 implies the remainder register can be shifted either direction.)

fig 3.9

fig 3.10

fig 3.11

Answer

3.22

question

• What decimal number does the bit pattern 0×0C000000 represent if it is a floating point number? Use the IEEE 754 standard.

Answer

3.26

question

• Write down the binary bit pattern to represent
  $-1.5625\times {10}^{-1}$ assuming a format similar to that employed by the DEC PDP-8 (the left most 12 bits are the exponent stored as a two’s complement number, and the rightmost 24 bits are the fraction stored as a two’s complement number). No hidden 1 is used. Comment on how the range and accuracy of this 36-bit pattern compares to the single and double precision IEEE 754 standards.

Answer

3.27

question

• IEEE 754-2008 contains a half precision that is only 16 bits wide. Th e left most bit is still the sign bit, the exponent is 5 bits wide and has a bias of 15, and the mantissa is 10 bits long. A hidden 1 is assumed. Write down the bit pattern to represent
  $-1.5625\times {10}^{-1}$ assuming a version of this format, which uses an excess-16 format to store the exponent. Comment on how the range and accuracy of this 16-bit floating point format compares to the single precision IEEE 754 standard.

Answer

3.28

question

• The Hewlett-Packard 2114, 2115, and 2116 used a format with the left most 16 bits being the fraction stored in two’s complement format, followed by another 16-bit field which had the left most 8 bits as an extension of the fraction (making the fraction 24 bits long), and the rightmost 8 bits representing the exponent. However, in an interesting twist, the exponent was stored in sign magnitude format with the sign bit on the far right! Write down the bit pattern to represent
  $-1.5625\times {10}^{-1}$ assuming this format. No hidden 1 is used. Comment on how the range and accuracy of this 32-bit pattern compares to the single precision IEEE 754 standard.

Answer

3.29

question

• Calculate the sum of
  $2.6125\times {10}^1$ and
  $4.150390625\times {10}^{-1}$ by hand, assuming A and B are stored in the 16-bit half precision 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.

answer

3.32

question

• Calculate

  $3.984375\times {10}^{-1}+3.4375\times {10}^{-1}+1.771\times {10}^3$ 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. Show all the steps, and write your answer in both the 16-bit floating point format and in decimal.

answer

3.36

question

• Calculate
  $3.41796875\times {10}^3\times (6.34765625\times 103\times 1.05625\times {10}^2)$ 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. Show all the steps, and write your answer in both the 16-bit floating point format and in decimal.

Answer

3.38

question

• Calculate
  $1.666015625\times {10}^0\times (1.9760\times {10}^4-1.9744\times {10}^4)$ 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. Show all the steps, and write your answer in both the 16-bit floating point format and in decimal.

Answer

3.42

question

• What do you get if you add -1/4 to itself 4 times? What is
  $1/4\times 4$? Are they the same? What should they be?

Answer

3.44

question

• Write down the bit pattern in the fraction 1/3 assuming a floating point format that uses Binary Coded Decimal (base 10) numbers in the fraction instead of base 2. Assume there are 24 bits, and you do not need to normalize. Is this representation exact?

Answer

3.46

question

• Write down the bit pattern assuming that we are using base 30 numbers in the fraction instead of base 2. (Base 16 numbers use the symbols 0–9 and A–F. Base 30 numbers would use 0–9 and A–T.) Assume there are 20 bits, and you do not need to normalize. Is this representation exact?

Answer

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