#include <ctime>

#include <cstdlib>

#include <cassert>

#include <complex>

typedef double Real;

typedef std::complex<Real> Complex;

const Real epsilon = 1e-6;

const Real epsilonSqr = epsilon*epsilon;

inline Real magSqr(Complex x)

  { return x.real()*x.real() + x.imag()*x.imag(); }

inline bool isZero(Real x)

  { return fabs(x) < epsilon; }

inline bool isZero(Complex x)

  { return magSqr(x) <= epsilonSqr; }

inline Complex verify2(Complex r, Real a, Real b, Real c) {

  assert(isZero(a*r*r + b*r + c));

  return r;

}

inline Complex verify3(Complex r, Real a, Real b, Real c, Real d) {

  assert(isZero(a*r*r*r + b*r*r + c*r + d));

  return r;

}

inline Complex verify4(Complex r, Real a, Real b, Real c, Real d, Real e) {

  assert(isZero(a*r*r*r*r + b*r*r*r + c*r*r + d*r + e));

  return r;

}

int solve2(Complex *roots, Real a, Real b, Real c) {

  if (isZero(a)) {

    if (isZero(b)) return 0;

    roots[0] = verify2(-c/b, a, b, c);

    return 1;

  }

  Real D = b*b - a*c*4;

  Real k = Real(0.5)/a;

  Complex d = D < 0 ? Complex(0, sqrt(-D)) : Complex(sqrt(D));

  roots[0] = verify2((-b - d)*k, a, b, c);

  roots[1] = verify2((-b + d)*k, a, b, c);

  return 2;

}

int solve3(Complex *roots, Real a, Real b, Real c, Real d) {

  if (isZero(a)) return solve2(roots, b, c, d);

  // x = y - b/(3*a)

  // y*y*y + p*y + q = 0

  Real p = (3*a*c - b*b)/(3*a*a);

  Real q = (27*a*a*d - 9*a*b*c + 2*b*b*b)/(27*a*a*a);

  Real Q = p*p*p/27 + q*q/4;

  Complex Qs = Q < 0 ? Complex(0, sqrt(-Q)) : Complex(sqrt(Q));

  Complex A = pow(-q/2 + Qs, Real(1)/3);

  Complex B = pow(-q/2 - Qs, Real(1)/3);

  // choose complimentary B for A (A*B must be equal -p/3)

  Complex rot(Real(-0.5), sqrt(Real(3))/2);

  if (!isZero(A*B + p/3)) B *= rot;

  if (!isZero(A*B + p/3)) B *= rot;

  Complex Y = (A - B)*Complex(0, sqrt(Real(3)));

  Complex y0 = A + B;

  Complex y1 = (-y0 - Y)/Real(2);

  Complex y2 = (-y0 + Y)/Real(2);

  Real dd = b/(3*a);

  roots[0] = verify3(y0 - dd, a, b, c, d);

  roots[1] = verify3(y1 - dd, a, b, c, d);

  roots[2] = verify3(y2 - dd, a, b, c, d);

  return 3;

}

int solve4(Complex *roots, Real a, Real b, Real c, Real d, Real e) {

  if (isZero(a)) return solve3(roots, b, c, d, e);

  //printf("%g*x^4 + %g*x^3 + %g*x^2 + %g*x + %g = 0\n", a, b, c, d, e);

  // x = y - b/(4*a)

  // y^4 + p*y^2 + q*y + r = 0

  Real dd = b/(4*a);

  Real p = (8*a*c - 3*b*b)/(8*a*a);

  Real q = (8*a*a*d - 4*a*b*c + b*b*b)/(8*a*a*a);

  Real r = (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b)/(256*a*a*a*a);

  //printf(", y^4 + %g*y^2 + %g*y + %g = 0", p, q, r);

  if (isZero(q)) {

    // biquadratic equation

    // y^4 + p*y^2 + r = 0

    Complex y[2];

    solve2(y, 1, p, r);

    y[0] = sqrt(y[0]);

    y[1] = sqrt(y[1]);

    roots[0] = verify4(-y[0] - dd, a, b, c, d, e);

    roots[1] = verify4( y[0] - dd, a, b, c, d, e);

    roots[2] = verify4(-y[1] - dd, a, b, c, d, e);

    roots[3] = verify4( y[1] - dd, a, b, c, d, e);

    return 4;

  }

  // solve cubic equation

  // z*z*z + (p/2)*z*z + ((p*p - 4*r)/16)*z - q*q/64 = 0

  Real pp = p/2;

  Real qq = (p*p - 4*r)/16;

  Real rr = -q*q/64;

  //printf("z^3 + %g*z^2 + %g*z + %g = 0\n", pp, qq, rr);

  Complex z[3];

  solve3(z, 1, pp, qq, rr);

  z[0] = sqrt(z[0]);

  z[1] = sqrt(z[1]);

  z[2] = sqrt(z[2]);

  // we need to find signs combination where following is valid:

  // (+-z0)*(+-z1)*(+-z2) = -q/8

  Complex zzz = z[0]*z[1]*z[2];

  //printf(

  //  "sqrt(z): (%g, %g), (%g, %g), (%g, %g), zzz: (%g, %g)\n",

  //  z[0].real(), z[0].imag(),

  //  z[1].real(), z[1].imag(),

  //  z[2].real(), z[2].imag(),

  //  zzz.real(), zzz.imag() );

  assert(isZero(zzz.imag()));

  assert(isZero(fabs(zzz.real()) - fabs(q/8)));

  if ((zzz.real() > 0) == (q > 0))

    z[0] = -z[0];

  assert(isZero(z[0]*z[1]*z[2] + q/8));

  roots[0] = verify4( z[0] - z[1] - z[2] - dd, a, b, c, d, e);

  roots[1] = verify4(-z[0] + z[1] - z[2] - dd, a, b, c, d, e);

  roots[2] = verify4(-z[0] - z[1] + z[2] - dd, a, b, c, d, e);

  roots[3] = verify4( z[0] + z[1] + z[2] - dd, a, b, c, d, e);

  return 4;

}

int main() {

  srand(time(NULL));

  bool failed = false;

  for(int i = 0; i < 1000000; ++i) {

    Real a = rand()%20 - 10;

    Real b = rand()%20 - 10;

    Real c = rand()%20 - 10;

    Real d = rand()%20 - 10;

    Real e = rand()%20 - 10;

    printf("%6d: %g*x^4 + %g*x^3 + %g*x^2 + %g*x + %g = 0", i, a, b, c, d, e);

    int vcnt = a ? 4

             : b ? 3

             : c ? 2

             : d ? 1 : 0;

    Complex roots[4];

    int cnt = solve4(roots, a, b, c, d, e);

    bool f = cnt != vcnt;

    for(int i = 0; i < cnt; ++i) {

      const Complex &x = roots[i];

      printf(", (%g, %g)", x.real(), x.imag());

      Complex r = a*x*x*x*x + b*x*x*x + c*x*x + d*x + e;

      if (!isZero(r))

        { printf(" ![%g, %g]", r.real(), r.imag()); f = true; }

    }

    printf(", %s\n", f ? "FAILED" : "passed");

    if (f) {

      printf("^^^^^^^^^^\n");

      failed = true;

      break;

    }

  }

  printf("\n%s\n\n", failed ? "FAILED" : "success");

  return failed;

}