Mathematics

Unary minus operator.

Examples

julia> -1
-1
julia> -(2)
-2
julia> -[1 2; 3 4]
2×2 Array{Int64,2}:
 -1  -2
 -3  -4

source

— Function.

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25
julia> +(1, 20, 4)
25

source

dt::Date + t::Time -> DateTime

The addition of a Date with a Time produces a DateTime. The hour, minute, second, and millisecond parts of the Time are used along with the year, month, and day of the Date to create the new DateTime. Non-zero microseconds or nanoseconds in the Time type will result in an InexactError being thrown.

Base.:- — Method.

-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1
julia> -(2, 4.5)
-2.5

Base.:* — Method.

*(x, y...)

Multiplication operator. x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...).

Examples

julia> 2 * 7 * 8
112
julia> *(2, 7, 8)
112

Base.:/ — Function.

/(x, y)

Right division operator: multiplication of x by the inverse of y on the right. Gives floating-point results for integer arguments.

Examples

julia> 1/2
0.5
julia> 4/2
2.0
julia> 4.5/2
2.25

Base.:\ — Method.

\(x, y)

Left division operator: multiplication of y by the inverse of x on the left. Gives floating-point results for integer arguments.

Examples

julia> 3 \ 6
2.0
julia> inv(3) * 6
2.0
julia> A = [4 3; 2 1]; x = [5, 6];
julia> A \ x
2-element Array{Float64,1}:
  6.5
 -7.0
julia> inv(A) * x
2-element Array{Float64,1}:
  6.5
 -7.0

Base.:^ — Method.

^(x, y)

Exponentiation operator. If x is a matrix, computes matrix exponentiation.

If y is an Int literal (e.g. 2 in x^2 or -3 in x^-3), the Julia code x^y is transformed by the compiler to Base.literal_pow(^, x, Val(y)), to enable compile-time specialization on the value of the exponent. (As a default fallback we have Base.literal_pow(^, x, Val(y)) = ^(x,y), where usually ^ == Base.^ unless ^ has been defined in the calling namespace.)

julia> 3^5
243
julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4
julia> A^3
2×2 Array{Int64,2}:
 37   54
 81  118

Base.fma — Function.

Computes x*y+z without rounding the intermediate result x*y. On some systems this is significantly more expensive than x*y+z. fma is used to improve accuracy in certain algorithms. See .

source

— Function.

muladd(x, y, z)

Combined multiply-add: computes x*y+z, but allowing the add and multiply to be merged with each other or with surrounding operations for performance. For example, this may be implemented as an fma if the hardware supports it efficiently. The result can be different on different machines and can also be different on the same machine due to constant propagation or other optimizations. See .

Examples

julia> muladd(3, 2, 1)
7
julia> 3 * 2 + 1
7

source

— Method.

inv(x)

Return the multiplicative inverse of x, such that x*inv(x) or inv(x)*x yields one(x) (the multiplicative identity) up to roundoff errors.

If x is a number, this is essentially the same as one(x)/x, but for some types inv(x) may be slightly more efficient.

Examples

julia> inv(2)
0.5
julia> inv(1 + 2im)
0.2 - 0.4im
julia> inv(1 + 2im) * (1 + 2im)
1.0 + 0.0im
julia> inv(2//3)
3//2

Julia 1.2

inv(::Missing) requires at least Julia 1.2.

Base.div — Function.

div(x, y)
÷(x, y)

The quotient from Euclidean division. Computes x/y, truncated to an integer.

Examples

julia> 9 ÷ 4
2
julia> -5 ÷ 3
-1

Base.fld — Function.

fld(x, y)

Largest integer less than or equal to x/y.

Examples

julia> fld(7.3,5.5)
1.0

Base.cld — Function.

cld(x, y)

Smallest integer larger than or equal to x/y.

Examples

julia> cld(5.5,2.2)
3.0

Base.mod — Function.

mod(x::Integer, r::AbstractUnitRange)

Find y in the range r such that $x ≡ y (mod n)$, where n = length(r), i.e. y = mod(x - first(r), n) + first(r).

See also >>, .

source

>>>(B::BitVector, n) -> BitVector

Unsigned right bitshift operator, B >>> n. Equivalent to B >> n. See for details and examples.

source

— Function.

