# Expressions

This chapter explains the meaning of the elements of expressions in Python.

Syntax Notes: In this and the following chapters, extended BNF notation will be used to describe syntax, not lexical analysis. When (one alternative of) a syntax rule has the form

and no semantics are given, the semantics of this form of name are the same as for othername.

## Arithmetic conversions

When a description of an arithmetic operator below uses the phrase "the numeric arguments are converted to a common type," the arguments are coerced using the coercion rules listed at :ref:coercion-rules. If both arguments are standard numeric types, the following coercions are applied:

• If either argument is a complex number, the other is converted to complex;
• otherwise, if either argument is a floating point number, the other is converted to floating point;
• otherwise, if either argument is a long integer, the other is converted to long integer;
• otherwise, both must be plain integers and no conversion is necessary.

Some additional rules apply for certain operators (e.g., a string left argument to the '%' operator). Extensions can define their own coercions.

## Atoms

Atoms are the most basic elements of expressions. The simplest atoms are identifiers or literals. Forms enclosed in reverse quotes or in parentheses, brackets or braces are also categorized syntactically as atoms. The syntax for atoms is:

### Identifiers (Names)

An identifier occurring as an atom is a name. See section :ref:identifiers for lexical definition and section :ref:naming for documentation of naming and binding.

When the name is bound to an object, evaluation of the atom yields that object. When a name is not bound, an attempt to evaluate it raises a :exc:NameError exception.

Private name mangling: When an identifier that textually occurs in a class definition begins with two or more underscore characters and does not end in two or more underscores, it is considered a :dfn:private name of that class. Private names are transformed to a longer form before code is generated for them. The transformation inserts the class name in front of the name, with leading underscores removed, and a single underscore inserted in front of the class name. For example, the identifier __spam occurring in a class named Ham will be transformed to _Ham__spam. This transformation is independent of the syntactical context in which the identifier is used. If the transformed name is extremely long (longer than 255 characters), implementation defined truncation may happen. If the class name consists only of underscores, no transformation is done.

### Literals

Python supports string literals and various numeric literals:

Evaluation of a literal yields an object of the given type (string, integer, long integer, floating point number, complex number) with the given value. The value may be approximated in the case of floating point and imaginary (complex) literals. See section :ref:literals for details.

All literals correspond to immutable data types, and hence the object's identity is less important than its value. Multiple evaluations of literals with the same value (either the same occurrence in the program text or a different occurrence) may obtain the same object or a different object with the same value.

### Parenthesized forms

A parenthesized form is an optional expression list enclosed in parentheses:

A parenthesized expression list yields whatever that expression list yields: if the list contains at least one comma, it yields a tuple; otherwise, it yields the single expression that makes up the expression list.

An empty pair of parentheses yields an empty tuple object. Since tuples are immutable, the rules for literals apply (i.e., two occurrences of the empty tuple may or may not yield the same object).

Note that tuples are not formed by the parentheses, but rather by use of the comma operator. The exception is the empty tuple, for which parentheses are required --- allowing unparenthesized "nothing" in expressions would cause ambiguities and allow common typos to pass uncaught.

### List displays

A list display is a possibly empty series of expressions enclosed in square brackets:

A list display yields a new list object. Its contents are specified by providing either a list of expressions or a list comprehension. When a comma- separated list of expressions is supplied, its elements are evaluated from left to right and placed into the list object in that order. When a list comprehension is supplied, it consists of a single expression followed by at least one :keyword:for clause and zero or more :keyword:for or :keyword:if clauses. In this case, the elements of the new list are those that would be produced by considering each of the :keyword:for or :keyword:if clauses a block, nesting from left to right, and evaluating the expression to produce a list element each time the innermost block is reached [1].

### Generator expressions

A generator expression is a compact generator notation in parentheses:

A generator expression yields a new generator object. It consists of a single expression followed by at least one :keyword:for clause and zero or more :keyword:for or :keyword:if clauses. The iterating values of the new generator are those that would be produced by considering each of the :keyword:for or :keyword:if clauses a block, nesting from left to right, and evaluating the expression to yield a value that is reached the innermost block for each iteration.

Variables used in the generator expression are evaluated lazily when the :meth:__next__ method is called for generator object (in the same fashion as normal generators). However, the leftmost :keyword:for clause is immediately evaluated so that error produced by it can be seen before any other possible error in the code that handles the generator expression. Subsequent :keyword:for clauses cannot be evaluated immediately since they may depend on the previous :keyword:for loop. For example: (x*y for x in range(10) for y in bar(x)).

