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2. Problem : Rational Number Library
Now that we know about the rounding issues of floating-point arithmetic, we’d like
to implement a library that will allow for exact computation with rational numbers.
This library should be easy and intuitive to use. You want to be able to write code as
straight-forward as you can see in this sample:
which produces the following as output.
auto a = Rational{ 1, 3 }; // the number 1/3
auto b = Rational{ -6, 7 }; // the number -6I7
std::cout «« "a * b = " «« a «« " * " «« b «« " = " «« a * b «« "n";
std::cout «« "a I ( a + b / a ) = " «« a / ( a + b / a ) «« "n";
a * b = 1I3 * -6I7 = -2/7
a / ( a + b / a ) = -7/47
Of course, we’ll be able to do much more impressive things. For example, here is
a snippet of code that computes the golden ratio ϕ to over 15 decimal digits of
accuracy using a very readable implementation of its continued fraction expansion.
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3. const int ITER = 40;
auto phi = Rational{ 1 }; II set equal to 1 to start off with
for( int i = 0; i « ITER; ++i ) {
phi = 1 I ( 1 + phi );
}
std::cout «« std::setprecision( 15 ); II look up «iomanip>
std::cout «« "phi = " «« ( 1 + phi ).to double() «« "n";
This prints out phi = 1.61803398874989, which happens to be 15 completely accurate
digits.
You can find the zipped project at provided in the file rational.zip as a basis for your
program. As in the other project problem, the folder contains an includeI directory,
our standard project Makefile, and a srcI directory. All of our header (.h) files will
be found in includeI while all of our source (.cpp) f iles will be in srcI. The C++
files you should find are rational.h, gcd.h, rational.cpp, gcd.cpp, and test.cpp; you
will be modifying rational.h and rational.cpp. Like in the previous project problem,
you can modify the Makefile or test.cpp to run your own tests, but your changes
there will be overwritten when you upload to the grader.
We are provided a header file describing the interface for the Rational class.
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4. #ifndef 6S096 RATIONAL H
#define 6S096 RATIONAL H
#include «cstdint>
#include «iosfwd>
#include «stdexcept>
class Rational {
intmax t num, den;
public:
enum sign type { POSITIVE, NEGATIVE };
Rational ( ) : num{0}, den{1} {}
Rational ( intmax t numer ) : num{numer}, den{1} {}
Rational ( intmax t numer, intmax t denom ) : num{numer}, den{denom} {
normalize(); // reduces the Rational to lowest terms
}
inline intmax t num() const { return num; }
inline intmax t den() const { return den; }
// you'll implement all these in rational.cpp
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5. void normalize( );
float to float() const;
double to double() const;
sign type sign() const;
Rational inverse() const;
};
// An example multiplication operator
inline Rational operator*( const Rational &a, const Rational &b ) {
return Rational{ a.num() * b.num(), a.den() * b.den() };
}
// Functions you'll be implementing:
std::ostream& operator««( std::ostream& os, const Rational &ratio );
// ... and so on
Notice the include guards at the top, protecting us from including the file multiple
times. An expla nation of some of the constructs we use is in order. You’ll notice
we use the intmax t type from the «cstdint> library. This type represents the
maximum width integer type for our particular architecture (basically, we want to
use the largest integers available). On my 64-bit computer, this means that we’lll
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6. f ind sizeof(Rational) to be 16 since it comprises two 64-bit (8-byte) integers.
Look up an enum. We use this to define a type sign type in the scope of our
class. Outside the class, we can refer to the type as Rational::sign type and its two
values as Rational::POSITIVE and Rational::NEGATIVE.
Contained in the header file is also a class that extends std::domain error to create
a special type of exception for our Rational class. Look up the meaning of explicit;
why do we use it in this particular constructor definition?
Read through the code carefully and make sure you understand it. If you have any
questions about the provided code or don’t know why something is structured the way it
is, please ask about it on Piazza.
This is the list of functions you should fill out. In the header file, you have a
number of inline functions to complete, as shown here.
