Index: Lib/random.py
===================================================================
--- Lib/random.py	(revision 82067)
+++ Lib/random.py	(working copy)
@@ -168,9 +168,7 @@
             raise ValueError("non-integer arg 1 for randrange()")
         if stop is default:
             if istart > 0:
-                if istart >= maxwidth:
-                    return self._randbelow(istart)
-                return int(self.random() * istart)
+                return self._randbelow(istart)
             raise ValueError("empty range for randrange()")
 
         # stop argument supplied.
@@ -192,9 +190,7 @@
             # a long, but we're supposed to return an int (for backward
             # compatibility).
 
-            if width >= maxwidth:
-                return int(istart + self._randbelow(width))
-            return int(istart + int(self.random()*width))
+            return istart + self._randbelow(width)
         if step == 1:
             raise ValueError("empty range for randrange() (%d,%d, %d)" % (istart, istop, width))
 
@@ -212,9 +208,7 @@
         if n <= 0:
             raise ValueError("empty range for randrange()")
 
-        if n >= maxwidth:
-            return istart + istep*self._randbelow(n)
-        return istart + istep*int(self.random() * n)
+        return istart + istep*self._randbelow(n)
 
     def randint(self, a, b):
         """Return random integer in range [a, b], including both end points.
@@ -222,7 +216,7 @@
 
         return self.randrange(a, b+1)
 
-    def _randbelow(self, n, _log=_log, int=int, _maxwidth=1<<BPF,
+    def _randbelow(self, n, int=int, divmod=divmod, _maxwidth=1<<BPF,
                    _Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):
         """Return a random int in the range [0,n)
 
@@ -230,6 +224,20 @@
         by a single call to the underlying generator.
         """
 
+        # For small n, just rely on random() to provide BPF random bits.
+        if n <= _maxwidth:
+            # Generate random integer in [0, rmax), where rmax is the
+            # largest multiple of n not exceeding _maxwidth.
+            quot, rem = divmod(_maxwidth, n)
+            rmax = _maxwidth - rem
+            r = int(self.random() * _maxwidth)
+            while r >= rmax:
+                r = int(self.random() * _maxwidth)
+            # Could use r % n here instead.  That would be fine for MT, but
+            # would behave poorly for PRNGs with low entropy in the low-order
+            # bits of random().  r // quot is safer in general.
+            return r // quot
+
         try:
             getrandbits = self.getrandbits
         except AttributeError:
@@ -239,14 +247,13 @@
             # has not been overridden or if a new getrandbits() was supplied.
             # This assures that the two methods correspond.
             if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:
-                k = int(1.00001 + _log(n-1, 2.0))   # 2**k > n-1 > 2**(k-2)
+                k = (n-1).bit_length()              # 2**k > n-1 >= 2**(k-1)
                 r = getrandbits(k)
                 while r >= n:
                     r = getrandbits(k)
                 return r
-        if n >= _maxwidth:
-            _warn("Underlying random() generator does not supply \n"
-                "enough bits to choose from a population range this large")
+        _warn("Underlying random() generator does not supply \n"
+              "enough bits to choose from a population range this large")
         return int(self.random() * n)
 
 ## -------------------- sequence methods  -------------------