Some ways of calculating pi and e that I came up with. None of these are original; these are just rediscoveries of existing methods.

e.py

# Approximate e using bisection.

import math

ITERS = 25
H = 2 ** (-26)
l = 2
m = 3
r = 4

for i in range(ITERS):
    e = m ** H
    d = (e - 1 / e) / H / 2
    if d > 1:
        r = m
    elif d < 1:
        l = m
    else:
        break
    m = (l + r) / 2

print(m, "was calculated by e.py.")
print(math.e, "is the real value of e.")

pi1.py

# Calculate pi by uniformly placing points inside a square and seeing whether
# they are inside they are inside the corresponding circle.

import math

SIZE = 10000
SIZE_SQUARED = SIZE * SIZE
x = 0
y = SIZE
inside = 0

while y >= 0:
    if x * x + y * y < SIZE_SQUARED:
        x += 1
    else:
        inside += x
        y -= 1

print(inside / SIZE_SQUARED * 4, "was calculated by pi1.py.")
print(math.pi, "is the real value of pi.")

pi2.py

# Calculate pi by approximating a circle using triangles and applying the half
# angle formula for sine. Susceptible to cancellation error due to subtraction.

import math

# s goes to 0 really fast, so ITERS must be small.
ITERS = 10
s = 1
m = 2

for i in range(ITERS):
    s = math.sqrt((1 - math.sqrt(1 - s * s)) / 2)
    m *= 2

print(s * m, "was calculated by pi2.py.")
print(math.pi, "is the real value of pi.")

pi3.py

# An improvement on pi2.py to help get rid of the cancellation error.

import math

ITERS = 20
s = 1
m = 2

for i in range(ITERS):
    s = s / (1 + math.sqrt(1 - s)) / 2
    m *= 2

print(math.sqrt(s) * m, "was calculated by pi3.py.")
print(math.pi, "is the real value of pi.")

pi4.py

# Approximate pi using the Taylor Series expansion of sqrt(1 - x ^ 2).

import math

TERMS = 1000
sum = 5 / 6
d = 1
b = 8

for n in range(3, TERMS):
    sum -= d / b / (2 * n - 1)
    d *= 2 * n - 3
    b *= 2 * n

print(sum * 4, "was calculated by pi4.py.")
print(math.pi, "is the real value of pi.")