Assignment 4
Calculator
CSE 13S, Winter 2024
1 Introduction
It is common knowledge that computer scientists have very few hobbies. One of the hobbies that almost every computer science major has is making worse versions of software that already exists. While there are many existing calculator applications, in this assignment, not only will you be making a basic scientific calculator, you will also be making a subset of the functions in <math .h> . You will also be learning more about how computers work, and how they approximate mathematical values. Finally, you will get a chance to test your debugging and testing skills.
2 Helpful information
Reverse Polish Notation
In your academic career, you have probably used a lot of calculators, and most of them have probably had different ways to input an expression. Because it is hard to interpret and implement a calculator that inputs an expression the way that one might write it, we will use the standard Reverse Polish Notation. You can read more about it here, or in your C textbook. RPN always takes numeric values first, followed by an operator to apply. For example, the expression 2 + 2 = would be entered as 2 2 + [enter].
We can also chain these to make more complicated expressions. For example, 2 2 + 1 - is the expression (2+2) − 1. You’ll notice that there is no need for parentheses in RPN, as it is always explicit which numbers are provided to which operator. With enough operators and a big enough buffer to hold previous results, it is possible to create a working, useful calculator.
RPN calculators are often implemented using a Stack ADT, which we will discuss more in the next section.
The stack ADT
To understand a Stack, we must first understand what an ADT is. ADT stands for "abstract data type". ADT’s are simple instructions that describe the input and output of a data type, without giving any rules for how they will be implemented internally. For example, a stack ADT could be implemented with a singly linked list, a doubly linked list, an array, or many other things. Depending on the use case, each of these has its advantages, but they are all stacks. Instead of describing implementation, ADT’s only describe functionality.
A good example of a stack is a stack of pancakes on a plate. There are only two things you can do to the stack: add one pancake to the top of the stack or remove a pancake from the top of the stack. Notice that there is noway to see or modify elements in the middle of the stack without removing all elements above it first.
A stack in its most basic form has 2 operations: Push and pop. Many versions of the stack ADT extend this functionality. To pushto the stack is to add an element to the top of the stack. To pop from the stack is to remove the top element and crucially, return it to the user. Obviously, popping from an empty stack will return an error.
For simplicity, our stack implementation has a fixed capacity. This means that we decide the maximum number of elements it can contain at compile time, its capacity. This is not to be confused with its size, which is the number of elements in the stack. If a stack is at is capacity, it is not able to push an element, but it can still pop an element.
Some extensions of the stack include having a function that checks if the stack is empty (has a size of zero elements) or full (has a size of capacity elements if it is memory restricted), Printing the elements in the stack, emptying the stack, or a "peek" function, that returns a copy of the top element of the stack. The reason that peek is not part of the base stack ADT is that a peek can be performed by popping an element, returning a copy to the user, and pushing the same element back to the stack.
Libraries
Please read Ben’s guide to compilation for more info.
Command Line Options and Arguments
Please read Ben’s Guide to Options for more info.
Math
We have already covered the fact that the numeric data types in C are imprecise. You can verify this by adding .1 and .2. While the output may look normal and correct at 16 digits, we are quickly reminded of the limitations of technology at 20 digits: the result is 0.30000000000000004! This occurs because floating- point numbers can only precisely represent values that can be written as fractions with a denominator that is a power of 2. Just like how 1/3 cannot be written exactly in decimal, the binary representation of fractions like 1/10 have patterns that should carry on infinitely, but are limited by the precision of numeric types.
This limitation is combined with the fact that one can not store the exact value of irrational numbers on a computer (or anywhere, really). Because we are already so limited, computers use approximations of standard math functions as well. One well known method to approximate a function is the Taylor Series. This method approximates any function as a polynomial. As we increase the degree of the polynomial, the function becomes more accurate. The Taylor series uses a starting point, and repeatedly finds the derivatives, adding a new term with higher degree each time. This process increases the precision.
We can generalize the Taylor Series further. The Maclaurin Expansion is the Taylor Series centered around x = 0. Because we are only approximating functions between x = 0 and x = 2π, we can use the Maclaurin expansion. To understand how the Maclaurin expansion of sin x works, see this link.
The formula for the Taylor Series can be written as
Our first natural question should be "How can a computer take a sum to ∞ ? Wouldn’t that take forever?". This is where we must approximate. As we sum more terms, we get an increasingly precise answer. The next natural question is "How do we know where to stop?" We can pick a very small number, ϵ ("epsilon"), and continue our summation until our the absolute value of next term is no longer greater than epsilon. All the series that we use add smaller and smaller terms in order to refine a more and more accurate answer, so when the absolute value of the term is very small we know that our approximation is very accurate. For this assignment we set ϵ = 10 −14, which is defined in mathlib .h.
