import math

def opt1(a, b, eps):
    N = 2
    while N > 0:
        h = (b - a)/N
        sq1 = Squres(a, b, N)
        sq2 = Squres(a, b, 2*N)
        #print(sq1, sq2)
        R = (sq2 - sq1)/3
        #print(R)
        #if (abs(R) - eps <= 0.001):
        if (abs(R) <= eps):
            print(N, sq2, sq2 + R)
            break;
        else:
            N= N*2

def Squres(a, h, N):
     Xs = []
     h = (b-a)/N

     for i in range(0, N+1):
        Xs.append(a + h*i)
     sum = 0
     #print(Xs, h, N)
     for i in range(0, N):
         sum += math.exp(Xs[i] + h/2)
     return sum*h

def opt2(a, b, eps):
    N = 2
    while (N > 0) :
        h = (b - a)/N
        tr1 = trap(a, b, h, N)
        tr2 = trap(a, b, h/2, 2*N)
        #print(tr1, tr2)

        R = (tr2 - tr1)/3
        #print(R)
        #if (abs(R) - eps <= 0.001):
        if (abs(R) <= eps):
            print(N, tr2, tr2 + R)
            break;
        else:
            N= N*2

def trap (a, b, h, N):
    ys = []
    for i in range(0, N+1):
        ys.append(math.exp(a + h*i))
    sum = (math.exp(a) + math.exp(b))/2
    for i in range(1, N):
         sum += ys[i]
    return sum*h


if __name__ == "__main__":

    N = 10
    a = 0
    b = 1
    eps = 0.001
    h = (b - a)/N
    #print(h)

    xs = []
    ys = []

    for i in range(0, N+1):
        xs.append(a + h*i)
        ys.append(math.exp(a + h*i))

    opt1(a, b, eps)
    opt2(a, b, eps)

    #print(trap(a, b, xs, ys, h, N))


8 1.7180021920526605 1.7182817010716516
8 1.7188411285799947 1.7182819740518924