Implement a rational arithmetic package and a client program that uses it to compute rational approximations to e. The purpose of this assignment is to learn about type definitions, packaging interfaces and implementations, separate compilation, and arithmetic overflow.

Rational numbers are numbers that can be represented as the ratio of two integers, i.e., any number p/q where p and q are integers is a rational number. C approximates noninteger rational numbers using floats or doubles, but these types are imprecise representations. For this assignment, you will represent rational numbers with a C structure that comprises two integers:

struct { int p; int q; }

typedef struct { int p; int q; } Rational;

int x, y;
Rational a, b;
a = RATinit(1, 3);
b = RATadd(a, RATinit(1, 4));
x = b.p; y = b.q;

First, make a file named RAT.h that contains the following code:

typedef struct { int p; int q; } Rational;
Rational RATinit(int, int);
    void RATshow(Rational);
Rational RATadd(Rational, Rational);
Rational RATmul(Rational, Rational);

Second, make a file named RAT.c and write straightforward implementations of all the functions. The first line of this file should include the interface, as follows:

#include "RAT.h"

Third, write a program named e.c that uses your package to compute a rational approximation to e, using the series expansion e = 1 + 1/1! + 1/2! +1/3! + 1/4! + 1/5! + 1/6! + ... Print out the value that you get after each term is added to the approximation. The first few terms are

1/1
2/1
5/2
32/12

A straightforward implementation of the rational arithmetic interface is easy, but has two flaws that will jump out at you as you add terms to the list above to get more accurate approximations. First, in the fourth line in the example in the previous paragraph, both numerator and denominator are divisible by 4, so we would prefer to see the result 8/3 instead of 32/12. You can fix this problem with Euclid's algorithm (see Sedgewick, p. 191), using it to change your package to adhere to the convention that functions only returns rational numbers whose numerator and denominator are relatively prime. The second problem, which is more subtle, is overflow: The products formed by the rational arithmetic functions might overflow, which leads to erroneous results, even if the result can be represented. For example, a/b × c/d is equal the reduced value of ac/bd, and either ac or bd might overflow before the reduction, even though the reduced answer might represent no problem (for example, suppose that a and d are equal and huge). You can't avoid overflow, but you can stave it off by exercising sound judgement when you use Euclid's algorithm in the implementations of the rational arithmetic functions.

The fourth part of this assignment is to develop an improved implementation RATbetter.c that handles a wider class of rationals than the straightforward implementation. Explain the approach that you use in your readme file, and don't go overboard. Note that you do not have to change the interface and the client at all: they can use either implementation. However, presumably the client will get more accurate answers with the improved implementation. Set the client to print out the maximum number of terms that you can get without having overflow.

As usual, turn in your code and your documentation with the command

/u/cs126/bin/submit 3 readme RAT.h RAT.c e.c RATbetter.c