.. _man-integers-and-floating-point-numbers:

*************************************
 Integers and Floating-Point Numbers  
*************************************

Integers and floating-point values are the basic building blocks of
arithmetic and computation. Built-in representations of such values are
called numeric primitives, while representations of integers and
floating-point numbers as immediate values in code are known as numeric
literals. For example, ``1`` is an integer literal, while ``1.0`` is a
floating-point literal; their binary in-memory representations as
objects are numeric primitives.

Julia provides a broad range of primitive numeric types, and a full complement
of arithmetic and bitwise operators as well as standard mathematical functions
are defined over them. These map directly onto numeric types and operations
that are natively supported on modern computers, thus allowing Julia to take
full advantage of computational resources. Additionally, Julia provides
software support for :ref:`man-arbitrary-precision-arithmetic`, which can
handle operations on numeric values that cannot be represented effectively in
native hardware representations, but at the cost of relatively slower
performance.

The following are Julia's primitive numeric types:

-  **Integer types:**

===========  =======  ==============  ============== ==================
Type         Signed?  Number of bits  Smallest value Largest value
-----------  -------  --------------  -------------- ------------------
``Int8``           ✓       8            -2^7             2^7 - 1
``Uint8``                  8             0               2^8 - 1
``Int16``          ✓       16           -2^15            2^15 - 1
``Uint16``                 16            0               2^16 - 1
``Int32``          ✓       32           -2^31            2^31 - 1
``Uint32``                 32            0               2^32 - 1
``Int64``          ✓       64           -2^63            2^63 - 1
``Uint64``                 64            0               2^64 - 1
``Int128``         ✓       128           -2^127          2^127 - 1
``Uint128``                128           0               2^128 - 1
``Bool``         N/A       8           ``false`` (0)  ``true`` (1)
``Char``         N/A       32          ``'\0'``       ``'\Uffffffff'``
===========  =======  ==============  ============== ==================

``Char`` natively supports representation of
`Unicode characters <http://en.wikipedia.org/wiki/Unicode>`_; see
:ref:`man-strings` for more details.

-  **Floating-point types:**

=========== ========= ==============
Type        Precision Number of bits
----------- --------- --------------
``Float16`` half_          16
``Float32`` single_        32
``Float64`` double_        64
=========== ========= ==============

.. _half: http://en.wikipedia.org/wiki/Half-precision_floating-point_format
.. _single: http://en.wikipedia.org/wiki/Single_precision_floating-point_format
.. _double: http://en.wikipedia.org/wiki/Double_precision_floating-point_format

Additionally, full support for :ref:`man-complex-and-rational-numbers` is built
on top of these primitive numeric types. All numeric types interoperate
naturally without explicit casting, thanks to a flexible, user-extensible
:ref:`type promotion system <man-conversion-and-promotion>`.

Integers
--------

Literal integers are represented in the standard manner:

.. doctest::

    julia> 1
    1

    julia> 1234
    1234

The default type for an integer literal depends on whether the target
system has a 32-bit architecture or a 64-bit architecture::

    # 32-bit system:
    julia> typeof(1)
    Int32

    # 64-bit system:
    julia> typeof(1)
    Int64

The Julia internal variable ``WORD_SIZE`` indicates whether the target system
is 32-bit or 64-bit.::

    # 32-bit system:
    julia> WORD_SIZE
    32

    # 64-bit system:
    julia> WORD_SIZE
    64
 
Julia also defines the types ``Int`` and ``Uint``, which are aliases for the
system's signed and unsigned native integer types respectively.::

    # 32-bit system:
    julia> Int
    Int32
    julia> Uint
    Uint32


    # 64-bit system:
    julia> Int
    Int64
    julia> Uint
    Uint64

Larger integer literals that cannot be represented using only 32 bits
but can be represented in 64 bits always create 64-bit integers,
regardless of the system type::

    # 32-bit or 64-bit system:
    julia> typeof(3000000000)
    Int64

Unsigned integers are input and output using the ``0x`` prefix and hexadecimal
(base 16) digits ``0-9a-f`` (the capitalized digits ``A-F`` also work for input).
The size of the unsigned value is determined by the number of hex digits used:

.. doctest::

    julia> 0x1
    0x01

    julia> typeof(ans)
    Uint8

    julia> 0x123
    0x0123

    julia> typeof(ans)
    Uint16

    julia> 0x1234567
    0x01234567

    julia> typeof(ans)
    Uint32

    julia> 0x123456789abcdef
    0x0123456789abcdef

    julia> typeof(ans)
    Uint64

This behavior is based on the observation that when one uses unsigned
hex literals for integer values, one typically is using them to
represent a fixed numeric byte sequence, rather than just an integer
value.