The parentheses can be omitted on calls with only one argument. See section :ref:calls for the detail.

### Dictionary displays

A dictionary display is a possibly empty series of key/datum pairs enclosed in curly braces:

A dictionary display yields a new dictionary object.

The key/datum pairs are evaluated from left to right to define the entries of the dictionary: each key object is used as a key into the dictionary to store the corresponding datum.

Restrictions on the types of the key values are listed earlier in section :ref:types. (To summarize, the key type should be hashable, which excludes all mutable objects.) Clashes between duplicate keys are not detected; the last datum (textually rightmost in the display) stored for a given key value prevails.

### Yield expressions

The :keyword:yield expression is only used when defining a generator function, and can only be used in the body of a function definition. Using a :keyword:yield expression in a function definition is sufficient to cause that definition to create a generator function instead of a normal function.

When a generator function is called, it returns an iterator known as a generator. That generator then controls the execution of a generator function. The execution starts when one of the generator's methods is called. At that time, the execution proceeds to the first :keyword:yield expression, where it is suspended again, returning the value of :token:expression_list to generator's caller. By suspended we mean that all local state is retained, including the current bindings of local variables, the instruction pointer, and the internal evaluation stack. When the execution is resumed by calling one of the generator's methods, the function can proceed exactly as if the :keyword:yield expression was just another external call. The value of the :keyword:yield expression after resuming depends on the method which resumed the execution.

All of this makes generator functions quite similar to coroutines; they yield multiple times, they have more than one entry point and their execution can be suspended. The only difference is that a generator function cannot control where should the execution continue after it yields; the control is always transfered to the generator's caller.

The following generator's methods can be used to control the execution of a generator function:

Here is a simple example that demonstrates the behavior of generators and generator functions:

>>> def echo(value=None):
...     print "Execution starts when 'next()' is called for the first time."
...     try:
...         while True:
...             try:
...                 value = (yield value)
...             except GeneratorExit:
...                 # never catch GeneratorExit
...                 raise
...             except Exception, e:
...                 value = e
...     finally:
...         print "Don't forget to clean up when 'close()' is called."
...
>>> generator = echo(1)
>>> print generator.next()
Execution starts when 'next()' is called for the first time.
1
>>> print generator.next()
None
>>> print generator.send(2)
2
>>> generator.throw(TypeError, "spam")
TypeError('spam',)
>>> generator.close()
Don't forget to clean up when 'close()' is called.


## Primaries

Primaries represent the most tightly bound operations of the language. Their syntax is:

### Attribute references

An attribute reference is a primary followed by a period and a name:

The primary must evaluate to an object of a type that supports attribute references, e.g., a module, list, or an instance. This object is then asked to produce the attribute whose name is the identifier. If this attribute is not available, the exception :exc:AttributeError is raised. Otherwise, the type and value of the object produced is determined by the object. Multiple evaluations of the same attribute reference may yield different objects.

### Subscriptions

A subscription selects an item of a sequence (string, tuple or list) or mapping (dictionary) object:

The primary must evaluate to an object of a sequence or mapping type.

If the primary is a mapping, the expression list must evaluate to an object whose value is one of the keys of the mapping, and the subscription selects the value in the mapping that corresponds to that key. (The expression list is a tuple except if it has exactly one item.)

If the primary is a sequence, the expression (list) must evaluate to a plain integer. If this value is negative, the length of the sequence is added to it (so that, e.g., x[-1] selects the last item of x.) The resulting value must be a nonnegative integer less than the number of items in the sequence, and the subscription selects the item whose index is that value (counting from zero).

A string's items are characters. A character is not a separate data type but a string of exactly one character.

### Slicings

A slicing selects a range of items in a sequence object (e.g., a string, tuple or list). Slicings may be used as expressions or as targets in assignment or :keyword:del statements. The syntax for a slicing:

There is ambiguity in the formal syntax here: anything that looks like an expression list also looks like a slice list, so any subscription can be interpreted as a slicing. Rather than further complicating the syntax, this is disambiguated by defining that in this case the interpretation as a subscription takes priority over the interpretation as a slicing (this is the case if the slice list contains no proper slice nor ellipses). Similarly, when the slice list has exactly one short slice and no trailing comma, the interpretation as a simple slicing takes priority over that as an extended slicing.

