2204.sagews

def sin_plus3(A,d):
    #print A,d
    #print exact_cos(d)
    A_next = 1/(2*exact_cos3(d))*A
    return A_next

def exact_sin3(d):
    A = 3
    SinA = 1/2*sqrt(3)
    while A < d:
        SinA = sin_plus3(SinA,A+3)
        A = A*2
        #print A
    return SinA

def sin_plus2(A,d):
    #print A,d
    #print exact_cos(d)
    A_next = 1/(2*exact_cos2(d))*A
    return A_next

def exact_sin2(d):
    A = 4
    SinA = 1/2*sqrt(2)
    while A < d:
        SinA = sin_plus2(SinA,A+2)
        A = A*2
        #print A
    return SinA

def cos_plus(A):
    A_next = 1/2*sqrt(2)*sqrt(A+1)
    return A_next

def exact_cos3(d):
    A = 3
    CosA = 1/2
    while A < d:
        CosA = cos_plus(CosA)
        A = A*2
    return CosA

def exact_cos2(d):
    A = 4
    CosA = 1/2*sqrt(2)
    while A < d:
        CosA = cos_plus(CosA)
        A = A*2
    return CosA

def approx_pi(A):
    #A = 24
    SinA = exact_sin3(A)
    Sin2A = exact_sin3(A/2)
    Sin3A = exact_sin2(A/3)
    Sin4A = exact_sin3(A/4)
    CosA = exact_cos3(A)
    Cos2A = exact_cos3(A/2)
    Cos3A = exact_cos2(A/3)
    Cos4A = exact_cos3(A/4)
    #
    func = (144/125*SinA+76/125*Sin2A+3568/42875*Sin3A+761/343000*Sin4A)/(1+32/25*CosA+8/25*Cos2A+32/1225*Cos3A+1/2450*Cos4A)
    approx = A*func
    #print "een exacte benaderinge voor pi is: "
    #print approx
    print
    print "een benadering voor pi is: "
    print n(approx,digits=50)
    print
    print "de correcte waarde van pi is: "
    print n(pi,digits=50)

approx_pi(24)

een benadering voor pi is: 3.1415926535897932384626350421304661802460770552332 de correcte waarde van pi is: 3.1415926535897932384626433832795028841971693993751