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Number Comparison

Square Roots

Project Goals

This assignment invites you to implement a program that features multiple algorithms for computing the square root of a number. You will implement two algorithms that use iteration constructs to approximate the square root of a number. For a given step size and a tolerance level saying how close the approximation must be, the exhaustive algorithm check if each possible answer is within a specified tolerance for the square root's approximation. In contrast, the efficient algorithm will use bisection search to rapidly prune the search to the best possible approximation of the number's square root. In addition to adding source code and documentation to the provided Python files, you will conduct an experiment to determine which algorithm is the fastest and estimate by how much it is faster. As you enhance your technical skills and explore the experimental evaluation of algorithms, you will continue to program with tools such as VS Code and a terminal window and the Python programming language and the Poetry package manager.

Project Access

If you are a student enrolled in a Computer Science class at Allegheny College, you can access this assignment by clicking the link provided to you in Discord. Once you click this link it will create a GitHub repository that you can clone to your computer by following the general-purpose instructions in the description of the technical skills. Specifically, you will need to use the git clone command to download the project from GitHub to your computer. Now you are ready to add source code and documentation to the project!


If you are an emerging proactive programmer who is not enrolled in a Computer Science class at Allegheny College, you can still work on this assignment! To get started, you should click the "Use this template" button in the square-roots-starter GitHub repository and create your own version of this project's source code. After creating your GitHub repository, you can follow all of the other steps!

Expected Output

This project invites you to implement a square root calculation program called squareroot. The program accepts through its command-line a single integer value for which it will compute the square root using either an efficient or an exhaustive approach. Finally, the squareroot program offers a --profile flag that causes it to use the Pyinstrument package to collect timing information about the efficiency of the chosen method. If you use Poetry to run the program with the command poetry run squareroot --number 25000 --approach exhaustive --profile it produces the following output:

🧮 Attempting to calculate the square root of 25000!

✨ Was this search for the square root successful? Yes
✨ How many guesses did it take to compute the solution? 1581139 The best approximation for the square root of 25000 is 158.1139000041253

🔬 Here's profile data from performing square root computation on 25000!

  _     ._   __/__   _ _  _  _ _/_   Recorded: 10:25:39  Samples:  507
 /_//_/// /_\ / //_// / //_'/ //     Duration: 0.508     CPU time: 0.508
/   _/                      v4.0.3

Program: squareroot --number 25000 --approach exhaustive --profile

0.507 primality  squareroot/
└─ 0.507 compute_square_root_exhaustive  squareroot/
   ├─ 0.396 [self]
   └─ 0.111 abs  <built-in>:0
         [2 frames hidden]  <built-in>

If you run the program with the command poetry run squareroot --number 25000 --approach efficient --profile then it produces output like:

🧮 Attempting to calculate the square root of 25000!

✨ Was this search for the square root successful? Yes
✨ How many guesses did it take to compute the solution? 27 The best approximation for the square root of 25000 is 158.11389312148094

🔬 Here's profile data from performing square root computation on 25000!

  _     ._   __/__   _ _  _  _ _/_   Recorded: 10:29:41  Samples:  0
 /_//_/// /_\ / //_// / //_'/ //     Duration: 0.000     CPU time: 0.000
/   _/                      v4.0.3

Program: squareroot --number 25000 --approach efficient --profile

No samples were recorded.

It is worth noting that the exhaustive algorithm took 1581139 guesses to approximate the square root of 25000 while the efficient algorithm only needed 27! This shows that efficient's bisection search algorithm is significantly faster than the exhaustive approach. Interestingly, the Pyinstrument package reports that it could not record any performance data for squareroot when the program runs in efficient mode. Why do you think that this is the case? Is there any way to overcome this issue by, for instance, reconfiguring Pyinstrument or changing the input that you pass to the program? Overall, how much faster is efficient in comparison to exhaustive? You can use the equations and suggestions in the engineering effort about primality testing to calculate the percentage change in execution time when running squareroot in efficient mode instead of the exhaustive mode. Finally, to understand why efficient is faster than exhaustive you should study each function's source code and the textbook's content.

To learn more about how to run this program, you can type the command poetry run squareroot --help to see the following output showing how to use squareroot:

Usage: squareroot [OPTIONS]

  Use iteration to perform square root computation of a number and then
  perform profiling to capture execution time.

  --number INTEGER                [default: 5]
  --profile / --no-profile        [default: False]
  --approach [exhaustive|efficient]
                                  [default: efficient]
  --install-completion            Install completion for the current shell.
  --show-completion               Show completion for the current shell, to
                                  copy it or customize the installation.

  --help                          Show this message and exit.

Please note that the provided source code does not contain all of the functionality to produce this output. As explained in the next section, you are invited to add the missing features and ensure that squareroot produces the expected output. Once the program is working correctly, it should produce output similar to that shown in this section.


Don't forget that if you want to run the squareroot program you must use your terminal to first go into the GitHub repository containing this project and then go into the squareroot/ directory that houses the project's code. Finally, remember that before running the program you must run poetry install to add the dependencies.