The semantics for a simple slicing are as follows. The primary must evaluate to a sequence object. The lower and upper bound expressions, if present, must evaluate to plain integers; defaults are zero and the sys.maxint, respectively. If either bound is negative, the sequence's length is added to it. The slicing now selects all items with index k such that i <= k < j where i and j are the specified lower and upper bounds. This may be an empty sequence. It is not an error if i or j lie outside the range of valid indexes (such items don't exist so they aren't selected).

The semantics for an extended slicing are as follows. The primary must evaluate to a mapping object, and it is indexed with a key that is constructed from the slice list, as follows. If the slice list contains at least one comma, the key is a tuple containing the conversion of the slice items; otherwise, the conversion of the lone slice item is the key. The conversion of a slice item that is an expression is that expression. The conversion of a proper slice is a slice object (see section :ref:types) whose :attr:start, :attr:stop and :attr:step attributes are the values of the expressions given as lower bound, upper bound and stride, respectively, substituting None for missing expressions.

### Calls

A call calls a callable object (e.g., a function) with a possibly empty series of arguments:

A trailing comma may be present after the positional and keyword arguments but does not affect the semantics.

The primary must evaluate to a callable object (user-defined functions, built-in functions, methods of built-in objects, class objects, methods of class instances, and certain class instances themselves are callable; extensions may define additional callable object types). All argument expressions are evaluated before the call is attempted. Please refer to section :ref:function for the syntax of formal parameter lists.

If keyword arguments are present, they are first converted to positional arguments, as follows. First, a list of unfilled slots is created for the formal parameters. If there are N positional arguments, they are placed in the first N slots. Next, for each keyword argument, the identifier is used to determine the corresponding slot (if the identifier is the same as the first formal parameter name, the first slot is used, and so on). If the slot is already filled, a :exc:TypeError exception is raised. Otherwise, the value of the argument is placed in the slot, filling it (even if the expression is None, it fills the slot). When all arguments have been processed, the slots that are still unfilled are filled with the corresponding default value from the function definition. (Default values are calculated, once, when the function is defined; thus, a mutable object such as a list or dictionary used as default value will be shared by all calls that don't specify an argument value for the corresponding slot; this should usually be avoided.) If there are any unfilled slots for which no default value is specified, a :exc:TypeError exception is raised. Otherwise, the list of filled slots is used as the argument list for the call.

If there are more positional arguments than there are formal parameter slots, a :exc:TypeError exception is raised, unless a formal parameter using the syntax *identifier is present; in this case, that formal parameter receives a tuple containing the excess positional arguments (or an empty tuple if there were no excess positional arguments).

If any keyword argument does not correspond to a formal parameter name, a :exc:TypeError exception is raised, unless a formal parameter using the syntax **identifier is present; in this case, that formal parameter receives a dictionary containing the excess keyword arguments (using the keywords as keys and the argument values as corresponding values), or a (new) empty dictionary if there were no excess keyword arguments.

If the syntax *expression appears in the function call, expression must evaluate to a sequence. Elements from this sequence are treated as if they were additional positional arguments; if there are postional arguments x1,...,*xN* , and expression evaluates to a sequence y1,...,*yM*, this is equivalent to a call with M+N positional arguments x1,...,*xN*,*y1*,...,*yM*.

A consequence of this is that although the *expression syntax appears after any keyword arguments, it is processed before the keyword arguments (and the **expression argument, if any -- see below). So:

>>> def f(a, b):
...  print a, b
...
>>> f(b=1, *(2,))
2 1
>>> f(a=1, *(2,))
Traceback (most recent call last):
File "<stdin>", line 1, in ?
TypeError: f() got multiple values for keyword argument 'a'
>>> f(1, *(2,))
1 2


It is unusual for both keyword arguments and the *expression syntax to be used in the same call, so in practice this confusion does not arise.

If the syntax **expression appears in the function call, expression must evaluate to a mapping, the contents of which are treated as additional keyword arguments. In the case of a keyword appearing in both expression and as an explicit keyword argument, a :exc:TypeError exception is raised.

Formal parameters using the syntax *identifier or **identifier cannot be used as positional argument slots or as keyword argument names.

A call always returns some value, possibly None, unless it raises an exception. How this value is computed depends on the type of the callable object.

If it is---

a user-defined function:
The code block for the function is executed, passing it the argument list. The first thing the code block will do is bind the formal parameters to the arguments; this is described in section :ref:function. When the code block executes a :keyword:return statement, this specifies the return value of the function call.
a built-in function or method:
The result is up to the interpreter; see the Python Library Reference (XXX reference: ../lib/built-in-funcs.html) for the descriptions of built-in functions and methods.
a class object:
A new instance of that class is returned.
a class instance method:
The corresponding user-defined function is called, with an argument list that is one longer than the argument list of the call: the instance becomes the first argument.
a class instance:
The class must define a :meth:__call__ method; the effect is then the same as if that method was called.