Range operator. a:b constructs a range from a to b with a step size of 1 (a UnitRange) , and a:s:b is similar but uses a step size of s (a ).

: is also used in indexing to select whole dimensions and for Symbol literals, as in e.g. :hello.

(:)(I::CartesianIndex, J::CartesianIndex)

Construct CartesianIndices from two CartesianIndex.

Julia 1.1

This method requires at least Julia 1.1.

Examples

julia> I = CartesianIndex(2,1);

julia> J = CartesianIndex(3,3);

julia> I:J
2×3 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
 CartesianIndex(2, 1)  CartesianIndex(2, 2)  CartesianIndex(2, 3)
 CartesianIndex(3, 1)  CartesianIndex(3, 2)  CartesianIndex(3, 3)

Base.range — Function.

range(start[, stop]; length, stop, step=1)

Given a starting value, construct a range either by length or from start to stop, optionally with a given step (defaults to 1, a ). One of length or stop is required. If length, stop, and step are all specified, they must agree.

If length and stop are provided and step is not, the step size will be computed automatically such that there are length linearly spaced elements in the range (a LinRange).

If step and stop are provided and length is not, the overall range length will be computed automatically such that the elements are step spaced (a ).

stop may be specified as either a positional or keyword argument.

Julia 1.1

stop as a positional argument requires at least Julia 1.1.

Examples

julia> range(1, length=100)
1:100

julia> range(1, stop=100)
1:100

julia> range(1, step=5, length=100)
1:5:496

julia> range(1, step=5, stop=100)
1:5:96

julia> range(1, 10, length=101)
1.0:0.09:10.0

julia> range(1, 100, step=5)
1:5:96

source

— Type.

Base.OneTo(n)

Define an AbstractUnitRange that behaves like 1:n, with the added distinction that the lower limit is guaranteed (by the type system) to be 1.

source

— Type.

StepRangeLen{T,R,S}(ref::R, step::S, len, [offset=1]) where {T,R,S}
StepRangeLen(       ref::R, step::S, len, [offset=1]) where {  R,S}

A range r where r[i] produces values of type T (in the second form, T is deduced automatically), parameterized by a reference value, a step, and the length. By default ref is the starting value r[1], but alternatively you can supply it as the value of r[offset] for some other index 1 <= offset <= len. In conjunction with TwicePrecision this can be used to implement ranges that are free of roundoff error.

source

— Function.

==(x, y)

Generic equality operator. Falls back to ===. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding. For collections, == is generally called recursively on all contents, though other properties (like the shape for arrays) may also be taken into account.

This operator follows IEEE semantics for floating-point numbers: 0.0 == -0.0 and NaN != NaN.

The result is of type Bool, except when one of the operands is , in which case missing is returned (three-valued logic). For collections, missing is returned if at least one of the operands contains a missing value and all non-missing values are equal. Use or === to always get a Bool result.

Implementation

New numeric types should implement this function for two arguments of the new type, and handle comparison to other types via promotion rules where possible.

falls back to ==, so new methods of == will be used by the Dict type to compare keys. If your type will be used as a dictionary key, it should therefore also implement .

source

==(x)

Create a function that compares its argument to x using , i.e. a function equivalent to y -> y == x.

The returned function is of type Base.Fix2{typeof(==)}, which can be used to implement specialized methods.

source

==(a::AbstractString, b::AbstractString) -> Bool

Test whether two strings are equal character by character (technically, Unicode code point by code point).

Examples

julia> "abc" == "abc"
true

julia> "abc" == "αβγ"
false

Base.:!= — Function.

!=(x, y)
≠(x,y)

Not-equals comparison operator. Always gives the opposite answer as .

Implementation

New types should generally not implement this, and rely on the fallback definition !=(x,y) = !(x==y) instead.

Examples

julia> 3 != 2
true

julia> "foo" ≠ "foo"
false

source

!=(x)

Create a function that compares its argument to x using , i.e. a function equivalent to y -> y != x. The returned function is of type Base.Fix2{typeof(!=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source

— Function.

!==(x, y)
≢(x,y)

Always gives the opposite answer as ===.

