Abstract
The current division (/) operator has an ambiguous meaning for
numerical arguments: it returns the floor of the mathematical
result of division if the arguments are ints or longs, but it
returns a reasonable approximation of the division result if the
arguments are floats or complex. This makes expressions expecting
float or complex results error-prone when integers are not
expected but possible as inputs.
We propose to fix this by introducing different operators for
different operations: x/y to return a reasonable approximation of
the mathematical result of the division ("true division"), x//y to
return the floor ("floor division"). We call the current, mixed
meaning of x/y "classic division".
Because of severe backwards compatibility issues, not to mention a
major flamewar on c.l.py, we propose the following transitional
measures (starting with Python 2.2):
- Classic division will remain the default in the Python 2.x
series; true division will be standard in Python 3.0.
- The // operator will be available to request floor division
unambiguously.
- The future division statement, spelled "from __future__ import
division", will change the / operator to mean true division
throughout the module.
- A command line option will enable run-time warnings for classic
division applied to int or long arguments; another command line
option will make true division the default.
- The standard library will use the future division statement and
the // operator when appropriate, so as to completely avoid
classic division.
Motivation
The classic division operator makes it hard to write numerical
expressions that are supposed to give correct results from
arbitrary numerical inputs. For all other operators, one can
write down a formula such as x*y**2 + z, and the calculated result
will be close to the mathematical result (within the limits of
numerical accuracy, of course) for any numerical input type (int,
long, float, or complex). But division poses a problem: if the
expressions for both arguments happen to have an integral type, it
implements floor division rather than true division.
The problem is unique to dynamically typed languages: in a
statically typed language like C, the inputs, typically function
arguments, would be declared as double or float, and when a call
passes an integer argument, it is converted to double or float at
the time of the call. Python doesn't have argument type
declarations, so integer arguments can easily find their way into
an expression.
The problem is particularly pernicious since ints are perfect
substitutes for floats in all other circumstances: math.sqrt(2)
returns the same value as math.sqrt(2.0), 3.14*100 and 3.14*100.0
return the same value, and so on. Thus, the author of a numerical
routine may only use floating point numbers to test his code, and
believe that it works correctly, and a user may accidentally pass
in an integer input value and get incorrect results.
Another way to look at this is that classic division makes it
difficult to write polymorphic functions that work well with
either float or int arguments; all other operators already do the
right thing. No algorithm that works for both ints and floats has
a need for truncating division in one case and true division in
the other.
The correct work-around is subtle: casting an argument to float()
is wrong if it could be a complex number; adding 0.0 to an
argument doesn't preserve the sign of the argument if it was minus
zero. The only solution without either downside is multiplying an
argument (typically the first) by 1.0. This leaves the value and
sign unchanged for float and complex, and turns int and long into
a float with the corresponding value.
It is the opinion of the authors that this is a real design bug in
Python, and that it should be fixed sooner rather than later.
Assuming Python usage will continue to grow, the cost of leaving
this bug in the language will eventually outweigh the cost of
fixing old code -- there is an upper bound to the amount of code
to be fixed, but the amount of code that might be affected by the
bug in the future is unbounded.
Another reason for this change is the desire to ultimately unify
Python's numeric model. This is the subject of PEP 228[0] (which
is currently incomplete). A unified numeric model removes most of
the user's need to be aware of different numerical types. This is
good for beginners, but also takes away concerns about different
numeric behavior for advanced programmers. (Of course, it won't
remove concerns about numerical stability and accuracy.)
In a unified numeric model, the different types (int, long, float,
complex, and possibly others, such as a new rational type) serve
mostly as storage optimizations, and to some extent to indicate
orthogonal properties such as inexactness or complexity. In a
unified model, the integer 1 should be indistinguishable from the
floating point number 1.0 (except for its inexactness), and both
should behave the same in all numeric contexts. Clearly, in a
unified numeric model, if a==b and c==d, a/c should equal b/d
(taking some liberties due to rounding for inexact numbers), and
since everybody agrees that 1.0/2.0 equals 0.5, 1/2 should also
equal 0.5. Likewise, since 1//2 equals zero, 1.0//2.0 should also
equal zero.
Variations
Aesthetically, x//y doesn't please everyone, and hence several
variations have been proposed. They are addressed here:
- x div y. This would introduce a new keyword. Since div is a
popular identifier, this would break a fair amount of existing
code, unless the new keyword was only recognized under a future
division statement. Since it is expected that the majority of
code that needs to be converted is dividing integers, this would
greatly increase the need for the future division statement.
