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// 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 core::cmp::Ordering;
use malachite_base::num::arithmetic::traits::ReciprocalSqrt;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::rounding_modes::RoundingMode::{self, *};
use malachite_nz::natural::arithmetic::float_extras::float_can_round;
use malachite_nz::platform::Limb;
impl Float {
/// Returns an approximation of $1/\sqrt{\tau}=1/\sqrt{2\pi}$, 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.)
///
/// $$
/// x = 1/\sqrt{\tau}+\varepsilon=1/\sqrt{2\pi}+\varepsilon.
/// $$
/// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{-p-1}$.
/// - If $m$ is `Nearest`, then $|\varepsilon| < 2^{-p-2}$.
///
/// 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 (one_over_sqrt_tau, o) = Float::one_over_sqrt_tau_prec_round(100, Floor);
/// assert_eq!(
/// one_over_sqrt_tau.to_string(),
/// "0.3989422804014326779399460599341"
/// );
/// assert_eq!(o, Less);
///
/// let (one_over_sqrt_tau, o) = Float::one_over_sqrt_tau_prec_round(100, Ceiling);
/// assert_eq!(
/// one_over_sqrt_tau.to_string(),
/// "0.3989422804014326779399460599344"
/// );
/// assert_eq!(o, Greater);
/// ```
pub fn one_over_sqrt_tau_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
let mut working_prec = prec + 10;
let mut increment = Limb::WIDTH;
loop {
let one_over_sqrt_tau = Self::tau_prec_round(working_prec, Floor)
.0
.reciprocal_sqrt();
if float_can_round(
one_over_sqrt_tau.significand_ref().unwrap(),
working_prec - 2,
prec,
rm,
) {
return Self::from_float_prec_round(one_over_sqrt_tau, prec, rm);
}
working_prec += increment;
increment = working_prec >> 1;
}
}
/// Returns an approximation of $1/\sqrt{\tau}=1/\sqrt{2\pi}$, 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.)
///
/// $$
/// x = 1/\sqrt{\tau}+\varepsilon=1/\sqrt{2\pi}+\varepsilon.
/// $$
/// - $|\varepsilon| < 2^{-p-2}$.
///
/// 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 (one_over_sqrt_tau, o) = Float::one_over_sqrt_tau_prec(1);
/// assert_eq!(one_over_sqrt_tau.to_string(), "0.5");
/// assert_eq!(o, Greater);
///
/// let (one_over_sqrt_tau, o) = Float::one_over_sqrt_tau_prec(10);
/// assert_eq!(one_over_sqrt_tau.to_string(), "0.3989");
/// assert_eq!(o, Less);
///
/// let (one_over_sqrt_tau, o) = Float::one_over_sqrt_tau_prec(100);
/// assert_eq!(
/// one_over_sqrt_tau.to_string(),
/// "0.3989422804014326779399460599344"
/// );
/// assert_eq!(o, Greater);
/// ```
#[inline]
pub fn one_over_sqrt_tau_prec(prec: u64) -> (Self, Ordering) {
Self::one_over_sqrt_tau_prec_round(prec, Nearest)
}
}