Our next simplification is turning this Taylor series into a Maclaurin Series. We can do that because the a in x − a is where we center our approximation, and that will always be x = 0. Now, we get the formula
It is possible to simplify this further. You can find a pattern in f (n)(0) which will allow you to simplify this significantly. That pattern is alternating as follows: 0, 1, 0, -1, 0, 1 ... which can be simplified further by removing the zeroes. This makes the final function for an approximation of sin
And the approximation of cosine
Computing tan x is much simpler. We know that the tan We can use this fact to find tan x without using a series. You should use this in your code. If you are concerned about dividing by zero, you should remember that our functions will create an approximation, and therefore will never actually equal zero.
Function Pointers
You already know that your C program can have pointers to data variables. When you declare char *s; double *d; then s is a pointer to a char, and d is a pointer to a double. Once your program sets these pointer variables to valid addresses, your program can “dereference” them using the * prefix operator. So *d can be used as a double value.
The C programming language also lets your program declare a pointer to a function. Such a pointer can be set to the address of function, optionally passed as a parameter, and then called, just as the function would be called.
An example is using the function apply_unary_operator(). You can call this function like apply_- unary_operator(sqrt);. You may recognize the parameter sqrt as is the name of a math function. In this context (with no parentheses after it), sqrt indicates a pointer to the sqrt() function.
Your program uses a function pointer as if it is a function name. That is, follow a function pointer with (), containing any necessary parameters, and whatever function the pointer is pointing to will be called.
In operators .h you will see two typedef lines. We use these because the programming-language syntax of function pointers is, let’s say, complicated. It is easier to declare two types binary_operator_fn and unary_operator_fn and then use these two types to declare function pointers.
So, looking at this declaration, bool apply_unary_operator(unary_operator_fn op);, the parameter op is of type unary_operator_fn. The typedef of unary_operator_fn says that op is a pointer to a function that accepts one double parameter and returns a double.
In operators .h you will see three curious array definitions. These are each 256 bytes long, so you can index them with a character. Each one has a few function pointers at various indices that you can use to run a function given a character input. You can read more about this in the operators section.
3 Your task
Workflow
1. Complete and submit your design.pdf draft by Monday, February 12th.
2. Implement and test the functions in the files operators .c, mathlib.c, and stack .c
3. Create your test file, tests .c, and the tests you will use. Test all the code you have written, and make sure that it is fully functional.
4. Complete your main function in calc .c and test your code
5. Fix any issues with your design draft, and finish the results section
6. Submit your final commit ID by Wednesday February 14th
Starter Files
The files we will provide to you to help you with this are as follows:
• asgn3.pdf: This file
• mathlib .h: A header file for implementing the math functions.
• messages .h: Contains the error and help messages you will need to printout.
• operators .h: Contains declarations for wrapper functions for most operators.
• stack .h: Contains the stack ADT function declarations
• calc: The reference program for this assignment.
Required functions
mathlib
In mathlib .c, you will be implementing the sin, cos and tan, and absolute value functions. Square root will be given to you.
double Abs(double x)
This function returns the absolute value of a double. You will not need to use math .h to complete this.
double Sqrt(double x)
This function returns the square root of a double.
double Sqrt(double x) { // Check domain. if (x < 0) { return nan("nan"); } double ld = 0.0; double new = 1.0; while (Abs(old - new) > EPSILON) { // Specifically, this is the Babylonian method--a simplification of // Newton's method possible only for Sqrt(x). ld = new; new = 0.5 * (old + (x / old)); } return new; }
For all of the following functions, you should normalize the input (which comes to you in radians) to something in the range (0, 2π)
double Sin(double x)
This function returns the sine of a double. The angle passed in to this function will be in radians. For the sake of accuracy, you should normalize the angle to something between zero and 2π You will not use sin from math .h to complete this.
double Cos(double x)
This function returns the cosine of a double. The angle passed in to this function will be in radians. You will not use cos from math .h to complete this.
double Tan(double x)
This function returns the tangent of a double. The angle passed in to this function will be in radians. You will not use sin, cos, or tan from math .h to complete this.
operators
In operators .c you will be implementing apply_binary_operator, along with a few small wrapper func- tions. parse_double and apply_unary_operator will be given to you.
operators .h contains three arrays of function pointers: for binary operators like addition, unary oper- ators like sine using your own functions, and unary operators using the standard math .h functions. These all have size 256 which means that you can index them using an ASCII character that the user input. For instance, binary_operators contains the entry [ ' + ' ] = operator_add. This means that the index ' + ' (remember, characters are numbers!) of this array is set to operator_add. So if the user inputs +, you could look it up in this array and find which function you should use to perform that operator. Since these arrays are global, the unspecified elements are set to zero or NULL, so you can use that to check if some character is avalid operator.
bool apply_binary_operator(binary_operator_fn op)
This function takes in an operator, and accesses the global stack. It then pops the first 2 elements on the stack, and calls the op function with those as its arguments (the first element popped is the right-hand side, and the next element popped is the left-hand side). Finally, it pushes the result to the stack.
