-(x)

Unary minus operator.

See also: abs, flipsign.

Examples

julia> -1
-1

julia> -(2)
-2

julia> -[1 2; 3 4]
2×2 Matrix{Int64}:
 -1  -2
 -3  -4

+(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

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.

-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5

*(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

/(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

\(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 Vector{Float64}:
  6.5
 -7.0

julia> inv(A) * x
2-element Vector{Float64}:
  6.5
 -7.0

^(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.) If y is a negative integer literal, then Base.literal_pow transforms the operation to inv(x)^-y by default, where -y is positive.

Examples

julia> 3^5
243

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> A^3
2×2 Matrix{Int64}:
 37   54
 81  118

fma(x, y, z)

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 muladd.

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 fma.

Examples

julia> muladd(3, 2, 1)
7

julia> 3 * 2 + 1
7

muladd(A, y, z)

Combined multiply-add, A*y .+ z, for matrix-matrix or matrix-vector multiplication. The result is always the same size as A*y, but z may be smaller, or a scalar.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; z=[0, 100];

julia> muladd(A, B, z)
2×2 Matrix{Float64}:
   3.0    3.0
 107.0  107.0

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

div(x, y)
÷(x, y)

The quotient from Euclidean (integer) division. Generally equivalent to a mathematical operation x/y without a fractional part.

See also: cld, fld, rem, divrem.

Examples

julia> 9 ÷ 4
2

julia> -5 ÷ 3
-1

julia> 5.0 ÷ 2
2.0

julia> div.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -1  -1  -1  0  0  0  0  0  1  1  1

fld(x, y)

Largest integer less than or equal to x/y. Equivalent to div(x, y, RoundDown).

See also div, cld, fld1.

Examples

julia> fld(7.3,5.5)
1.0

julia> fld.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -2  -2  -1  -1  -1  0  0  0  1  1  1

Because fld(x, y) implements strictly correct floored rounding based on the true value of floating-point numbers, unintuitive situations can arise. For example:

julia> fld(6.0,0.1)
59.0
julia> 6.0/0.1
60.0
julia> 6.0/big(0.1)
59.99999999999999666933092612453056361837965690217069245739573412231113406246995

What is happening here is that the true value of the floating-point number written as 0.1 is slightly larger than the numerical value 1/10 while 6.0 represents the number 6 precisely. Therefore the true value of 6.0 / 0.1 is slightly less than 60. When doing division, this is rounded to precisely 60.0, but fld(6.0, 0.1) always takes the floor or the true value, so the result is 59.0.

cld(x, y)

Smallest integer larger than or equal to x/y. Equivalent to div(x, y, RoundUp).

See also div, fld.

Examples

julia> cld(5.5,2.2)
3.0

julia> cld.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -1  -1  -1  0  0  0  1  1  1  2  2

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 mod1.

Examples

julia> mod(0, Base.OneTo(3))
3

julia> mod(3, 0:2)
0

mod(x, y)
rem(x, y, RoundDown)

The reduction of x modulo y, or equivalently, the remainder of x after floored division by y, i.e. x - y*fld(x,y) if computed without intermediate rounding.

The result will have the same sign as y, and magnitude less than abs(y) (with some exceptions, see note below).

See also: rem, div, fld, mod1, invmod.

julia> mod(8, 3)
2

julia> mod(9, 3)
0

julia> mod(8.9, 3)
2.9000000000000004

julia> mod(eps(), 3)
2.220446049250313e-16

julia> mod(-eps(), 3)
3.0

julia> mod.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 1  2  0  1  2  0  1  2  0  1  2

rem(x::Integer, T::Type{<:Integer}) -> T
mod(x::Integer, T::Type{<:Integer}) -> T
%(x::Integer, T::Type{<:Integer}) -> T

Find y::T such that x ≡ y (mod n), where n is the number of integers representable in T, and y is an integer in [typemin(T),typemax(T)]. If T can represent any integer (e.g. T == BigInt), then this operation corresponds to a conversion to T.

Examples

julia> 129 % Int8
-127

rem(x, y)
%(x, y)

Remainder from Euclidean division, returning a value of the same sign as x, and smaller in magnitude than y. This value is always exact.

See also: div, mod, mod1, divrem.

Examples

julia> x = 15; y = 4;

julia> x % y
3

julia> x == div(x, y) * y + rem(x, y)
true

julia> rem.(-5:5, 3)'
1×11 adjoint(::Vector{Int64}) with eltype Int64:
 -2  -1  0  -2  -1  0  1  2  0  1  2

rem2pi(x, r::RoundingMode)

Compute the remainder of x after integer division by 2π, with the quotient rounded according to the rounding mode r. In other words, the quantity

x - 2π*round(x/(2π),r)

without any intermediate rounding. This internally uses a high precision approximation of 2π, and so will give a more accurate result than rem(x,2π,r)

• if r == RoundNearest, then the result is in the interval $[-π, π]$. This will generally be the most accurate result. See also RoundNearest.

