Why is "1000000000000000 in range(1000000000000001)" so fast in Python 3?
Question:
It is my understanding that the range()
function, which is actually an object type in Python 3, generates its contents on the fly, similar to a generator.
This being the case, I would have expected the following line to take an inordinate amount of time because, in order to determine whether 1 quadrillion is in the range, a quadrillion values would have to be generated:
1_000_000_000_000_000 in range(1_000_000_000_000_001)
Furthermore: it seems that no matter how many zeroes I add on, the calculation more or less takes the same amount of time (basically instantaneous).
I have also tried things like this, but the calculation is still almost instant:
# count by tens
1_000_000_000_000_000_000_000 in range(0,1_000_000_000_000_000_000_001,10)
If I try to implement my own range function, the result is not so nice!
def my_crappy_range(N):
i = 0
while i < N:
yield i
i += 1
return
What is the range()
object doing under the hood that makes it so fast?
Martijn Pieters’s answer was chosen for its completeness, but also see abarnert’s first answer for a good discussion of what it means for range
to be a fullfledged sequence in Python 3, and some information/warning regarding potential inconsistency for __contains__
function optimization across Python implementations. abarnert’s other answer goes into some more detail and provides links for those interested in the history behind the optimization in Python 3 (and lack of optimization of xrange
in Python 2). Answers by poke and by wim provide the relevant C source code and explanations for those who are interested.
Answers:
The Python 3 range()
object doesn’t produce numbers immediately; it is a smart sequence object that produces numbers on demand. All it contains is your start, stop and step values, then as you iterate over the object the next integer is calculated each iteration.
The object also implements the object.__contains__
hook, and calculates if your number is part of its range. Calculating is a (near) constant time operation ^{*}. There is never a need to scan through all possible integers in the range.
From the range()
object documentation:
The advantage of the
range
type over a regularlist
ortuple
is that a range object will always take the same (small) amount of memory, no matter the size of the range it represents (as it only stores thestart
,stop
andstep
values, calculating individual items and subranges as needed).
So at a minimum, your range()
object would do:
class my_range:
def __init__(self, start, stop=None, step=1, /):
if stop is None:
start, stop = 0, start
self.start, self.stop, self.step = start, stop, step
if step < 0:
lo, hi, step = stop, start, step
else:
lo, hi = start, stop
self.length = 0 if lo > hi else ((hi  lo  1) // step) + 1
def __iter__(self):
current = self.start
if self.step < 0:
while current > self.stop:
yield current
current += self.step
else:
while current < self.stop:
yield current
current += self.step
def __len__(self):
return self.length
def __getitem__(self, i):
if i < 0:
i += self.length
if 0 <= i < self.length:
return self.start + i * self.step
raise IndexError('my_range object index out of range')
def __contains__(self, num):
if self.step < 0:
if not (self.stop < num <= self.start):
return False
else:
if not (self.start <= num < self.stop):
return False
return (num  self.start) % self.step == 0
This is still missing several things that a real range()
supports (such as the .index()
or .count()
methods, hashing, equality testing, or slicing), but should give you an idea.
I also simplified the __contains__
implementation to only focus on integer tests; if you give a real range()
object a noninteger value (including subclasses of int
), a slow scan is initiated to see if there is a match, just as if you use a containment test against a list of all the contained values. This was done to continue to support other numeric types that just happen to support equality testing with integers but are not expected to support integer arithmetic as well. See the original Python issue that implemented the containment test.
* Near constant time because Python integers are unbounded and so math operations also grow in time as N grows, making this a O(log N) operation. Since it’s all executed in optimised C code and Python stores integer values in 30bit chunks, you’d run out of memory before you saw any performance impact due to the size of the integers involved here.
To add to Martijn’s answer, this is the relevant part of the source (in C, as the range object is written in native code):
static int
range_contains(rangeobject *r, PyObject *ob)
{
if (PyLong_CheckExact(ob)  PyBool_Check(ob))
return range_contains_long(r, ob);
return (int)_PySequence_IterSearch((PyObject*)r, ob,
PY_ITERSEARCH_CONTAINS);
}
So for PyLong
objects (which is int
in Python 3), it will use the range_contains_long
function to determine the result. And that function essentially checks if ob
is in the specified range (although it looks a bit more complex in C).
If it’s not an int
object, it falls back to iterating until it finds the value (or not).
The whole logic could be translated to pseudoPython like this:
def range_contains (rangeObj, obj):
if isinstance(obj, int):
return range_contains_long(rangeObj, obj)
# default logic by iterating
return any(obj == x for x in rangeObj)
def range_contains_long (r, num):
if r.step > 0:
# positive step: r.start <= num < r.stop
cmp2 = r.start <= num
cmp3 = num < r.stop
else:
# negative step: r.start >= num > r.stop
cmp2 = num <= r.start
cmp3 = r.stop < num
# outside of the range boundaries
if not cmp2 or not cmp3:
return False
# num must be on a valid step inside the boundaries
return (num  r.start) % r.step == 0
Use the source, Luke!
