import decimal
import math


def f(X, Y):
    return decimal.Decimal((-X * Y) / (X * X - 9))


def Y(X):
    return decimal.Decimal(4 / (math.sqrt(X * X - 9)))


h = decimal.Decimal('0.05')
x = []
y = []
x.append(decimal.Decimal('5.00'))
y.append(decimal.Decimal('1.00'))


for i in range(0, 21):

    if i >= 1:
        k1 = f(x[i - 1], y[i - 1])
        k2 = f(x[i - 1] + h / 2, y[i - 1] + h * k1 / 2)
        k3 = f(x[i - 1] + h / 2, y[i - 1] + h * k2 / 2)
        k4 = f(x[i - 1] + h, y[i - 1] + h * k3)

        x.append(x[i - 1] + h)
        y.append(y[i - 1] + h * (k1 + 2 * k2 + 2 * k3 + k4) / 6)

    print("x{} =".format(i), x[i], "  RK4 y{} =".format(i), "%.10f" % y[i], end="")
    print("   참값 y{} =".format(i), "%.10f" % Y(x[i]), end="")
    print("   절대오차 {} =".format(i), "%.10f" % abs(y[i] - Y(x[i])), end="")
    print("   상대오차 {} =".format(i), "%.10f" % (abs(y[i] - Y(x[i])) * 100 / abs(Y(x[i]))), end="")
    print("%")

x0 = 5.00   RK4 y0 = 1.0000000000   참값 y0 = 1.0000000000   절대오차 0 = 0.0000000000   상대오차 0 = 0.0000000000%
x1 = 5.05   RK4 y1 = 0.9846573354   참값 y1 = 0.9846573354   절대오차 1 = 0.0000000000   상대오차 1 = 0.0000000031%
x2 = 5.10   RK4 y2 = 0.9698572899   참값 y2 = 0.9698572899   절대오차 2 = 0.0000000001   상대오차 2 = 0.0000000059%
x3 = 5.15   RK4 y3 = 0.9555687905   참값 y3 = 0.9555687906   절대오차 3 = 0.0000000001   상대오차 3 = 0.0000000083%
x4 = 5.20   RK4 y4 = 0.9417632186   참값 y4 = 0.9417632187   절대오차 4 = 0.0000000001   상대오차 4 = 0.0000000104%
x5 = 5.25   RK4 y5 = 0.9284141650   참값 y5 = 0.9284141651   절대오차 5 = 0.0000000001   상대오차 5 = 0.0000000123%
x6 = 5.30   RK4 y6 = 0.9154972141   참값 y6 = 0.9154972142   절대오차 6 = 0.0000000001   상대오차 6 = 0.0000000140%
x7 = 5.35   RK4 y7 = 0.9029897537   참값 y7 = 0.9029897539   절대오차 7 = 0.0000000001   상대오차 7 = 0.0000000155%
x8 = 5.40   RK4 y8 = 0.8908708062   참값 y8 = 0.8908708064   절대오차 8 = 0.0000000002   상대오차 8 = 0.0000000169%
x9 = 5.45   RK4 y9 = 0.8791208790   참값 y9 = 0.8791208791   절대오차 9 = 0.0000000002   상대오차 9 = 0.0000000181%
x10 = 5.50   RK4 y10 = 0.8677218311   참값 y10 = 0.8677218313   절대오차 10 = 0.0000000002   상대오차 10 = 0.0000000192%
x11 = 5.55   RK4 y11 = 0.8566567548   참값 y11 = 0.8566567549   절대오차 11 = 0.0000000002   상대오차 11 = 0.0000000202%
x12 = 5.60   RK4 y12 = 0.8459098687   참값 y12 = 0.8459098689   절대오차 12 = 0.0000000002   상대오차 12 = 0.0000000211%
x13 = 5.65   RK4 y13 = 0.8354664231   참값 y13 = 0.8354664233   절대오차 13 = 0.0000000002   상대오차 13 = 0.0000000219%
x14 = 5.70   RK4 y14 = 0.8253126139   참값 y14 = 0.8253126141   절대오차 14 = 0.0000000002   상대오차 14 = 0.0000000226%
x15 = 5.75   RK4 y15 = 0.8154355061   참값 y15 = 0.8154355063   절대오차 15 = 0.0000000002   상대오차 15 = 0.0000000233%
x16 = 5.80   RK4 y16 = 0.8058229638   참값 y16 = 0.8058229640   절대오차 16 = 0.0000000002   상대오차 16 = 0.0000000239%
x17 = 5.85   RK4 y17 = 0.7964635879   참값 y17 = 0.7964635881   절대오차 17 = 0.0000000002   상대오차 17 = 0.0000000244%
x18 = 5.90   RK4 y18 = 0.7873466590   참값 y18 = 0.7873466592   절대오차 18 = 0.0000000002   상대오차 18 = 0.0000000249%
x19 = 5.95   RK4 y19 = 0.7784620858   참값 y19 = 0.7784620860   절대오차 19 = 0.0000000002   상대오차 19 = 0.0000000254%
x20 = 6.00   RK4 y20 = 0.7698003587   참값 y20 = 0.7698003589   절대오차 20 = 0.0000000002   상대오차 20 = 0.0000000258%