The documentation for newton_raphson_iterate states that "Out-of-bounds steps revert to bisection of the current bounds", which should be robust, but the implementation fails to bisect correctly in some cases. The following program requests the root of a quartic with the sole root 1 in the bounding interval, but with an initial guess on the wrong side of an extremum that causes a step out of bounds. The returned result is approximately the endpoint of the interval farthest from the actual root, which should have been eliminated by a single bisection.

#include <boost/math/tools/roots.hpp>
#include <iostream>
#include <utility>

int main() {
  const auto quartic = [](const double x) {
    return std::make_pair((x * x - 1.0) * (x * x + 0.1),
                          2.0 * x * ((x * x - 1.0) + (x * x + 0.1)));
  };

  const double result =
      boost::math::tools::newton_raphson_iterate(quartic, 0.5, 0.1, 1.1, 5);
  std::cout << result << "\t" << quartic(result).first << "\n";
  return result;
}

Output:

0.10625 -0.110033