feanor-math 3.5.18

use std::f64::EPSILON;
use std::f64::consts::PI;

use crate::divisibility::{DivisibilityRing, Domain};
use crate::field::Field;
use crate::homomorphism::CanHomFrom;
use crate::impl_eq_based_self_iso;
use crate::integer::{IntegerRing, IntegerRingStore};
use crate::pid::{EuclideanRing, PrincipalIdealRing};
use crate::ring::*;
use crate::rings::approx_real::float::Real64;
use crate::rings::rational::RationalFieldBase;

/// An approximate implementation of the complex numbers `C`, using 64 bit floating
/// point numbers.
///
/// # Warning
///
/// Since floating point numbers do not exactly represent the complex numbers, and this crate
/// follows a mathematically precise approach, we cannot provide any function related to equality.
/// In particular, `Complex64Base.eq_el(a, b)` is not supported, and will panic.
/// Hence, this ring has only limited use within this crate, and is currently only used for
/// floating-point FFTs.
#[derive(Clone, Copy, PartialEq, Debug)]
pub struct Complex64Base;

/// An element of [`Complex64`].
#[derive(Clone, Copy, Debug)]
pub struct Complex64El(f64, f64);

/// [`RingStore`] corresponding to [`Complex64Base`]
pub type Complex64 = RingValue<Complex64Base>;

impl Complex64 {
    pub const RING: Self = RingValue::from(Complex64Base);
    pub const I: Complex64El = Complex64El(0.0, 1.0);
}

impl Complex64Base {
    pub fn abs(&self, Complex64El(re, im): Complex64El) -> f64 { (re * re + im * im).sqrt() }

    pub fn conjugate(&self, Complex64El(re, im): Complex64El) -> Complex64El { Complex64El(re, -im) }

    pub fn exp(&self, Complex64El(exp_re, exp_im): Complex64El) -> Complex64El {
        let angle = exp_im;
        let abs = exp_re.exp();
        Complex64El(abs * angle.cos(), abs * angle.sin())
    }

    pub fn closest_gaussian_int(&self, Complex64El(re, im): Complex64El) -> (i64, i64) {
        (re.round() as i64, im.round() as i64)
    }

    pub fn ln_main_branch(&self, Complex64El(re, im): Complex64El) -> Complex64El {
        Complex64El(self.abs(Complex64El(re, im)).ln(), im.atan2(re))
    }

    pub fn is_absolute_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, absolute_threshold: f64) -> bool {
        self.abs(self.sub(lhs, rhs)) < absolute_threshold
    }

    pub fn is_relative_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, relative_limit: f64) -> bool {
        self.is_absolute_approx_eq(lhs, rhs, self.abs(lhs) * relative_limit)
    }

    pub fn is_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, precision: u64) -> bool {
        let scaled_precision = precision as f64 * EPSILON;
        if self.is_absolute_approx_eq(lhs, self.zero(), scaled_precision) {
            self.is_absolute_approx_eq(rhs, self.zero(), scaled_precision)
        } else {
            self.is_relative_approx_eq(lhs, rhs, scaled_precision)
        }
    }

    pub fn from_f64(&self, x: f64) -> Complex64El { Complex64El(x, 0.0) }

    pub fn root_of_unity(&self, i: i64, n: i64) -> Complex64El {
        self.exp(self.mul(self.from_f64((i as f64 / n as f64) * (2.0 * PI)), Complex64::I))
    }

    pub fn re(&self, Complex64El(re, _im): Complex64El) -> f64 { re }

    pub fn im(&self, Complex64El(_re, im): Complex64El) -> f64 { im }
}

impl Complex64 {
    pub fn abs(&self, val: Complex64El) -> f64 { self.get_ring().abs(val) }

    pub fn conjugate(&self, val: Complex64El) -> Complex64El { self.get_ring().conjugate(val) }

    pub fn exp(&self, exp: Complex64El) -> Complex64El { self.get_ring().exp(exp) }

    pub fn closest_gaussian_int(&self, val: Complex64El) -> (i64, i64) { self.get_ring().closest_gaussian_int(val) }

    pub fn ln_main_branch(&self, val: Complex64El) -> Complex64El { self.get_ring().ln_main_branch(val) }