Even with a future statement, the general sentiment against
adding new keywords unless absolutely necessary argues against
this.
- div(x, y). This makes the conversion of old code much harder.
Replacing x/y with x//y or x div y can be done with a simple
query replace; in most cases the programmer can easily verify
that a particular module only works with integers so all
occurrences of x/y can be replaced. (The query replace is still
needed to weed out slashes occurring in comments or string
literals.) Replacing x/y with div(x, y) would require a much
more intelligent tool, since the extent of the expressions to
the left and right of the / must be analyzed before the
placement of the "div(" and ")" part can be decided.
- x \ y. The backslash is already a token, meaning line
continuation, and in general it suggests an "escape" to Unix
eyes. In addition (this due to Terry Reedy) this would make
things like eval("x\y") harder to get right.
Alternatives
In order to reduce the amount of old code that needs to be
converted, several alternative proposals have been put forth.
Here is a brief discussion of each proposal (or category of
proposals). If you know of an alternative that was discussed on
c.l.py that isn't mentioned here, please mail the second author.
- Let / keep its classic semantics; introduce // for true
division. This still leaves a broken operator in the language,
and invites to use the broken behavior. It also shuts off the
road to a unified numeric model a la PEP 228[0].
- Let int division return a special "portmanteau" type that
behaves as an integer in integer context, but like a float in a
float context. The problem with this is that after a few
operations, the int and the float value could be miles apart,
it's unclear which value should be used in comparisons, and of
course many contexts (like conversion to string) don't have a
clear integer or float preference.
- Use a directive to use specific division semantics in a module,
rather than a future statement. This retains classic division
as a permanent wart in the language, requiring future
generations of Python programmers to be aware of the problem and
the remedies.
- Use "from __past__ import division" to use classic division
semantics in a module. This also retains the classic division
as a permanent wart, or at least for a long time (eventually the
past division statement could raise an ImportError).
- Use a directive (or some other way) to specify the Python
version for which a specific piece of code was developed. This
requires future Python interpreters to be able to emulate
*exactly* several previous versions of Python, and moreover to
do so for multiple versions within the same interpreter. This
is way too much work. A much simpler solution is to keep
multiple interpreters installed. Another argument against this
is that the version directive is almost always overspecified:
most code written for Python X.Y, works for Python X.(Y-1) and
X.(Y+1) as well, so specifying X.Y as a version is more
constraining than it needs to be. At the same time, there's no
way to know at which future or past version the code will break.
API Changes
During the transitional phase, we have to support *three* division
operators within the same program: classic division (for / in
modules without a future division statement), true division (for /
in modules with a future division statement), and floor division
(for //). Each operator comes in two flavors: regular, and as an
augmented assignment operator (/= or //=).
The names associated with these variations are:
- Overloaded operator methods:
__div__(), __floordiv__(), __truediv__();
__idiv__(), __ifloordiv__(), __itruediv__().
- Abstract API C functions:
PyNumber_Divide(), PyNumber_FloorDivide(),
PyNumber_TrueDivide();
PyNumber_InPlaceDivide(), PyNumber_InPlaceFloorDivide(),
PyNumber_InPlaceTrueDivide().
- Byte code opcodes:
BINARY_DIVIDE, BINARY_FLOOR_DIVIDE, BINARY_TRUE_DIVIDE;
INPLACE_DIVIDE, INPLACE_FLOOR_DIVIDE, INPLACE_TRUE_DIVIDE.
- PyNumberMethod slots:
nb_divide, nb_floor_divide, nb_true_divide,
nb_inplace_divide, nb_inplace_floor_divide,
nb_inplace_true_divide.
The added PyNumberMethod slots require an additional flag in
tp_flags; this flag will be named Py_TPFLAGS_HAVE_NEWDIVIDE and
will be included in Py_TPFLAGS_DEFAULT.
The true and floor division APIs will look for the corresponding
slots and call that; when that slot is NULL, they will raise an
exception. There is no fallback to the classic divide slot.
In Python 3.0, the classic division semantics will be removed; the
classic division APIs will become synonymous with true division.
