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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 core::cmp::Ordering::*;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::One;
use malachite_base::rounding_modes::RoundingMode::{self, *};
use malachite_nz::platform::Limb;
impl Float {
/// Returns an approximation of Gauss's constant, 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.)
///
/// $$
/// x = G+\varepsilon=1/\mathrm{AGM}(1,\sqrt{2})+\varepsilon,
/// $$
/// where AGM is the arithmetic-geometric mean.
/// - 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 (gauss_constant, o) = Float::gauss_constant_prec_round(100, Floor);
/// assert_eq!(
/// gauss_constant.to_string(),
/// "0.834626841674073186281429732799"
/// );
/// assert_eq!(o, Less);
///
/// let (gauss_constant, o) = Float::gauss_constant_prec_round(100, Ceiling);
/// assert_eq!(gauss_constant.to_string(), "0.8346268416740731862814297328");
/// assert_eq!(o, Greater);
/// ```
pub fn gauss_constant_prec_round(prec: u64, rm: RoundingMode) -> (Self, Ordering) {
let mut working_prec = prec + 10;
let mut increment = Limb::WIDTH;
loop {
let sqrt_2_lo = Self::sqrt_2_prec_round(working_prec, Floor).0;
let mut sqrt_2_hi = sqrt_2_lo.clone();
sqrt_2_hi.increment();
let lo = Self::ONE
.agm_round(sqrt_2_hi, Ceiling)
.0
.reciprocal_round(Floor)
.0;
let hi = Self::ONE
.agm_round(sqrt_2_lo, Floor)
.0
.reciprocal_round(Ceiling)
.0;
let (gauss_constant_lo, mut o_lo) = Self::from_float_prec_round(lo, prec, rm);
let (gauss_constant_hi, mut o_hi) = Self::from_float_prec_round(hi, prec, rm);
if o_lo == Equal {
o_lo = o_hi;
}
if o_hi == Equal {
o_hi = o_lo;
}
if o_lo == o_hi && gauss_constant_lo == gauss_constant_hi {
return (gauss_constant_lo, o_lo);
}
working_prec += increment;
increment = working_prec >> 1;
}
}
/// Returns an approximation of Gauss's constant, $G=1/\mathrm{AGM}(1,\sqrt{2})$, 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.)
///
/// $$
/// x=G+\varepsilon=1/\mathrm{AGM}(1,\sqrt{2})+\varepsilon,
/// $$
/// where AGM is the arithmetic-geometric mean.
/// - $|\varepsilon| < 2^{-p-1}$.
///
/// 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 (gauss_constant, o) = Float::gauss_constant_prec(1);
/// assert_eq!(gauss_constant.to_string(), "1.0");
/// assert_eq!(o, Greater);
///
/// let (gauss_constant, o) = Float::gauss_constant_prec(10);
/// assert_eq!(gauss_constant.to_string(), "0.835");
/// assert_eq!(o, Greater);
///
/// let (gauss_constant, o) = Float::gauss_constant_prec(100);
/// assert_eq!(
/// gauss_constant.to_string(),
/// "0.834626841674073186281429732799"
/// );
/// assert_eq!(o, Less);
/// ```
#[inline]
pub fn gauss_constant_prec(prec: u64) -> (Self, Ordering) {
Self::gauss_constant_prec_round(prec, Nearest)
}
}