Pi with Machin's formula (Python)

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Machin's formula

A simple way to compute the mathematical constant π ≈ 3.14159 with any desired precision is due to John Machin. In 1706, he found the formula

which he used along with the Taylor series expansion of the arc cotangent function,

to calculate 100 decimals by hand. The formula is well suited for computer implementation, both to compute π with little coding effort (adding up the series term by term) and using more advanced strategies (such as binary splitting) for better speed.


In order to obtain n digits, we will use fixed-point arithmetic to compute π × 10n as a Python long.

High-precision arccot computation

To calculate arccot of an argument x, we start by dividing the number 1 (represented by 10n, which we provide as the argument unity) by x to obtain the first term. We then repeatedly divide by x2 and a counter value that runs over 3, 5, 7, ..., to obtain each next term. The summation is stopped at the first zero term, which in this fixed-point representation corresponds to a real value less than 10-n.

def arccot(x, unity):
    sum = xpower = unity // x
    n = 3
    sign = -1
    while 1:
        xpower = xpower // (x*x)
        term = xpower // n
        if not term:
        sum += sign * term
        sign = -sign
        n += 2
    return sum

Applying Machin's formula

Finally, the main function, which uses Machin's formula to compute π using the necessary level of precision:

def pi(digits):
    unity = 10**(digits + 10)
    pi = 4 * (4*arccot(5, unity) - arccot(239, unity))
    return pi // 10**10

To avoid rounding errors in the result, we use 10 guard digits internally during the calculation. We may now reproduce Machin's result:

>>> pi(100)

The program can be used to compute tens of thousands of digits in just a few seconds on a modern computer. (More sophisticated techniques are necessary to calculate millions or more digits in reasonable time, although in principle this program will also work.)

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