Chudnovsky Algorithm
The Chudnovsky algorithm is a fast method for calculating the digits of π.
import math
from gmpy2 import mpz
from time import time
import gmpy2
def pi_chudnovsky_bs(digits):
"""
Compute int(pi * 10**digits)
This is done using Chudnovsky's series with binary splitting
"""
C = 640320
C3_OVER_24 = C**3 // 24
def bs(a, b):
"""
Computes the terms for binary splitting the Chudnovsky infinite series
a(a) = +/- (13591409 + 545140134*a)
p(a) = (6*a-5)*(2*a-1)*(6*a-1)
b(a) = 1
q(a) = a*a*a*C3_OVER_24
returns P(a,b), Q(a,b) and T(a,b)
"""
if b - a == 1:
# Directly compute P(a,a+1), Q(a,a+1) and T(a,a+1)
if a == 0:
Pab = Qab = mpz(1)
else:
Pab = mpz((6*a-5)*(2*a-1)*(6*a-1))
Qab = mpz(a*a*a*C3_OVER_24)
Tab = Pab * (13591409 + 545140134*a) # a(a) * p(a)
if a & 1:
Tab = -Tab
else:
# Recursively compute P(a,b), Q(a,b) and T(a,b)
# m is the midpoint of a and b
m = (a + b) // 2
# Recursively calculate P(a,m), Q(a,m) and T(a,m)
Pam, Qam, Tam = bs(a, m)
# Recursively calculate P(m,b), Q(m,b) and T(m,b)
Pmb, Qmb, Tmb = bs(m, b)
# Now combine
Pab = Pam * Pmb
Qab = Qam * Qmb
Tab = Qmb * Tam + Pam * Tmb
return Pab, Qab, Tab
# how many terms to compute
DIGITS_PER_TERM = math.log10(C3_OVER_24/6/2/6)
N = int(digits/DIGITS_PER_TERM + 1)
# Calclate P(0,N) and Q(0,N)
P, Q, T = bs(0, N)
one_squared = mpz(10)**(2*digits)
sqrtC = gmpy2.isqrt(10005*one_squared)
return (Q*426880*sqrtC) // T
# The last 5 digits or pi for various numbers of digits
check_digits = {
100 : 70679,
1000 : 1989,
10000 : 75678,
100000 : 24646,
1000000 : 58151,
10000000 : 55897,
}
if __name__ == "__main__":
#raise SystemExit
for log10_digits in range(1,8): # 10자리 ~ 1000만자리
digits = 10**log10_digits
start =time()
pi = pi_chudnovsky_bs(digits)
f = open('d_' + str(digits), 'w')
f.write(str(pi))
f.close()
print("chudnovsky_gmpy_mpz_bs: digits",digits,"time",time()-start)
if digits in check_digits:
last_five_digits = pi % 100000
if check_digits[digits] == last_five_digits:
print("Last 5 digits %05d OK" % last_five_digits)
else:
print("Last 5 digits %05d wrong should be %05d" % (last_five_digits, check_digits[digits]))