Command Line Option
The -Q command line option takes a string argument that can take
four values: "old", "warn", "warnall", or "new". The default is
"old" in Python 2.2 but will change to "warn" in later 2.x
versions. The "old" value means the classic division operator
acts as described. The "warn" value means the classic division
operator issues a warning (a DeprecationWarning using the standard
warning framework) when applied to ints or longs. The "warnall"
value also issues warnings for classic division when applied to
floats or complex; this is for use by the fixdiv.py conversion
script mentioned below. The "new" value changes the default
globally so that the / operator is always interpreted as true
division. The "new" option is only intended for use in certain
educational environments, where true division is required, but
asking the students to include the future division statement in
all their code would be a problem.
This option will not be supported in Python 3.0; Python 3.0 will
always interpret / as true division.
(This option was originally proposed as -D, but that turned out to
be an existing option for Jython, hence the Q -- mnemonic for
Quotient. Other names have been proposed, like -Qclassic,
-Qclassic-warn, -Qtrue, or -Qold_division etc.; these seem more
verbose to me without much advantage. After all the term classic
division is not used in the language at all (only in the PEP), and
the term true division is rarely used in the language -- only in
__truediv__.)
Semantics of Floor Division
Floor division will be implemented in all the Python numeric
types, and will have the semantics of
a // b == floor(a/b)
except that the result type will be the common type into which a
and b are coerced before the operation.
Specifically, if a and b are of the same type, a//b will be of
that type too. If the inputs are of different types, they are
first coerced to a common type using the same rules used for all
other arithmetic operators.
In particular, if a and b are both ints or longs, the result has
the same type and value as for classic division on these types
(including the case of mixed input types; int//long and long//int
will both return a long).
For floating point inputs, the result is a float. For example:
3.5//2.0 == 1.0
For complex numbers, // raises an exception, since floor() of a
complex number is not allowed.
For user-defined classes and extension types, all semantics are up
to the implementation of the class or type.
Semantics of True Division
True division for ints and longs will convert the arguments to
float and then apply a float division. That is, even 2/1 will
return a float (2.0), not an int. For floats and complex, it will
be the same as classic division.
The 2.2 implementation of true division acts as if the float type
had unbounded range, so that overflow doesn't occur unless the
magnitude of the mathematical *result* is too large to represent
as a float. For example, after "x = 1L << 40000", float(x) raises
OverflowError (note that this is also new in 2.2: previously the
outcome was platform-dependent, most commonly a float infinity). But
x/x returns 1.0 without exception, while x/1 raises OverflowError.
Note that for int and long arguments, true division may lose
information; this is in the nature of true division (as long as
rationals are not in the language). Algorithms that consciously
use longs should consider using //, as true division of longs
retains no more than 53 bits of precision (on most platforms).
If and when a rational type is added to Python (see PEP 239[2]),
true division for ints and longs should probably return a
rational. This avoids the problem with true division of ints and
longs losing information. But until then, for consistency, float is
the only choice for true division.
The Future Division Statement
If "from __future__ import division" is present in a module, or if
-Qnew is used, the / and /= operators are translated to true
division opcodes; otherwise they are translated to classic
division (until Python 3.0 comes along, where they are always
translated to true division).
The future division statement has no effect on the recognition or
translation of // and //=.
See PEP 236[4] for the general rules for future statements.
(It has been proposed to use a longer phrase, like "true_division"
or "modern_division". These don't seem to add much information.)
Open Issues
We expect that these issues will be resolved over time, as more
feedback is received or we gather more experience with the initial
implementation.
- It has been proposed to call // the quotient operator, and the /
operator the ratio operator. I'm not sure about this -- for
some people quotient is just a synonym for division, and ratio
suggests rational numbers, which is wrong. I prefer the
terminology to be slightly awkward if that avoids unambiguity.
Also, for some folks "quotient" suggests truncation towards
zero, not towards infinity as "floor division" says explicitly.
- It has been argued that a command line option to change the
default is evil. It can certainly be dangerous in the wrong
hands: for example, it would be impossible to combine a 3rd
party library package that requires -Qnew with another one that
requires -Qold. But I believe that the VPython folks need a way
to enable true division by default, and other educators might
need the same. These usually have enough control over the
library packages available in their environment.
