PI

Here’s a #bash script I saved as “pi” somewhere in $PATH

#!/bin/bash
echo "scale=$1;a(1)*4" | bc -l

I demonstrate it this way:

$ for i in {1..20}; do pi $i; done

2.8 
3.12 
3.140 
3.1412 
3.14156 
3.141592 
3.1415924 
3.14159264 
3.141592652 
3.1415926532 
3.14159265356 
3.141592653588 
3.1415926535896 
3.14159265358976 
3.141592653589792 
3.1415926535897932 
3.14159265358979320 
3.141592653589793236 
3.1415926535897932384 
3.14159265358979323844

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π calculation using a Machin-like arctangent formula

#!/usr/bin/python3
from decimal import Decimal, getcontext
import sys
digits = int(sys.argv[1]) if len(sys.argv) > 1 else 50
getcontext().prec = digits + 5  # a few extra digits to avoid rounding errors

def arctan(x):
    """Compute arctan(1/x) using the Taylor series."""
    x = Decimal(x)
    x2 = x * x
    term = Decimal(1) / x
    total = term
    n = 1
    sign = -1
    while True:
        term = term / x2
        delta = term / (2 * n + 1)
        if delta == 0:
            break
        total += sign * delta
        sign *= -1
        n += 1
    return total

pi = 16 * arctan(5) - 4 * arctan(239)
pi = +pi  # unary plus applies current context precision
print(f"Pi to {digits} digits:\n{str(pi)[:digits+2]}")

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