malachite_nz/gaussian_integer/comparison/
cmp_abs.rs

1// Copyright © 2026 Mikhail Hogrefe
2//
3// This file is part of Malachite.
4//
5// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
6// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
7// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
8
9use crate::gaussian_integer::GaussianInteger;
10use core::cmp::Ordering::{self, Equal};
11use malachite_base::num::arithmetic::traits::AbsSquared;
12use malachite_base::num::comparison::traits::{OrdAbs, PartialOrdAbs};
13
14// If two component comparisons pull in the same direction (or one is a tie), they decide the
15// comparison of the sums of squares; only a strict conflict is indecisive.
16fn combine(x: Ordering, y: Ordering) -> Option<Ordering> {
17    if x == y || y == Equal {
18        Some(x)
19    } else if x == Equal {
20        Some(y)
21    } else {
22        None
23    }
24}
25
26impl PartialOrdAbs for GaussianInteger {
27    /// Compares the absolute values of two [`GaussianInteger`]s.
28    ///
29    /// See the documentation for the [`OrdAbs`] implementation.
30    #[inline]
31    fn partial_cmp_abs(&self, other: &Self) -> Option<Ordering> {
32        Some(self.cmp_abs(other))
33    }
34}
35
36impl OrdAbs for GaussianInteger {
37    /// Compares the absolute values of two [`GaussianInteger`]s.
38    ///
39    /// The absolute value of a complex number is its distance from the origin, so this is
40    /// equivalent to comparing squared absolute values:
41    ///
42    /// $$
43    /// f(x, y) = \operatorname{cmp}(|x|, |y|) = \operatorname{cmp}(|x|^2, |y|^2).
44    /// $$
45    ///
46    /// The squared absolute values are usually not actually computed: comparing the
47    /// [`Integer`](crate::integer::Integer) parts componentwise, either directly or crosswise,
48    /// often decides the ordering, and the [`AbsSquared`] fallback only runs when both pairings
49    /// strictly conflict.
50    ///
51    /// # Worst-case complexity
52    /// $T(n) = O(n \log n \log\log n)$
53    ///
54    /// $M(n) = O(n \log n)$
55    ///
56    /// where $T$ is time, $M$ is additional memory, and $n$ is the maximum number of significant
57    /// bits of the real and imaginary parts of `self` and `other`.
58    ///
59    /// # Examples
60    /// ```
61    /// use malachite_base::num::comparison::traits::{OrdAbs, PartialOrdAbs};
62    /// use malachite_nz::gaussian_integer::GaussianInteger;
63    /// use std::cmp::Ordering::*;
64    /// use std::str::FromStr;
65    ///
66    /// let x = GaussianInteger::from_str("2+2i").unwrap();
67    /// let y = GaussianInteger::from_str("3i").unwrap();
68    /// // |2+2i|^2 = 8 and |3i|^2 = 9
69    /// assert_eq!(x.cmp_abs(&y), Less);
70    /// assert!(x.lt_abs(&y));
71    ///
72    /// let x = GaussianInteger::from_str("1+2i").unwrap();
73    /// let y = GaussianInteger::from_str("-2+i").unwrap();
74    /// assert_eq!(x.cmp_abs(&y), Equal);
75    ///
76    /// let x = GaussianInteger::from_str("3").unwrap();
77    /// let y = GaussianInteger::from_str("2+2i").unwrap();
78    /// // |3|^2 = 9 and |2+2i|^2 = 8
79    /// assert_eq!(x.cmp_abs(&y), Greater);
80    /// ```
81    fn cmp_abs(&self, other: &Self) -> Ordering {
82        if let Some(o) = combine(
83            self.real.cmp_abs(&other.real),
84            self.imaginary.cmp_abs(&other.imaginary),
85        ) {
86            return o;
87        }
88        if let Some(o) = combine(
89            self.real.cmp_abs(&other.imaginary),
90            self.imaginary.cmp_abs(&other.real),
91        ) {
92            return o;
93        }
94        self.abs_squared().cmp(&other.abs_squared())
95    }
96}
malachite_nz/gaussian_integer/comparison/cmp_abs.rs

malachite_nz/gaussian_integer/comparison/
cmp_abs.rs
