.. _man-integers-and-floating-point-numbers: .. currentmodule:: Base ************************************* 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 ================ ======= ============== ============== ================== :class:`Int8` ✓ 8 -2^7 2^7 - 1 :class:`UInt8` 8 0 2^8 - 1 :class:`Int16` ✓ 16 -2^15 2^15 - 1 :class:`UInt16` 16 0 2^16 - 1 :class:`Int32` ✓ 32 -2^31 2^31 - 1 :class:`UInt32` 32 0 2^32 - 1 :class:`Int64` ✓ 64 -2^63 2^63 - 1 :class:`UInt64` 64 0 2^64 - 1 :class:`Int128` ✓ 128 -2^127 2^127 - 1 :class:`UInt128` 128 0 2^128 - 1 :class:`Bool` N/A 8 ``false`` (0) ``true`` (1) ================ ======= ============== ============== ================== - **Floating-point types:** ================ ========= ============== Type Precision Number of bits ================ ========= ============== :class:`Float16` half_ 16 :class:`Float32` single_ 32 :class:`Float64` double_ 64 ================ ========= ============== .. _half: https://en.wikipedia.org/wiki/Half-precision_floating-point_format .. _single: https://en.wikipedia.org/wiki/Single_precision_floating-point_format .. _double: https://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 `. 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 :const:`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 :class:`Int` and :class:`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 :data:`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 :func:`typemin` and :func:`typemax` functions: .. doctest:: julia> (typemin(Int32), typemax(Int32)) (-2147483648,2147483647) julia> for T in [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 :func:`typemin` and :func:`typemax` are always of the given argument type. (The above expression uses several features we have yet to introduce, including :ref:`for 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 `_. 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. Division errors ~~~~~~~~~~~~~~~ Integer division (the ``div`` function) has two exceptional cases: dividing by zero, and dividing the lowest negative number (:func:`typemin`) by -1. Both of these cases throw a :exc:`DivideError`. The remainder and modulus functions (``rem`` and ``mod``) throw a :exc:`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 The underscore ``_`` can be used as digit separator: .. doctest:: julia> 10_000, 0.000_000_005, 0xdead_beef, 0b1011_0010 (10000,5.0e-9,0xdeadbeef,0xb2) Floating-point zero ~~~~~~~~~~~~~~~~~~~ Floating-point numbers have `two zeros `_, 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 `_, 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 :func:`typemin` and :func:`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 `_. Julia provides :func:`eps`, 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 :func:`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 :func:`nextfloat` and :func:`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 `_:: 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 :const:`RoundNearest`, which rounds to the nearest representable value, with ties rounded towards the nearest value with an even least significant bit. .. warning:: Rounding is generally only correct for basic arithmetic functions (:func:`+`, :func:`-`, :func:`*`, :func:`/` and :func:`sqrt`) and type conversion operations. Many other functions assume the default :const:`RoundNearest` mode is set, and can give erroneous results when operating under other rounding modes. 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 `_; 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 `_ on the subject as well as his `introduction `_ 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 `_. - 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 `_. - 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 `_ of `William Kahan `_, commonly known as the "Father of Floating-Point". Of particular interest may be `An Interview with the Old Man of Floating-Point `_. .. _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) `_ and the `GNU MPFR Library `_, respectively. The :class:`BigInt` and :class:`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, and :func:`parse` can be use to construct them from :class:`AbstractString`\ s. Once created, they participate in arithmetic with all other numeric types thanks to Julia's :ref:`type promotion and conversion mechanism `: .. doctest:: julia> BigInt(typemax(Int64)) + 1 9223372036854775808 julia> parse(BigInt, "123456789012345678901234567890") + 1 123456789012345678901234567891 julia> parse(BigFloat, "1.23456789012345678901") 1.234567890123456789010000000000000000000000000000000000000000000000000000000004 julia> BigFloat(2.0^66) / 3 2.459565876494606882133333333333333333333333333333333333333333333333333333333344e+19 julia> factorial(BigInt(40)) 815915283247897734345611269596115894272000000000 However, type promotion between the primitive types above and :class:`BigInt`/:class:`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 The default precision (in number of bits of the significand) and rounding mode of :class:`BigFloat` operations can be changed globally by calling :func:`set_bigfloat_precision` and :func:`set_rounding`, and all further calculations will take these changes in account. Alternatively, the precision or the rounding can be changed only within the execution of a particular block of code by :func:`with_bigfloat_precision` or :func:`with_rounding`: .. doctest:: julia> with_rounding(BigFloat,RoundUp) do BigFloat(1) + parse(BigFloat, "0.1") end 1.100000000000000000000000000000000000000000000000000000000000000000000000000003 julia> with_rounding(BigFloat,RoundDown) do BigFloat(1) + parse(BigFloat, "0.1") end 1.099999999999999999999999999999999999999999999999999999999999999999999999999986 julia> with_bigfloat_precision(40) do BigFloat(1) + parse(BigFloat, "0.1") end 1.1000000000004 .. _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: MethodError: `call` has no method matching call(::Int64, ::Int64) Closest candidates are: BoundsError() BoundsError(!Matched::Any...) DivideError() ... julia> x(x+1) ERROR: MethodError: `call` has no method matching call(::Int64, ::Int64) Closest candidates are: BoundsError() BoundsError(!Matched::Any...) DivideError() ... Both 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 ====================== ===================================================== :func:`zero(x) ` Literal zero of type ``x`` or type of variable ``x`` :func:`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 `. Examples: .. doctest:: julia> zero(Float32) 0.0f0 julia> zero(1.0) 0.0 julia> one(Int32) 1 julia> one(BigFloat) 1.000000000000000000000000000000000000000000000000000000000000000000000000000000