malachite_float/constants/
sqrt_2_over_2.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::Float;
10use core::cmp::Ordering;
11use malachite_base::num::basic::traits::Two;
12use malachite_base::rounding_modes::RoundingMode::{self, *};
13
14impl Float {
15    /// Returns an approximation of half of the square root of 2, with the given precision and
16    /// rounded using the given [`RoundingMode`]. An [`Ordering`] is also returned, indicating
17    /// whether the rounded value is less than or greater than the exact value of the constant.
18    /// (Since the constant is irrational, the rounded value is never equal to the exact value.)
19    ///
20    /// $$
21    /// x = \sqrt{2}/2+\varepsilon=\sqrt{1/2}+\varepsilon=1/\sqrt{2}+\varepsilon.
22    /// $$
23    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p}$.
24    /// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
25    ///
26    /// The constant is irrational and algebraic.
27    ///
28    /// The output has precision `prec`.
29    ///
30    /// # Worst-case complexity
31    /// $T(n) = O(n \log n \log\log n)$
32    ///
33    /// $M(n) = O(n \log n)$
34    ///
35    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
36    ///
37    /// # Panics
38    /// Panics if `prec` is zero or if `rm` is `Exact`.
39    ///
40    /// # Examples
41    /// ```
42    /// use malachite_base::rounding_modes::RoundingMode::*;
43    /// use malachite_float::Float;
44    /// use std::cmp::Ordering::*;
45    ///
46    /// let (sqrt_2_over_2, o) = Float::sqrt_2_over_2_prec_round(100, Floor);
47    /// assert_eq!(
48    ///     sqrt_2_over_2.to_string(),
49    ///     "0.7071067811865475244008443621046"
50    /// );
51    /// assert_eq!(o, Less);
52    ///
53    /// let (sqrt_2_over_2, o) = Float::sqrt_2_over_2_prec_round(100, Ceiling);
54    /// assert_eq!(
55    ///     sqrt_2_over_2.to_string(),
56    ///     "0.7071067811865475244008443621054"
57    /// );
58    /// assert_eq!(o, Greater);
59    /// ```
60    #[inline]
61    pub fn sqrt_2_over_2_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
62        let (sqrt_2, o) = Self::sqrt_prec_round(Self::TWO, prec, rm);
63        (sqrt_2 >> 1u32, o)
64    }
65
66    /// Returns an approximation of half of the square root of 2, with the given precision and
67    /// rounded to the nearest [`Float`] of that precision. An [`Ordering`] is also returned,
68    /// indicating whether the rounded value is less than or greater than the exact value of the
69    /// constant. (Since the constant is irrational, the rounded value is never equal to the exact
70    /// value.)
71    ///
72    /// $$
73    /// x = \sqrt{2}/2+\varepsilon=\sqrt{1/2}+\varepsilon=1/\sqrt{2}+\varepsilon.
74    /// $$
75    /// - $|\varepsilon| < 2^{-p-1}$.
76    ///
77    /// The constant is irrational and algebraic.
78    ///
79    /// The output has precision `prec`.
80    ///
81    /// # Worst-case complexity
82    /// $T(n) = O(n \log n \log\log n)$
83    ///
84    /// $M(n) = O(n \log n)$
85    ///
86    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
87    ///
88    /// # Panics
89    /// Panics if `prec` is zero.
90    ///
91    /// # Examples
92    /// ```
93    /// use malachite_float::Float;
94    /// use std::cmp::Ordering::*;
95    ///
96    /// let (sqrt_2_over_2, o) = Float::sqrt_2_over_2_prec(1);
97    /// assert_eq!(sqrt_2_over_2.to_string(), "0.5");
98    /// assert_eq!(o, Less);
99    ///
100    /// let (sqrt_2_over_2, o) = Float::sqrt_2_over_2_prec(10);
101    /// assert_eq!(sqrt_2_over_2.to_string(), "0.707");
102    /// assert_eq!(o, Less);
103    ///
104    /// let (sqrt_2_over_2, o) = Float::sqrt_2_over_2_prec(100);
105    /// assert_eq!(
106    ///     sqrt_2_over_2.to_string(),
107    ///     "0.7071067811865475244008443621046"
108    /// );
109    /// assert_eq!(o, Less);
110    /// ```
111    #[inline]
112    pub fn sqrt_2_over_2_prec(prec: u64) -> (Self, Ordering) {
113        Self::sqrt_2_over_2_prec_round(prec, Nearest)
114    }
115}
malachite_float/constants/sqrt_2_over_2.rs

malachite_float/constants/
sqrt_2_over_2.rs
