ellip 1.2.3

/*
 * Ellip is licensed under The 3-Clause BSD, see LICENSE.
 * Copyright 2025 Sira Pornsiriprasert <code@psira.me>
 */

use num_traits::Float;

use crate::{
    bulirsch::{constants::BulirschConst, N_MAX_ITERATIONS},
    crate_util::{check, declare},
    ellipeinc, ellipf, ellippi, ellippiinc, StrErr,
};

/// Computes [incomplete elliptic integral of the third kind in Bulirsch's form](https://dlmf.nist.gov/19.2.E16).
/// ```text
///                    arctan(x)                                                   
///                   ⌠                             
///                   |                          dϑ
/// el3(x, kc, p)  =  ⎮  ───────────────────────────────────────────────────
///                   ⎮                              ______________________
///                   ⌡  ( cos²(ϑ) + p sin²(ϑ) ) ⋅ ╲╱ cos²(ϑ) + kc² sin²(ϑ)   
///                  0                                                   
/// ```
///
/// ## Parameters
/// - x: tangent of amplitude angle. x ∈ ℝ.
/// - kc: complementary modulus. kc ∈ ℝ, kc ≠ 0.
/// - p ∈ ℝ
///
/// ## Domain
/// - Returns error if:
///   - kc = 0,
///   - 1 + px² = 0,
///   - or |kc| > 1 for p < 0.
/// - Returns the Cauchy principal value when 1 + px² < 0
///
/// ## Graph
/// ![Bulirsch's Incomplete Elliptic Integral of the Third Kind](https://github.com/p-sira/ellip/blob/main/figures/el3.svg?raw=true)
///
/// [Interactive Plot](https://p-sira.github.io/ellippy/_static/figures/el3.html)
///
/// ## Special Cases
/// - el3(0, kc, p) = 0
/// - el3(∞, kc, p) = cel(kc, p, 1, 1) = Π(1-p, 1-kc²)
///
/// # Related Functions
/// With x = tan φ, kc² = 1 - m, and p = 1 - n,
/// - [ellippiinc](crate::ellippiinc)(φ, n, m) = [el3](crate::el3)(x, kc, p)
/// - [el3](crate::el3)(∞, kc, p) = [cel](crate::cel)(kc, p, 1, 1) = [ellippi](crate::ellippi)(n, m)
///
/// # Examples
/// ```
/// use ellip::{el3, util::assert_close};
/// use std::f64::consts::FRAC_PI_4;
///
/// assert_close(el3(FRAC_PI_4.tan(), 0.5, 1.0).unwrap(), 0.8512237490711854, 1e-15);
/// ```
///
/// # Notes
/// The default precision of the function is set according to the original literature by [Bulirsch](https://doi.org/10.1007/BF02165405)
/// for [f64] and [f32]. The precision can be modified in the function [el3_with_const] (requires `unstable` feature flag).
///
/// # References
/// - Bulirsch, R. “Numerical Calculation of Elliptic Integrals and Elliptic Functions. III.” Numerische Mathematik 13, no. 4 (August 1, 1969): 305–15. <https://doi.org/10.1007/BF02165405>.
/// - Carlson, B. C. “DLMF: Chapter 19 Elliptic Integrals.” Accessed February 19, 2025. <https://dlmf.nist.gov/19>.
pub fn el3<T: Float>(x: T, kc: T, p: T) -> Result<T, StrErr> {
    if core::mem::size_of::<T>() <= 4 {
        el3_with_const::<T, f32>(x, kc, p)
    } else {
        el3_with_const::<T, f64>(x, kc, p)
    }
}

/// Computes [el3]. Control the precision using [BulirschConst].
#[numeric_literals::replace_float_literals(T::from(literal).unwrap())]
#[inline]
pub fn el3_with_const<T: Float, C: BulirschConst<T>>(x: T, kc: T, p: T) -> Result<T, StrErr> {
    let m = 1.0 - kc * kc;
    let n = 1.0 - p;

    if kc.abs() < epsilon!() {
        return Err("el3: kc must not be zero.");
    }

    if x == 0.0 {
        return Ok(0.0);
    }

    // Handle special cases
    if x == inf!() {
        return ellippi(n, m);
    }

    if kc == 1.0 {
        // A&S 17.7.20:
        if p == 0.0 {
            return Ok(x);
        }
        if p > 0.0 {
            let vcr = p.sqrt();
            return Ok((vcr * x).atan() / vcr);
        } else {
            // v > 1:
            let vcr = (-p).sqrt();
            let arg = vcr * x;
            if arg.abs() == 1.0 {
                return Err("el3: 1 + px² cannot be zero.");
            }
            // Real Cauchy principal value on either side of the pole.
            let log_ratio = if arg.abs() < 1.0 {
                arg.ln_1p() - (-arg).ln_1p()
            } else {
                (1.0 / arg).ln_1p() - (-1.0 / arg).ln_1p()
            };
            return Ok(log_ratio / (2.0 * vcr));
        }
    }

