# CS3144 Exercise 4 Solved

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Problem 1:

The purpose of this question is to implement three functions for determining the bit contents of a given int. Recall that on our server, the compiler allocates 4 bytes (32 bits) for every int, but this size may be different in other systems. So you should not explicitly assume a specific size for int and use the sizeof() operator when appropriate. In your implementation of these functions, try to use simple expressions to compute the return value. In particular, do not use loops or function calls.

1. Complete the implementation of the function rightBits, such that it returns the right-most (least significant) bits of a given integer. The function receives two int parameters, n and numBits, and returns an int that is equal to the numBits least significant bits of n (). If numBits is non-positive, then the function should return 0, and if numBits is larger than the number of bits allocated for an int, then the function should return n.

# Execution examples: (on our server)

 • rightBits(7,2) returns 3 • rightBits(7,5) returns 7 • rightBits(7,80) returns 7 • rightBits(7,-1) returns 0 • rightBits(-1,2) returns 3        [ think why this is ] • rightBits(-1,7) returns 127 [ think why this is ] • rightBits(1,80) returns 1

Validation: Test your code by compiling it with the test code file

/share/ex_data/ex4/test_ex4.c using <QUESTION_ID>=1_1 (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_1_1.out.

Manual inspection: your code should be correct for different sizes of int (e.g., if int takes 2 bytes), and it should not use loops or function calls.

1. Complete the implementation of the function leftBits, such that it returns the left-most bits of a given integer. The function receives two int parameters, n and numBits, and returns an int that is equal to the numBits left-most bits of n. Note that these bits are 0 if n does not occupy the numBits left-most bits in the space allocated for an int. If numBits is non-positive, then the function should return 0, and if numBits is larger than the number of bits allocated for an int, then the function should return n.

# Execution examples: (on our server)

 • leftBits(7,31) returns 3        [ think why this is ] • leftBits(7,30) returns 1 • leftBits(7,29) returns 0 • leftBits(-1,2) returns 3 • leftBits(-1,7) returns 127 • leftBits(-1,80) returns -1 • leftBits(-1,-1) returns 0

Validation: Test your code by compiling it with the test code file

/share/ex_data/ex4/test_ex4.c using <QUESTION_ID>=1_2 (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_1_2.out.

Manual inspection: your code should be correct for different sizes of int (e.g., if int takes 2 bytes), and it should not use loops or function calls.

1. Complete the implementation of the function getBit, such that it returns a specific bit in a given number. The function receives two int parameters, n and bitInd, and returns the value of the bitIndth bit (0 or 1) of n. For this purpose, the least significant bit (the parity bit) is associated with index 0, and the most significant bit (the sign bit) has the maximum index (determined by the size of int). If bitInd is out of bounds, then the function should return 0.

# Execution examples: (on our server)

 • getBit(7,0) returns 1 • getBit(7,2) returns 1 • getBit(7,3) returns 0 • getBit(7,-12) returns 0 • getBit(-404,31) returns 1 [ think why this is ] • getBit(-404,0) returns 0 [ think why this is ] • getBit(-404,32) returns 0

Validation: Test your code by compiling it with the test code file

/share/ex_data/ex4/test_ex4.c using <QUESTION_ID>=1_3 (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_1_3.out.

Manual inspection: your code should be correct for different sizes of int (e.g., if int takes 2 bytes), and it should not use loops or function calls.

Final testing for Problem 1: Execute script /share/ex_data/ex4/test_ex4.1 from the directory containing your newFloat.c source file.

Problem 2:

The purpose of this question is to implement a floating-point representation for large integers. This “new float” uses the same space as int (4 bytes on our server), but it enables representation of much larger integers, by sacrificing some accuracy.

Our “new float” is similar in spirit to float, but it simpler in two main ways: (1) it does not represent negative numbers, and (2) it only represents integers. On the other hand, a nice feature of our “new float” is that we can control the number of bits allocated to the exponent. The representation is described in detail below. Read this carefully before you start implementing.

Specification of “new float”:

• The e left-most bits hold the binary representation of the exponent, and the remaining m bits hold the binary representation of the mantissa. Both exponent and mantissa are considered as non-negative integers here.
• If the exponent (as determined by the e left bits) is equal to EXP, and the mantissa (as determined by the m right bits) is equal to MAN, then the value of the “new float” is considered to be MAN×2EXP.
• Integer values that can be represented by m bits or less (meaning that they are in the range [0, 2m-1]) are represented with complete accuracy using the mantissa bits and setting the exponent to 0.
• Integer values that are represented by n>m bits (meaning that they are larger than 2m-1), are represented by setting the mantissa to the m most significant bits of the number, and setting the exponent to nm. This implies the following: o When the exponent is non-zero, the left-most bit of the mantissa is 1.

o If an integer number is represented by n>m bits and one of its nm least significant bits is non-zero, then its representation as a “new float” will be approximate and not accurate.