Recall that the variable ``ans`` is set to the value of the last expression
evaluated in an interactive session. This does not occur when Julia code is
run in other ways.

Binary and octal literals are also supported:

.. doctest::

    julia> 0b10
    0x02

    julia> typeof(ans)
    Uint8

    julia> 0o10
    0x08

    julia> typeof(ans)
    Uint8

The minimum and maximum representable values of primitive numeric types
such as integers are given by the ``typemin`` and ``typemax`` functions:

.. doctest::

    julia> (typemin(Int32), typemax(Int32))
    (-2147483648,2147483647)

    julia> for T = {Int8,Int16,Int32,Int64,Int128,Uint8,Uint16,Uint32,Uint64,Uint128}
             println("$(lpad(T,7)): [$(typemin(T)),$(typemax(T))]")
           end
       Int8: [-128,127]
      Int16: [-32768,32767]
      Int32: [-2147483648,2147483647]
      Int64: [-9223372036854775808,9223372036854775807]
     Int128: [-170141183460469231731687303715884105728,170141183460469231731687303715884105727]
      Uint8: [0,255]
     Uint16: [0,65535]
     Uint32: [0,4294967295]
     Uint64: [0,18446744073709551615]
    Uint128: [0,340282366920938463463374607431768211455]

The values returned by ``typemin`` and ``typemax`` are always of the
given argument type. (The above expression uses several features we have
yet to introduce, including :ref:`for loops <man-loops>`,
:ref:`man-strings`, and :ref:`man-string-interpolation`,
but should be easy enough to understand for users with some existing
programming experience.)


Overflow behavior
~~~~~~~~~~~~~~~~~

In Julia, exceeding the maximum representable value of a given type results in
a wraparound behavior:

.. doctest::

    julia> x = typemax(Int64)
    9223372036854775807
    
    julia> x + 1
    -9223372036854775808

    julia> x + 1 == typemin(Int64)
    true

Thus, arithmetic with Julia integers is actually a form of `modular arithmetic
<http://en.wikipedia.org/wiki/Modular_arithmetic>`_. This reflects the
characteristics of the underlying arithmetic of integers as implemented on
modern computers. In applications where overflow is possible, explicit checking
for wraparound produced by overflow is essential; otherwise, the ``BigInt`` type
in :ref:`man-arbitrary-precision-arithmetic` is recommended instead.

To minimize the practical impact of this overflow, integer addition,
subtraction, multiplication, and exponentiation operands are promoted
to ``Int`` or ``Uint`` from narrower integer types.  (However,
divisions, remainders, and bitwise operations do not promote narrower
types.)

Division errors
~~~~~~~~~~~~~~~

Integer division (the ``div`` function) has two exceptional cases: dividing by
zero, and dividing the lowest negative number (``typemin``) by -1. Both of
these cases throw a ``DivideError``. The remainder and modulus functions
(``rem`` and ``mod``) throw a ``DivideError`` when their second argument is
zero.

Floating-Point Numbers
----------------------

Literal floating-point numbers are represented in the standard formats:

.. doctest::

    julia> 1.0
    1.0

    julia> 1.
    1.0

    julia> 0.5
    0.5

    julia> .5
    0.5

    julia> -1.23
    -1.23

    julia> 1e10
    1.0e10

    julia> 2.5e-4
    0.00025

The above results are all ``Float64`` values. Literal ``Float32`` values can
be entered by writing an ``f`` in place of ``e``:

.. doctest::

    julia> 0.5f0
    0.5f0

    julia> typeof(ans)
    Float32

    julia> 2.5f-4
    0.00025f0

Values can be converted to ``Float32`` easily:

.. doctest::

    julia> float32(-1.5)
    -1.5f0

    julia> typeof(ans)
    Float32

Hexadecimal floating-point literals are also valid, but only as ``Float64`` values:

.. doctest::

    julia> 0x1p0
    1.0

    julia> 0x1.8p3
    12.0

    julia> 0x.4p-1
    0.125

    julia> typeof(ans)
    Float64

Half-precision floating-point numbers are also supported (``Float16``), but
only as a storage format. In calculations they'll be converted to ``Float32``:

.. doctest::

    julia> sizeof(float16(4.))
    2

    julia> 2*float16(4.)
    8.0f0


Floating-point zero
~~~~~~~~~~~~~~~~~~~

Floating-point numbers have `two zeros
<http://en.wikipedia.org/wiki/Signed_zero>`_, positive zero and negative zero.
They are equal to each other but have different binary representations, as can
be seen using the ``bits`` function: :

.. doctest::

    julia> 0.0 == -0.0
    true
    
    julia> bits(0.0)
    "0000000000000000000000000000000000000000000000000000000000000000"
    
    julia> bits(-0.0)
    "1000000000000000000000000000000000000000000000000000000000000000"

.. _man-special-floats:

Special floating-point values
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There are three specified standard floating-point values that do not
correspond to any point on the real number line:

=========== =========== ===========  ================= =================================================================
Special value                        Name              Description 
-----------------------------------  ----------------- -----------------------------------------------------------------
``Float16`` ``Float32`` ``Float64``
=========== =========== ===========  ================= =================================================================
``Inf16``   ``Inf32``    ``Inf``     positive infinity a value greater than all finite floating-point values
``-Inf16``  ``-Inf32``   ``-Inf``    negative infinity a value less than all finite floating-point values
``NaN16``   ``NaN32``    ``NaN``     not a number      a value not ``==`` to any floating-point value (including itself)
=========== =========== ===========  ================= =================================================================

For further discussion of how these non-finite floating-point values are
ordered with respect to each other and other floats, see
:ref:`man-numeric-comparisons`. By the
`IEEE 754 standard <http://en.wikipedia.org/wiki/IEEE_754-2008>`_, these
floating-point values are the results of certain arithmetic operations:

.. doctest::

    julia> 1/Inf
    0.0

    julia> 1/0
    Inf

    julia> -5/0
    -Inf

    julia> 0.000001/0
    Inf

    julia> 0/0
    NaN

    julia> 500 + Inf
    Inf

    julia> 500 - Inf
    -Inf

    julia> Inf + Inf
    Inf

    julia> Inf - Inf
    NaN

    julia> Inf * Inf
    Inf

    julia> Inf / Inf
    NaN

    julia> 0 * Inf
    NaN

The ``typemin`` and ``typemax`` functions also apply to floating-point
types:

.. doctest::

    julia> (typemin(Float16),typemax(Float16))
    (-Inf16,Inf16)

    julia> (typemin(Float32),typemax(Float32))
    (-Inf32,Inf32)

    julia> (typemin(Float64),typemax(Float64))
    (-Inf,Inf)


Machine epsilon
~~~~~~~~~~~~~~~

Most real numbers cannot be represented exactly with floating-point numbers,
and so for many purposes it is important to know the distance between two
adjacent representable floating-point numbers, which is often known as
`machine epsilon <http://en.wikipedia.org/wiki/Machine_epsilon>`_.

Julia provides the ``eps`` function, which gives the distance between ``1.0``
and the next larger representable floating-point value:

.. doctest::

    julia> eps(Float32)
    1.1920929f-7

    julia> eps(Float64)
    2.220446049250313e-16

    julia> eps() # same as eps(Float64)
    2.220446049250313e-16

These values are ``2.0^-23`` and ``2.0^-52`` as ``Float32`` and ``Float64``
values, respectively. The ``eps`` function can also take a
floating-point value as an argument, and gives the absolute difference
between that value and the next representable floating point value. That
is, ``eps(x)`` yields a value of the same type as ``x`` such that
``x + eps(x)`` is the next representable floating-point value larger
than ``x``:

.. doctest::

    julia> eps(1.0)
    2.220446049250313e-16

    julia> eps(1000.)
    1.1368683772161603e-13

    julia> eps(1e-27)
    1.793662034335766e-43

    julia> eps(0.0)
    5.0e-324

The distance between two adjacent representable floating-point numbers is not
constant, but is smaller for smaller values and larger for larger values. In
other words, the representable floating-point numbers are densest in the real
number line near zero, and grow sparser exponentially as one moves farther away
from zero. By definition, ``eps(1.0)`` is the same as ``eps(Float64)`` since
``1.0`` is a 64-bit floating-point value.

Julia also provides the ``nextfloat`` and ``prevfloat`` functions which return
the next largest or smallest representable floating-point number to the
argument respectively: :

.. doctest::

    julia> x = 1.25f0
    1.25f0
    
    julia> nextfloat(x)
    1.2500001f0
    
    julia> prevfloat(x)
    1.2499999f0
    
    julia> bits(prevfloat(x))
    "00111111100111111111111111111111"
    
    julia> bits(x)
    "00111111101000000000000000000000"
    
    julia> bits(nextfloat(x))
    "00111111101000000000000000000001"

This example highlights the general principle that the adjacent representable
floating-point numbers also have adjacent binary integer representations.