Adding Functionality

If you study the file squareroot/squareroot/ you will see that it has many TODO markers that designate the parts of the program that you need to implement before squareroot will produce correct output. In summary, you should implement the following functions for the squareroot program:

  • def compute_square_root_exhaustive(x: int, epsilon: float = 0.01) -> Tuple[bool, float, int]
  • def compute_square_root_efficient(x: int, epsilon: float = 0.01) -> Tuple[bool, float, int]

Importantly, you will notice that both compute_square_root_efficient and compute_square_root_exhaustive accept the same types of inputs and produce the same types of outputs. In particular, the parameter called x is the number whose square root the function will compute and epsilon is the tolerance parameter describing how close the approximation of x's square root must be. The notation Tuple[bool, float, int] that describes the output of these functions shows that they each return three values. The first variable in the return value is a bool indicating whether or not the function found an answer within the tolerance of epsilon. Finally, the second returned variable is a float for the calculated value of the square root and the third one is an int for the number of guesses that the algorithm took.

You will also notice that there are some TODO markers in the squareroot function of the main module. In the scope of the conditional logic statement if approach.value == SquareRootCalculationingApproach.efficient on lines 1 through 7, the program should call the function compute_square_root_efficient depending on whether the profile variable specified on the command-line is True or False. If profile is True, then the program should use Pyinstrument to measure its execution time, as illustrated on lines 3 through 5. However, if profile is False, then the program should only call the compute_square_root_efficient as shown on line 7. As lines 9 through 15 show, the function should take analogous steps for its exhaustive mode, calling the compute_square_root_exhaustive instead of compute_square_root_efficient.

if approach.value == SquareRootCalculationingApproach.efficient:
    if profile:
        square_root_tuple = compute_square_root_efficient(number)
        square_root_tuple = compute_square_root_efficient(number)
# use the exhaustive square root computation algorithm
elif approach.value == SquareRootCalculationingApproach.exhaustive:
    if profile:
        square_root_tuple = compute_square_root_exhaustive(number)
        square_root_tuple = compute_square_root_exhaustive(number)

Once you have correctly resolved all of the TODO markers in the squareroot program, it should produce the expected output described in the previous section. With that said, please bear in mind that, when running squareroot with the --profile flag it will produce different profiling data depending on the performance of your computer.

Running Checks

If you study the source code in the pyproject.toml file you will see that it includes a section that specifies different executable tasks like lint. If you are in the squareroot/ directory that contains the pyproject.toml file and the poetry.lock file, the tasks in this section make it easy to run commands like poetry run task lint to automatically run all of the linters designed to check the Python source code in your program and its test suite. You can also use the command poetry run task black to confirm that your source code adheres to the industry-standard format defined by the black tool. If it does not adhere to the standard then you can run the command poetry run black squareroot tests or, alternatively, poetry run task fixformat, and it will automatically reformat the code!

Along with running tasks like poetry run task lint, you can leverage the relevant instructions in the technical skills to run the command gatorgrade --config config/gatorgrade.yml to check your work. If your work meets the baseline requirements and adheres to the best practices that proactive programmers adopt you will see that all the checks pass when you run gatorgrade. You can study the config/gatorgrade.yml file in your repository to learn how the GatorGrade program runs GatorGrader to automatically check your program and technical writing.

If your program has all of the anticipated functionality, you can run the command poetry run task test and see that the test suite passes and produces output like this:

collected 3 items

tests/ ....

Don't forget that when you commit source code or technical writing to your GitHub repository for this project, it will trigger the run of a GitHub Actions workflow. If you are a student at Allegheny College, then running this workflow consumes build minutes for the course's organization! As such, you should only commit to your repository once you have made substantive changes to your project and you are ready to confirm its correctness. Before you commit to your repository, you can still run checks on your own computer by using the GatorGrade program to run GatorGrader.

Project Reflection

Once you have finished all of the previous technical tasks, you can use a text editor to answer all of the questions in the writing/ file. For instance, you should provide the output of the Python program in a fenced code block, explain the meaning of the Python source code segments that you implemented and tested, compare and contrast the performance of different implementations of the square root calculation, and answer all of the other questions about your experiences in completing this project.

Project Assessment

Since this is a programming project, it is aligned with the applying and analyzing levels of Bloom's taxonomy. You can learn more about how a proactive programming expert will assess your work by examining the assessment strategy. From the start to the end of this project you may make an unlimited number of reattempts at submitting source code and technical writing that meet all aspects of the project's specification.

Seeking Assistance

Emerging proactive programmers who have questions about this project are invited to ask them in either the GitHub discussions forum or the Proactive Programmers Discord server. Before you ask your question, please read the advice concerning how to best participate in the Proactive Programmers community. If you find a mistake in this project, please describe it and propose a solution by creating an issue in the GitHub Issue Tracker.

Updated: 2023-03-17   Created: 2021-09-16
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