An example:

To demonstrate this specification, consider the specific example of the large integer

182,400. We would like to represent this integer using a “new float” in a system where int takes up only two bytes (we assume this for simplicity). If we assume that e=4 and m=12, then 182,400 is represented as a “new float” as follows:

Because the exponent is 6 and the mantissa is 2,850, the value represented by this

“new float” is 2,850×26 = 2,850×64 = 182,400. Note that this integer is larger than 65,535=216-1, so it cannot be represented by simple binary integer representation using two bytes. As specified above, the left-most bit of the mantissa is 1 (indicated by the red arrow). Note that the same integer value can also be represented as 182,400=1,425×27, but setting the mantissa to 1,425 (shifting it one bit to the right) will result in a 0 in the left-most bit of the mantissa, which violates the standard requirement mentioned above. Finally, note that this “new float” represents all values in the range [182,400, 182,463], because the binary integer representations of numbers in this range takes up n=18 bits, and their m=12 most significant bits are identical to the mantissa specified in the figure above.

“New float” versus float: (… for those of you who are curious)

Our “new float” is actually very similar to the standard IEEE floating point representation used in C (and accepted widely by nearly all computer applications). Two obvious differences are that float also represents negative values (by allocating one bit for the sign), and fractional values (by allowing negative exponents). Another subtle difference has to do with the lead bit of the mantissa (the one indicated by an arrow in the figure in the previous page). In order to guarantee a standard representation, we require this bit to be set to 1 whenever the exponent is greater than 0. This is clearly a bit wasteful. In float this lead bit is implicitly assumed to be ‘1’ without it actually being stored in memory, allowing the representation to gain an additional mantissa bit “for free.”

Implementation notes:

The source file newFloat.c. contains five functions that deal with “new floats”. In all functions, the “new float” value is represented by a simple int. We implemented one of these functions, newFloatToDouble, which takes an int representing a

“new float” and returns its actual value as a double. You are encouraged to read this function for your reference. You should implement the remaining 4 functions.

The number of exponent and mantissa bits (e and m) are set using the symbolic constants NUM_EXP_BITS and NUM_MAN_BITS at the top of the file:

#define NUM_EXP_BITS 4

#define NUM_MAN_BITS (sizeof(int)*8 – NUM_EXP_BITS)

Use these two symbolic constants in your code when you wish to refer to e and m. Symbolic constants are not variables in your programs. They are “text replacement’ directives for the compiler. The compiler will replace every instance of

NUM_EXP_BITS in your code with 4, and every instance of NUM_MAN_BITS with

(sizeof(int)*8 – NUM_EXP_BITS), which evaluates on our server to 28

(= 4*8 – 4). We discuss symbolic constants in detail later in the course.

Note that if you want to change the number of exponent bits, you need to change the value of NUM_EXP_BITS from 4 to something else, and then you need to recompile the program. You are encouraged to try this and see how it affects the results of your tests in Problems 2.3 and 2.4.

1. Complete the implementation of the function getExponent, such that it returns the integer value of the exponent, given a new float. Do this by replacing the 0 in the return statement with the appropriate expression. You may invoke functions that you implemented in Problem 1 above.

Validation: Test your code by compiling it with the test code file /share/ex_data/ex4/test_ex4.c       using <QUESTION_ID>=2_1    (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_2_1.out.

Manual inspection: your code should contain a single line with a return statement.

1. Complete the implementation of the function getMantissa, such that it returns the integer value of the mantissa, given a new float. Do this by replacing the 0 in the return statement with the appropriate expression. You may invoke functions that you implemented in Problem 1 above.

Validation: Test your code by compiling it with the test code file /share/ex_data/ex4/test_ex4.c       using <QUESTION_ID>=2_2    (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_2_2.out.

Manual inspection: your code should contain a single line with a return statement.

1. Complete the implementation of the function getNewFloat, which creates a “new float” (as an int). The function receives two int parameters: mantissa, exponent, and returns the appropriate “new float”. Follow these guidelines to ensure that your implementation adheres to the specification from page 4:
• If mantissa takes up less than NUM_MAN_BITS bits and exponent is non-zero, then you need to adjust the values of the actual mantissa and exponent to represent the same value in the standard form (such that the left-most bit of the mantissa is 1 or the exponent is 0).
• If mantissa takes up more than NUM_MAN_BITS bits, then the actual mantissa should consist of the NUM_MAN_BITS most significant bits of mantissa, and the actual exponent should be adjusted accordingly.
• If the exponent (as specified in the parameter or after the adjustments mentioned above) takes up more than NUM_EXP_BITS, then the value is too large to be represented by “new float” and you should return the maximum

“new float”.   Hint: the maximum “new float” has a simple representation as a signed int. Figure this out and simply return this int.

