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Assignment 1

Instructions

• PHY410: Do problems 1–3
• PHY 505: Do problems 1-4

Accept the assignment from "Classroom 50" website: https://classroom50.org/ubsuny/compphys-fall26/assignments/assignment-1/accept. This will create a new repository for you on github, titled something like github.com/ubsuny/ubsuny/compphys-fall26-assignment-1-username. The repository is located in the ubsuny github group, but it is your personal repository, and only you can view it. Technical detail: the repository is actually not a fork, but rather a brand new repository created by directly copying files from a "template repository."

The assignment requires two uploads:

1. Submit a writeup/lab report to UBLearns. The writeup should contain your complete solutions to the problems in writing. Feel free to use any format: latex, markdown, etc. For assignments using Jupyter notebooks, you can also do your writeup inside the notebook using Markdown cells, export the notebook to HTML/PDF/etc. (this may require some additional setup), and upload the exported notebook.
2. Upload your code through Classroom 50. This just means to git push any code you wrote to your GitHub repository. You can push as many times as you like before the due date (in fact, this is a good idea, as it serves as a backup; nothing will be graded before the due date).

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Problem 1

25 points

What are the two's complement representations for the following numbers? Use N=16 bits.

• 10
• 436
• 1024
• -13
• -1023
• -1024

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Problem 2

25 points

Suppose I want to compute the following series:

$$ \begin{aligned} f_0 &= 2 \\ f_n &= f_{n-1}^2 \end{aligned} $$

What is the maximum $n$ that can be stored in the following C++ data types, assuming that an int is 4 bytes, a long int is 8 bytes, and each byte stores 4 bits? Make sure to explain your reasoning in your answer.

• int
• long int
• unsigned long int

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Problem 3

25 points

A small C++ program has been provided that demonstrates the series from Problem 2 for the int type. The program prints out each element in the series, thus allowing you to see directly what happens when your data type is exceeded.

Importantly, the series is implement using a simple recursive function that ensures that the data type always stays the same. You might be tempted to simplify the program, using the fact that the series is simply $f_n = 2^{2^n}$ and exploiting the built-in pow() function. However, if you check the documentation, you'lll see that pow() implcitly converts the inputs to floating point numbers, which would ruin your experiment for the desired type!

Problem 3a

10 points

Compile and execute the program. For example:

g++ Problem3.cpp -o Problem3
./Problem3

(MacOS users can use clang++ instead of g++; your pixi environment probably installed clang++ and not g++.)

Does the program find the boundary where overflow occurs? If not, modify the program accordingly. In your writeup, include the full output of the program.

Problem 3b

15 points

Does the boundary match what your answer from Problem 2? Explain why or why not.

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Problem 4

25 points

PHY505 students only

PHY410 students are welcome to submit, but it will not count towards the grade.

Problem 4a

15 points

Copy Problem3.cpp to Problem4.cpp and modify the new program to test additional data types:

• int
• long int
• unsigned long int

Make sure to extend the $f_n$ series far enough to find the overflow boundary. In your writeup, include the output of your program and describe the observed differences between the data types. Also, make sure to git add/commit/push your new .cpp file.

Problem 4b

10 points

Modify the program (or write a new program) to find the actual maximum value for each data type, on your own computer. You might want to make an educated guess as a starting point (see https://en.cppreference.com/cpp/language/types), and focus your search on the known boundaries. Demonstrate the overflow behavior with std::cout statements, as in Problem 3.

In your writeup, include the program output, state the maximum value for each type, and determine the number of bits used for each type. Make sure to git add/commit/push your C++ code.