## The power operator

The power operator binds more tightly than unary operators on its left; it binds less tightly than unary operators on its right. The syntax is:

Thus, in an unparenthesized sequence of power and unary operators, the operators are evaluated from right to left (this does not constrain the evaluation order for the operands).

The power operator has the same semantics as the built-in :func:pow function, when called with two arguments: it yields its left argument raised to the power of its right argument. The numeric arguments are first converted to a common type. The result type is that of the arguments after coercion.

With mixed operand types, the coercion rules for binary arithmetic operators apply. For int and long int operands, the result has the same type as the operands (after coercion) unless the second argument is negative; in that case, all arguments are converted to float and a float result is delivered. For example, 10**2 returns 100, but 10**-2 returns 0.01. (This last feature was added in Python 2.2. In Python 2.1 and before, if both arguments were of integer types and the second argument was negative, an exception was raised).

Raising 0.0 to a negative power results in a :exc:ZeroDivisionError. Raising a negative number to a fractional power results in a :exc:ValueError.

## Unary arithmetic operations

All unary arithmetic (and bit-wise) operations have the same priority:

The unary - (minus) operator yields the negation of its numeric argument.

The unary + (plus) operator yields its numeric argument unchanged.

The unary ~ (invert) operator yields the bit-wise inversion of its plain or long integer argument. The bit-wise inversion of x is defined as -(x+1). It only applies to integral numbers.

In all three cases, if the argument does not have the proper type, a :exc:TypeError exception is raised.

## Binary arithmetic operations

The binary arithmetic operations have the conventional priority levels. Note that some of these operations also apply to certain non-numeric types. Apart from the power operator, there are only two levels, one for multiplicative operators and one for additive operators:

The * (multiplication) operator yields the product of its arguments. The arguments must either both be numbers, or one argument must be an integer (plain or long) and the other must be a sequence. In the former case, the numbers are converted to a common type and then multiplied together. In the latter case, sequence repetition is performed; a negative repetition factor yields an empty sequence.

The / (division) and // (floor division) operators yield the quotient of their arguments. The numeric arguments are first converted to a common type. Plain or long integer division yields an integer of the same type; the result is that of mathematical division with the 'floor' function applied to the result. Division by zero raises the :exc:ZeroDivisionError exception.

The % (modulo) operator yields the remainder from the division of the first argument by the second. The numeric arguments are first converted to a common type. A zero right argument raises the :exc:ZeroDivisionError exception. The arguments may be floating point numbers, e.g., 3.14%0.7 equals 0.34 (since 3.14 equals 4*0.7 + 0.34.) The modulo operator always yields a result with the same sign as its second operand (or zero); the absolute value of the result is strictly smaller than the absolute value of the second operand [2].

The integer division and modulo operators are connected by the following identity: x == (x/y)*y + (x%y). Integer division and modulo are also connected with the built-in function :func:divmod: divmod(x, y) == (x/y, x%y). These identities don't hold for floating point numbers; there similar identities hold approximately where x/y is replaced by floor(x/y) or floor(x/y) - 1 [3].

In addition to performing the modulo operation on numbers, the % operator is also overloaded by string and unicode objects to perform string formatting (also known as interpolation). The syntax for string formatting is described in the Python Library Reference (XXX reference: ../lib/typesseq-strings.html), section "Sequence Types".

The + (addition) operator yields the sum of its arguments. The arguments must either both be numbers or both sequences of the same type. In the former case, the numbers are converted to a common type and then added together. In the latter case, the sequences are concatenated.

The - (subtraction) operator yields the difference of its arguments. The numeric arguments are first converted to a common type.

## Shifting operations

The shifting operations have lower priority than the arithmetic operations:

These operators accept plain or long integers as arguments. The arguments are converted to a common type. They shift the first argument to the left or right by the number of bits given by the second argument.

A right shift by n bits is defined as division by pow(2,n). A left shift by n bits is defined as multiplication with pow(2,n); for plain integers there is no overflow check so in that case the operation drops bits and flips the sign if the result is not less than pow(2,31) in absolute value. Negative shift counts raise a :exc:ValueError exception.

