venv/lib/python3.11/site-packages/scipy/special/xsf/cephes/ellie.h

/* Translated into C++ by SciPy developers in 2024.
 * Original header with Copyright information appears below.
 */

/*                                                     ellie.c
 *
 *     Incomplete elliptic integral of the second kind
 *
 *
 *
 * SYNOPSIS:
 *
 * double phi, m, y, ellie();
 *
 * y = ellie( phi, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *                 phi
 *                  -
 *                 | |
 *                 |                   2
 * E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
 *                 |
 *               | |
 *                -
 *                 0
 *
 * of amplitude phi and modulus m, using the arithmetic -
 * geometric mean algorithm.
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random arguments with phi in [-10, 10] and m in
 * [0, 1].
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE     -10,10      150000       3.3e-15     1.4e-16
 */

/*
 * Cephes Math Library Release 2.0:  April, 1987
 * Copyright 1984, 1987, 1993 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */
/* Copyright 2014, Eric W. Moore */

/*     Incomplete elliptic integral of second kind     */
#pragma once

#include "../config.h"
#include "const.h"
#include "ellpe.h"
#include "ellpk.h"
#include "unity.h"

namespace xsf {
namespace cephes {
    namespace detail {

        /* To calculate legendre's incomplete elliptical integral of the second kind for
         * negative m, we use a power series in phi for small m*phi*phi, an asymptotic
         * series in m for large m*phi*phi* and the relation to Carlson's symmetric
         * integrals, R_F(x,y,z) and R_D(x,y,z).
         *
         * E(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0)
         *             - m * sin(phi)^3 * R_D(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) / 3
         *
         *           = R_F(c-1, c-m, c) - m * R_D(c-1, c-m, c) / 3
         *
         * where c = csc(phi)^2. We use the second form of this for (approximately)
         * phi > 1/(sqrt(DBL_MAX) ~ 1e-154, where csc(phi)^2 overflows. Elsewhere we
         * use the first form, accounting for the smallness of phi.
         *
         * The algorithm used is described in Carlson, B. C. Numerical computation of
         * real or complex elliptic integrals. (1994) https://arxiv.org/abs/math/9409227
         * Most variable names reflect Carlson's usage.
         *
         * In this routine, we assume m < 0 and  0 > phi > pi/2.
         */
        XSF_HOST_DEVICE inline double ellie_neg_m(double phi, double m) {
            double x, y, z, x1, y1, z1, ret, Q;
            double A0f, Af, Xf, Yf, Zf, E2f, E3f, scalef;
            double A0d, Ad, seriesn, seriesd, Xd, Yd, Zd, E2d, E3d, E4d, E5d, scaled;
            int n = 0;
            double mpp = (m * phi) * phi;

            if (-mpp < 1e-6 && phi < -m) {
                return phi + (mpp * phi * phi / 30.0 - mpp * mpp / 40.0 - mpp / 6.0) * phi;
            }

            if (-mpp > 1e6) {
                double sm = std::sqrt(-m);
                double sp = std::sin(phi);
                double cp = std::cos(phi);

                double a = -cosm1(phi);
                double b1 = std::log(4 * sp * sm / (1 + cp));
                double b = -(0.5 + b1) / 2.0 / m;
                double c = (0.75 + cp / sp / sp - b1) / 16.0 / m / m;
                return (a + b + c) * sm;
            }

            if (phi > 1e-153 && m > -1e200) {
                double s = std::sin(phi);
                double csc2 = 1.0 / s / s;
                scalef = 1.0;
                scaled = m / 3.0;
                x = 1.0 / std::tan(phi) / std::tan(phi);
                y = csc2 - m;
                z = csc2;
            } else {
                scalef = phi;
                scaled = mpp * phi / 3.0;
                x = 1.0;
                y = 1 - mpp;
                z = 1.0;
            }

            if (x == y && x == z) {
                return (scalef + scaled / x) / std::sqrt(x);
            }

            A0f = (x + y + z) / 3.0;
            Af = A0f;
            A0d = (x + y + 3.0 * z) / 5.0;
            Ad = A0d;
            x1 = x;
            y1 = y;
            z1 = z;
            seriesd = 0.0;
            seriesn = 1.0;
            /* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps,
             * it is ~338.38. */

            /* N.B. This will evaluate its arguments multiple times. */
            Q = 400.0 * std::fmax(std::abs(A0f - x), std::fmax(std::abs(A0f - y), std::abs(A0f - z)));

            while (Q > std::abs(Af) && Q > std::abs(Ad) && n <= 100) {
                double sx = std::sqrt(x1);
                double sy = std::sqrt(y1);
                double sz = std::sqrt(z1);
                double lam = sx * sy + sx * sz + sy * sz;
                seriesd += seriesn / (sz * (z1 + lam));
                x1 = (x1 + lam) / 4.0;
                y1 = (y1 + lam) / 4.0;
                z1 = (z1 + lam) / 4.0;
                Af = (x1 + y1 + z1) / 3.0;
                Ad = (Ad + lam) / 4.0;
                n += 1;
                Q /= 4.0;
                seriesn /= 4.0;
            }