• if r == RoundToZero, then the result is in the interval $[0, 2π]$ if x is positive,. or $[-2π, 0]$ otherwise. See also RoundToZero.

• if r == RoundDown, then the result is in the interval $[0, 2π]$. See also RoundDown.

• if r == RoundUp, then the result is in the interval $[-2π, 0]$. See also RoundUp.

Examples

julia> rem2pi(7pi/4, RoundNearest)
-0.7853981633974485

julia> rem2pi(7pi/4, RoundDown)
5.497787143782138

mod2pi(x)

Modulus after division by 2π, returning in the range $[0,2π)$.

This function computes a floating point representation of the modulus after division by numerically exact 2π, and is therefore not exactly the same as mod(x,2π), which would compute the modulus of x relative to division by the floating-point number 2π.

Examples

julia> mod2pi(9*pi/4)
0.7853981633974481

divrem(x, y, r::RoundingMode=RoundToZero)

The quotient and remainder from Euclidean division. Equivalent to (div(x,y,r), rem(x,y,r)). Equivalently, with the default value of r, this call is equivalent to (x÷y, x%y).

See also: fldmod, cld.

Examples

julia> divrem(3,7)
(0, 3)

julia> divrem(7,3)
(2, 1)

fldmod(x, y)

The floored quotient and modulus after division. A convenience wrapper for divrem(x, y, RoundDown). Equivalent to (fld(x,y), mod(x,y)).

See also: fld, cld, fldmod1.

fld1(x, y)

Flooring division, returning a value consistent with mod1(x,y)

See also mod1, fldmod1.

Examples

julia> x = 15; y = 4;

julia> fld1(x, y)
4

julia> x == fld(x, y) * y + mod(x, y)
true

julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
true

mod1(x, y)

Modulus after flooring division, returning a value r such that mod(r, y) == mod(x, y) in the range $(0, y]$ for positive y and in the range $[y,0)$ for negative y.

See also fld1, fldmod1.

Examples

julia> mod1(4, 2)
2

julia> mod1(4, 3)
1

fldmod1(x, y)

Return (fld1(x,y), mod1(x,y)).

See also fld1, mod1.

//(num, den)

Divide two integers or rational numbers, giving a Rational result.

Examples

julia> 3 // 5
3//5

julia> (3 // 5) // (2 // 1)
3//10

rationalize([T<:Integer=Int,] x; tol::Real=eps(x))

Approximate floating point number x as a Rational number with components of the given integer type. The result will differ from x by no more than tol.

Examples

julia> rationalize(5.6)
28//5

julia> a = rationalize(BigInt, 10.3)
103//10

julia> typeof(numerator(a))
BigInt

numerator(x)

Numerator of the rational representation of x.

Examples

julia> numerator(2//3)
2

julia> numerator(4)
4

denominator(x)

Denominator of the rational representation of x.

Examples

julia> denominator(2//3)
3

julia> denominator(4)
1

<<(x, n)

Left bit shift operator, x << n. For n >= 0, the result is x shifted left by n bits, filling with 0s. This is equivalent to x * 2^n. For n < 0, this is equivalent to x >> -n.

Examples

julia> Int8(3) << 2
12

julia> bitstring(Int8(3))
"00000011"

julia> bitstring(Int8(12))
"00001100"

See also >>, >>>, exp2, ldexp.

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

Left bit shift operator, B << n. For n >= 0, the result is B with elements shifted n positions backwards, filling with false values. If n < 0, elements are shifted forwards. Equivalent to B >> -n.

Examples

julia> B = BitVector([true, false, true, false, false])
5-element BitVector:
 1
 0
 1
 0
 0

julia> B << 1
5-element BitVector:
 0
 1
 0
 0
 0

julia> B << -1
5-element BitVector:
 0
 1
 0
 1
 0

>>(x, n)

Right bit shift operator, x >> n. For n >= 0, the result is x shifted right by n bits, where n >= 0, filling with 0s if x >= 0, 1s if x < 0, preserving the sign of x. This is equivalent to fld(x, 2^n). For n < 0, this is equivalent to x << -n.

Examples

julia> Int8(13) >> 2
3

julia> bitstring(Int8(13))
"00001101"

julia> bitstring(Int8(3))
"00000011"

julia> Int8(-14) >> 2
-4

julia> bitstring(Int8(-14))
"11110010"

julia> bitstring(Int8(-4))
"11111100"

See also >>>, <<.