## Binary bit-wise operations

Each of the three bitwise operations has a different priority level:

The & operator yields the bitwise AND of its arguments, which must be plain or long integers. The arguments are converted to a common type.

The ^ operator yields the bitwise XOR (exclusive OR) of its arguments, which must be plain or long integers. The arguments are converted to a common type.

The | operator yields the bitwise (inclusive) OR of its arguments, which must be plain or long integers. The arguments are converted to a common type.

## Comparisons

Unlike C, all comparison operations in Python have the same priority, which is lower than that of any arithmetic, shifting or bitwise operation. Also unlike C, expressions like a < b < c have the interpretation that is conventional in mathematics:

Comparisons yield boolean values: True or False.

Comparisons can be chained arbitrarily, e.g., x < y <= z is equivalent to x < y and y <= z, except that y is evaluated only once (but in both cases z is not evaluated at all when x < y is found to be false).

Formally, if a, b, c, ..., y, z are expressions and opa, opb, ..., opy are comparison operators, then a opa b opb c ...*y opy z* is equivalent to a opa b :keyword:and b opb c :keyword:and ... y opy z, except that each expression is evaluated at most once.

Note that a opa b opb c doesn't imply any kind of comparison between a and c, so that, e.g., x < y > z is perfectly legal (though perhaps not pretty).

The operators <, >, ==, >=, <=, and != compare the values of two objects. The objects need not have the same type. If both are numbers, they are converted to a common type. Otherwise, objects of different types always compare unequal, and are ordered consistently but arbitrarily. You can control comparison behavior of objects of non-builtin types by defining a __cmp__ method or rich comparison methods like __gt__, described in section :ref:specialnames.

(This unusual definition of comparison was used to simplify the definition of operations like sorting and the :keyword:in and :keyword:not in operators. In the future, the comparison rules for objects of different types are likely to change.)

Comparison of objects of the same type depends on the type:

• Numbers are compared arithmetically.

• Strings are compared lexicographically using the numeric equivalents (the result of the built-in function :func:ord) of their characters. Unicode and 8-bit strings are fully interoperable in this behavior.

• Tuples and lists are compared lexicographically using comparison of corresponding elements. This means that to compare equal, each element must compare equal and the two sequences must be of the same type and have the same length.

If not equal, the sequences are ordered the same as their first differing elements. For example, cmp([1,2,x], [1,2,y]) returns the same as cmp(x,y). If the corresponding element does not exist, the shorter sequence is ordered first (for example, [1,2] < [1,2,3]).

• Mappings (dictionaries) compare equal if and only if their sorted (key, value) lists compare equal. [4] Outcomes other than equality are resolved consistently, but are not otherwise defined. [5]

• Most other objects of builtin types compare unequal unless they are the same object; the choice whether one object is considered smaller or larger than another one is made arbitrarily but consistently within one execution of a program.

The operators :keyword:in and :keyword:not in test for set membership. x in s evaluates to true if x is a member of the set s, and false otherwise. x not in s returns the negation of x in s. The set membership test has traditionally been bound to sequences; an object is a member of a set if the set is a sequence and contains an element equal to that object. However, it is possible for an object to support membership tests without being a sequence. In particular, dictionaries support membership testing as a nicer way of spelling key in dict; other mapping types may follow suit.

For the list and tuple types, x in y is true if and only if there exists an index i such that x == y[i] is true.

For the Unicode and string types, x in y is true if and only if x is a substring of y. An equivalent test is y.find(x) != -1. Note, x and y need not be the same type; consequently, u'ab' in 'abc' will return True. Empty strings are always considered to be a substring of any other string, so "" in "abc" will return True.

For user-defined classes which define the :meth:__contains__ method, x in y is true if and only if y.__contains__(x) is true.

For user-defined classes which do not define :meth:__contains__ and do define :meth:__getitem__, x in y is true if and only if there is a non-negative integer index i such that x == y[i], and all lower integer indices do not raise :exc:IndexError exception. (If any other exception is raised, it is as if :keyword:in raised that exception).

The operator :keyword:not in is defined to have the inverse true value of :keyword:in.

The operators :keyword:is and :keyword:is not test for object identity: x is y is true if and only if x and y are the same object. x is not y yields the inverse truth value.

## Boolean operations

Boolean operations have the lowest priority of all Python operations:

In the context of Boolean operations, and also when expressions are used by control flow statements, the following values are interpreted as false: False, None, numeric zero of all types, and empty strings and containers (including strings, tuples, lists, dictionaries, sets and frozensets). All other values are interpreted as true.