    // Didn't improve the accuracy
    // if n = 0.0 {
    //     // A&S 17.7.18 & 19
    //     return if m = 0.0 {
    //         Ok(phi)
    //     } else {
    //         ellipf(phi, m)
    //     };
    // }

    // This identity is exact only for p = 0. Small p can still give large p*x².
    if p == 0.0 && m != 1.0 {
        let phi = x.atan();
        // http://functions.wolfram.com/08.06.03.0008.01
        let sp2 = phi.sin() * phi.sin();
        let mut result = (1.0 - m * sp2).sqrt() * x - ellipeinc(phi, m)?;
        result = result / (1.0 - m);
        result = result + ellipf(phi, m)?;
        return Ok(result);
    }

    if kc.abs() < C::lim_kc_p() && p > C::lim_kc_p() {
        let phi = x.atan();
        return ellippiinc(phi, n, m);
    }

    let x_abs = x.abs();

    // real
    declare!(mut [c, d, de, e, f, fa, g, h, hh]);
    declare!(mut [pm, pz, q, r, s = kc, t, u, v, w, y, ye = T::zero(), z]);
    let zd;

    // int
    declare!(mut [k = 0, l, mm, nn]);

    // bool
    declare!(mut [bo, bk = false]);

    hh = x * x;
    f = p * hh;
    t = s * s;
    pm = t * 0.5;
    e = hh * t;
    z = f.abs();
    r = p.abs();
    h = 1.0 + hh;

    // small
    if e < 0.1 && z < 0.1 && t < 1.0 && r < 1.0 {
        declare!(mut [rb, ra, rr] = [T::zero(); MAX_ND]);

        for k in 2..=C::ND {
            let k_float = T::from(k).unwrap();
            rb[k - 2] = 0.5 / k_float;
            ra[k - 2] = 1.0 - rb[k - 2];
        }

        zd = 0.5 / (T::from(C::ND).unwrap() + 1.0);
        s = p + pm;

        for k in 0..C::ND - 2 {
            rr[k] = s;
            pm = pm * t * ra[k];
            s = s * p + pm;
        }
        s = s * zd;
        u = s;
        bo = false;
        for k in (0..C::ND - 2).rev() {
            u = u + (rr[k] - u) * rb[k];
            bo = !bo;
            let v = if bo { -u } else { u };
            s = s * hh + v;
        }

        if bo {
            s = -s;
        }
        u = 0.5 * (u + 1.0);
        return Ok((u - s * h) * h.sqrt() * x + u * x.asinh());
    }

    w = 1.0 + f;
    if w == 0.0 {
        return Err("el3: 1 + px² cannot be zero.");
    }

    let p1 = p;
    s = s.abs();
    y = x_abs;
    g = if p == 1.0 { C::cb() } else { p1 - 1.0 };
    f = p1 - t;
    if f == 0.0 {
        f = C::cb() * t;
    }
    let am = 1.0 - t;
    let ap = 1.0 + e;
    r = p1 * h;
    fa = g / (f * p1);
    bo = fa > 0.0;
    fa = fa.abs();
    pz = (g * f).abs();
    de = pz.sqrt();
    q = p1.abs().sqrt();

    pm = pm.min(0.5);
    pm = p1 - pm;

    if pm >= 0.0 {
        u = (r * ap).sqrt();
        v = y * de;
        if g < 0.0 {
            v = -v;
        }
        d = 1.0 / q;
        c = 1.0;
    } else {
        u = (h * ap * pz).sqrt();
        ye = y * q;
        v = am * ye;
        q = -de / g;
        d = -am / de;
        c = 0.0;
        pz = ap - r;
    }

    if bo {
        r = v / u;
        z = 1.0;
        k = 1;
        if pm < 0.0 {
            h = y * (h / ap / fa).sqrt();
            h = 1.0 / h - h;
            z = h - r - r;
            r = 2.0 + r * h;
            // if r == 0.0 {
            //     r = C::cb();
            // }
            // if z == 0.0 {
            //     z = h * C::cb();
            // }
            z = r / z;
            r = z;
            w = pz;
        }
        u = u / w;
        v = v / w;
    } else {
        t = u + v.abs();
        bk = true;
        if p1 < 0.0 {
            de = v / pz;
            ye = u * ye;
            ye = ye + ye;
            u = t / pz;
            v = (-f - g * e) / t;
            t = pz * w.abs();
            z = (hh * r * f - g * ap + ye) / t;
            ye = ye / t;
        } else {
            de = v / w;
            ye = 0.0;
            u = (e + p1) / t;
            v = t / w;
            z = 1.0;
        }
        if s > 1.0 {
            h = u;
            u = v;
            v = h;
        }
    }