There is no case in which this will make the stack too big. It is possible that there are not two elements to pop. If that is the case, this function will return an false, and not print anything (error handling should be done in calc.c). It will return true on success.
If, for example, the stack contains (bottom to top) "2, 4, 1", and we apply the "+" operator, the stack will end with "2, 5".
bool apply_unary_operator(unary_operator_fn op)
This function takes in an operator, and accesses the global stack. It then applies the operator to the first element on the stack and pushes the result to the stack.
There is no case in which this will make the stack too big. It is possible that there are not enough elements to pop. If that is the case, this function will return an error, and not print anything.
bool apply_unary_operator(unary_operator_fn op) { // Make sure that the stack is not empty if (stack_size < 1) { return false; } double x; // It should be impossible for stack_pop to fail here, since we have checked // above that the stack is not empty. The assert() function causes your // program to exit if the condition passed is false. It is often useful for // "sanity checks" like this: if stack_pop fails, either our assumption that // it can't fail was false, or the implementation of stack_pop has a bug. // Either way, we'd like to know immediately instead of silently ignoring an // error. assert(stack_pop(&x)); // Calculate the value we should push to the stack double result = op(x); // Similar to above: stack_push should never fail here, because we just // popped an element, so we are not increasing the stack size past the size // it had originally. assert(stack_push(result)); return true; }
double operator_add(double lhs, double rhs)
This function takes the sum of the doubles lefthand side (lhs) and right hand side ( rhs) and returns it.
Yes, this function is as simple as it sounds (as are the next three). We need these basic math operators to be functions so that we can store them in arrays of function pointers.
double operator_sub(double lhs, double rhs)
This function takes the difference of the doubles lefthand side (lhs) and right hand side ( rhs) and returns it. (lhs − rhs)
double operator_mul(double lhs, double rhs)
This function multiplies the doubles lefthand side (lhs) and right hand side ( rhs) and returns it.
double operator_div(double lhs, double rhs)
This function divides the double lefthand side (lhs) by right hand side ( rhs) and returns it. (rhs/lhs)
bool parse_double(const char *s, double *d)
This function attempts to parse a double-precision floating point number from the string s. If successful, it stores the number in the location pointed to by d and returns true. If the string is not a valid number, it returns false.
Here is the implementation of this function. It uses the library function strtod. endptr is set by strtod to point to the end of the number that was parsed from the input string. We check if endptr is the same as s to see if parsing failed; if they are the same, it means there was not a valid number in the string s therefore the number ends immediately at the start of the string.
bool parse_double(const char *s, double *d) { char *endptr; double result = strtod(s, &endptr); if (endptr != s) { *d = result; return true; } else { return false; } }
stack
In stack .c, you will be implementing the stack ADT functions.
The stack that you will be building contains the following data:
• an int for storing the size (not capacity!)
• an array of STACK_CAPACITY doubles, which you should initialize to zero.
These variables are defined in stack .h as extern, meaning that you can access them in any file that includes stack.h, but must only define and initialize them once in stack .c. It’s very similar to how functions are declared in header files and defined in the corresponding C file. You may modify them as you see fit. As much as possible, you should avoid touching the stack outside of stack .c. You will be implementing functions to push, pull, peek, pop, and clear the stack. We have provided a function to print the stack.
bool stack_push(double item)
Pushes item to the top of the stack. Updates stack_size. Returns true if a push was possible, and false if the stack is at capacity. If the stack is at capacity, the stack and size should not be modified.
bool stack_peek(double *item)
Without modifying the stack, copies the first item of the stack to the address pointed to by item. Returns false, and does not modify *item if the stack is empty, and true otherwise.
Why do this function and stack_pop take pointer arguments? We’d really like these functions to return two values: whether a value was popped/peeked successfully, and the value itself. But C functions can only return one value. To solve this, we make the success state the actual return value, and handle the numeric value through a pointer. Whoever calls this function should set up a variable in which they want the peeked value to go, and then pass the address of that variable into stack_peek. This will allow stack_peek to change the contents of that variable to the value that was peeked. See our implementation of apply_unary_operator for an example of how this looks in C code.
bool stack_pop(double *item)
Stores the first item of the stack in the memory location pointed to by item. Returns false, and does not modify item if the stack is empty, and true otherwise. Updates stack_size. Note that you do not need to change the element in the stack array, as it will get overridden when stack_push is called.
void stack_clear(void)
Sets the size of the stack to zero. Values in the stack do not need to be modified. See stack_pop for more info.
void stack_print(void)
This function prints every element in the stack to stdout, separated by spaces. It does not print a newline after. We give you the code for this function to ensure that your results are printed the same way as ours:
void stack_print(void) { // make sure we don't print stack[0] when it is empty if (stack_size == 0) { return } // print the first element with 10 decimal places printf("%.10f", stack[0]); // print the remaining elements (if any), with a space before each one for (int i = 1; i < stack_size; i++) { printf(" %.10f", stack[i]); } }
Assignment specifics CLI options
In this assignment (and future ones), we will be using atool called getopt. It will help us process command line arguments from argc and argv. This program will support the following arguments:
• m: Uses trig functions from libm.