Examples

julia> a = [1, 2]; b = [1, 2];

julia> a ≢ b
true

julia> a ≢ a
false

Base.:< — Function.

<(x, y)

Less-than comparison operator. Falls back to . Because of the behavior of floating-point NaN values, this operator implements a partial order.

Implementation

New numeric types with a canonical partial order should implement this function for two arguments of the new type. Types with a canonical total order should implement isless instead. (x < y) | (x == y)

Examples

julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false

<(x)

Create a function that compares its argument to x using <, i.e. a function equivalent to y -> y < x. The returned function is of type Base.Fix2{typeof(<)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

Base.:<= — Function.

<=(x, y)
≤(x,y)

Less-than-or-equals comparison operator. Falls back to (x < y) | (x == y).

Examples

julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false

<=(x)

Create a function that compares its argument to x using <=, i.e. a function equivalent to y -> y <= x. The returned function is of type Base.Fix2{typeof(<=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

Base.:> — Function.

>(x, y)

Greater-than comparison operator. Falls back to y < x.

Implementation

Generally, new types should implement instead of this function, and rely on the fallback definition >(x, y) = y < x.

Examples

julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true

source

>(x)

Create a function that compares its argument to x using , i.e. a function equivalent to y -> y > x. The returned function is of type Base.Fix2{typeof(>)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source

— Function.

>=(x, y)
≥(x,y)

Greater-than-or-equals comparison operator. Falls back to y <= x.

Examples

julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true

source

>=(x)

Create a function that compares its argument to x using , i.e. a function equivalent to y -> y >= x. The returned function is of type Base.Fix2{typeof(>=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source

— Function.

cmp(x,y)

Return -1, 0, or 1 depending on whether x is less than, equal to, or greater than y, respectively. Uses the total order implemented by isless.

Examples

julia> cmp(1, 2)
-1

julia> cmp(2, 1)
1

julia> cmp(2+im, 3-im)
ERROR: MethodError: no method matching isless(::Complex{Int64}, ::Complex{Int64})
[...]

source

Return -1, 0, or 1 depending on whether x is less than, equal to, or greater than y, respectively. The first argument specifies a less-than comparison function to use.

cmp(a::AbstractString, b::AbstractString) -> Int

Compare two strings. Return 0 if both strings have the same length and the character at each index is the same in both strings. Return -1 if a is a prefix of b, or if a comes before b in alphabetical order. Return 1 if b is a prefix of a, or if b comes before a in alphabetical order (technically, lexicographical order by Unicode code points).

julia> cmp("abc", "abc")
0

julia> cmp("ab", "abc")
-1

julia> cmp("abc", "ab")
1

julia> cmp("ab", "ac")
-1

julia> cmp("ac", "ab")
1

julia> cmp("α", "a")
1

julia> cmp("b", "β")
-1

source

— Function.

~(x)

Bitwise not.

Examples

julia> ~4
-5

julia> ~10
-11

julia> ~true
false

source

— Function.

&(x, y)

Bitwise and. Implements three-valued logic, returning if one operand is missing and the other is true.

Examples

julia> 4 & 10
0

julia> 4 & 12
4

julia> true & missing
missing

julia> false & missing
false

source

— Function.

|(x, y)

Bitwise or. Implements three-valued logic, returning if one operand is missing and the other is false.

Examples

julia> 4 | 10
14

julia> 4 | 1
5

julia> true | missing
true

julia> false | missing
missing

source

— Function.

xor(x, y)
⊻(x, y)

Bitwise exclusive or of x and y. Implements three-valued logic, returning if one of the arguments is missing.

The infix operation a ⊻ b is a synonym for xor(a,b), and ⊻ can be typed by tab-completing \xor or \veebar in the Julia REPL.

Examples

julia> xor(true, false)
true

julia> xor(true, true)
false

julia> xor(true, missing)
missing

julia> false ⊻ false
false

julia> [true; true; false] .⊻ [true; false; false]
3-element BitArray{1}:
 0
 1
 0

source

— Function.

!(x)

Boolean not. Implements three-valued logic, returning if x is missing.

