bisect Error Analysis

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

using namespace std;

// This function if is 0 is the solutioon of sqrt(2) - n = 0
double f(double n){
    return n*n - 2;
    //return n*n*n*n - n*n;
}

// function that returns the "sign" of the value -1, 0 for 0 or, 1 for positive
template <typename T>
int sgn(T val) {
    return (T(0) < val) - (val < T(0));
}

// contains the result of the function bysect, the iteration count and the
// resulting number to calculate the error
struct BisectResult{
    int itr_count;
    double result;
};

// type of function pointer f(x)
typedef double (*fptr_t) (double);

// calculates how many iterations to reach the given tolerance
BisectResult bisect(double lower, double upper, double tolerance, fptr_t func){
    int itr_max = 10000;
    int itr_count = 0;
    BisectResult bs_res;

    double half = (upper + lower) / 2.0;
    double res;

    do{
        res = func(half);
        bs_res.result = half;
        bs_res.itr_count = itr_count;

        if( sgn<double>(res) == sgn<double>(f(lower)) ){
            lower = half;
        }
        else{
            upper = half;
        }

        half = (lower + upper) / 2;

        ++itr_count;

        if(itr_count >= itr_max){
            throw runtime_error("Function bisect: Max iteration reached without convergence");
        }

    } while(abs(res) > tolerance);

    return bs_res;
}

// calculate the error given the expected result
// 0 as expected result is a pain tho
double calc_error(double expected_result, double experimental_result){
    double err = (expected_result - experimental_result) / expected_result;
    err = err * 100;
    return err;

}


int main()
{
    //------------------ user inputs --------------------------------------

    // define the search interval
    double lower = 0, upper = 2.0;

    //------------------ algorithm starts --------------------------------------
    BisectResult br;
    for(double tolerance = 0.1; tolerance >= 0.00000001; tolerance/=10){
            cout << "------------------------------" << endl;
            cout << "tolerance: " << tolerance << endl;
            br = bisect(lower, upper, tolerance, f);
            cout << "iterations: " << br.itr_count << " result: " << br.result << endl;
            cout << "error: " << calc_error(sqrt(2), br.result) << "%" << endl;
    }

    return 0;
}