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(x > 1) + + This calculates it in fixed point, using the value for one passed in + """ + power = one // x # the +/- 1/x**n part of the term + total = power # the total so far + x_squared = x * x # precalculate x**2 + divisor = 1 # the 1,3,5,7 part of the divisor + while 1: + power = - power // x_squared + divisor += 2 + power += divisor // 2 # round the division + delta = power // divisor + if delta == 0: + break + total += delta + return total + +def arctan_euler(x, one=1000000): + """ + Calculate arctan(1/x) using euler's accelerated formula + + arctan(1/x) = x / (1+x**2) + + (2/3) * x / (1+x**2)**2 + + (2*4/(3*5)) * x / (1+x**2)**3 + + (2*4*6/(3*5*7)) * x / (1+x**2)**4 + + ... + + This calculates it in fixed point, using the value for one passed in + """ + x_squared = x * x + x_squared_plus_1 = x_squared + 1 + term = (x * one) // x_squared_plus_1 + total = term + two_n = 2 + while 1: + divisor = (two_n+1) * x_squared_plus_1 + term *= two_n + term += divisor // 2 # round the division + term = term // divisor + if term == 0: + break + total += term + two_n += 2 + return total + +def pi_machin(one): + return 4*(4*arctan(5, one) - arctan(239, one)) +def pi_machin_euler(one): + return 4*(4*arctan_euler(5, one) - arctan_euler(239, one)) + +def pi_ferguson(one): + return 4*(3*arctan(4, one) + arctan(20, one) + arctan(1985, one)) + +def pi_hutton(one): + return 4*(2*arctan(3, one) + arctan(7, one)) + +def pi_gauss(one): + return 4*(12*arctan(18, one) + 8*arctan(57, one) - 5*arctan(239, one)) +def pi_gauss_euler(one): + return 4*(12*arctan_euler(18, one) + 8*arctan_euler(57, one) - 5*arctan_euler(239, one)) + +if __name__ == "__main__": + for log10_digits in range(1,7): + digits = 10**log10_digits + one = 10**digits + + start =time() + pi = pi_machin(one) + #print(pi) + print("machin: digits",digits,"time",time()-start) + + start =time() + pi = pi_machin_euler(one) + #print(pi) + print("machin euler: digits",digits,"time",time()-start) + + start =time() + pi = pi_gauss(one) + #print(pi) + print("gauss: digits",digits,"time",time()-start) + + start =time() + pi = pi_gauss_euler(one) + #print(pi) + print("gauss euler: digits",digits,"time",time()-start) + #print(pi_ferguson(one)) + #print(pi_hutton(one)) + #print(pi_gauss(one)) -- 2.30.2

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