malachite_float/float/constants/
one_over_pi.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 malachite_base::num::arithmetic::traits::Reciprocal;
18use malachite_base::num::basic::integers::PrimitiveInt;
19use malachite_base::rounding_modes::RoundingMode::{self, *};
20use malachite_nz::natural::arithmetic::float_extras::float_can_round;
21use malachite_nz::platform::Limb;
22
23impl Float {
24    /// Returns an approximation of $1/\pi$, with the given precision and rounded using the given
25    /// [`RoundingMode`]. An [`Ordering`] is also returned, indicating whether the rounded value is
26    /// 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    /// $$
30    /// x = 1/\pi+\varepsilon.
31    /// $$
32    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
33    /// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-2}$.
34    ///
35    /// The constant is irrational and transcendental.
36    ///
37    /// The output has precision `prec`.
38    ///
39    /// # Worst-case complexity
40    /// $T(n) = O(n (\log n)^2 \log\log n)$
41    ///
42    /// $M(n) = O(n (\log n)^2)$
43    ///
44    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
45    ///
46    /// # Panics
47    /// Panics if `prec` is zero or if `rm` is `Exact`.
48    ///
49    /// # Examples
50    /// ```
51    /// use malachite_base::rounding_modes::RoundingMode::*;
52    /// use malachite_float::Float;
53    /// use std::cmp::Ordering::*;
54    ///
55    /// let (one_over_pi, o) = Float::one_over_pi_prec_round(100, Floor);
56    /// assert_eq!(
57    ///     one_over_pi.to_string(),
58    ///     "0.31830988618379067153776752674490"
59    /// );
60    /// assert_eq!(o, Less);
61    ///
62    /// let (one_over_pi, o) = Float::one_over_pi_prec_round(100, Ceiling);
63    /// assert_eq!(
64    ///     one_over_pi.to_string(),
65    ///     "0.31830988618379067153776752674529"
66    /// );
67    /// assert_eq!(o, Greater);
68    /// ```
69    pub fn one_over_pi_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
70        let mut working_prec = prec + 10;
71        let mut increment = Limb::WIDTH;
72        loop {
73            let one_over_pi = Self::pi_prec_round(working_prec, Floor).0.reciprocal();
74            // See algorithms.tex. Since we rounded down when computing pi, the absolute error of
75            // the inverse is bounded by (c_w + 2c_uk_u)ulp(log_e(2)) <= 4ulp(pi).
76            if float_can_round(
77                one_over_pi.significand_ref().unwrap(),
78                working_prec - 2,
79                prec,
80                rm,
81            ) {
82                return Self::from_float_prec_round(one_over_pi, prec, rm);
83            }
84            working_prec += increment;
85            increment = working_prec >> 1;
86        }
87    }
88
89    /// Returns an approximation of $1/\pi$, with the given precision and rounded to the nearest
90    /// [`Float`] of that precision. An [`Ordering`] is also returned, indicating whether the
91    /// rounded value is less than or greater than the exact value of the constant. (Since the
92    /// constant is irrational, the rounded value is never equal to the exact value.)
93    ///
94    /// $$
95    /// x = 1/\pi+\varepsilon.
96    /// $$
97    /// - $|\varepsilon| < 2^{-p-1}$.
98    ///
99    /// The constant is irrational and transcendental.
100    ///
101    /// The output has precision `prec`.
102    ///
103    /// # Worst-case complexity
104    /// $T(n) = O(n (\log n)^2 \log\log n)$
105    ///
106    /// $M(n) = O(n (\log n)^2)$
107    ///
108    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
109    ///
110    /// # Panics
111    /// Panics if `prec` is zero.
112    ///
113    /// # Examples
114    /// ```
115    /// use malachite_float::Float;
116    /// use std::cmp::Ordering::*;
117    ///
118    /// let (one_over_pi, o) = Float::one_over_pi_prec(1);
119    /// assert_eq!(one_over_pi.to_string(), "0.25");
120    /// assert_eq!(o, Less);
121    ///
122    /// let (one_over_pi, o) = Float::one_over_pi_prec(10);
123    /// assert_eq!(one_over_pi.to_string(), "0.31836");
124    /// assert_eq!(o, Greater);
125    ///
126    /// let (one_over_pi, o) = Float::one_over_pi_prec(100);
127    /// assert_eq!(
128    ///     one_over_pi.to_string(),
129    ///     "0.31830988618379067153776752674490"
130    /// );
131    /// assert_eq!(o, Less);
132    /// ```
133    #[inline]
134    pub fn one_over_pi_prec(prec: u64) -> (Self, Ordering) {
135        Self::one_over_pi_prec_round(prec, Nearest)
136    }
137}
malachite_float/float/constants/one_over_pi.rs

malachite_float/float/constants/
one_over_pi.rs
