M | Terminal Cruise

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]}")

Today is piday 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