text: #include <math.h>
  #include <iostream>
  using namespace std;
  #include "precomp.hpp"
  
  #include "polynom_solver.h"
  
  int solve_deg2(double a, double b, double c, double & x1, double & x2)
  {
    double delta = b * b - 4 * a * c;
  
    if (delta < 0) return 0;
  
    double inv_2a = 0.5 / a;
  
    if (delta == 0) {
      x1 = -b * inv_2a;
      x2 = x1;
      return 1;
    }
  
    double sqrt_delta = sqrt(delta);
    x1 = (-b + sqrt_delta) * inv_2a;
    x2 = (-b - sqrt_delta) * inv_2a;
    return 2;
  }
  
  
  /// Reference : Eric W. Weisstein. "Cubic Equation." From MathWorld--A Wolfram Web Resource.
  /// http://mathworld.wolfram.com/CubicEquation.html
  /// \return Number of real roots found.
  int solve_deg3(double a, double b, double c, double d,
                 double & x0, double & x1, double & x2)
  {
    if (a == 0) {
      // Solve second order sytem
      if (b == 0)	{
        // Solve first order system
        if (c == 0)
      return 0;
  
        x0 = -d / c;
        return 1;
      }
  
      x2 = 0;
      return solve_deg2(b, c, d, x0, x1);
    }
  
    // Calculate the normalized form x^3 + a2 * x^2 + a1 * x + a0 = 0
    double inv_a = 1. / a;
    double b_a = inv_a * b, b_a2 = b_a * b_a;
    double c_a = inv_a * c;
    double d_a = inv_a * d;
  
    // Solve the cubic equation
    double Q = (3 * c_a - b_a2) / 9;
    double R = (9 * b_a * c_a - 27 * d_a - 2 * b_a * b_a2) / 54;
    double Q3 = Q * Q * Q;
    double D = Q3 + R * R;
    double b_a_3 = (1. / 3.) * b_a;
  
    if (Q == 0) {
      if(R == 0) {
        x0 = x1 = x2 = - b_a_3;
        return 3;
      }
      else {
        x0 = pow(2 * R, 1 / 3.0) - b_a_3;
        return 1;
      }
    }
  
    if (D <= 0) {
      // Three real roots
      double theta = acos(R / sqrt(-Q3));
      double sqrt_Q = sqrt(-Q);
      x0 = 2 * sqrt_Q * cos(theta             / 3.0) - b_a_3;
      x1 = 2 * sqrt_Q * cos((theta + 2 * CV_PI)/ 3.0) - b_a_3;
      x2 = 2 * sqrt_Q * cos((theta + 4 * CV_PI)/ 3.0) - b_a_3;
  
      return 3;
    }
  
    // D > 0, only one real root
    double AD = pow(fabs(R) + sqrt(D), 1.0 / 3.0) * (R > 0 ? 1 : (R < 0 ? -1 : 0));
    double BD = (AD == 0) ? 0 : -Q / AD;
  
    // Calculate the only real root
    x0 = AD + BD - b_a_3;
  
    return 1;
  }
  
  /// Reference : Eric W. Weisstein. "Quartic Equation." From MathWorld--A Wolfram Web Resource.
  /// http://mathworld.wolfram.com/QuarticEquation.html
  /// \return Number of real roots found.
  int solve_deg4(double a, double b, double c, double d, double e,
                 double & x0, double & x1, double & x2, double & x3)
  {
    if (a == 0) {
      x3 = 0;
      return solve_deg3(b, c, d, e, x0, x1, x2);
    }
  
    // Normalize coefficients
    double inv_a = 1. / a;
    b *= inv_a; c *= inv_a; d *= inv_a; e *= inv_a;
    double b2 = b * b, bc = b * c, b3 = b2 * b;
  
    // Solve resultant cubic
    double r0, r1, r2;
    int n = solve_deg3(1, -c, d * b - 4 * e, 4 * c * e - d * d - b2 * e, r0, r1, r2);
    if (n == 0) return 0;
  
    // Calculate R^2
    double R2 = 0.25 * b2 - c + r0, R;
    if (R2 < 0)
      return 0;
  
    R = sqrt(R2);
    double inv_R = 1. / R;
  
    int nb_real_roots = 0;
  
    // Calculate D^2 and E^2
    double D2, E2;
    if (R < 10E-12) {
      double temp = r0 * r0 - 4 * e;
      if (temp < 0)
        D2 = E2 = -1;
      else {
        double sqrt_temp = sqrt(temp);
        D2 = 0.75 * b2 - 2 * c + 2 * sqrt_temp;
        E2 = D2 - 4 * sqrt_temp;
      }
    }
    else {
      double u = 0.75 * b2 - 2 * c - R2,
        v = 0.25 * inv_R * (4 * bc - 8 * d - b3);
      D2 = u + v;
      E2 = u - v;
    }
  
    double b_4 = 0.25 * b, R_2 = 0.5 * R;
    if (D2 >= 0) {
      double D = sqrt(D2);
      nb_real_roots = 2;
      double D_2 = 0.5 * D;
      x0 = R_2 + D_2 - b_4;
      x1 = x0 - D;
    }
  
    // Calculate E^2
    if (E2 >= 0) {
      double E = sqrt(E2);
      double E_2 = 0.5 * E;
      if (nb_real_roots == 0) {
        x0 = - R_2 + E_2 - b_4;
        x1 = x0 - E;
        nb_real_roots = 2;
      }
      else {
        x2 = - R_2 + E_2 - b_4;
        x3 = x2 - E;
        nb_real_roots = 4;
      }
    }
  
    return nb_real_roots;
  }
  