The operator :keyword:not yields True if its argument is false, False otherwise.

The expression x if C else y first evaluates C (not x); if C is true, x is evaluated and its value is returned; otherwise, y is evaluated and its value is returned.

The expression x and y first evaluates x; if x is false, its value is returned; otherwise, y is evaluated and the resulting value is returned.

The expression x or y first evaluates x; if x is true, its value is returned; otherwise, y is evaluated and the resulting value is returned.

(Note that neither :keyword:and nor :keyword:or restrict the value and type they return to False and True, but rather return the last evaluated argument. This is sometimes useful, e.g., if s is a string that should be replaced by a default value if it is empty, the expression s or 'foo' yields the desired value. Because :keyword:not has to invent a value anyway, it does not bother to return a value of the same type as its argument, so e.g., not 'foo' yields False, not ''.)

## Lambdas

Lambda forms (lambda expressions) have the same syntactic position as expressions. They are a shorthand to create anonymous functions; the expression lambda arguments: expression yields a function object. The unnamed object behaves like a function object defined with

def name(arguments):
return expression


See section :ref:function for the syntax of parameter lists. Note that functions created with lambda forms cannot contain statements or annotations.

## Expression lists

An expression list containing at least one comma yields a tuple. The length of the tuple is the number of expressions in the list. The expressions are evaluated from left to right.

The trailing comma is required only to create a single tuple (a.k.a. a singleton); it is optional in all other cases. A single expression without a trailing comma doesn't create a tuple, but rather yields the value of that expression. (To create an empty tuple, use an empty pair of parentheses: ().)

## Evaluation order

Python evaluates expressions from left to right. Notice that while evaluating an assignment, the right-hand side is evaluated before the left-hand side.

In the following lines, expressions will be evaluated in the arithmetic order of their suffixes:

expr1, expr2, expr3, expr4
(expr1, expr2, expr3, expr4)
{expr1: expr2, expr3: expr4}
expr1 + expr2 * (expr3 - expr4)
func(expr1, expr2, *expr3, **expr4)
expr3, expr4 = expr1, expr2


## Summary

The following table summarizes the operator precedences in Python, from lowest precedence (least binding) to highest precedence (most binding). Operators in the same box have the same precedence. Unless the syntax is explicitly given, operators are binary. Operators in the same box group left to right (except for comparisons, including tests, which all have the same precedence and chain from left to right --- see section :ref:comparisons -- and exponentiation, which groups from right to left).

Operator Description
:keyword:lambda Lambda expression
:keyword:or Boolean OR
:keyword:and Boolean AND
:keyword:not x Boolean NOT
:keyword:in, :keyword:not :keyword:in Membership tests
:keyword:is, :keyword:is not Identity tests
<, <=, >, >=, !=, == Comparisons
| Bitwise OR
^ Bitwise XOR
& Bitwise AND
<<, >> Shifts
*, /, % Multiplication, division, remainder
+x, -x Positive, negative
~x Bitwise not
** Exponentiation
x.attribute Attribute reference
x[index] Subscription
x[index:index] Slicing
f(arguments...) Function call
(expressions...) Binding or tuple display
[expressions...] List display
{key:datum...} Dictionary display

Footnotes

 [1] In Python 2.3, a list comprehension "leaks" the control variables of each for it contains into the containing scope. However, this behavior is deprecated, and relying on it will not work once this bug is fixed in a future release
 [2] While abs(x%y) < abs(y) is true mathematically, for floats it may not be true numerically due to roundoff. For example, and assuming a platform on which a Python float is an IEEE 754 double-precision number, in order that -1e-100 % 1e100 have the same sign as 1e100, the computed result is -1e-100 + 1e100, which is numerically exactly equal to 1e100. Function :func:fmod in the :mod:math module returns a result whose sign matches the sign of the first argument instead, and so returns -1e-100 in this case. Which approach is more appropriate depends on the application.
 [3] If x is very close to an exact integer multiple of y, it's possible for floor(x/y) to be one larger than (x-x%y)/y due to rounding. In such cases, Python returns the latter result, in order to preserve that divmod(x,y)[0] * y + x % y be very close to x.
 [4] The implementation computes this efficiently, without constructing lists or sorting.
 [5] Earlier versions of Python used lexicographic comparison of the sorted (key, value) lists, but this was very expensive for the common case of comparing for equality. An even earlier version of Python compared dictionaries by identity only, but this caused surprises because people expected to be able to test a dictionary for emptiness by comparing it to {}.