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Chapter 3: 43. (page 241)

Write down the bit pattern in the fraction of value 1/3 assuming a floating-point format that uses binary numbers in the fraction. Assume there are 24 bits, and you do not need to normalize the value of 1/3. Is this representation exact?

Short Answer

Expert verified

The value of 1/3 is 0101 0101 0101 0101 0101 0101 and it is not exact.

Step by step solution

01

Determine the IEEE 754 floating-point format.

Floating-point is the number obtained by fractional division. The value of the fraction will be the number with the decimal point. The number will be converted to a binary number. The sign will be presented as a separate bit. The floating point will have the scientific notation with a single digit to the left of the decimal point.

02

Determine the bit pattern of the fraction value 1/3. 

The given number is 1/3.

The decimal value of the given number is 0.333333.

The above number can be written as

Now, converting the number to a binary value, we have

333333 = 01010101010101010101

The above value is not exact because the value does not represent the exponent and it is not normalized.

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Most popular questions from this chapter

As discussed in the text, one possible performance enhancement is to do a shift and add instead of actual multiplication. Since 9×6, for example, can be written 2×2×2+1×6 , we can calculate9×6 by shifting 6 to the left 3 times and then adding 6 to that result. Show the best way to calculate 0×33×0×55 using shifts and adds/subtracts. Assume both inputs are 8 bit unsigned integers.

Calculate 1.666015625×100×1.9760×104+1.666015625×100×-1.9744×104 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.

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.

Calculate the sum of 2.6125×101and4.150390625×10-1 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

Write down the binary bit pattern to represent -1.5625×10-1assuming 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.
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