Rounding modes
~~~~~~~~~~~~~~

If a number doesn't have an exact floating-point representation, it must be
rounded to an appropriate representable value, however, if wanted, the manner
in which this rounding is done can be changed according to the rounding modes
presented in the `IEEE 754 standard <http://en.wikipedia.org/wiki/IEEE_754-2008>`_::
    

    julia> 1.1 + 0.1
    1.2000000000000002

    julia> with_rounding(Float64,RoundDown) do
           1.1 + 0.1
           end
    1.2

The default mode used is always ``RoundNearest``, which rounds to the nearest
representable value, with ties rounded towards the nearest value with an even
least significant bit.

Background and References
~~~~~~~~~~~~~~~~~~~~~~~~~

Floating-point arithmetic entails many subtleties which can be surprising to
users who are unfamiliar with the low-level implementation details. However,
these subtleties are described in detail in most books on scientific
computation, and also in the following references:

- The definitive guide to floating point arithmetic is the `IEEE 754-2008
  Standard <http://standards.ieee.org/findstds/standard/754-2008.html>`_;
  however, it is not available for free online.
- For a brief but lucid presentation of how floating-point numbers are
  represented, see John D. Cook's `article
  <http://www.johndcook.com/blog/2009/04/06/anatomy-of-a-floating-point-number/>`_
  on the subject as well as his `introduction
  <http://www.johndcook.com/blog/2009/04/06/numbers-are-a-leaky-abstraction/>`_
  to some of the issues arising from how this representation differs in
  behavior from the idealized abstraction of real numbers.
- Also recommended is Bruce Dawson's `series of blog posts on floating-point
  numbers <http://randomascii.wordpress.com/2012/05/20/thats-not-normalthe-performance-of-odd-floats/>`_.
- For an excellent, in-depth discussion of floating-point numbers and issues of
  numerical accuracy encountered when computing with them, see David Goldberg's
  paper `What Every Computer Scientist Should Know About Floating-Point
  Arithmetic
  <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.244&rep=rep1&type=pdf>`_.
- For even more extensive documentation of the history of, rationale for,
  and issues with floating-point numbers, as well as discussion of many other
  topics in numerical computing, see the `collected writings
  <http://www.cs.berkeley.edu/~wkahan/>`_ of `William Kahan
  <http://en.wikipedia.org/wiki/William_Kahan>`_, commonly known as the "Father
  of Floating-Point". Of particular interest may be `An Interview with the Old
  Man of Floating-Point
  <http://www.cs.berkeley.edu/~wkahan/ieee754status/754story.html>`_.

.. _man-arbitrary-precision-arithmetic:

Arbitrary Precision Arithmetic
------------------------------

To allow computations with arbitrary-precision integers and floating point numbers, 
Julia wraps the `GNU Multiple Precision Arithmetic Library (GMP) <https://gmplib.org>`_ and the `GNU MPFR Library <http://www.mpfr.org>`_, respectively. 
The `BigInt` and `BigFloat` types are available in Julia for arbitrary precision 
integer and floating point numbers respectively. 

Constructors exist to create these types from primitive numerical types, or from ``String``. 
Once created, they participate in arithmetic with all other numeric types thanks to Julia's 
:ref:`type promotion and conversion mechanism <man-conversion-and-promotion>`. :

.. doctest::

    julia> BigInt(typemax(Int64)) + 1
    9223372036854775808

    julia> BigInt("123456789012345678901234567890") + 1
    123456789012345678901234567891

    julia> BigFloat("1.23456789012345678901")
    1.234567890123456789010000000000000000000000000000000000000000000000000000000004e+00 with 256 bits of precision

    julia> BigFloat(2.0^66) / 3
    2.459565876494606882133333333333333333333333333333333333333333333333333333333344e+19 with 256 bits of precision

    julia> factorial(BigInt(40))
    815915283247897734345611269596115894272000000000

However, type promotion between the primitive types above and
`BigInt`/`BigFloat` is not automatic and must be explicitly stated. :