Examples

julia> !true
false

julia> !false
true

julia> !missing
missing

julia> .![true false true]
1×3 BitArray{2}:
 0  1  0

source

!f::Function

Predicate function negation: when the argument of ! is a function, it returns a function which computes the boolean negation of f.

Examples

julia> str = "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"
"∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"

julia> filter(isletter, str)
"εδxyδfxfyε"

julia> filter(!isletter, str)
"∀  > 0, ∃  > 0: |-| <  ⇒ |()-()| < "

&& — Keyword.

x && y

Short-circuiting boolean AND.

|| — Keyword.

x || y

Short-circuiting boolean OR.

Mathematical Functions

Base.isapprox — Function.

isapprox(x, y; rtol::Real=atol>0 ? 0 : √eps, atol::Real=0, nans::Bool=false, norm::Function)

Inexact equality comparison: true if norm(x-y) <= max(atol, rtol*max(norm(x), norm(y))). The default atol is zero and the default rtol depends on the types of x and y. The keyword argument nans determines whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, if an atol > 0 is not specified, rtol defaults to the square root of of the type of x or y, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an atol > 0 is supplied, rtol defaults to zero.

x and y may also be arrays of numbers, in which case norm defaults to the usual norm function in LinearAlgebra, but may be changed by passing a norm::Function keyword argument. (For numbers, norm is the same thing as abs.) When x and y are arrays, if norm(x-y) is not finite (i.e. ±Inf or NaN), the comparison falls back to checking whether all elements of x and y are approximately equal component-wise.

The binary operator ≈ is equivalent to isapprox with the default arguments, and x ≉ y is equivalent to !isapprox(x,y).

Note that x ≈ 0 (i.e., comparing to zero with the default tolerances) is equivalent to x == 0 since the default atol is 0. In such cases, you should either supply an appropriate atol (or use norm(x) ≤ atol) or rearrange your code (e.g. use x ≈ y rather than x - y ≈ 0). It is not possible to pick a nonzero atol automatically because it depends on the overall scaling (the “units”) of your problem: for example, in x - y ≈ 0, atol=1e-9 is an absurdly small tolerance if x is the radius of the Earth in meters, but an absurdly large tolerance if x is the in meters.

Examples

julia> 0.1 ≈ (0.1 - 1e-10)
true

julia> isapprox(10, 11; atol = 2)
true

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])
true

julia> 1e-10 ≈ 0
false

julia> isapprox(1e-10, 0, atol=1e-8)
true

source

— Method.

sin(x)

Compute sine of x, where x is in radians.

source

— Method.

cos(x)

Compute cosine of x, where x is in radians.

source

— Method.

sincos(x)

Simultaneously compute the sine and cosine of x, where the x is in radians.

source

— Method.

tan(x)

Compute tangent of x, where x is in radians.

source

— Function.

sind(x)

Compute sine of x, where x is in degrees.

source

— Function.

cosd(x)

Compute cosine of x, where x is in degrees.

source

— Function.

tand(x)

Compute tangent of x, where x is in degrees.

source

— Function.

sinpi(x)

Compute $\sin(\pi x)$ more accurately than sin(pi*x), especially for large x.

source

— Function.

cospi(x)

Compute $\cos(\pi x)$ more accurately than cos(pi*x), especially for large x.

source

— Method.

sinh(x)

Compute hyperbolic sine of x.

source

— Method.

cosh(x)

Compute hyperbolic cosine of x.

source

— Method.

tanh(x)

Compute hyperbolic tangent of x.

source

— Method.

asin(x)

Compute the inverse sine of x, where the output is in radians.

source

— Method.

acos(x)

Compute the inverse cosine of x, where the output is in radians

source

— Method.

atan(y)
atan(y, x)

Compute the inverse tangent of y or y/x, respectively.

For one argument, this is the angle in radians between the positive x-axis and the point (1, y), returning a value in the interval $[-\pi/2, \pi/2]$.

For two arguments, this is the angle in radians between the positive x-axis and the point (x, y), returning a value in the interval $[-\pi, \pi]$. This corresponds to a standard atan2 function.

Base.Math.asind — Function.

asind(x)

Compute the inverse sine of x, where the output is in degrees.

Base.Math.acosd — Function.

acosd(x)

Compute the inverse cosine of x, where the output is in degrees.