    pub fn is_absolute_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, absolute_threshold: f64) -> bool {
        self.get_ring().is_absolute_approx_eq(lhs, rhs, absolute_threshold)
    }

    pub fn is_relative_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, relative_limit: f64) -> bool {
        self.get_ring().is_relative_approx_eq(lhs, rhs, relative_limit)
    }

    pub fn is_approx_eq(&self, lhs: Complex64El, rhs: Complex64El, precision: u64) -> bool {
        self.get_ring().is_approx_eq(lhs, rhs, precision)
    }

    pub fn from_f64(&self, x: f64) -> Complex64El { self.get_ring().from_f64(x) }

    pub fn root_of_unity(&self, i: i64, n: i64) -> Complex64El { self.get_ring().root_of_unity(i, n) }

    pub fn re(&self, x: Complex64El) -> f64 { self.get_ring().re(x) }

    pub fn im(&self, x: Complex64El) -> f64 { self.get_ring().im(x) }
}

impl RingBase for Complex64Base {
    type Element = Complex64El;

    fn clone_el(&self, val: &Self::Element) -> Self::Element { *val }

    fn add_assign(&self, Complex64El(lhs_re, lhs_im): &mut Self::Element, Complex64El(rhs_re, rhs_im): Self::Element) {
        *lhs_re += rhs_re;
        *lhs_im += rhs_im;
    }

    fn negate_inplace(&self, Complex64El(re, im): &mut Self::Element) {
        *re = -*re;
        *im = -*im;
    }

    fn mul_assign(&self, Complex64El(lhs_re, lhs_im): &mut Self::Element, Complex64El(rhs_re, rhs_im): Self::Element) {
        let new_im = *lhs_re * rhs_im + *lhs_im * rhs_re;
        *lhs_re = *lhs_re * rhs_re - *lhs_im * rhs_im;
        *lhs_im = new_im;
    }

    fn from_int(&self, value: i32) -> Self::Element { Complex64El(value as f64, 0.0) }

    fn eq_el(&self, _: &Self::Element, _: &Self::Element) -> bool {
        panic!("Cannot provide equality on approximate rings")
    }

    fn pow_gen<R: IntegerRingStore>(&self, x: Self::Element, power: &El<R>, integers: R) -> Self::Element
    where
        R::Type: IntegerRing,
    {
        self.exp(self.mul(
            self.ln_main_branch(x),
            Complex64El(integers.to_float_approx(power), 0.0),
        ))
    }

    fn is_commutative(&self) -> bool { true }

    fn is_noetherian(&self) -> bool { true }

    fn is_approximate(&self) -> bool { true }

    fn dbg_within<'a>(
        &self,
        Complex64El(re, im): &Self::Element,
        out: &mut std::fmt::Formatter<'a>,
        env: EnvBindingStrength,
    ) -> std::fmt::Result {
        if env >= EnvBindingStrength::Product {
            write!(out, "({} + {}i)", re, im)
        } else {
            write!(out, "{} + {}i", re, im)
        }
    }

    fn dbg<'a>(&self, value: &Self::Element, out: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        self.dbg_within(value, out, EnvBindingStrength::Weakest)
    }

    fn characteristic<I: IntegerRingStore + Copy>(&self, ZZ: I) -> Option<El<I>>
    where
        I::Type: IntegerRing,
    {
        Some(ZZ.zero())
    }
}

impl_eq_based_self_iso! { Complex64Base }

impl Domain for Complex64Base {}

impl DivisibilityRing for Complex64Base {
    fn checked_left_div(&self, lhs: &Self::Element, rhs: &Self::Element) -> Option<Self::Element> {
        let abs_sqr = self.abs(*rhs) * self.abs(*rhs);
        let Complex64El(res_re, res_im) = self.mul(*lhs, self.conjugate(*rhs));
        return Some(Complex64El(res_re / abs_sqr, res_im / abs_sqr));
    }
}

impl PrincipalIdealRing for Complex64Base {
    fn checked_div_min(&self, lhs: &Self::Element, rhs: &Self::Element) -> Option<Self::Element> {
        self.checked_left_div(lhs, rhs)
    }

    fn extended_ideal_gen(
        &self,
        _lhs: &Self::Element,
        _rhs: &Self::Element,
    ) -> (Self::Element, Self::Element, Self::Element) {
        panic!("Since Complex64 is only approximate, this cannot be implemented properly")
    }
}

