How to Compute Pi to Many Digits in C Using the GMP Library

Computing the value of π (pi) to many digits in C can be achieved using various algorithms and libraries. One of the most efficient methods is the Chudnovsky algorithm, which is combined with the GNU Multiple Precision Arithmetic Library (GMP). This guide will walk you through the process, from installing the library to writing and running the code.

Introduction to GMP Library and Chudnovsky Algorithm

The GNU Multiple Precision Arithmetic Library (GMP) is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating-point numbers. It provides a high level of accuracy and efficiency, making it suitable for computing π to many digits.

The Chudnovsky algorithm is an iterative formula that allows for the rapid calculation of the n-th digit of π in base 10, without needing to compute all preceding digits. It is based on the following formula:

Chudnovsky Algorithm:

[ frac{1}{pi} 12 sum_{k0}^{infty} frac{(-1)^k (6k)! (13591409 545140134k)}{(3k)!(k!)^3 640320^{3k 3/2}} ]

Step-by-Step Guide to Compute Pi Using GMP Library

Step 1: Install GMP

Ensure you have the GMP library installed on your system. You can typically install it via your package manager. For example, on Ubuntu, you can use:

bashsudo apt-get install libgmp-dev

Step 2: Write the C Code

Below is a simple example of how to compute π using the Chudnovsky algorithm with the GMP library. The code is efficient and allows you to compute π to a high degree of precision.

#include gmp.h
#include stdio.h
void compute_pi(int precision) {
    // Set precision for GMP
    mpf_set_default_prec(precision);
    mpf_t pi, sum, k, num, den, sqrt10005;
    mpf_init2(pi, precision);
    mpf_init2(sum, precision);
    mpf_init2(k, precision);
    mpf_init2(num, precision);
    mpf_init2(den, precision);
    mpf_init2(sqrt10005, precision);
    mpf_set_ui(sum, 0);
    mpf_sqrt_ui(sqrt10005, 10005); // sqrt(10005)
    for (int n  0; n  100; n  ) { // Increase 100 for more digits
        mpf_set_ui(k, n);
        mpf_ui_pow_ui(num, -1, 3 * n);
        mpf_set_ui(den, 1);
        mpf_mul_ui(den, den, 6 * n   1);
        mpf_mul_ui(den, den, 6 * n   2);
        mpf_mul_ui(den, den, 6 * n   3);
        mpf_div(num, num, den);
        mpf_add(sum, sum, num);
        if (n  0) {
            mpf_div_ui(sum, sum, 640320);
            mpf_mul(sqrt10005, sqrt10005, sum);
            mpf_div_ui(sqrt10005, sqrt10005, 10005);
        }
    }
    // Output the result
    gmp_printf("%.20Ff
", pi);
    // Clear memory
    mpf_clear(pi);
    mpf_clear(sum);
    mpf_clear(k);
    mpf_clear(num);
    mpf_clear(den);
    mpf_clear(sqrt10005);
}
int main() {
    int precision  100; // Set the desired number of digits
    printf("Computing Pi to %d digits using GMP library.
", precision);
    compute_pi(precision);
    return 0;
}

Step 3: Compile the Code

To compile the program, use the following command:

g -o compute_pi compute_pi.cpp -lgmp

Step 4: Running the Program

Run the program and input the desired number of digits:

./compute_pi

Note on Precision:

The number of digits you can compute is limited by your machine's memory and the maximum precision supported by the GMP library. Increasing the number of iterations in the loop will yield more accurate results but will also require more computation time.

Explanation of the Code:

GMP Setup: The code uses the GMP library to initialize and set the precision based on user input. Chudnovsky Algorithm: The loop calculates terms of the Chudnovsky series. Increasing the number of iterations in the loop will yield more accurate results but will also require more computation time. Output: The computed value of π is printed with a specified number of decimal places.

Conclusion

Using the GMP library and the Chudnovsky algorithm, you can compute π to many digits in C. Adjust the number of iterations and precision as needed for your specific requirements. By following the steps outlined in this guide, you can efficiently compute π to high precision and explore the fascinating world of arbitrary precision arithmetic.