#include <stdio.h>
#include <math.h>

static double h=1.0/64;
double rk2step(double x,double y){
    double k1=h*f(x,y);
    return y+(k1+h*f(x+h,y+k1))/2;
}
// Runge-Kutta Order Four from page 288
double rk4step(double t,double w){
    double k1=h*f(t,w);
    double k2=h*f(t+h/2,w+k1/2);
    double k3=h*f(t+h/2,w+k2/2);
    double k4=h*f(t+h,w+k3);
    return w+(k1+2*(k2+k3)+k4)/6;
}

int main(){
    double x0=0,xn=10;
    double y0=2,tol=1e-6;
    double x=x0,y=y0,esofar=0;
    for(;;){
        if(x+h>xn) h=xn-x;
        double t1=rk2step(x,y);
        double t2=rk4step(x,y);
        double epsilon=(xn-x)/h*(tol-esofar);
        if(fabs(t1-t2)>epsilon){
            h/=2;
        } else {
            y=t2; x+=h;
            esofar+=fabs(t1-t2);
        }
        if(x==xn) break;
    }
    return 0;
}