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
// 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::integer::Integer;
use crate::natural::Natural;
use core::cmp::Ordering::Less;
use malachite_base::num::arithmetic::traits::{BalancedMod, BalancedModAssign, ModEuclidean};
use malachite_base::num::comparison::traits::OrdDouble;
// The Euclidean remainder already lies in [0, |m|), so the balanced one is that remainder when it
// is at most half the modulus, and that remainder less the modulus otherwise. A remainder of
// exactly half the modulus stays positive, which puts the endpoint at the top of the range.
fn balanced_mod_helper(x: &Integer, m: &Integer) -> Integer {
let r: Natural = x.mod_euclidean(m);
let abs_m = m.unsigned_abs_ref();
// `r <= abs_m >> 1` is exactly `2r <= abs_m`: for an integer `r`, `r <= floor(x)` iff `r <= x`.
// Phrasing it as the latter lets `cmp_double` answer it without building either value.
if abs_m.cmp_double(&r) == Less {
Integer::from(r) - Integer::from(abs_m)
} else {
Integer::from(r)
}
}
impl BalancedMod<Self> for Integer {
type Output = Self;
/// Divides an [`Integer`] by another [`Integer`], returning the balanced remainder: the
/// representative of `self` modulo `other` that is closest to zero, taking both [`Integer`]s by
/// value.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$ and $r \equiv x \bmod y$, which
/// determine it uniquely. A remainder of exactly $|y|/2$ is positive. Only the magnitude of
/// `other` matters, so negating it leaves the result unchanged.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedMod;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(Integer::from(23).balanced_mod(Integer::from(10)), 3);
/// // 7 is more than half of 10, so the representative closest to zero is negative
/// assert_eq!(Integer::from(27).balanced_mod(Integer::from(10)), -3);
/// // only the magnitude of the modulus matters
/// assert_eq!(Integer::from(27).balanced_mod(Integer::from(-10)), -3);
/// ```
#[inline]
fn balanced_mod(self, other: Self) -> Self {
balanced_mod_helper(&self, &other)
}
}
impl BalancedMod<&Self> for Integer {
type Output = Self;
/// Divides an [`Integer`] by another [`Integer`], returning the balanced remainder: the
/// representative of `self` modulo `other` that is closest to zero, taking the first
/// [`Integer`] by value and the second by reference.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$ and $r \equiv x \bmod y$, which
/// determine it uniquely. A remainder of exactly $|y|/2$ is positive. Only the magnitude of
/// `other` matters, so negating it leaves the result unchanged.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedMod;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!(Integer::from(23).balanced_mod(&Integer::from(10)), 3);
/// // 7 is more than half of 10, so the representative closest to zero is negative
/// assert_eq!(Integer::from(27).balanced_mod(&Integer::from(10)), -3);
/// // only the magnitude of the modulus matters
/// assert_eq!(Integer::from(27).balanced_mod(&Integer::from(-10)), -3);
/// ```
#[inline]
fn balanced_mod(self, other: &Self) -> Self {
balanced_mod_helper(&self, other)
}
}
impl BalancedMod<Integer> for &Integer {
type Output = Integer;
/// Divides an [`Integer`] by another [`Integer`], returning the balanced remainder: the
/// representative of `self` modulo `other` that is closest to zero, taking the first
/// [`Integer`] by reference and the second by value.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$ and $r \equiv x \bmod y$, which
/// determine it uniquely. A remainder of exactly $|y|/2$ is positive. Only the magnitude of
/// `other` matters, so negating it leaves the result unchanged.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedMod;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!((&Integer::from(23)).balanced_mod(Integer::from(10)), 3);
/// // 7 is more than half of 10, so the representative closest to zero is negative
/// assert_eq!((&Integer::from(27)).balanced_mod(Integer::from(10)), -3);
/// // only the magnitude of the modulus matters
/// assert_eq!((&Integer::from(27)).balanced_mod(Integer::from(-10)), -3);
/// ```
#[inline]
fn balanced_mod(self, other: Integer) -> Integer {
balanced_mod_helper(self, &other)
}
}
impl BalancedMod<&Integer> for &Integer {
type Output = Integer;
/// Divides an [`Integer`] by another [`Integer`], returning the balanced remainder: the
/// representative of `self` modulo `other` that is closest to zero, taking both [`Integer`]s by
/// reference.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$ and $r \equiv x \bmod y$, which
/// determine it uniquely. A remainder of exactly $|y|/2$ is positive. Only the magnitude of
/// `other` matters, so negating it leaves the result unchanged.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedMod;
/// use malachite_nz::integer::Integer;
///
/// assert_eq!((&Integer::from(23)).balanced_mod(&Integer::from(10)), 3);
/// // 7 is more than half of 10, so the representative closest to zero is negative
/// assert_eq!((&Integer::from(27)).balanced_mod(&Integer::from(10)), -3);
/// // only the magnitude of the modulus matters
/// assert_eq!((&Integer::from(27)).balanced_mod(&Integer::from(-10)), -3);
/// ```
#[inline]
fn balanced_mod(self, other: &Integer) -> Integer {
balanced_mod_helper(self, other)
}
}
impl BalancedModAssign<Self> for Integer {
/// Divides an [`Integer`] by another [`Integer`], replacing the first [`Integer`] by the
/// balanced remainder: the representative of `self` modulo `other` that is closest to zero. The
/// [`Integer`] on the right-hand side is taken by value.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$; a remainder of exactly $|y|/2$ is
/// positive.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedModAssign;
/// use malachite_nz::integer::Integer;
///
/// let mut x = Integer::from(27);
/// x.balanced_mod_assign(Integer::from(10));
/// assert_eq!(x, -3);
/// ```
#[inline]
fn balanced_mod_assign(&mut self, other: Self) {
*self = balanced_mod_helper(self, &other);
}
}
impl BalancedModAssign<&Self> for Integer {
/// Divides an [`Integer`] by another [`Integer`], replacing the first [`Integer`] by the
/// balanced remainder: the representative of `self` modulo `other` that is closest to zero. The
/// [`Integer`] on the right-hand side is taken by reference.
///
/// The remainder $r$ satisfies $-|y|/2 < r \leq |y|/2$; a remainder of exactly $|y|/2$ is
/// positive.
///
/// # 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 `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::arithmetic::traits::BalancedModAssign;
/// use malachite_nz::integer::Integer;
///
/// let mut x = Integer::from(27);
/// x.balanced_mod_assign(&Integer::from(10));
/// assert_eq!(x, -3);
/// ```
#[inline]
fn balanced_mod_assign(&mut self, other: &Self) {
*self = balanced_mod_helper(self, other);
}
}