malachite_float/float/constants/
prime_constant.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::factorization::primes::prime_indicator_sequence_less_than_or_equal_to;
12use malachite_base::rounding_modes::RoundingMode::{self, *};
13
14impl Float {
15    /// Returns an approximation of the prime constant, with the given precision and rounded using
16    /// the given [`RoundingMode`]. An [`Ordering`] is also returned, indicating whether the rounded
17    /// value is less than or greater than the exact value of the constant. (Since the constant is
18    /// irrational, the rounded value is never equal to the exact value.)
19    ///
20    /// The prime constant is the real number whose $n$th bit is prime if and only if $n$ is prime.
21    /// That is,
22    /// $$
23    /// \rho = \sum_{p\ \text{prime}\}2^{-p}+\varepsilon.
24    /// $$
25    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
26    /// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-2}$.
27    ///
28    /// The constant is irrational. It is unknown whether it is transcendental; see
29    /// <https://mathoverflow.net/questions/114905>.
30    ///
31    /// The output has precision `prec`.
32    ///
33    /// # Worst-case complexity
34    /// $T(n) = O(n)$
35    ///
36    /// $M(n) = O(n)$
37    ///
38    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
39    ///
40    /// # Panics
41    /// Panics if `prec` is zero or if `rm` is `Exact`.
42    ///
43    /// # Examples
44    /// ```
45    /// use malachite_base::rounding_modes::RoundingMode::*;
46    /// use malachite_float::Float;
47    /// use std::cmp::Ordering::*;
48    ///
49    /// let (pc, o) = Float::prime_constant_prec_round(100, Floor);
50    /// assert_eq!(pc.to_string(), "0.41468250985111166024810962215420");
51    /// assert_eq!(o, Less);
52    ///
53    /// let (pc, o) = Float::prime_constant_prec_round(100, Ceiling);
54    /// assert_eq!(pc.to_string(), "0.41468250985111166024810962215460");
55    /// assert_eq!(o, Greater);
56    /// ```
57    #[inline]
58    pub fn prime_constant_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
59        // Strictly speaking, this call violates the preconditions for
60        // `non_dyadic_from_bits_prec_round`, because the iterator passed in is finite. But since we
61        // know exactly how many bits `non_dyadic_from_bits_prec_round` will read, we can get away
62        // with this.
63        Self::non_dyadic_from_bits_prec_round(
64            prime_indicator_sequence_less_than_or_equal_to(if rm == Nearest {
65                prec + 2
66            } else {
67                prec + 1
68            }),
69            prec,
70            rm,
71        )
72    }
73
74    /// Returns an approximation of the prime constant, with the given precision and rounded to the
75    /// nearest [`Float`] of that precision. An [`Ordering`] is also returned, indicating whether
76    /// the rounded value is less than or greater than the exact value of the constant. (Since the
77    /// constant is irrational, the rounded value is never equal to the exact value.)
78    ///
79    /// The prime constant is the real number whose $n$th bit is prime if and only if $n$ is prime.
80    /// That is,
81    /// $$
82    /// \rho = \sum_{p\ \text{prime}\}2^{-p}+\varepsilon.
83    /// $$
84    /// - $|\varepsilon| < 2^{-p-2}$.
85    ///
86    /// The constant is irrational. It is unknown whether it is transcendental; see
87    /// <https://mathoverflow.net/questions/114905>.
88    ///
89    /// The output has precision `prec`.
90    ///
91    /// # Worst-case complexity
92    /// $T(n) = O(n)$
93    ///
94    /// $M(n) = O(n)$
95    ///
96    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
97    ///
98    /// # Panics
99    /// Panics if `prec` is zero.
100    ///
101    /// # Examples
102    /// ```
103    /// use malachite_float::Float;
104    /// use std::cmp::Ordering::*;
105    ///
106    /// let (pc, o) = Float::prime_constant_prec(1);
107    /// assert_eq!(pc.to_string(), "0.50");
108    /// assert_eq!(o, Greater);
109    ///
110    /// let (pc, o) = Float::prime_constant_prec(10);
111    /// assert_eq!(pc.to_string(), "0.41455");
112    /// assert_eq!(o, Less);
113    ///
114    /// let (pc, o) = Float::prime_constant_prec(100);
115    /// assert_eq!(pc.to_string(), "0.41468250985111166024810962215420");
116    /// assert_eq!(o, Less);
117    /// ```
118    #[inline]
119    pub fn prime_constant_prec(prec: u64) -> (Self, Ordering) {
120        Self::prime_constant_prec_round(prec, Nearest)
121    }
122}
malachite_float/float/constants/prime_constant.rs

malachite_float/float/constants/
prime_constant.rs