#include <stdexcept>
// ...near the bottom of the file
class bad rational : public std::domain error {
public:
explicit bad rational() : std::domain error( "Bad rational: zero denom." ) {}
};
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7. inline Rational operator+( const Rational &a, const Rational &b ) {/* ... */}
inline Rational operator-( const Rational &a, const Rational &b ) {/* ... */}
inline Rational operatorI( const Rational &a, const Rational &b ) {/* ... */}
inline bool operator«( const Rational &lhs, const Rational &rhs ) {/* ... */}
inline bool operator==( const Rational &lhs, const Rational &rhs ) {/* ... */}
There are also a number of functions declared in the header file which you
should write an implemeatation for in the source file rational.cpp. Those declarations
are:
class Rational {
// ...snipped
public:
// ...snipped
void normalize();
float to float() const;
double to double() const;
sign type sign() const;
};
std::ostream& operator««( std::ostream& os, const Rational &ratio );
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8. Look in the existing rational.cpp file for extensive descriptions of the required
functionality. In gcd.h, you can find the declaration for two functions which you
may find useful: fast computation of the greatest common divisor and least
common multiple of two integers. Feel free to include this header and use these
functions in your code as needed.
#ifndef 6S096 GCD H
#define 6S096 GCD H
#include «cstdint>
intmax t gcd( intmax t a,
intmax t b ); intmax t lcm( intmax t a, intmax t b );
#endif // 6S096 GCD H
Input Format
Not applicable; your library will be compiled into a testing suite, your
implemented functions will be called by the program, and the behavior checked
for correctness. For example, here is a potential test:
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9. #include "rational.h"
#include «cassert>
void test addition() {
// Let's try adding 1/3 + 2/15 = 7/15.
auto sum = Rational{ 1, 3 } + Rational{ 2, 15 }
assert( sum.num() == 7 );
assert( sum.den() == 15 );
}
You are strongly encouraged to write your own tests in test.cpp so that you can
try out your imple mentation code before submitting it to the online grader.
Output Format
Not applicable.
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10. Code : Rational Number Library
#ifndef _6S096_RATIONAL_H
#define _6S096_RATIONAL_H
#include <cstdint>
#include <iosfwd>
#include <stdexcept>
class Rational {
intmax_t _num, _den;
public:
enum sign_type { POSITIVE, NEGATIVE };
Rational() : _num{0}, _den{1} {}
Rational( intmax_t numer ) : _num{numer}, _den{1} {}
Rational( intmax_t numer, intmax_t denom ) : _num{numer}, _den{denom} { normalize(); }
inline intmax_t num() const { return _num; }
inline intmax_t den() const { return _den; }
void normalize();
float to_float()const;
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11. double to_double()const;
sign_type sign() const;
Rational inverse() const;
};
std::ostream& operator<<( std::ostream& os, const Rational &ratio );
inline bool operator==( const Rational &lhs, const Rational &rhs ) { return lhs.num() *
rhs.den() == rhs.num() * lhs.den();
}
inline bool operator<( const Rational &lhs, const Rational &rhs ) {
if( lhs.sign() == Rational::POSITIVE && rhs.sign() == Rational::POSITIVE ) {
return lhs.num() * rhs.den() < rhs.num() * lhs.den();
} else if( lhs.sign() == Rational::NEGATIVE && rhs.sign() == Rational::NEGATIVE ) {
return lhs.num() * rhs.den() > rhs.num() * lhs.den();
} else {
return lhs.sign() == Rational::NEGATIVE;
}
}
inline Rational operator*( const Rational &a, const Rational &b ) {
return Rational{ a.num() * b.num(), a.den() * b.den() };
}
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12. inline Rational operator+( const Rational &a, const Rational &b ) {
return Rational{ a.num() * b.den() + b.num() * a.den(), a.den() * b.den() };
}
inline Rational operator-( const Rational &a, const Rational &b ) {
return Rational{ a.num() * b.den() - b.num() * a.den(), a.den() * b.den() };
}