    y = 1.0 / y;
    e = s;
    nn = 1;
    t = 1.0;
    l = 0;
    mm = 0;

    for _ in 0..N_MAX_ITERATIONS {
        y = y - e / y;
        // if y == 0.0 {
        //     y = e.sqrt() * C::cb();
        // }
        f = c;
        c = d / q + c;
        g = e / q;
        d = f * g + d;
        d = d + d;
        q = g + q;
        g = t;
        t = s + t;
        nn = nn + nn;
        mm = mm + mm;
        if bo {
            if z < 0.0 {
                mm += k;
            }
            k = if r < 0.0 { -1 } else { 1 };
            h = e / (u * u + v * v);
            u = u * (1.0 + h);
            v = v * (1.0 - h);
        } else {
            r = u / v;
            h = z * r;
            z = h * z;
            hh = e / v;

            de = de / u;
            ye = ye * (h + 1.0 / h) + de * (1.0 + r);
            de = de * (u - hh);
            bk = ye.abs() < 1.0;

            // Removed crack function
        }
        if (g - s).abs() > C::ca() * g {
            if bo {
                g = (1.0 / r - r) * 0.5;
                hh = u + v * g;
                h = g * u - v;
                // if hh == 0.0 {
                //     hh = u * C::cb();
                // }
                // if h == 0.0 {
                //     h = v * C::cb();
                // }
                z = r * h;
                r = hh / h;
            } else {
                u = u + e / u;
                v = v + hh;
            }
            s = e.sqrt();
            s = s + s;
            e = s * t;
            l *= 2;
            if y < 0.0 {
                l += 1;
            }
            continue;
        }
        if y < 0.0 {
            l += 1;
        }
        e = (t / y).atan() + pi!() * T::from(l).unwrap();
        e = e * (c * t + d) / (t * (t + q));

        if bo {
            h = v / (t + u);
            z = 1.0 - r * h;
            h = r + h;
            // if z == 0.0 {
            //     z = C::cb();
            // }
            // if z < 0.0 {
            //     mm += if h < 0.0 { -1 } else { 1 };
            // }
            s = (h / z).atan() + T::from(mm).unwrap() * pi!();
        } else {
            s = if bk {
                ye.asinh()
            } else {
                z.ln() + T::from(mm).unwrap() * ln_2!()
            };
            s = s * 0.5;
        }
        e = (e + fa.sqrt() * s) / T::from(nn).unwrap();
        let ans = x.signum() * e;
        if ans.is_finite() {
            return Ok(ans);
        }
        check!(@nan, el3, [x, kc, p]);
        break;
    }

    let phi = x.atan();
    let ans = ellippiinc(phi, n, m);
    if ans.is_ok() {
        return ans;
    }
    Err("el3: Failed to converge.")
}

const MAX_ND: usize = 50;

#[cfg(not(feature = "test_force_fail"))]
#[cfg(test)]
mod tests {
    use super::*;
    use crate::{assert_close, compare_test_data_wolfram};

    #[test]
    fn test_el3() {
        compare_test_data_wolfram!("el3_data.csv", el3, 3, 3e-12);
        compare_test_data_wolfram!("el3_pv.csv", el3, 3, 5e-15);
    }

    #[test]
    fn test_el3_references() {
        // Test computed values from the reference
        // Bulirsch, “Numerical Calculation of Elliptic Integrals and Elliptic Functions III.”
        fn test_reference(x: f64, kc: f64, p: f64, expected: f64) {
            assert_close!(expected, el3(x, kc, p).unwrap(), 2.0 * f64::EPSILON);
        }

        test_reference(1.3, 0.11, 4.21, 6.6220785847015254e-1);
        test_reference(1.3, 0.11, 0.82, 1.1307046442074609);
        test_reference(1.3, 0.92, 0.71, 1.0058286266977115);
        test_reference(1.3, 0.92, 0.23, 1.1884070823345123);
        test_reference(1.3, 0.12, -0.11, 1.7259650355348878);
        test_reference(1.3, 0.12, -2.11, 2.4416814520721179e-1);
        test_reference(1.3, 0.40, 0.1600001, 1.4004165258366944);
        test_reference(1.3, 1.0e-10, 0.82, 1.1341505395282723);
        test_reference(1.3e-10, 1.0e-10, 1.0e-10, 1.3e-10);
        test_reference(1.6, 1.90, 9.81, 3.8572324379967252e-1);
        test_reference(1.6, 1.90, 1.22, 7.6656179311956402e-1);
        test_reference(1.6, 1.90, 0.87, 8.3210591112618096e-1);
        test_reference(1.6, 1.90, 0.21, 1.0521272221906806);
        test_reference(1.6, 1.50, 2.24999, 7.0057431688357934e-1);
        test_reference(1.6, 1e10, 1.20, 2.3734774669772208e-9);
        test_reference(-1.6, 1e10, 1.20, -2.3734774669772208e-9);
        test_reference(1.0, 0.31, 9.90e-2, 1.0903577921777398);