Execution examples: (on our server, with NUM_EXP_BITS set to 4)

• newFloatToDouble( getNewFloat(6,0) )      returns 6.0

(actual mantissa is 6 and actual exponent is 0)

• newFloatToDouble( getNewFloat(3,1) )      returns 6.0

(actual mantissa is 6 and actual exponent is 0)

• newFloatToDouble( getNewFloat(1000000,20) ) returns 1.048576e12

(actual mantissa is 256,000,000 and actual exponent is 12)

• newFloatToDouble( getNewFloat(1023456789,3) ) returns 8.187654e+09 (actual mantissa is 255,864,197 and actual exponent is 5)
• newFloatToDouble( getNewFloat(1000000,30) ) returns 8.796093e+12

(actual exponent is out of bounds, so function returns the maximum “new float”, which under these circumstances is 8.796093e+12)

Validation: Test your code by compiling it with the test code file

/share/ex_data/ex4/test_ex4.c using <QUESTION_ID>=2_3 (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_2_3.out.

1. Complete the implementation of the function newFloatSum, which computes the sum of two “new floats.” The function receives two int parameters, newFloatNum1 and newFloatNum2, and returns the “new float” that represents their sum. The sum can be computed as follows. Denote by 𝑀𝐴𝑁1 and 𝑀𝐴𝑁2 the two mantissas and by 𝐸𝑋𝑃1 and 𝐸𝑋𝑃2 the two exponents, and assume (without loss of generality) that 𝐸𝑋𝑃1 ≥ 𝐸𝑋𝑃2 (otherwise you can switch their roles). Then, the sum of the two “new floats” can be represented as follows:

𝑀𝐴𝑁1 × 2𝐸𝑋𝑃1 + 𝑀𝐴𝑁2 × 2𝐸𝑋𝑃2  =  (𝑀𝐴𝑁1 + 𝑀𝐴𝑁2/2𝐸𝑋𝑃1−𝐸𝑋𝑃2) × 2𝐸𝑋𝑃1

This equation implies that by setting 𝐸𝑋𝑃 = 𝐸𝑋𝑃1 and 𝑀𝐴𝑁 = 𝑀𝐴𝑁1 +

𝑀𝐴𝑁2/2𝐸𝑋𝑃1−𝐸𝑋𝑃2, the sum can be represented as 𝑀𝐴𝑁 × 2𝐸𝑋𝑃. However, since 𝑀𝐴𝑁 may be larger than 2𝑚 − 1, the actual exponent and mantissa may need to be adjusted. Further, note that if the adjusted exponent takes up more than

NUM_EXP_BITS, then the sum is too large to be represented by “new float” and the function should return the maximum “new float” (see Problem 2.3 above). If one of the two parameters does not correspond to a valid “new float”, the function should also return the maximum “new float.” You may invoke functions that you implemented in Problems 1 and 2.1-2.3.

Execution examples: (on our server, with NUM_EXP_BITS set to 4)

• If a=b=6         then newFloatSum(a,b)   returns 12
• If a=6 and b=100 then newFloatSum(a,b)   returns 106
• If a=-10 and b=10 then newFloatSum(a,b)   returns -10 (a represents a very large number (~8.8e+12) and b represents a much smaller number, so a+b ends up being rounded down to a.)
• If a=-2 and b=1<<14 then newFloatSum(a,b) returns -2 (a represents a very large number (~8.8e+12) and b represents a much smaller number, so a+b ends up being rounded down to a.)

Cases where the maximum “new float” is returned:

• If a=-2 and b=1<<15 (a represents a very large number (~8.8e+12) and b is sufficiently large such that a+b equals the maximum “new float”)
• If a=-2 and b=1<<16 (a+b is out of bounds)
• If a=(8<<28)|300 and b=100 (a is an invalid “new float” because it has a positive exponent (8) and the left-most bit of its mantissa (300) is 0)
• If a=getNewFloat(1023456789,3) and b=(5<<28)|1000 (b is an invalid “new float” because it has a positive exponent (5) and the left-most bit of its mantissa (1000) is 0)

Validation: Test your code by compiling it with the test code file

/share/ex_data/ex4/test_ex4.c using <QUESTION_ID>=2_4 (see instructions on page 1). Execute the resulting program and compare your output with /share/ex_data/ex4/test_ex4_2_4.out.

Final testing for Problem 2: Execute script /share/ex_data/ex4/test_ex4.2 from the directory containing your newFloat.c source file.

Submission Instructions:

1. After you validated and tested your solution, make sure that your ~/exercises/ex4/ directory contains the following C source file, which includes your implementation:
• newFloat.c
1. your ~/exercises/ex4/ directory should also contain a PARTNER file with the user id of the non-submitting partner. The non-submitting partner should also add a PARTNER file containing the user id of the submitting partner.
2. Check your solution by running check_ex ex4. The script should be executed from the account of the submitting partner, and it may be run from any directory. Clean execution of this script guarantees you 80% of the assignment’s grade.
3. Once you are satisfied with your solution, you may submit it by running submit_ex ex4. The script should be executed from the account of the submitting partner, and it may be run from any directory. You may modify your submission any time before the deadline (11/6 @ 21:00) by running submit_ex ex4 -o from any location.
4. For more information on the submission process, see the Homework submission instructions file on the course website.