inline Rational operator/( const Rational &a, const Rational &b ) {
return a * b.inverse();
}
class bad_rational : public std::domain_error {
public:
explicit bad_rational() : std::domain_error("Bad rational: zero denominator" ) {}
};
#endif // _6S096_RATIONAL_H
Here is the source code file rational.cpp:
#include "rational.h"
#include "gcd.h"
#include <stdexcept>
#include <ostream>
#include <iostream>
#include <cmath> cpphomeworkhelp.com

13. Rational Rational::inverse() const {
return Rational{ _den, _num };
}
Rational::sign_type Rational::sign()const {
return _num >= 0 ? POSITIVE : NEGATIVE;
}
std::ostream& operator<<( std::ostream& os, const Rational &ratio ) {
if( ratio == 0 ) {
os << "0";
} else {
if( ratio.sign() == Rational::NEGATIVE ) {
os << "-";
}
os << std::abs( ratio.num() ) << "/" << std::abs( ratio.den() );
}
return os;
}
void Rational::normalize() {
if( _den == 0 ) {
throw bad_rational();
} if( _num == 0 ) {
_den = 1; return;
}
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14. auto g = gcd( std::abs( _num ), std::abs( _den ) );
_num /= g; _den /= g;
if( _den < 0 ) {
_num = -_num;
_den = -_den;
}
}
float Rational::to_float() const {
return static_cast<float>( _num ) / static_cast<float>( _den );
}
double Rational::to_double()const {
return static_cast<double>( _num ) / static_cast<double>( _den );
}
Below is the output using the test data:
rational:
1: OK [0.007 seconds] OK! add
2: OK [0.006 seconds] OK! mult
3: OK [0.009 seconds] OK! add1024
4: OK [0.014 seconds] OK! add1024
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15. 5: OK [0.158 seconds] OK! add32768
6: OK [0.007 seconds] OK! op<<
7: OK [0.289 seconds] OK! div65536 in 0.280000 s
8: OK [0.006 seconds] OK! phi, 0.000000e+00
9: OK [0.006 seconds] OK! (Bad rational: zero denominator)
10: OK [0.006 seconds] OK! xyz
11: OK [0.007 seconds] OK! pow2
12: OK [0.006 seconds] OK! X1z
#ifndef _6S096_GCD_H
#define _6S096_GCD_H
#include <cstdint>
intmax_t gcd( intmax_t a, intmax_t b );
intmax_t lcm( intmax_t a, intmax_t b );
#endif // _6S096_GCD_H
#ifndef _6S096_RATIONAL_H
#define _6S096_RATIONAL_H
#include <cstdint>
#include <iosfwd>
#include <stdexcept> cpphomeworkhelp.com

16. class Rational {
intmax_t _num, _den;
public:
enum sign_type { POSITIVE, NEGATIVE };
Rational() : _num{0}, _den{1} {}
Rational( intmax_t numer ) : _num{numer}, _den{1} {}
Rational( intmax_t numer, intmax_t denom ) : _num{numer}, _den{denom} {
normalize(); }
inline intmax_t num() const { return _num; }
inline intmax_t den() const { return _den; }
// You should implement all of these functions in rational.cpp
void normalize();
float to_float() const;
double to_double() const;
sign_type sign() const;
const Rational inverse() const;
};
std::ostream& operator<<( std::ostream &os, const Rational &ratio );
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17. inline bool operator==( const Rational &lhs, const Rational &rhs ) {
// You should implement
}
inline bool operator<( const Rational &lhs, const Rational &rhs ) {
// You should implement
}
// This one is completed for you. We multiply two Rationals
// by creating a new Rational as shown:
//
// a.num() b.num() a.num() * b.num()
// ------- * ------- = -----------------
// a.den() b.den() a.den() * b.den()
//
// Our constructor should automatically reduce it to lowest terms,
// e.g. 1/2 * 2/3 = 1/3 automatically.
inline Rational operator*( const Rational &a, const Rational &b ) {
return Rational{ a.num() * b.num(), a.den() * b.den() };
}
inline Rational operator+( const Rational &a, const Rational &b ) {
// You should implement
}
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18. inline Rational operator-( const Rational &a, const Rational &b ) {
// You should implement
}
inline Rational operator/( const Rational &a, const Rational &b ) {
// You should implement
}
class bad_rational : public std::domain_error {
public:
explicit bad_rational() : std::domain_error( "Bad rational: zero denominator" ) {}
};
#endif // _6S096_RATIONAL_H
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