# Why is "1000000000000000 in range(1000000000000001)" so fast in Python 3?

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 regular`list`

or`tuple`

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 the`start`

,`stop`

and`step`

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 non-integer 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 30-bit chunks, you’d run out of memory before you saw any performance impact due to the size of the integers involved here.

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] # indexable3>>> len(a) # sized5>>> 3 in a # membershipTrue>>> 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 non-integral 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 3.0-3.1 *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`

)?

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`

and`stop`

, 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 intrange_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 intrange_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 line:

`/* 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)`