Approximate Pi

approx-pi.cpp

// A program I made after Calculus II class because I was
// itching to apply the idea of using series to estimate certain
// mathematical constants. I use the Leibniz series knowing that
// there are better, faster-converging series out there.
//
// This program currently uses only built-in C++ data types rather
// than arbitrary precision numbers. As such, there are likely errors
// introduced to the calculation inherent to that data representation.
//
// Program output assumes Unicode for Greek/mathematical characters

#include <iostream>
#include <iomanip>

using namespace std;

/**
* Uses Leibniz Series to calculate Pi
* @param unsigned long int terms - Number of terms for series
* @return long double - Estimation of Pi
*/
long double estimatePi(unsigned long int terms) {
    unsigned long int i = 0;
    long double sum = 0;
    long double term = 0;
    while (i < terms) {
        // The denominator is a way of alternating between addition and
        // subtration for each term in the series without using branching logic
        // My assumption is that mod/multiply will be much faster even on CPUs with branch prediction. 
        sum += static_cast<int>(1 + (-2)*(i % 2)) / static_cast<long double>(2*i + 1);
        i++;
    }
    return sum * 4; // Series gives pi/4, so multiply by 4
}

/**
* Calculates error for the Leibniz Series given # of terms
* @param unsigned long int terms - Number of terms in the partial sum
* @return long double - Absolute value of the error
*/
long double calcError(unsigned long int terms) {
    return static_cast<long double>(4)/( 2*terms + 3);
}

// Harness to demonstrate that more iterations results in higher accuracy
int main()
{
    // Greet user
    cout << "Estimate \u03C0 using the Leibniz series!\n\n";

    // Declare and initialize variables
    unsigned long int terms = 1;      // Number of terms in the series
    long double error = 0.25;         // Error in calculation
    long double pi = 0.25;            // Estimate of pi constant

    // Priming prompt/read for terms
    cout << "Enter # of terms (enter non-number to quit): ";
    cin >> terms;

    // Keep prompting and calculating until the user quits
    while (cin) {
        // Echo back terms entered
        cout << "Calculating from " << terms << " terms . . . \n";

        // Run calculations
        pi = estimatePi(terms);
        error = calcError(terms); // Get error for calculation

        // Display Pi estimate
        cout << "\u03C0 \u2248" << ' '
            << setprecision(10) << pi << endl;
        // Display error calculation
        cout << "error \u2248 \u00B1" << error << "\n\n";

        // Prompt for another term count
        cout << "Enter # of terms (enter non-number to quit): ";
        cin >> terms;
    }
    cout << "Goodbye!\n";
    return 0;
}