diff --git a/maths/pi_generator.py b/maths/pi_generator.py new file mode 100644 index 000000000..dcd218aae --- /dev/null +++ b/maths/pi_generator.py @@ -0,0 +1,94 @@ +def calculate_pi(limit: int) -> str: + """ + https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 + Leibniz Formula for Pi + + The Leibniz formula is the special case arctan 1 = 1/4 Pi . + Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. + + Convergence (https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80#Convergence) + + We cannot try to prove against an interrupted, uncompleted generation. + https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80#Unusual_behaviour + The errors can in fact be predicted; + but those calculations also approach infinity for accuracy. + + Our output will always be a string since we can defintely store all digits in there. + For simplicity' sake, let's just compare against known values and since our outpit + is a string, we need to convert to float. + + >>> import math + >>> float(calculate_pi(15)) == math.pi + True + + Since we cannot predict errors or interrupt any infinite alternating + series generation since they approach infinity, + or interrupt any alternating series, we are going to need math.isclose() + + >>> math.isclose(float(calculate_pi(50)), math.pi) + True + + >>> math.isclose(float(calculate_pi(100)), math.pi) + True + + Since math.pi-constant contains only 16 digits, here some test with preknown values: + + >>> calculate_pi(50) + '3.14159265358979323846264338327950288419716939937510' + >>> calculate_pi(80) + '3.14159265358979323846264338327950288419716939937510582097494459230781640628620899' + + To apply the Leibniz formula for calculating pi, + the variables q, r, t, k, n, and l are used for the iteration process. + """ + q = 1 + r = 0 + t = 1 + k = 1 + n = 3 + l = 3 + decimal = limit + counter = 0 + + result = "" + + """ + We will avoid using yield since we otherwise get a Generator-Object, + which we can't just compare against anything. We would have to make a list out of it + after the generation, so we will just stick to plain return logic: + """ + while counter != decimal + 1: + if 4 * q + r - t < n * t: + result += str(n) + if counter == 0: + result += "." + + if decimal == counter: + break + + counter += 1 + nr = 10 * (r - n * t) + n = ((10 * (3 * q + r)) // t) - 10 * n + q *= 10 + r = nr + else: + nr = (2 * q + r) * l + nn = (q * (7 * k) + 2 + (r * l)) // (t * l) + q *= k + t *= l + l += 2 + k += 1 + n = nn + r = nr + return result + + +def main() -> None: + print(f"{calculate_pi(50) = }") + import doctest + + doctest.testmod() + + +if __name__ == "__main__": + main()