            Xf = (A0f - x) / Af / (1 << 2 * n);
            Yf = (A0f - y) / Af / (1 << 2 * n);
            Zf = -(Xf + Yf);

            E2f = Xf * Yf - Zf * Zf;
            E3f = Xf * Yf * Zf;

            ret = scalef * (1.0 - E2f / 10.0 + E3f / 14.0 + E2f * E2f / 24.0 - 3.0 * E2f * E3f / 44.0) / sqrt(Af);

            Xd = (A0d - x) / Ad / (1 << 2 * n);
            Yd = (A0d - y) / Ad / (1 << 2 * n);
            Zd = -(Xd + Yd) / 3.0;

            E2d = Xd * Yd - 6.0 * Zd * Zd;
            E3d = (3 * Xd * Yd - 8.0 * Zd * Zd) * Zd;
            E4d = 3.0 * (Xd * Yd - Zd * Zd) * Zd * Zd;
            E5d = Xd * Yd * Zd * Zd * Zd;

            ret -= scaled *
                   (1.0 - 3.0 * E2d / 14.0 + E3d / 6.0 + 9.0 * E2d * E2d / 88.0 - 3.0 * E4d / 22.0 -
                    9.0 * E2d * E3d / 52.0 + 3.0 * E5d / 26.0) /
                   (1 << 2 * n) / Ad / sqrt(Ad);
            ret -= 3.0 * scaled * seriesd;
            return ret;
        }

    } // namespace detail

    XSF_HOST_DEVICE inline double ellie(double phi, double m) {
        double a, b, c, e, temp;
        double lphi, t, E, denom, npio2;
        int d, mod, sign;

        if (std::isnan(phi) || std::isnan(m))
            return std::numeric_limits<double>::quiet_NaN();
        if (m > 1.0)
            return std::numeric_limits<double>::quiet_NaN();
        ;
        if (std::isinf(phi))
            return phi;
        if (std::isinf(m))
            return -m;
        if (m == 0.0)
            return (phi);
        lphi = phi;
        npio2 = std::floor(lphi / M_PI_2);
        if (std::fmod(std::abs(npio2), 2.0) == 1.0)
            npio2 += 1;
        lphi = lphi - npio2 * M_PI_2;
        if (lphi < 0.0) {
            lphi = -lphi;
            sign = -1;
        } else {
            sign = 1;
        }
        a = 1.0 - m;
        E = ellpe(m);
        if (a == 0.0) {
            temp = std::sin(lphi);
            goto done;
        }
        if (a > 1.0) {
            temp = detail::ellie_neg_m(lphi, m);
            goto done;
        }

        if (lphi < 0.135) {
            double m11 = (((((-7.0 / 2816.0) * m + (5.0 / 1056.0)) * m - (7.0 / 2640.0)) * m + (17.0 / 41580.0)) * m -
                          (1.0 / 155925.0)) *
                         m;
            double m9 = ((((-5.0 / 1152.0) * m + (1.0 / 144.0)) * m - (1.0 / 360.0)) * m + (1.0 / 5670.0)) * m;
            double m7 = ((-m / 112.0 + (1.0 / 84.0)) * m - (1.0 / 315.0)) * m;
            double m5 = (-m / 40.0 + (1.0 / 30)) * m;
            double m3 = -m / 6.0;
            double p2 = lphi * lphi;

            temp = ((((m11 * p2 + m9) * p2 + m7) * p2 + m5) * p2 + m3) * p2 * lphi + lphi;
            goto done;
        }
        t = std::tan(lphi);
        b = std::sqrt(a);
        /* Thanks to Brian Fitzgerald <[email protected]>
         * for pointing out an instability near odd multiples of pi/2.  */
        if (std::abs(t) > 10.0) {
            /* Transform the amplitude */
            e = 1.0 / (b * t);
            /* ... but avoid multiple recursions.  */
            if (std::abs(e) < 10.0) {
                e = std::atan(e);
                temp = E + m * std::sin(lphi) * std::sin(e) - ellie(e, m);
                goto done;
            }
        }
        c = std::sqrt(m);
        a = 1.0;
        d = 1;
        e = 0.0;
        mod = 0;

        while (std::abs(c / a) > detail::MACHEP) {
            temp = b / a;
            lphi = lphi + atan(t * temp) + mod * M_PI;
            denom = 1 - temp * t * t;
            if (std::abs(denom) > 10 * detail::MACHEP) {
                t = t * (1.0 + temp) / denom;
                mod = (lphi + M_PI_2) / M_PI;
            } else {
                t = std::tan(lphi);
                mod = static_cast<int>(std::floor((lphi - std::atan(t)) / M_PI));
            }
            c = (a - b) / 2.0;
            temp = std::sqrt(a * b);
            a = (a + b) / 2.0;
            b = temp;
            d += d;
            e += c * std::sin(lphi);
        }

        temp = E / ellpk(1.0 - m);
        temp *= (std::atan(t) + mod * M_PI) / (d * a);
        temp += e;

    done:

        if (sign < 0)
            temp = -temp;
        temp += npio2 * E;
        return (temp);
    }

} // namespace cephes
} // namespace xsf