Base.Math.atand — Function.

atand(y)
atand(y,x)

Compute the inverse tangent of y or y/x, respectively, where the output is in degrees.

Base.Math.sec — Method.

sec(x)

Compute the secant of x, where x is in radians.

Base.Math.csc — Method.

csc(x)

Compute the cosecant of x, where x is in radians.

Base.Math.cot — Method.

cot(x)

Compute the cotangent of x, where x is in radians.

Base.Math.secd — Function.

secd(x)

Compute the secant of x, where x is in degrees.

Base.Math.cscd — Function.

cscd(x)

Compute the cosecant of x, where x is in degrees.

Base.Math.cotd — Function.

cotd(x)

Compute the cotangent of x, where x is in degrees.

Base.Math.asec — Method.

asec(x)

Compute the inverse secant of x, where the output is in radians.

Base.Math.acsc — Method.

acsc(x)

Compute the inverse cosecant of x, where the output is in radians.

Base.Math.acot — Method.

acot(x)

Compute the inverse cotangent of x, where the output is in radians.

Base.Math.asecd — Function.

asecd(x)

Compute the inverse secant of x, where the output is in degrees.

Base.Math.acscd — Function.

acscd(x)

Compute the inverse cosecant of x, where the output is in degrees.

Base.Math.acotd — Function.

acotd(x)

Compute the inverse cotangent of x, where the output is in degrees.

Base.Math.sech — Method.

sech(x)

Compute the hyperbolic secant of x.

Base.Math.csch — Method.

csch(x)

Compute the hyperbolic cosecant of x.

Base.Math.coth — Method.

coth(x)

Compute the hyperbolic cotangent of x.

Base.asinh — Method.

asinh(x)

Compute the inverse hyperbolic sine of x.

Base.acosh — Method.

acosh(x)

Compute the inverse hyperbolic cosine of x.

Base.atanh — Method.

atanh(x)

Compute the inverse hyperbolic tangent of x.

Base.Math.asech — Method.

asech(x)

Compute the inverse hyperbolic secant of x.

Base.Math.acsch — Method.

acsch(x)

Compute the inverse hyperbolic cosecant of x.

Base.Math.acoth — Method.

acoth(x)

Compute the inverse hyperbolic cotangent of x.

Base.Math.sinc — Function.

sinc(x)

Compute $\sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.

Base.Math.cosc — Function.

cosc(x)

Compute $\cos(\pi x) / x - \sin(\pi x) / (\pi x^2)$ if $x \neq 0$, and $0$ if $x = 0$. This is the derivative of sinc(x).

Base.Math.deg2rad — Function.

deg2rad(x)

Convert x from degrees to radians.

Examples

julia> deg2rad(90)
1.5707963267948966

Base.Math.rad2deg — Function.

rad2deg(x)

Convert x from radians to degrees.

Examples

julia> rad2deg(pi)
180.0

Base.Math.hypot — Function.

hypot(x, y)

Compute the hypotenuse $\sqrt{|x|^2+|y|^2}$ avoiding overflow and underflow.

This code is an implementation of the algorithm described in: An Improved Algorithm for hypot(a,b) by Carlos F. Borges The article is available online at ArXiv at the link

Examples

julia> a = Int64(10)^10;

julia> hypot(a, a)
1.4142135623730951e10

julia> √(a^2 + a^2) # a^2 overflows
ERROR: DomainError with -2.914184810805068e18:
sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
Stacktrace:
[...]

julia> hypot(3, 4im)
5.0

source

hypot(x...)

Compute the hypotenuse $\sqrt{\sum |x_i|^2}$ avoiding overflow and underflow.

Examples

julia> hypot(-5.7)
5.7

julia> hypot(3, 4im, 12.0)
13.0

Base.log — Method.

log(x)

Compute the natural logarithm of x. Throws for negative Real arguments. Use complex negative arguments to obtain complex results.

Examples

julia> log(2)
0.6931471805599453

julia> log(-3)
ERROR: DomainError with -3.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

Base.log — Method.

log(b,x)

Compute the base b logarithm of x. Throws for negative arguments.

