std::sqrt(std::valarray)

#include <cassert>
#include <complex>
#include <cstddef>
#include <iostream>
#include <numbers>
#include <valarray>
 
using CD = std::complex <double>;
using VA = std::valarray <CD>;
 
// return all n complex roots out of a given complex number x
VA root(CD x, unsigned n)
{
 const double mag = std::pow (std::abs(x), 1.0 / n);
 const double step = 2.0 * std::numbers::pi / n;
 double phase = std::arg (x) / n;
 VA v(n);
 for (std::size_t i{}; i != n; ++i, phase += step)
 v[i] = std::polar (mag, phase);
 return v;
}
 
// return n complex roots of each element in v; in the output valarray first
// goes the sequence of all n roots of v[0], then all n roots of v[1], etc.
VA root(VA v, unsigned n)
{
 VA o(v.size() * n);
 VA t(n);
 for (std::size_t i = 0; i != v.size(); ++i)
 {
 t = root(v[i], n);
 for (unsigned j = 0; j != n; ++j)
 o[n * i + j] = t[j];
 }
 return o;
}
 
// floating-point numbers comparator that tolerates given rounding error
inline bool is_equ(CD x, CD y, double tolerance = 0.000'000'001)
{
 return std::abs(std::abs(x) - std::abs(y)) < tolerance;
}
 
int main()
{
 // input coefficients for polynomial x3 + p·x + q
 const VA p{1, 2, 3, 4, 5, 6, 7, 8};
 const VA q{1, 2, 3, 4, 5, 6, 7, 8};
 
 // the solver
 const VA d = std::sqrt (std::pow (q / 2, 2) + std::pow (p / 3, 3));
 const VA u = root(-q / 2 + d, 3);
 const VA n = root(-q / 2 - d, 3);
 
 // allocate memory for roots: 3 * number of input cubic polynomials
 VA x[3];
 for (std::size_t t = 0; t != 3; ++t)
 x[t].resize(p.size());
 
 auto is_proper_root = [](CD a, CD b, CD p) { return is_equ(a * b + p / 3.0, 0.0); };
 
 // sieve out 6 out of 9 generated roots, leaving only 3 proper roots (per polynomial)
 for (std::size_t i = 0; i != p.size(); ++i)
 for (std::size_t j = 0, r = 0; j != 3; ++j)
 for (std::size_t k = 0; k != 3; ++k)
 if (is_proper_root(u[3 * i + j], n[3 * i + k], p[i]))
 x[r++][i] = u[3 * i + j] + n[3 * i + k];
 
 std::cout << "Depressed cubic equation: Root 1: \t\t Root 2: \t\t Root 3:\n";
 for (std::size_t i = 0; i != p.size(); ++i)
 {
 std::cout << "x3 + " << p[i] << "·x + " << q[i] << " = 0 "
 << std::fixed << x[0][i] << " " << x[1][i] << " " << x[2][i]
 << std::defaultfloat << '\n';
 
 assert (is_equ(std::pow (x[0][i], 3) + x[0][i] * p[i] + q[i], 0.0));
 assert (is_equ(std::pow (x[1][i], 3) + x[1][i] * p[i] + q[i], 0.0));
 assert (is_equ(std::pow (x[2][i], 3) + x[2][i] * p[i] + q[i], 0.0));
 }
}

Depressed cubic equation: Root 1: Root 2: Root 3:
x3 + (1,0)·x + (1,0) = 0 (-0.682328,0.000000) (0.341164,1.161541) (0.341164,-1.161541)
x3 + (2,0)·x + (2,0) = 0 (-0.770917,0.000000) (0.385458,1.563885) (0.385458,-1.563885)
x3 + (3,0)·x + (3,0) = 0 (-0.817732,0.000000) (0.408866,1.871233) (0.408866,-1.871233)
x3 + (4,0)·x + (4,0) = 0 (-0.847708,0.000000) (0.423854,2.130483) (0.423854,-2.130483)
x3 + (5,0)·x + (5,0) = 0 (-0.868830,0.000000) (0.434415,2.359269) (0.434415,-2.359269)
x3 + (6,0)·x + (6,0) = 0 (-0.884622,0.000000) (0.442311,2.566499) (0.442311,-2.566499)
x3 + (7,0)·x + (7,0) = 0 (-0.896922,0.000000) (0.448461,2.757418) (0.448461,-2.757418)
x3 + (8,0)·x + (8,0) = 0 (-0.906795,0.000000) (0.453398,2.935423) (0.453398,-2.935423)