- For classes to have to support all three of __div__(),
__floordiv__() and __truediv__() seems painful; and what to do
in 3.0? Maybe we only need __div__() and __floordiv__(), or
maybe at least true division should try __truediv__() first and
__div__() second.
Resolved Issues
- Issue: For very large long integers, the definition of true
division as returning a float causes problems, since the range of
Python longs is much larger than that of Python floats. This
problem will disappear if and when rational numbers are supported.
Resolution: For long true division, Python uses an internal
float type with native double precision but unbounded range, so
that OverflowError doesn't occur unless the quotient is too large
to represent as a native double.
- Issue: In the interim, maybe the long-to-float conversion could be
made to raise OverflowError if the long is out of range.
Resolution: This has been implemented, but, as above, the
magnitude of the inputs to long true division doesn't matter; only
the magnitude of the quotient matters.
- Issue: Tim Peters will make sure that whenever an in-range float
is returned, decent precision is guaranteed.
Resolution: Provided the quotient of long true division is
representable as a float, it suffers no more than 3 rounding
errors: one each for converting the inputs to an internal float
type with native double precision but unbounded range, and
one more for the division. However, note that if the magnitude
of the quotient is too *small* to represent as a native double,
0.0 is returned without exception ("silent underflow").
FAQ
Q. When will Python 3.0 be released?
A. We don't plan that long ahead, so we can't say for sure. We
want to allow at least two years for the transition. If Python
3.0 comes out sooner, we'll keep the 2.x line alive for
backwards compatibility until at least two years from the
release of Python 2.2. In practice, you will be able to
continue to use the Python 2.x line for several years after
Python 3.0 is released, so you can take your time with the
transition. Sites are expected to have both Python 2.x and
Python 3.x installed simultaneously.
Q. Why isn't true division called float division?
A. Because I want to keep the door open to *possibly* introducing
rationals and making 1/2 return a rational rather than a
float. See PEP 239[2].
Q. Why is there a need for __truediv__ and __itruediv__?
A. We don't want to make user-defined classes second-class
citizens. Certainly not with the type/class unification going
on.
Q. How do I write code that works under the classic rules as well
as under the new rules without using // or a future division
statement?
A. Use x*1.0/y for true division, divmod(x, y)[0] for int
division. Especially the latter is best hidden inside a
function. You may also write float(x)/y for true division if
you are sure that you don't expect complex numbers. If you
know your integers are never negative, you can use int(x/y) --
while the documentation of int() says that int() can round or
truncate depending on the C implementation, we know of no C
implementation that doesn't truncate, and we're going to change
the spec for int() to promise truncation. Note that classic
division (and floor division) round towards negative infinity,
while int() rounds towards zero, giving different answers for
negative numbers.
Q. How do I specify the division semantics for input(), compile(),
execfile(), eval() and exec?
A. They inherit the choice from the invoking module. PEP 236[4]
now lists this as a resolved problem, referring to PEP 264[5].
Q. What about code compiled by the codeop module?
A. This is dealt with properly; see PEP 264[5].
Q. Will there be conversion tools or aids?
A. Certainly. While these are outside the scope of the PEP, I
should point out two simple tools that will be released with
Python 2.2a3: Tools/scripts/finddiv.py finds division operators
(slightly smarter than "grep /") and Tools/scripts/fixdiv.py
can produce patches based on run-time analysis.
Q. Why is my question not answered here?
A. Because we weren't aware of it. If it's been discussed on
c.l.py and you believe the answer is of general interest,
please notify the second author. (We don't have the time or
inclination to answer every question sent in private email,
hence the requirement that it be discussed on c.l.py first.)
Implementation
Essentially everything mentioned here is implemented in CVS and
will be released with Python 2.2a3; most of it was already
released with Python 2.2a2.
References
[0] PEP 228, Reworking Python's Numeric Model
http://www.python.org/dev/peps/pep-0228/
[1] PEP 237, Unifying Long Integers and Integers, Zadka,
http://www.python.org/dev/peps/pep-0237/
[2] PEP 239, Adding a Rational Type to Python, Zadka,
http://www.python.org/dev/peps/pep-0239/
[3] PEP 240, Adding a Rational Literal to Python, Zadka,
http://www.python.org/dev/peps/pep-0240/
[4] PEP 236, Back to the __future__, Peters,
http://www.python.org/dev/peps/pep-0236/
[5] PEP 264, Future statements in simulated shells
http://www.python.org/dev/peps/pep-0236/
Copyright
This document has been placed in the public domain.