1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
// Copyright © 2026 Mikhail Hogrefe
//
// Uses code adopted from the GNU MPFR Library.
//
// Copyright 1999, 2001-2024 Free Software Foundation, Inc.
//
// Contributed by the AriC and Caramba projects, INRIA.
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::Float;
use alloc::vec;
use core::cmp::Ordering;
use core::mem::swap;
use malachite_base::num::arithmetic::traits::CeilingLogBase2;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::{One, Zero};
use malachite_base::num::conversion::traits::WrappingFrom;
use malachite_base::rounding_modes::RoundingMode::{self, *};
use malachite_nz::integer::Integer;
use malachite_nz::natural::arithmetic::float_extras::float_can_round;
use malachite_nz::platform::Limb;
// Auxiliary function: Compute the terms from n1 to n2 (excluded) 3 / 4 * sum((-1) ^ n * n! ^ 2 / 2
// ^ n / (2 * n + 1)!, n = n1...n2 - 1).s
//
// Numerator is T[0], denominator is Q[0], Compute P[0] only when need_P is non-zero.
//
// Need 1 + ceil(log(n2 - n1) / log(2)) cells in T[], P[], Q[].
//
// This is S from const_log2.c, MPFR 4.2.0.
fn sum(t: &mut [Integer], p: &mut [Integer], q: &mut [Integer], n1: u64, n2: u64, need_p: bool) {
if n2 == n1 + 1 {
p[0] = if n1 == 0 {
const { Integer::const_from_unsigned(3) }
} else {
-Integer::from(n1)
};
q[0] = ((Integer::from(n1) << 1u32) + Integer::ONE) << 2u32;
t[0].clone_from(&p[0]);
} else {
let m = (n1 >> 1) + (n2 >> 1) + (n1 & 1 & n2);
sum(t, p, q, n1, m, true);
let (t_head, t_tail) = t.split_first_mut().unwrap();
let (p_head, p_tail) = p.split_first_mut().unwrap();
let (q_head, q_tail) = q.split_first_mut().unwrap();
sum(t_tail, p_tail, q_tail, m, n2, need_p);
*t_head *= &q_tail[0];
t_tail[0] *= &*p_head;
*t_head += &t_tail[0];
if need_p {
*p_head *= &p_tail[0];
}
*q_head *= &q_tail[0];
// remove common trailing zeros if any
let mut tz = t_head.trailing_zeros().unwrap();
if tz != 0 {
let mut qz = q_head.trailing_zeros().unwrap();
if qz < tz {
tz = qz;
}
if need_p {
qz = p_head.trailing_zeros().unwrap();
if qz < tz {
tz = qz;
}
}
// now tz = min(val(T), val(Q), val(P))
if tz != 0 {
*t_head >>= tz;
*q_head >>= tz;
if need_p {
*p_head >>= tz;
}
}
}
}
}
impl Float {
/// Returns an approximation of the natural logarithm of 2, with the given precision and rounded
/// using the given [`RoundingMode`]. An [`Ordering`] is also returned, indicating whether the
/// rounded value is less than or greater than the exact value of the constant. (Since the
/// constant is irrational, the rounded value is never equal to the exact value.)
///
/// $$
/// x = \ln 2+\varepsilon.
/// $$
/// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p}$.
/// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
///
/// The constant is irrational and transcendental.
///
/// The output has precision `prec`.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n (\log n)^2)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
///
/// # Panics
/// Panics if `prec` is zero or if `rm` is `Exact`.
///
/// # Examples
/// ```
/// use malachite_base::rounding_modes::RoundingMode::*;
/// use malachite_float::Float;
/// use std::cmp::Ordering::*;
///
/// let (l2, o) = Float::ln_2_prec_round(100, Floor);
/// assert_eq!(l2.to_string(), "0.693147180559945309417232121458");
/// assert_eq!(o, Less);
///
/// let (l2, o) = Float::ln_2_prec_round(100, Ceiling);
/// assert_eq!(l2.to_string(), "0.693147180559945309417232121459");
/// assert_eq!(o, Greater);
/// ```
///
/// This is mpfr_const_log2_internal from const_log2.c, MPFR 4.2.0.
pub fn ln_2_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
let mut working_prec = prec + prec.ceiling_log_base_2() + 3;
let mut increment = Limb::WIDTH;
loop {
let big_n = working_prec / 3 + 1;
// the following are needed for error analysis (see algorithms.tex)
assert!(working_prec >= 3 && big_n >= 2);
let lg_big_n = usize::wrapping_from(big_n.ceiling_log_base_2()) + 1;
let mut scratch = vec![Integer::ZERO; 3 * lg_big_n];
split_into_chunks_mut!(scratch, lg_big_n, [t, p], q);
sum(t, p, q, 0, big_n, false);
let mut t0 = Integer::ZERO;
let mut q0 = Integer::ZERO;
swap(&mut t0, &mut t[0]);
swap(&mut q0, &mut q[0]);
let ln_2 = Self::from_integer_prec(t0, working_prec).0
/ Self::from_integer_prec(q0, working_prec).0;
if float_can_round(ln_2.significand_ref().unwrap(), working_prec - 2, prec, rm) {
return Self::from_float_prec_round(ln_2, prec, rm);
}
working_prec += increment;
increment = working_prec >> 1;
}
}
/// Returns an approximation of the natural logarithm of 2, with the given precision and rounded
/// to the nearest [`Float`] of that precision. An [`Ordering`] is also returned, indicating
/// whether the rounded value is less than or greater than the exact value of the constant.
/// (Since the constant is irrational, the rounded value is never equal to the exact value.)
///
/// $$
/// x = \ln 2+\varepsilon.
/// $$
/// - $|\varepsilon| < 2^{-p-1}$.
///
/// The constant is irrational and transcendental.
///
/// The output has precision `prec`.
///
/// # Worst-case complexity
/// $T(n) = O(n (\log n)^2 \log\log n)$
///
/// $M(n) = O(n (\log n)^2)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `prec`.
///
/// # Panics
/// Panics if `prec` is zero.
///
/// # Examples
/// ```
/// use malachite_float::Float;
/// use std::cmp::Ordering::*;
///
/// let (l2, o) = Float::ln_2_prec(1);
/// assert_eq!(l2.to_string(), "0.5");
/// assert_eq!(o, Less);
///
/// let (l2, o) = Float::ln_2_prec(10);
/// assert_eq!(l2.to_string(), "0.693");
/// assert_eq!(o, Greater);
///
/// let (l2, o) = Float::ln_2_prec(100);
/// assert_eq!(l2.to_string(), "0.693147180559945309417232121458");
/// assert_eq!(o, Less);
/// ```
#[inline]
pub fn ln_2_prec(prec: u64) -> (Self, Ordering) {
Self::ln_2_prec_round(prec, Nearest)
}
}