malachite_float/constants/
gauss_constant.rs

1// Copyright © 2026 Mikhail Hogrefe
2//
3// Uses code adopted from the GNU MPFR Library.
4//
5//      Copyright 1999, 2001-2024 Free Software Foundation, Inc.
6//
7//      Contributed by the AriC and Caramba projects, INRIA.
8//
9// This file is part of Malachite.
10//
11// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
12// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
13// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
14
15use crate::Float;
16use core::cmp::Ordering;
17use core::cmp::Ordering::*;
18use malachite_base::num::basic::integers::PrimitiveInt;
19use malachite_base::num::basic::traits::One;
20use malachite_base::rounding_modes::RoundingMode::{self, *};
21use malachite_nz::platform::Limb;
22
23impl Float {
24    /// Returns an approximation of Gauss's constant, with the given precision and rounded using the
25    /// given [`RoundingMode`]. An [`Ordering`] is also returned, indicating whether the rounded
26    /// value is less than or greater than the exact value of the constant. (Since the constant is
27    /// irrational, the rounded value is never equal to the exact value.)
28    ///
29    /// $$
30    /// x = G+\varepsilon=1/\mathrm{AGM}(1,\sqrt{2})+\varepsilon,
31    /// $$
32    /// where AGM is the arithmetic-geometric mean.
33    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p}$.
34    /// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
35    ///
36    /// The constant is irrational and transcendental.
37    ///
38    /// The output has precision `prec`.
39    ///
40    /// # Worst-case complexity
41    /// $T(n) = O(n (\log n)^2 \log\log n)$
42    ///
43    /// $M(n) = O(n (\log n)^2)$
44    ///
45    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
46    ///
47    /// # Panics
48    /// Panics if `prec` is zero or if `rm` is `Exact`.
49    ///
50    /// # Examples
51    /// ```
52    /// use malachite_base::rounding_modes::RoundingMode::*;
53    /// use malachite_float::Float;
54    /// use std::cmp::Ordering::*;
55    ///
56    /// let (gauss_constant, o) = Float::gauss_constant_prec_round(100, Floor);
57    /// assert_eq!(
58    ///     gauss_constant.to_string(),
59    ///     "0.834626841674073186281429732799"
60    /// );
61    /// assert_eq!(o, Less);
62    ///
63    /// let (gauss_constant, o) = Float::gauss_constant_prec_round(100, Ceiling);
64    /// assert_eq!(gauss_constant.to_string(), "0.8346268416740731862814297328");
65    /// assert_eq!(o, Greater);
66    /// ```
67    pub fn gauss_constant_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
68        let mut working_prec = prec + 10;
69        let mut increment = Limb::WIDTH;
70        loop {
71            let sqrt_2_lo = Self::sqrt_2_prec_round(working_prec, Floor).0;
72            let mut sqrt_2_hi = sqrt_2_lo.clone();
73            sqrt_2_hi.increment();
74            let lo = Self::ONE
75                .agm_round(sqrt_2_hi, Ceiling)
76                .0
77                .reciprocal_round(Floor)
78                .0;
79            let hi = Self::ONE
80                .agm_round(sqrt_2_lo, Floor)
81                .0
82                .reciprocal_round(Ceiling)
83                .0;
84            let (gauss_constant_lo, mut o_lo) = Self::from_float_prec_round(lo, prec, rm);
85            let (gauss_constant_hi, mut o_hi) = Self::from_float_prec_round(hi, prec, rm);
86            if o_lo == Equal {
87                o_lo = o_hi;
88            }
89            if o_hi == Equal {
90                o_hi = o_lo;
91            }
92            if o_lo == o_hi && gauss_constant_lo == gauss_constant_hi {
93                return (gauss_constant_lo, o_lo);
94            }
95            working_prec += increment;
96            increment = working_prec >> 1;
97        }
98    }
99
100    /// Returns an approximation of Gauss's constant, $G=1/\mathrm{AGM}(1,\sqrt{2})$, with the given
101    /// precision and rounded to the nearest [`Float`] of that precision. An [`Ordering`] is also
102    /// returned, indicating whether the rounded value is less than or greater than the exact value
103    /// of the constant. (Since the constant is irrational, the rounded value is never equal to the
104    /// exact value.)
105    ///
106    /// $$
107    /// x=G+\varepsilon=1/\mathrm{AGM}(1,\sqrt{2})+\varepsilon,
108    /// $$
109    /// where AGM is the arithmetic-geometric mean.
110    /// - $|\varepsilon| < 2^{-p-1}$.
111    ///
112    /// The constant is irrational and transcendental.
113    ///
114    /// The output has precision `prec`.
115    ///
116    /// # Worst-case complexity
117    /// $T(n) = O(n (\log n)^2 \log\log n)$
118    ///
119    /// $M(n) = O(n (\log n)^2)$
120    ///
121    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
122    ///
123    /// # Panics
124    /// Panics if `prec` is zero.
125    ///
126    /// # Examples
127    /// ```
128    /// use malachite_float::Float;
129    /// use std::cmp::Ordering::*;
130    ///
131    /// let (gauss_constant, o) = Float::gauss_constant_prec(1);
132    /// assert_eq!(gauss_constant.to_string(), "1.0");
133    /// assert_eq!(o, Greater);
134    ///
135    /// let (gauss_constant, o) = Float::gauss_constant_prec(10);
136    /// assert_eq!(gauss_constant.to_string(), "0.835");
137    /// assert_eq!(o, Greater);
138    ///
139    /// let (gauss_constant, o) = Float::gauss_constant_prec(100);
140    /// assert_eq!(
141    ///     gauss_constant.to_string(),
142    ///     "0.834626841674073186281429732799"
143    /// );
144    /// assert_eq!(o, Less);
145    /// ```
146    #[inline]
147    pub fn gauss_constant_prec(prec: u64) -> (Self, Ordering) {
148        Self::gauss_constant_prec_round(prec, Nearest)
149    }
150}
malachite_float/constants/gauss_constant.rs

malachite_float/constants/
gauss_constant.rs