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

Right bit shift operator, B >> n. For n >= 0, the result is B with elements shifted n positions forward, filling with false values. If n < 0, elements are shifted backwards. Equivalent to B << -n.

Examples

julia> B = BitVector([true, false, true, false, false])
5-element BitVector:
 1
 0
 1
 0
 0

julia> B >> 1
5-element BitVector:
 0
 1
 0
 1
 0

julia> B >> -1
5-element BitVector:
 0
 1
 0
 0
 0

>>>(x, n)

Unsigned right bit shift operator, x >>> n. For n >= 0, the result is x shifted right by n bits, where n >= 0, filling with 0s. For n < 0, this is equivalent to x << -n.

For Unsigned integer types, this is equivalent to >>. For Signed integer types, this is equivalent to signed(unsigned(x) >> n).

Examples

julia> Int8(-14) >>> 2
60

julia> bitstring(Int8(-14))
"11110010"

julia> bitstring(Int8(60))
"00111100"

BigInts are treated as if having infinite size, so no filling is required and this is equivalent to >>.

See also >>, <<.

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

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

bitrotate(x::Base.BitInteger, k::Integer)

bitrotate(x, k) implements bitwise rotation. It returns the value of x with its bits rotated left k times. A negative value of k will rotate to the right instead.

See also: <<, circshift, BitArray.

julia> bitrotate(UInt8(114), 2)
0xc9

julia> bitstring(bitrotate(0b01110010, 2))
"11001001"

julia> bitstring(bitrotate(0b01110010, -2))
"10011100"

julia> bitstring(bitrotate(0b01110010, 8))
"01110010"

(:)(start, [step], stop)

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 StepRange).

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

(:)(start::CartesianIndex, [step::CartesianIndex], stop::CartesianIndex)

Construct CartesianIndices from two CartesianIndex and an optional step.

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)

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

range(start, stop, length)
range(start, stop; length, step)
range(start; length, stop, step)
range(;start, length, stop, step)

Construct a specialized array with evenly spaced elements and optimized storage (an AbstractRange) from the arguments. Mathematically a range is uniquely determined by any three of start, step, stop and length. Valid invocations of range are:

• Call range with any three of start, step, stop, length.
• Call range with two of start, stop, length. In this case step will be assumed

to be one. If both arguments are Integers, a UnitRange will be returned.

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

julia> range(stop=10, length=5)
6:10

julia> range(stop=10, step=1, length=5)
6:1:10

julia> range(start=1, step=1, stop=10)
1:1:10

If length is not specified and stop - start is not an integer multiple of step, a range that ends before stop will be produced.

julia> range(1, 3.5, step=2)
1.0:2.0:3.0

Special care is taken to ensure intermediate values are computed rationally. To avoid this induced overhead, see the LinRange constructor.

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.

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

A range r where r[i] produces values of type T (in the first 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.

==(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 missing, 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 isequal 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.

isequal 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 hash.

If some type defines ==, isequal, and isless then it should also implement < to ensure consistency of comparisons.

!=(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

!=(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.

!==(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

<(x, y)

Less-than comparison operator. Falls back to isless. 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.

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.

<=(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.

>(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

>(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.

>=(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

>=(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.

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})
[...]

cmp(<, x, y)

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).

Examples

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

~(x)

Bitwise not.

See also: !, &, |.

Examples

julia> ~4
-5

julia> ~10
-11

julia> ~true
false

x & y

Bitwise and. Implements three-valued logic, returning missing if one operand is missing and the other is true. Add parentheses for function application form: (&)(x, y).

See also: |, xor, &&.

Examples

julia> 4 & 10
0

julia> 4 & 12
4

julia> true & missing
missing

julia> false & missing
false

x | y

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

See also: &, xor, ||.

Examples

julia> 4 | 10
14

julia> 4 | 1
5

julia> true | missing
true

julia> false | missing
missing

xor(x, y)
⊻(x, y)

Bitwise exclusive or of x and y. Implements three-valued logic, returning missing 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 BitVector:
 0
 1
 0

nand(x, y)
⊼(x, y)

Bitwise nand (not and) of x and y. Implements three-valued logic, returning missing if one of the arguments is missing.

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

Examples

julia> nand(true, false)
true

julia> nand(true, true)
false

julia> nand(true, missing)
missing

julia> false ⊼ false
true

julia> [true; true; false] .⊼ [true; false; false]
3-element BitVector:
 0
 1
 1

nor(x, y)
⊽(x, y)

Bitwise nor (not or) of x and y. Implements three-valued logic, returning missing if one of the arguments is missing.

The infix operation a ⊽ b is a synonym for nor(a,b), and ⊽ can be typed by tab-completing \nor or \veebar in the Julia REPL.