In CPython, range(...).__contains__
(a method wrapper) will eventually delegate to a simple calculation which checks if the value can possibly be in the range. The reason for the speed here is we’re using mathematical reasoning about the bounds, rather than a direct iteration of the range object. To explain the logic used:
 Check that the number is between
start
andstop
, and  Check that the stride value doesn’t "step over" our number.
For example, 994
is in range(4, 1000, 2)
because:
4 <= 994 < 1000
, and(994  4) % 2 == 0
.
The full C code is included below, which is a bit more verbose because of memory management and reference counting details, but the basic idea is there:
static int
range_contains_long(rangeobject *r, PyObject *ob)
{
int cmp1, cmp2, cmp3;
PyObject *tmp1 = NULL;
PyObject *tmp2 = NULL;
PyObject *zero = NULL;
int result = 1;
zero = PyLong_FromLong(0);
if (zero == NULL) /* MemoryError in int(0) */
goto end;
/* Check if the value can possibly be in the range. */
cmp1 = PyObject_RichCompareBool(r>step, zero, Py_GT);
if (cmp1 == 1)
goto end;
if (cmp1 == 1) { /* positive steps: start <= ob < stop */
cmp2 = PyObject_RichCompareBool(r>start, ob, Py_LE);
cmp3 = PyObject_RichCompareBool(ob, r>stop, Py_LT);
}
else { /* negative steps: stop < ob <= start */
cmp2 = PyObject_RichCompareBool(ob, r>start, Py_LE);
cmp3 = PyObject_RichCompareBool(r>stop, ob, Py_LT);
}
if (cmp2 == 1  cmp3 == 1) /* TypeError */
goto end;
if (cmp2 == 0  cmp3 == 0) { /* ob outside of range */
result = 0;
goto end;
}
/* Check that the stride does not invalidate ob's membership. */
tmp1 = PyNumber_Subtract(ob, r>start);
if (tmp1 == NULL)
goto end;
tmp2 = PyNumber_Remainder(tmp1, r>step);
if (tmp2 == NULL)
goto end;
/* result = ((int(ob)  start) % step) == 0 */
result = PyObject_RichCompareBool(tmp2, zero, Py_EQ);
end:
Py_XDECREF(tmp1);
Py_XDECREF(tmp2);
Py_XDECREF(zero);
return result;
}
static int
range_contains(rangeobject *r, PyObject *ob)
{
if (PyLong_CheckExact(ob)  PyBool_Check(ob))
return range_contains_long(r, ob);
return (int)_PySequence_IterSearch((PyObject*)r, ob,
PY_ITERSEARCH_CONTAINS);
}
The "meat" of the idea is mentioned in the comment lines:
/* positive steps: start <= ob < stop */
/* negative steps: stop < ob <= start */
/* result = ((int(ob)  start) % step) == 0 */
As a final note – look at the range_contains
function at the bottom of the code snippet. If the exact type check fails then we don’t use the clever algorithm described, instead falling back to a dumb iteration search of the range using _PySequence_IterSearch
! You can check this behaviour in the interpreter (I’m using v3.5.0 here):
>>> x, r = 1000000000000000, range(1000000000000001)
>>> class MyInt(int):
... pass
...
>>> x_ = MyInt(x)
>>> x in r # calculates immediately :)
True
>>> x_ in r # iterates for ages.. :(
^Quit (core dumped)
The fundamental misunderstanding here is in thinking that range
is a generator. It’s not. In fact, it’s not any kind of iterator.
You can tell this pretty easily:
>>> a = range(5)
>>> print(list(a))
[0, 1, 2, 3, 4]
>>> print(list(a))
[0, 1, 2, 3, 4]
If it were a generator, iterating it once would exhaust it:
>>> b = my_crappy_range(5)
>>> print(list(b))
[0, 1, 2, 3, 4]
>>> print(list(b))
[]
What range
actually is, is a sequence, just like a list. You can even test this:
>>> import collections.abc
>>> isinstance(a, collections.abc.Sequence)
True
This means it has to follow all the rules of being a sequence:
>>> a[3] # indexable
3
>>> len(a) # sized
5
>>> 3 in a # membership
True
>>> reversed(a) # reversible
<range_iterator at 0x101cd2360>
>>> a.index(3) # implements 'index'
3
>>> a.count(3) # implements 'count'
1
The difference between a range
and a list
is that a range
is a lazy or dynamic sequence; it doesn’t remember all of its values, it just remembers its start
, stop
, and step
, and creates the values on demand on __getitem__
.
(As a side note, if you print(iter(a))
, you’ll notice that range
uses the same listiterator
type as list
. How does that work? A listiterator
doesn’t use anything special about list
except for the fact that it provides a C implementation of __getitem__
, so it works fine for range
too.)
Now, there’s nothing that says that Sequence.__contains__
has to be constant time—in fact, for obvious examples of sequences like list
, it isn’t. But there’s nothing that says it can’t be. And it’s easier to implement range.__contains__
to just check it mathematically ((val  start) % step
, but with some extra complexity to deal with negative steps) than to actually generate and test all the values, so why shouldn’t it do it the better way?
