#include <iostream>
#include <iomanip>
#include <cmath>


static const double rel_error= 1E-12; //calculate 12 significant figures
//you can adjust rel_error to trade off between accuracy and speed
//but don't ask for > 15 figures (assuming usual 52 bit mantissa in a double)


double erf(double x)
//erf(x) = 2/sqrt(pi)*integral(exp(-t^2),t,0,x)
// = 2/sqrt(pi)*[x - x^3/3 + x^5/5*2! - x^7/7*3! + ...]
// = 1-erfc(x)
{
  static const double two_sqrtpi= 1.128379167095512574; // 2/sqrt(pi)
  if (fabs(x) > 2.2) {
    return 1.0 - erfc(x); //use continued fraction when fabs(x) > 2.2
}
  double sum= x, term= x, xsqr= x*x;
  int j= 1;
  do {
    term*= xsqr/j;
    sum-= term/(2*j+1);
    ++j;
    term*= xsqr/j;
    sum+= term/(2*j+1);
    ++j;
  } while (fabs(term/sum) > rel_error);
  return two_sqrtpi*sum;
}


double erfc(double x)
  //erfc(x) = 2/sqrt(pi)*integral(exp(-t^2),t,x,inf)
  // = exp(-x^2)/sqrt(pi) * [1/x+ (1/2)/x+ (2/2)/x+ (3/2)/x+ (4/2)/x+ ...]
  // = 1-erf(x)
  //expression inside [] is a continued fraction so '+' means add to denominator  only
{
  static const double one_sqrtpi= 0.564189583547756287; // 1/sqrt(pi)
  if (fabs(x) < 2.2) {
    return 1.0 - erf(x); //use series when fabs(x) < 2.2
  }
  if (std::signbit(x)) { //continued fraction only valid for x>0
    return 2.0 - erfc(-x);
  }
  double a=1, b=x; //last two convergent numerators
  double c=x, d=x*x+0.5; //last two convergent denominators
  double q1, q2= b/d; //last two convergents (a/c and b/d)
  double n= 1.0, t;
  do {
    t= a*n+b*x;
    a= b;
    b= t;
    t= c*n+d*x;
    c= d;
    d= t;
    n+= 0.5;
    q1= q2;
    q2= b/d;
  } while (fabs(q1-q2)/q2 > rel_error);
  return one_sqrtpi*exp(-x*x)*q2;
}