6.4.3 Rational Numbers

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186

/*

Perform operations on rational numbers internally represented as two integers: a numerator and a
denominator. The template integer type must support streamed input/output, comparisons, and
arithmetic operations. Overflow is not checked for in internal operations: comparisons and
arithmetic cross-multiply numerators and denominators, so instantiate with a wider integer type
(such as `__int128`, or even `BigInt`) if the values may grow large.

- `Rational<Int>(n)` constructs a rational number with numerator `n` and denominator 1.
- `Rational<Int>(n, d)` constructs a rational number with numerator `n` and denominator `d`.
- `operator>>` inputs a rational number using the next integer from the stream as the numerator and
  1 as the denominator.
- `operator<<` outputs the rational number as a string consisting of possibly a minus sign followed
  by the numerator, followed by a slash, followed by the denominator.
- `v.to_string()`, `v.to_llong()`, `v.to_double()`, and `v.to_ldouble()` return the rational `v`
  converted to an `std::string`, `int64_t`, `double`, and `long double` respectively.
- `v.abs()`, `abs(v)`, `v.floor()`, and `v.ceil()` return the absolute value, floor, and ceiling of
  rational `v`.
- Operators `<`, `>`, `<=`, `>=`, `==`, `!=`, `+`, `-`, `*`, `/`, `%`, `++`, `--`, `+=`, `-=`, `*=`,
  `/=`, and `%=` are defined analogous to those on numerical primitives. The comparisons and binary
  arithmetic are hidden friends, so a raw integer operand works on either side.

Time Complexity:
- O(log(n + d)) per call to constructor `Rational(n, d)`.
- O(1) per call to all other operations, assuming that corresponding operations on the template
  integer type are O(1) as well.

Space Complexity:
- O(1) for storage of the rational number.
- O(1) auxiliary space for all operations.

*/

#include <cstdint>
#include <istream>
#include <ostream>
#include <sstream>
#include <string>

template<class Int>
class Rational {
  Int num, den;

 public:
  Rational() : num(0), den(1) {}
  Rational(const Int &n) : num(n), den(1) {}

  template<class T1, class T2>
  Rational(const T1 &n, const T2 &d) : num(n), den(d) {
    if (den == 0) {
      throw std::runtime_error("Division by zero in Rational.");
    }
    if (den < 0) {
      num = -num;
      den = -den;
    }
    Int a(num < 0 ? -num : num), b(den), tmp;
    while (a != 0 && b != 0) {
      tmp = a % b;
      a = b;
      b = tmp;
    }
    Int gcd = (b == 0) ? a : b;
    num /= gcd;
    den /= gcd;
  }

  friend std::istream &operator>>(std::istream &in, Rational &r) {
    in >> r.num;
    r.den = 1;
    return in;
  }

  friend std::ostream &operator<<(std::ostream &out, const Rational &r) {
    out << r.num << "/" << r.den;
    return out;
  }

  std::string to_string() const {
    std::stringstream ss;
    ss << num << "/" << den;
    return ss.str();
  }

  // to_llong/to_double/to_ldouble round-trip through a stringstream so that any Int supporting
  // streamed I/O (e.g. a big-integer type) converts, even without a direct cast to the target type.
  int64_t to_llong() const {
    std::stringstream ss;
    ss << num << " " << den;
    int64_t n, d;
    ss >> n >> d;
    return n / d;
  }

  double to_double() const {
    std::stringstream ss;
    ss << num << " " << den;
    double n, d;
    ss >> n >> d;
    return n / d;
  }

  long double to_ldouble() const {
    long double n, d;
    std::stringstream ss;
    ss << num << " " << den;
    ss >> n >> d;
    return n / d;
  }

  // The comparison and binary arithmetic operators are hidden friends, so a raw integer operand on
  // either side converts through the implicit constructor.
  friend bool operator<(const Rational &a, const Rational &b) {
    return a.num * b.den < b.num * a.den;
  }

  friend bool operator==(const Rational &a, const Rational &b) {
    return a.num == b.num && a.den == b.den;
  }

  // clang-format off
  friend bool operator>(const Rational &a, const Rational &b) { return b < a; }
  friend bool operator<=(const Rational &a, const Rational &b) { return !(b < a); }
  friend bool operator>=(const Rational &a, const Rational &b) { return !(a < b); }
  friend bool operator!=(const Rational &a, const Rational &b) { return !(a == b); }
  // clang-format on

  Rational abs() const { return Rational(num < 0 ? -num : num, den); }
  friend Rational abs(const Rational &r) { return r.abs(); }
  Int floor() const { return num < 0 ? -((-num + den - 1) / den) : num / den; }
  Int ceil() const { return num < 0 ? -(-num / den) : (num + den - 1) / den; }

  friend Rational operator+(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den + b.num * a.den, a.den * b.den);
  }

  friend Rational operator-(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den - b.num * a.den, a.den * b.den);
  }

  friend Rational operator*(const Rational &a, const Rational &b) {
    return Rational(a.num * b.num, a.den * b.den);
  }

  friend Rational operator/(const Rational &a, const Rational &b) {
    return Rational(a.num * b.den, a.den * b.num);
  }

  friend Rational operator%(const Rational &a, const Rational &b) {
    return a - b * Rational(a.num * b.den / (b.num * a.den), 1);
  }

  // clang-format off
  Rational operator-() const { return Rational(-num, den); }
  Rational operator++(int) { Rational t(*this); operator++(); return t; }
  Rational operator--(int) { Rational t(*this); operator--(); return t; }
  Rational &operator++() { *this = *this + 1; return *this; }
  Rational &operator--() { *this = *this - 1; return *this; }
  Rational &operator+=(const Rational &r) { *this = *this + r; return *this; }
  Rational &operator-=(const Rational &r) { *this = *this - r; return *this; }
  Rational &operator*=(const Rational &r) { *this = *this * r; return *this; }
  Rational &operator/=(const Rational &r) { *this = *this / r; return *this; }
  Rational &operator%=(const Rational &r) { *this = *this % r; return *this; }
  // clang-format on
};

/*** Example Usage ***/

#include <cassert>
#include <cmath>

#define EQ(a, b) (fabs((a) - (b)) <= 1e-9)

int main() {
  using Rational = Rational<int64_t>;
  assert(Rational(-21, 1) % 2 == -1);
  Rational r(Rational(-53, 10) % Rational(-17, 10));
  assert(EQ(r.to_ldouble(), fmod(-5.3, -1.7)));
  assert(r.to_string() == "-1/5");

  // Raw integers work on either side of comparisons and arithmetic.
  assert(2 + Rational(1, 2) == Rational(5, 2));
  assert(1 < Rational(3, 2) && Rational(3, 2) < 2);
  assert(2 % Rational(3, 4) == Rational(1, 2));
  return 0;
}