.. doctest::

    julia> x = typemin(Int64)
    -9223372036854775808
    
    julia> x = x - 1
    9223372036854775807
    
    julia> typeof(x)
    Int64

    julia> y = BigInt(typemin(Int64))
    -9223372036854775808
    
    julia> y = y - 1
    -9223372036854775809
    
    julia> typeof(y)
    BigInt (constructor with 10 methods)

The default precision (in number of bits of the significand) and rounding
mode of `BigFloat` operations can be changed, and all further calculations 
will take these changes in account:

.. doctest::

    julia> with_rounding(BigFloat,RoundUp) do
           BigFloat(1) + BigFloat("0.1")
           end
    1.100000000000000000000000000000000000000000000000000000000000000000000000000003e+00 with 256 bits of precision

    julia> with_rounding(BigFloat,RoundDown) do
           BigFloat(1) + BigFloat("0.1")
           end
    1.099999999999999999999999999999999999999999999999999999999999999999999999999986e+00 with 256 bits of precision

    julia> with_bigfloat_precision(40) do
           BigFloat(1) + BigFloat("0.1")
           end
    1.1000000000004e+00 with 40 bits of precision


.. _man-numeric-literal-coefficients:

Numeric Literal Coefficients
----------------------------

To make common numeric formulas and expressions clearer, Julia allows
variables to be immediately preceded by a numeric literal, implying
multiplication. This makes writing polynomial expressions much cleaner:

.. doctest::

    julia> x = 3
    3

    julia> 2x^2 - 3x + 1
    10

    julia> 1.5x^2 - .5x + 1
    13.0

It also makes writing exponential functions more elegant:

.. doctest::

    julia> 2^2x
    64

The precedence of numeric literal coefficients is the same as that of unary
operators such as negation. So ``2^3x`` is parsed as ``2^(3x)``, and
``2x^3`` is parsed as ``2*(x^3)``.

Numeric literals also work as coefficients to parenthesized
expressions:

.. doctest::

    julia> 2(x-1)^2 - 3(x-1) + 1
    3

Additionally, parenthesized expressions can be used as coefficients to
variables, implying multiplication of the expression by the variable:

.. doctest::

    julia> (x-1)x
    6

Neither juxtaposition of two parenthesized expressions, nor placing a
variable before a parenthesized expression, however, can be used to
imply multiplication:

.. doctest::

    julia> (x-1)(x+1)
    ERROR: type: apply: expected Function, got Int64

    julia> x(x+1)
    ERROR: type: apply: expected Function, got Int64

Both of these expressions are interpreted as function application: any
expression that is not a numeric literal, when immediately followed by a
parenthetical, is interpreted as a function applied to the values in
parentheses (see :ref:`man-functions` for more about functions).
Thus, in both of these cases, an error occurs since the left-hand value
is not a function.

The above syntactic enhancements significantly reduce the visual noise
incurred when writing common mathematical formulae. Note that no
whitespace may come between a numeric literal coefficient and the
identifier or parenthesized expression which it multiplies.

Syntax Conflicts
~~~~~~~~~~~~~~~~

Juxtaposed literal coefficient syntax may conflict with two numeric literal
syntaxes: hexadecimal integer literals and engineering notation for
floating-point literals. Here are some situations where syntactic
conflicts arise:

-  The hexadecimal integer literal expression ``0xff`` could be
   interpreted as the numeric literal ``0`` multiplied by the variable
   ``xff``.
-  The floating-point literal expression ``1e10`` could be interpreted
   as the numeric literal ``1`` multiplied by the variable ``e10``, and
   similarly with the equivalent ``E`` form.

In both cases, we resolve the ambiguity in favor of interpretation as a
numeric literals:

-  Expressions starting with ``0x`` are always hexadecimal literals.
-  Expressions starting with a numeric literal followed by ``e`` or
   ``E`` are always floating-point literals.

Literal zero and one
--------------------

Julia provides functions which return literal 0 and 1 corresponding to a
specified type or the type of a given variable.

===========  =====================================================
Function     Description
-----------  -----------------------------------------------------
``zero(x)``  Literal zero of type ``x`` or type of variable ``x``
``one(x)``   Literal one of type ``x`` or type of variable ``x``
===========  =====================================================

These functions are useful in :ref:`man-numeric-comparisons` to avoid overhead
from unnecessary :ref:`type conversion <man-conversion-and-promotion>`.

Examples:

.. doctest::

    julia> zero(Float32)
    0.0f0
    
    julia> zero(1.0)
    0.0

    julia> one(Int32)
    1

    julia> one(BigFloat)
    1e+00 with 256 bits of precision