• h: Prints the help message,
Errors
This is the first assignment in which we will be using stderr. When we use printf, everything we print is sent to standard out, aka stdio. We can use the fprintf function to choose where we print anything (see the man pages for more). All errors must be printed to stderr. If the user enters an invalid option, the help message will be printed to stderr, and the program will quit.
While stdout and stder r are both printed to your terminal by default, they are different. You can tell them apart using your shell’s redirection operators: > redirects stdout to a file and 2> redirects stderr to a file. Combined with < to prove stdin from an existing file, you could run a command like ./calc < input.txt > out.txt 2> err.txt, which redirects stdout and stderr to different files so that you can make sure they are split correctly.
You must use all errors in messages .h correctly. If you encounter an error with starting the program (like an invalid option), you must return a nonzero exit code from main (you can run echo $? immediately after your program to check what the exit code was). If you encounter any other error (like empty stack or invalid operator), you must print the respective error message and continue running the program. If your program quits normally due to EOF (which you can accomplish by pressing Ctrl+D), it should exit with code 0 (even if some of the expressions that the user input were not valid).
Using math.h
In your mathlib .c file, you can only use 3 things from math .h. Those are: M_PI (The value of π), fmod, which allows you to use the modulo operator for floating point values, and nan, which the provided Sqrt implementation uses for its return value when the input is negative. All other usage of math.h in mathlib .c will be considered an attempt to cheat. Obviously, you may use math .h in calc .c or any other file.
Parsing user input
When you start the calculator, you will print a > character to standard err (stderr). You need to read one line of input at a time, and then split it into words separated by the space character and process each word (which may be a number, like 5.3, or an operator like + or r) one by one. Reading a line of input may be done with the function fgets on a suitably sized buffer (1024 bytes is plenty). Keep in mind that fgets will store a newline character at the end of your buffer which you’ll have to remove.
Once you’ve parsed a line of input into expr, you should use the strtok_ r function according to the following template to split it into words:
// saveptr is a variable that strtok_r will use to track its state as we // call it multiple times char *saveptr; // you can set error to true to stop trying to parse the rest of the expression bool error = false; // strtok_r splits the input string (expr) on any delimiter character in a // sequence (for us, only spaces) // it stores its own state in saveptr // it returns a pointer to the next token, or NULL if we have processed the entire string const char *token = strtok_r(expr, " ", &saveptr); // loop until we finish parsing the string or we encounter an error while (token != NULL && !error) { // process token // possibly set error to true // then, at the very end of your loop... // (note that we pass NULL instead of expr to indicate we are still processing the same string) token = strtok_r(NULL, " ", &saveptr); }
You will then apply the operator specified, and use stack_print to print all the contents of the stack to stdout.
Submission
These are the files we require from you:
• Makefile: This should have rules for all, calc, tests and all .o files with the following com- piler flags: -Werror -Wall -Wextra -Wconversion -Wdouble-promotion -Wstrict-prototypes - pedantic. You may use other flags for the purposes of debugging, but you must remove them for your final submission. make all should make both tests and calc binaries. You must also have targets for make clean and make format.
• design.pdf: Instructions can be found in the report section( 3).
• mathlib .h: This file is given. It contains function declarations for the math functions you will be implementing.
• messages .h: This file is given. It contains some error messages you need to print.
• operators .h: This file contains some function declarations. You should not need to modify this file.
• stack .h: This file contains function declarations for the Stack ADT
• calc .c: This file should contain your main program.
• tests .c: This file should contain a main function that tests other code in this assignment.
• mathlib .c: This file should implement all required math functions using their approximations.
• operators .c: This file should contain the basic operators in their wrapper functions.
• stack.c: This file will contain your implementation of the stack ADT.
• The tests folder, and associated tests
You must also submit any additional code you wrote to create the graphs. If any of this code is written in C, we recommend adding a rule to your Makefile to compile this code.
All header files may be modified to add more functions or variables, but existing function declarations may not be modified.
All header files and source files must be clang-formatted by running make format before you submit them.
Remember that you may use code from lecture, the textbook, this pdf, my guide, or sections, but they must be cited (at least) as a comment in your code and in your design.pdf. We do not require any particular citation style, but you must include the name of the author, and the resource location (page, slide, file... location). If you chose to use code from any of these resources, you should also understand how this code works.
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