Examples

julia> log(4,8)
1.5

julia> log(4,2)
0.5

julia> log(-2, 3)
ERROR: DomainError with -2.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(2, -3)
ERROR: DomainError with -3.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

Note

If b is a power of 2 or 10, or log10 should be used, as these will typically be faster and more accurate. For example,

julia> log(100,1000000)
2.9999999999999996

julia> log10(1000000)/2
3.0

Base.log2 — Function.

log2(x)

Compute the logarithm of x to base 2. Throws for negative Real arguments.

Examples

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362

julia> log2(-2)
ERROR: DomainError with -2.0:
NaN result for non-NaN input.
Stacktrace:
 [1] nan_dom_err at ./math.jl:325 [inlined]
[...]

Base.log10 — Function.

log10(x)

Compute the logarithm of x to base 10. Throws for negative Real arguments.

Examples

julia> log10(100)
2.0

julia> log10(2)
0.3010299956639812

julia> log10(-2)
ERROR: DomainError with -2.0:
NaN result for non-NaN input.
Stacktrace:
 [1] nan_dom_err at ./math.jl:325 [inlined]
[...]

Base.log1p — Function.

log1p(x)

Accurate natural logarithm of 1+x. Throws for Real arguments less than -1.

Examples

julia> log1p(-0.5)
-0.6931471805599453

julia> log1p(0)
0.0

julia> log1p(-2)
ERROR: DomainError with -2.0:
log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

Base.Math.frexp — Function.

frexp(val)

Return (x,exp) such that x has a magnitude in the interval $[1/2, 1)$ or 0, and val is equal to $x \times 2^{exp}$.

Base.exp — Method.

exp(x)

Compute the natural base exponential of x, in other words $e^x$.

Examples

julia> exp(1.0)
2.718281828459045

Base.exp2 — Function.

exp2(x)

Compute the base 2 exponential of x, in other words $2^x$.

Examples

julia> exp2(5)
32.0

Base.exp10 — Function.

exp10(x)

Compute the base 10 exponential of x, in other words $10^x$.

Examples

julia> exp10(2)
100.0

exp10(x)

Compute $10^x$.

Examples

julia> exp10(2)
100.0

julia> exp10(0.2)
1.5848931924611136

source

— Function.

ldexp(x, n)

Compute $x \times 2^n$.

Examples

julia> ldexp(5., 2)
20.0

source

— Function.

modf(x)

Return a tuple (fpart, ipart) of the fractional and integral parts of a number. Both parts have the same sign as the argument.

Examples

julia> modf(3.5)
(0.5, 3.0)

julia> modf(-3.5)
(-0.5, -3.0)

source

— Function.

expm1(x)

Accurately compute $e^x-1$.

source

— Method.

round([T,] x, [r::RoundingMode])
round(x, [r::RoundingMode]; digits::Integer=0, base = 10)
round(x, [r::RoundingMode]; sigdigits::Integer, base = 10)

Rounds the number x.

Without keyword arguments, x is rounded to an integer value, returning a value of type T, or of the same type of x if no T is provided. An InexactError will be thrown if the value is not representable by T, similar to .

If the digits keyword argument is provided, it rounds to the specified number of digits after the decimal place (or before if negative), in base base.

If the sigdigits keyword argument is provided, it rounds to the specified number of significant digits, in base base.

The RoundingMode r controls the direction of the rounding; the default is , which rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer. Note that round may give incorrect results if the global rounding mode is changed (see rounding).

Examples

julia> round(1.7)
2.0

julia> round(Int, 1.7)
2

julia> round(1.5)
2.0

julia> round(2.5)
2.0

julia> round(pi; digits=2)
3.14

julia> round(pi; digits=3, base=2)
3.125

julia> round(123.456; sigdigits=2)
120.0

julia> round(357.913; sigdigits=4, base=2)
352.0

Note

Rounding to specified digits in bases other than 2 can be inexact when operating on binary floating point numbers. For example, the value represented by 1.15 is actually less than 1.15, yet will be rounded to 1.2.

Examples

julia> x = 1.15
1.15

julia> @sprintf "%.20f" x
"1.14999999999999991118"

julia> x < 115//100
true

julia> round(x, digits=1)
1.2

Extensions

To extend round to new numeric types, it is typically sufficient to define Base.round(x::NewType, r::RoundingMode).

source

— Type.

