import math

def fx(x):
    if x == 0:
        x = 1e-10  # Avoid division by zero
    return math.sin(x)/x

def Romberg(a, b, err):
    t00 = (b-a)*(fx(a)+fx(b))/2
    t01 = t10 = t11 = t20 = t21 = t30 = t31 = 0
    k = 1
    while True:
        tmp = 0
        for i in range(1, 2**(k-1)+1):
            tmp += fx(a + (b-a)*(2*i-1)/(2**k))
        t01 = t00/2+(b-a)*tmp/(2**k)
        
        if k>1:
            t11 = (4*t01-t00)/3
            
            if k>2:
                t21 = (16*t11-t10)/15
                
                if k>3:
                    t31 = (64*t21-t20)/63
                    if abs(t31-t30) < err:
                        break
                    t30 = t31
                else:
                    t30 = (64*t21-t20)/63
                t20 = t21
            else:
                t20 = (16*t11-t10)/15
            t10 = t11
        else:
            t10 = (4*t01-t00)/3
        t00 = t01
        k += 1
    return t31, k


if __name__ == "__main__":
    a = 0
    b = 1
    err = 0.5e-6
    result, k = Romberg(a, b, err)
    print(f"Result: {result}, k={k}")