pi.sagews

Different Approximations of $\pi$

# Sage's value
n(pi, digits = 100)

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

# using mpmath - http://code.google.com/p/mpmath/
import mpmath as mm
mm.mp.dps = 100
print mm.pi
print 2*mm.asinh(1j).imag
print mm.gamma(0.5)**2
print mm.sqrt(6*mm.zeta(2))
print mm.quad(lambda x: 4*mm.sqrt(1-x**2), [0, 1])
print mm.quad(lambda x: mm.exp(-x**2), [-mm.inf, mm.inf]) ** 2
print mm.nsum(lambda n: 4*(-1)**n/(2*n+1), [0, mm.inf])
print mm.limit(lambda n: 2**(4*n+1)*mm.factorial(n)**4/(2*n+1)/mm.factorial(2*n)**2, mm.inf)
print mm.findroot(mm.sin, 3.14)

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

# dropping points
import numpy as np
N = 10000
x, y = 2 * np.random.rand(2, N) - 1
inside = np.sum(np.sqrt(x^2 + y^2) <= 1)
list_plot(zip(x, y), color='gray', size = 1) + circle((0,0), 1, color= 'green')
print("pi ~ %f" % (float(inside) / N * 4))

pi ~ 3.135200