impl EuclideanRing for Complex64Base {
    fn euclidean_div_rem(&self, _lhs: Self::Element, _rhs: &Self::Element) -> (Self::Element, Self::Element) {
        panic!("Since Complex64 is only approximate, this cannot be implemented properly")
    }

    fn euclidean_deg(&self, _: &Self::Element) -> Option<usize> {
        panic!("Since Complex64 is only approximate, this cannot be implemented properly")
    }
}

impl Field for Complex64Base {
    fn div(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element {
        self.checked_left_div(lhs, rhs).unwrap()
    }
}

impl RingExtension for Complex64Base {
    type BaseRing = Real64;

    fn base_ring(&self) -> &Self::BaseRing { &Real64::RING }

    fn from(&self, x: El<Self::BaseRing>) -> Self::Element { self.from_f64(x) }

    fn mul_assign_base(&self, lhs: &mut Self::Element, rhs: &El<Self::BaseRing>) {
        lhs.0 *= *rhs;
        lhs.1 *= *rhs;
    }
}

impl<I: ?Sized + IntegerRing> CanHomFrom<I> for Complex64Base {
    type Homomorphism = ();

    fn has_canonical_hom(&self, _from: &I) -> Option<Self::Homomorphism> { Some(()) }

    fn map_in(&self, from: &I, el: <I as RingBase>::Element, hom: &Self::Homomorphism) -> Self::Element {
        self.map_in_ref(from, &el, hom)
    }

    fn map_in_ref(&self, from: &I, el: &<I as RingBase>::Element, _hom: &Self::Homomorphism) -> Self::Element {
        self.from_f64(from.to_float_approx(el))
    }
}

impl<I> CanHomFrom<RationalFieldBase<I>> for Complex64Base
where
    I: IntegerRingStore,
    I::Type: IntegerRing,
{
    type Homomorphism = <Self as CanHomFrom<I::Type>>::Homomorphism;

    fn has_canonical_hom(&self, from: &RationalFieldBase<I>) -> Option<Self::Homomorphism> {
        self.has_canonical_hom(from.base_ring().get_ring())
    }

    fn map_in(
        &self,
        from: &RationalFieldBase<I>,
        el: <RationalFieldBase<I> as RingBase>::Element,
        hom: &Self::Homomorphism,
    ) -> Self::Element {
        self.map_in_ref(from, &el, hom)
    }

    fn map_in_ref(
        &self,
        from: &RationalFieldBase<I>,
        el: &<RationalFieldBase<I> as RingBase>::Element,
        hom: &Self::Homomorphism,
    ) -> Self::Element {
        self.div(
            &self.map_in_ref(from.base_ring().get_ring(), from.num(el), hom),
            &self.map_in_ref(from.base_ring().get_ring(), from.den(el), hom),
        )
    }
}

#[test]
fn test_pow() {
    let CC = Complex64::RING;
    let i = Complex64::I;
    assert!(CC.is_approx_eq(CC.negate(i), CC.pow(i, 3), 1));
    assert!(!CC.is_approx_eq(CC.negate(i), CC.pow(i, 1024 + 3), 1));
    assert!(CC.is_approx_eq(CC.negate(i), CC.pow(i, 1024 + 3), 100));
    assert!(CC.is_approx_eq(
        CC.exp(CC.mul(CC.from_f64(PI / 4.0), i)),
        CC.mul(CC.add(CC.one(), i), CC.from_f64(2.0f64.powf(-0.5))),
        1
    ));

    let seventh_root_of_unity = CC.exp(CC.mul(i, CC.from_f64(2.0 * PI / 7.0)));
    assert!(CC.is_approx_eq(CC.pow(seventh_root_of_unity, 7 * 100 + 1), seventh_root_of_unity, 1000));
}

#[test]
fn test_mul() {
    let CC = Complex64::RING;
    let i = Complex64::I;
    assert!(CC.is_approx_eq(CC.mul(i, i), CC.from_f64(-1.0), 1));
    assert!(CC.is_approx_eq(CC.mul(i, CC.negate(i)), CC.from_f64(1.0), 1));
    assert!(CC.is_approx_eq(CC.mul(CC.add(i, CC.one()), i), CC.sub(i, CC.one()), 1));
}