But there doesn’t seem to be anything in the language that guarantees this will happen. As Ashwini Chaudhari points out, if you give it a nonintegral value, instead of converting to integer and doing the mathematical test, it will fall back to iterating all the values and comparing them one by one. And just because CPython 3.2+ and PyPy 3.x versions happen to contain this optimization, and it’s an obvious good idea and easy to do, there’s no reason that IronPython or NewKickAssPython 3.x couldn’t leave it out. (And in fact, CPython 3.03.1 didn’t include it.)
If range
actually were a generator, like my_crappy_range
, then it wouldn’t make sense to test __contains__
this way, or at least the way it makes sense wouldn’t be obvious. If you’d already iterated the first 3 values, is 1
still in
the generator? Should testing for 1
cause it to iterate and consume all the values up to 1
(or up to the first value >= 1
)?
The other answers explained it well already, but I’d like to offer another experiment illustrating the nature of range objects:
>>> r = range(5)
>>> for i in r:
print(i, 2 in r, list(r))
0 True [0, 1, 2, 3, 4]
1 True [0, 1, 2, 3, 4]
2 True [0, 1, 2, 3, 4]
3 True [0, 1, 2, 3, 4]
4 True [0, 1, 2, 3, 4]
As you can see, a range
object is an object that remembers its range and can be used many times (even while iterating over it), not just a onetime generator.
If you’re wondering why this optimization was added to range.__contains__
, and why it wasn’t added to xrange.__contains__
in 2.7:
First, as Ashwini Chaudhary discovered, issue 1766304 was opened explicitly to optimize [x]range.__contains__
. A patch for this was accepted and checked in for 3.2, but not backported to 2.7 because "xrange
has behaved like this for such a long time that I don’t see what it buys us to commit the patch this late." (2.7 was nearly out at that point.)
Meanwhile:
Originally, xrange
was a notquitesequence object. As the 3.1 docs say:
Range objects have very little behavior: they only support indexing, iteration, and the
len
function.
This wasn’t quite true; an xrange
object actually supported a few other things that come automatically with indexing and len
,^{*} including __contains__
(via linear search). But nobody thought it was worth making them full sequences at the time.
Then, as part of implementing the Abstract Base Classes PEP, it was important to figure out which builtin types should be marked as implementing which ABCs, and xrange
/range
claimed to implement collections.Sequence
, even though it still only handled the same "very little behavior". Nobody noticed that problem until issue 9213. The patch for that issue not only added index
and count
to 3.2’s range
, it also reworked the optimized __contains__
(which shares the same math with index
, and is directly used by count
).^{**} This change went in for 3.2 as well, and was not backported to 2.x, because "it’s a bugfix that adds new methods". (At this point, 2.7 was already past rc status.)
So, there were two chances to get this optimization backported to 2.7, but they were both rejected.
_{* In fact, you even get iteration for free with indexing alone, but in 2.3 xrange objects got a custom iterator.}
_{** The first version actually reimplemented it, and got the details wrong—e.g., it would give you MyIntSubclass(2) in range(5) == False. But Daniel Stutzbach’s updated version of the patch restored most of the previous code, including the fallback to the generic, slow _PySequence_IterSearch that pre3.2 range.__contains__ was implicitly using when the optimization doesn’t apply.}
It’s all about a lazy approach to the evaluation and some extra optimization of range
.
Values in ranges don’t need to be computed until real use, or even further due to extra optimization.
By the way, your integer is not such big, consider sys.maxsize
sys.maxsize in range(sys.maxsize)
is pretty fast
due to optimization – it’s easy to compare given integers just with min and max of range.
but:
Decimal(sys.maxsize) in range(sys.maxsize)
is pretty slow.
(in this case, there is no optimization in range
, so if python receives unexpected Decimal, python will compare all numbers)
You should be aware of an implementation detail but should not be relied upon, because this may change in the future.
TL;DR
The object returned by range()
is actually a range
object. This object implements the iterator interface so you can iterate over its values sequentially, just like a generator, list, or tuple.
But it also implements the __contains__
interface which is actually what gets called when an object appears on the righthand side of the in
operator. The __contains__()
method returns a bool
of whether or not the item on the lefthand side of the in
is in the object. Since range
objects know their bounds and stride, this is very easy to implement in O(1).
 Due to optimization, it is very easy to compare given integers just with min and max range.
 The reason that the range() function is so fast in Python3 is that here we use mathematical reasoning for the bounds, rather than a direct iteration of the range object.
 So for explaining the logic here:
 Check whether the number is between the start and stop.
 Check whether the step precision value doesn’t go over our number.

Take an example, 997 is in range(4, 1000, 3) because:
4 <= 997 < 1000, and (997  4) % 3 == 0.
Try x1 in (i for i in range(x))
for large x
values, which uses a generator comprehension to avoid invoking the range.__contains__
optimisation.
TLDR;
the range
is an arithmetic series so it can very easily calculate whether the object is there. It could even get the index of it if it were list like really quickly.
__contains__
method compares directly with the start and end of the range