        // Use wolfram value at 16-digit precision instead since the values provided by
        // reference were calculated using 36-bit Matissa float (f32 and f64) making them inaccurate.
        test_reference(1.6, 1.90, -0.21, 1.473043981995436);
        test_reference(1.6, 1.90, -4.30, 0.2547695139719361);
        assert_close!(0.3950170978760504, el3(1.6, 1.01e1, -1.0e-5).unwrap(), 1e-9);
    }

    #[test]
    fn test_el3_special_cases() {
        use crate::cel;
        use std::f64::{INFINITY, NAN};
        // x = 0: el3(0, kc, p) = 0
        assert_eq!(el3(0.0, 0.5, 0.5).unwrap(), 0.0);
        // kc = 0: should return Err
        assert_eq!(el3(0.5, 0.0, 0.5), Err("el3: kc must not be zero."));

        let complete_el3 = el3(INFINITY, 0.5, 0.5).unwrap();
        // x = inf: el3(inf, kc, p) = cel(kc, p, 1, 1)
        assert_close!(complete_el3, cel(0.5, 0.5, 1.0, 1.0).unwrap(), 1e-15);
        // x = inf: el3(inf, kc, p) = Π(n, m)
        assert_close!(complete_el3, ellippi(0.5, 0.75).unwrap(), 1e-15);

        // kc = 1, p > 0: el3(x, 1, p) = atan(sqrt(p) * x) / sqrt(p)
        assert_eq!(
            el3(4.0, 1.0, 0.5).unwrap(),
            (0.5.sqrt() * 4.0).atan() / 0.5.sqrt()
        );
        // kc = 1, p <= 0: el3(x, 1, p) = (ln(1+vx) - ln(1-vx)) / (2v); v = sqrt(-p)
        // Corrected while fixing audit finding A15:
        // https://github.com/p-sira/ellip/pull/129
        assert_close!(0.5225504573804798, el3(0.5, 1.0, -0.5).unwrap(), 1e-15);
        // 1 + px² = 0: should return Err
        assert_eq!(el3(1.0, 0.5, -1.0), Err("el3: 1 + px² cannot be zero."));
        // x = nan, kc = nan, or p = nan: should return Err
        assert_eq!(el3(NAN, 0.5, 0.5), Err("el3: Arguments cannot be NAN."));
        assert_eq!(el3(0.5, NAN, 0.5), Err("el3: Arguments cannot be NAN."));
        assert_eq!(el3(0.5, 0.5, NAN), Err("el3: Arguments cannot be NAN."));
    }

    // Regression for audit finding A3: https://github.com/p-sira/ellip/pull/117
    #[test]
    fn test_small_nonzero_characteristic() {
        for x in [1e6, -1e6] {
            assert_close!(
                el3(x, 0.5, 1e-11).unwrap(),
                x.signum() * 799749.3224180657,
                3e-14
            );
        }
        assert_close!(el3(1.0, 0.5, 0.0).unwrap(), 1.1006047874101814, 3e-14);
    }

    // Regression for audit finding A7: https://github.com/p-sira/ellip/pull/121
    #[test]
    fn test_elementary_real_values() {
        for x in [1.0, -1.0, 2.0, -2.0] {
            assert_eq!(el3(x, 1.0, 0.0).unwrap(), x);
        }
        for x in [2.0, -2.0] {
            assert_close!(
                el3(x, 1.0, -1.0).unwrap(),
                x.signum() * 0.5493061443340548,
                2e-15
            );
        }
        assert!(el3(1.0, 1.0, -1.0).is_err());
        assert_close!(el3(1.0, 1.0, 1e-20).unwrap(), 1.0, 1e-15);
    }
}

#[cfg(feature = "test_force_fail")]
crate::test_force_unreachable! {
    use crate::bulirsch::constants::DefaultPrecision;
    assert_eq!(el3_with_const::<f64, DefaultPrecision>(0.5, 0.5, 0.5), Err("el3: Failed to converge."));
}