Number of solutions depend on untold rounding rules.
For exaple, let our algorithm gave comlex root
1+0.0000001*i. What descision, include 1 to real roots or not. For me right answer may vary in root numbers but contain in it's 0.000001 - neighbornhood ideal mathamatical set of roots.

It seems that with type double and 16 righs digits in results the problem impossible to solve. For examle if we try compair solutions x1=0.999988 and x2=0.999982 for equation f(x)=(x-1)^4=x^4-4*x^3+6*x^2-4*x+1=0 computer
says that x2 is better. If we have 30-digits floats we would correct answer by optimization near initial approximation. If somebody knows internet resources with such module I shall be grateful.