malachite_float/constants/
log_2_e.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 the base-2 logarithm of $e$, with the given precision and
25    /// rounded using the given [`RoundingMode`]. An [`Ordering`] is also returned, indicating
26    /// whether the rounded value is less than or greater than the exact value of the constant.
27    /// (Since the constant is irrational, the rounded value is never equal to the exact value.)
28    ///
29    /// $$
30    /// x = \log_2 e+\varepsilon.
31    /// $$
32    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p+1}$.
33    /// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p}$.
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 (log_2_e, o) = Float::log_2_e_prec_round(100, Floor);
56    /// assert_eq!(log_2_e.to_string(), "1.442695040888963407359924681001");
57    /// assert_eq!(o, Less);
58    ///
59    /// let (log_2_e, o) = Float::log_2_e_prec_round(100, Ceiling);
60    /// assert_eq!(log_2_e.to_string(), "1.442695040888963407359924681003");
61    /// assert_eq!(o, Greater);
62    /// ```
63    pub fn log_2_e_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
64        let mut working_prec = prec + 10;
65        let mut increment = Limb::WIDTH;
66        loop {
67            let log_2_e = Self::ln_2_prec_round(working_prec, Floor).0.reciprocal();
68            // See algorithms.tex. Since we rounded down when computing ln_2, the absolute error of
69            // the inverse is bounded by (c_w + 2c_uk_u)ulp(log_e(2)) <= 4ulp(log_e(2)).
70            if float_can_round(
71                log_2_e.significand_ref().unwrap(),
72                working_prec - 2,
73                prec,
74                rm,
75            ) {
76                return Self::from_float_prec_round(log_2_e, prec, rm);
77            }
78            working_prec += increment;
79            increment = working_prec >> 1;
80        }
81    }
82
83    /// Returns an approximation of the base-2 logarithm of $e$, with the given precision and
84    /// rounded to the nearest [`Float`] of that precision. An [`Ordering`] is also returned,
85    /// indicating whether the rounded value is less than or greater than the exact value of the
86    /// constant. (Since the constant is irrational, the rounded value is never equal to the exact
87    /// value.)
88    ///
89    /// $$
90    /// x = \log_2 e+\varepsilon.
91    /// $$
92    /// - $|\varepsilon| < 2^{-p}$.
93    ///
94    /// The constant is irrational and transcendental.
95    ///
96    /// The output has precision `prec`.
97    ///
98    /// # Worst-case complexity
99    /// $T(n) = O(n (\log n)^2 \log\log n)$
100    ///
101    /// $M(n) = O(n (\log n)^2)$
102    ///
103    /// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
104    ///
105    /// # Panics
106    /// Panics if `prec` is zero.
107    ///
108    /// # Examples
109    /// ```
110    /// use malachite_float::Float;
111    /// use std::cmp::Ordering::*;
112    ///
113    /// let (log_2_e, o) = Float::log_2_e_prec(1);
114    /// assert_eq!(log_2_e.to_string(), "1.0");
115    /// assert_eq!(o, Less);
116    ///
117    /// let (log_2_e, o) = Float::log_2_e_prec(10);
118    /// assert_eq!(log_2_e.to_string(), "1.443");
119    /// assert_eq!(o, Greater);
120    ///
121    /// let (log_2_e, o) = Float::log_2_e_prec(100);
122    /// assert_eq!(log_2_e.to_string(), "1.442695040888963407359924681003");
123    /// assert_eq!(o, Greater);
124    /// ```
125    #[inline]
126    pub fn log_2_e_prec(prec: u64) -> (Self, Ordering) {
127        Self::log_2_e_prec_round(prec, Nearest)
128    }
129}
malachite_float/constants/log_2_e.rs

malachite_float/constants/
log_2_e.rs
