Fix issue with big integers (outlined in readme)

pairing/main.py

@@ -1,12 +1,38 @@
 import math
 
 
+def improved_i_sqrt(n):
+    assert n >= 0
+    if n == 0:
+        return 0
+    i = n.bit_length() >> 1    # i = floor( (1 + floor(log_2(n))) / 2 )
+    m = 1 << i    # m = 2^i
+    #
+    # Fact: (2^(i + 1))^2 > n, so m has at least as many bits
+    # as the floor of the square root of n.
+    #
+    # Proof: (2^(i+1))^2 = 2^(2i + 2) >= 2^(floor(log_2(n)) + 2)
+    # >= 2^(ceil(log_2(n) + 1) >= 2^(log_2(n) + 1) > 2^(log_2(n)) = n. QED.
+    #
+    while (m << i) > n: # (m<<i) = m*(2^i) = m*m
+        m >>= 1
+        i -= 1
+    d = n - (m << i) # d = n-m^2
+    for k in range(i-1, -1, -1):
+        j = 1 << k
+        new_diff = d - (((m<<1) | j) << k) # n-(m+2^k)^2 = n-m^2-2*m*2^k-2^(2k)
+        if new_diff >= 0:
+            d = new_diff
+            m |= j
+    return m
+
+
 def pair(k1, k2, safe=True):
     """
     Cantor pairing function
     http://en.wikipedia.org/wiki/Pairing_function#Cantor_pairing_function
     """
-    z = int(0.5 * (k1 + k2) * (k1 + k2 + 1) + k2)
+    z = (k1 + k2) * (k1 + k2 + 1) // 2 + k2
     if safe and (k1, k2) != depair(z):
         raise ValueError("{} and {} cannot be paired".format(k1, k2))
     return z
@@ -17,9 +43,9 @@ def depair(z):
     Inverse of Cantor pairing function
     http://en.wikipedia.org/wiki/Pairing_function#Inverting_the_Cantor_pairing_function
     """
-    w = math.floor((math.sqrt(8 * z + 1) - 1)/2)
-    t = (w**2 + w) / 2
-    y = int(z - t)
-    x = int(w - y)
+    w = (improved_i_sqrt(8 * z + 1) - 1)//2
+    t = (w**2 + w) // 2
+    y = z - t
+    x = w - y
     # assert z != pair(x, y, safe=False):
     return x, y

test.py

@@ -2,23 +2,29 @@
 from __future__ import print_function
 from pairing import pair, depair
 import timeit
+import random
 
 
 def test_pair(a, b):
     assert depair(pair(a, b)) == (a, b)
     return pair(a, b)
 
 
-def run_tests():
+def run_tests(num_random_tests = 0):
     test_pair(22, 33)
     test_pair(2**8, 2**8)
     test_pair(2**16, 2**16)
-
-    try:
-        test_pair(2**52, 2**52)
-    except ValueError:
-        pass  # long integers suffer from some imprecision at this size
-
+    test_pair(2**52, 2**52)
+    test_pair(2**52, 1)
+    test_pair(2**52, 2)
+    test_pair(2**52, 55555555)
+    for i in range(0, num_random_tests):
+        test_pair(random.getrandbits(512), random.getrandbits(512))
+        test_pair(random.getrandbits(512), 0)
+        test_pair(0, random.getrandbits(512))
+        test_pair(random.getrandbits(512), 1)
+        test_pair(1, random.getrandbits(512))
+
     try:
         test_pair(-1, -1)
     except ValueError:
@@ -28,7 +34,7 @@ def run_tests():
 
 
 if __name__ == '__main__':
-    print(run_tests())
+    print(run_tests(2000))
 
     print("Benchmarking...")
     i = 20000