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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
// Copyright © 2026 Mikhail Hogrefe
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::gaussian_integer::GaussianInteger;
use crate::gaussian_integer::arithmetic::div_rem::{div_rem_ref_ref, div_rem_val_ref};
use core::mem::take;
use core::ops::{Rem, RemAssign};
impl Rem<Self> for GaussianInteger {
type Output = Self;
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`], taking both by value, and
/// returns the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// f(x, y) = x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let x = GaussianInteger::from_str("5+3i").unwrap();
/// let y = GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!((x % y).to_string(), "-1");
/// ```
#[inline]
fn rem(self, other: Self) -> Self {
div_rem_val_ref(self, &other).1
}
}
impl Rem<&Self> for GaussianInteger {
type Output = Self;
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`], taking the first by value and
/// the second by reference, and returns the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// f(x, y) = x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let x = GaussianInteger::from_str("5+3i").unwrap();
/// let y = GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!((x % &y).to_string(), "-1");
/// ```
#[inline]
fn rem(self, other: &Self) -> Self {
div_rem_val_ref(self, other).1
}
}
impl Rem<GaussianInteger> for &GaussianInteger {
type Output = GaussianInteger;
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`], taking the first by reference
/// and the second by value, and returns the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// f(x, y) = x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let x = GaussianInteger::from_str("5+3i").unwrap();
/// let y = GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!((&x % y).to_string(), "-1");
/// ```
#[inline]
fn rem(self, other: GaussianInteger) -> GaussianInteger {
div_rem_ref_ref(self, &other).1
}
}
impl Rem<&GaussianInteger> for &GaussianInteger {
type Output = GaussianInteger;
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`], taking both by reference, and
/// returns the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// f(x, y) = x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let x = GaussianInteger::from_str("5+3i").unwrap();
/// let y = GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!((&x % &y).to_string(), "-1");
/// ```
#[inline]
fn rem(self, other: &GaussianInteger) -> GaussianInteger {
div_rem_ref_ref(self, other).1
}
}
impl RemAssign<Self> for GaussianInteger {
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`] in place, taking the
/// [`GaussianInteger`] on the right-hand side by value and replacing the first
/// [`GaussianInteger`] with the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// x \gets x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let mut x = GaussianInteger::from_str("5+3i").unwrap();
/// x %= GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!(x.to_string(), "-1");
/// ```
#[inline]
fn rem_assign(&mut self, other: Self) {
*self = div_rem_val_ref(take(self), &other).1;
}
}
impl RemAssign<&Self> for GaussianInteger {
/// Divides a [`GaussianInteger`] by another [`GaussianInteger`] in place, taking the
/// [`GaussianInteger`] on the right-hand side by reference and replacing the first
/// [`GaussianInteger`] with the remainder.
///
/// The quotient (which is not returned) is the Gaussian integer nearest to the exact quotient,
/// with each part rounded to the nearest integer and ties rounded up, and the remainder is what
/// is left over. The quotient and remainder satisfy $x = qy + r$ and $N(r) \leq N(y) / 2$,
/// where $N$ is the norm. To get both at once, use
/// [`div_rem`](malachite_base::num::arithmetic::traits::DivRem::div_rem).
///
/// $$
/// x \gets x - qy, \quad \text{where } q = \left \lfloor \frac{x \bar{y}}{N(y)} +
/// \frac{1 + i}{2} \right \rfloor
/// $$
/// and the floor is taken on each part.
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
/// bits of the real and imaginary parts of `self` and `other`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_nz::gaussian_integer::GaussianInteger;
/// use std::str::FromStr;
///
/// // (2+i)(3) + (-1) = 5+3i
/// let mut x = GaussianInteger::from_str("5+3i").unwrap();
/// x %= &GaussianInteger::from_str("2+i").unwrap();
/// assert_eq!(x.to_string(), "-1");
/// ```
#[inline]
fn rem_assign(&mut self, other: &Self) {
*self = div_rem_val_ref(take(self), other).1;
}
}