A type used for controlling the rounding mode of floating point operations (via rounding/ functions), or as optional arguments for rounding to the nearest integer (via the round function).

(default)
RoundNearestTiesAway
RoundToZero
(BigFloat only)
RoundDown

Base.Rounding.RoundNearest — Constant.

RoundNearest

The default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.

Base.Rounding.RoundNearestTiesAway — Constant.

RoundNearestTiesAway

Rounds to nearest integer, with ties rounded away from zero (C/C++ behaviour).

source

— Constant.

RoundNearestTiesUp

Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript round behaviour).

Base.Rounding.RoundToZero — Constant.

RoundToZero

using this rounding mode is an alias for trunc.

Base.Rounding.RoundFromZero — Constant.

RoundFromZero

Rounds away from zero. This rounding mode may only be used with T == BigFloat inputs to .

Examples

julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06

source

— Constant.

RoundUp

round using this rounding mode is an alias for .

source

— Constant.

RoundDown

round using this rounding mode is an alias for .

source

— Method.

round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]])
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; digits=, base=10)
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; sigdigits=, base=10)

Return the nearest integral value of the same type as the complex-valued z to z, breaking ties using the specified RoundingModes. The first is used for rounding the real components while the second is used for rounding the imaginary components.

Example

julia> round(3.14 + 4.5im)
3.0 + 4.0im

source

— Function.

ceil([T,] x)
ceil(x; digits::Integer= [, base = 10])
ceil(x; sigdigits::Integer= [, base = 10])

ceil(x) returns the nearest integral value of the same type as x that is greater than or equal to x.

ceil(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for round.

Base.floor — Function.

floor([T,] x)
floor(x; digits::Integer= [, base = 10])
floor(x; sigdigits::Integer= [, base = 10])

floor(x) returns the nearest integral value of the same type as x that is less than or equal to x.

floor(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for .

source

— Function.

trunc([T,] x)
trunc(x; digits::Integer= [, base = 10])
trunc(x; sigdigits::Integer= [, base = 10])

trunc(x) returns the nearest integral value of the same type as x whose absolute value is less than or equal to x.

trunc(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for round.

Base.unsafe_trunc — Function.

unsafe_trunc(T, x)

Return the nearest integral value of type T whose absolute value is less than or equal to x. If the value is not representable by T, an arbitrary value will be returned.

Base.min — Function.

min(x, y, ...)

Return the minimum of the arguments. See also the function to take the minimum element from a collection.

Examples

julia> min(2, 5, 1)
1

source

— Function.

max(x, y, ...)

Return the maximum of the arguments. See also the maximum function to take the maximum element from a collection.

Examples

julia> max(2, 5, 1)
5

Base.minmax — Function.

minmax(x, y)

Return (min(x,y), max(x,y)). See also: that returns (minimum(x), maximum(x)).

Examples

julia> minmax('c','b')
('b', 'c')

source

— Function.

clamp(x, lo, hi)

Return x if lo <= x <= hi. If x > hi, return hi. If x < lo, return lo. Arguments are promoted to a common type.

Examples

julia> clamp.([pi, 1.0, big(10.)], 2., 9.)
3-element Array{BigFloat,1}:
 3.141592653589793238462643383279502884197169399375105820974944592307816406286198
 2.0
 9.0

julia> clamp.([11,8,5],10,6) # an example where lo > hi
3-element Array{Int64,1}:
  6
  6
 10

source

— Function.

clamp!(array::AbstractArray, lo, hi)

Restrict values in array to the specified range, in-place. See also clamp.

Base.abs — Function.

abs(x)

The absolute value of x.

When abs is applied to signed integers, overflow may occur, resulting in the return of a negative value. This overflow occurs only when abs is applied to the minimum representable value of a signed integer. That is, when x == typemin(typeof(x)), abs(x) == x < 0, not -x as might be expected.

Examples

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808

Base.Checked.checked_abs — Function.

Base.checked_abs(x)

Calculates abs(x), checking for overflow errors where applicable. For example, standard two’s complement signed integers (e.g. Int) cannot represent abs(typemin(Int)), thus leading to an overflow.