Examples

julia> nor(true, false)
false

julia> nor(true, true)
false

julia> nor(true, missing)
false

julia> false ⊽ false
true

julia> [true; true; false] .⊽ [true; false; false]
3-element BitVector:
 0
 0
 1

!(x)

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

See also ~ for bitwise not.

Examples

julia> !true
false

julia> !false
true

julia> !missing
missing

julia> .![true false true]
1×3 BitMatrix:
 0  1  0

!f::Function

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

See also ∘.

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: |-| <  ⇒ |()-()| < "

x && y

Short-circuiting boolean AND.

See also &, the ternary operator ? :, and the manual section on control flow.

Examples

julia> x = 3;

julia> x > 1 && x < 10 && x isa Int
true

julia> x < 0 && error("expected positive x")
false

x || y

Short-circuiting boolean OR.

See also: |, xor, &&.

Examples

julia> pi < 3 || ℯ < 3
true

julia> false || true || println("neither is true!")
true

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

Inexact equality comparison. Two numbers compare equal if their relative distance or their absolute distance is within tolerance bounds: isapprox returns 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 eps 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.

The norm keyword defaults to abs for numeric (x,y) and to LinearAlgebra.norm for arrays (where an alternative norm choice is sometimes useful). 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 radius of a Hydrogen atom in meters.

Examples

julia> isapprox(0.1, 0.15; atol=0.05)
true

julia> isapprox(0.1, 0.15; rtol=0.34)
true

julia> isapprox(0.1, 0.15; rtol=0.33)
false

julia> 0.1 + 1e-10 ≈ 0.1
true

julia> 1e-10 ≈ 0
false

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

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0]) # using `norm`
true

isapprox(x; kwargs...) / ≈(x; kwargs...)

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

The keyword arguments supported here are the same as those in the 2-argument isapprox.

sin(x)

Compute sine of x, where x is in radians.

See also [sind], [sinpi], [sincos], [cis].

cos(x)

Compute cosine of x, where x is in radians.

See also [cosd], [cospi], [sincos], [cis].

sincos(x)

Simultaneously compute the sine and cosine of x, where x is in radians, returning a tuple (sine, cosine).

See also cis, sincospi, sincosd.

tan(x)

Compute tangent of x, where x is in radians.

sind(x)

Compute sine of x, where x is in degrees. If x is a matrix, x needs to be a square matrix.

cosd(x)

Compute cosine of x, where x is in degrees. If x is a matrix, x needs to be a square matrix.

tand(x)

Compute tangent of x, where x is in degrees. If x is a matrix, x needs to be a square matrix.

sincosd(x)

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

sinpi(x)

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

See also sind, cospi, sincospi.

cospi(x)

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

sincospi(x)

Simultaneously compute sinpi(x) and cospi(x) (the sine and cosine of π*x, where x is in radians), returning a tuple (sine, cosine).

See also: cispi, sincosd, sinpi.

sinh(x)

Compute hyperbolic sine of x.

cosh(x)

Compute hyperbolic cosine of x.

tanh(x)

Compute hyperbolic tangent of x.

asin(x)

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

acos(x)

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

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. Note that by convention atan(0.0,x) is defined as $\pi$ and atan(-0.0,x) is defined as $-\pi$ when x < 0.

asind(x)

Compute the inverse sine of x, where the output is in degrees. If x is a matrix, x needs to be a square matrix.

acosd(x)

Compute the inverse cosine of x, where the output is in degrees. If x is a matrix, x needs to be a square matrix.

atand(y)
atand(y,x)

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

sec(x)

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

csc(x)

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

cot(x)

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

secd(x)

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

cscd(x)

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

cotd(x)

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

asec(x)

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

acsc(x)

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

acot(x)

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

asecd(x)

Compute the inverse secant of x, where the output is in degrees. If x is a matrix, x needs to be a square matrix.

acscd(x)

Compute the inverse cosecant of x, where the output is in degrees. If x is a matrix, x needs to be a square matrix.

acotd(x)

Compute the inverse cotangent of x, where the output is in degrees. If x is a matrix, x needs to be a square matrix.

sech(x)

Compute the hyperbolic secant of x.

csch(x)

Compute the hyperbolic cosecant of x.

coth(x)

Compute the hyperbolic cotangent of x.

asinh(x)

Compute the inverse hyperbolic sine of x.

acosh(x)

Compute the inverse hyperbolic cosine of x.

atanh(x)

Compute the inverse hyperbolic tangent of x.

asech(x)

Compute the inverse hyperbolic secant of x.

acsch(x)

Compute the inverse hyperbolic cosecant of x.

acoth(x)

Compute the inverse hyperbolic cotangent of x.

sinc(x)

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

See also cosc, its derivative.

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).

deg2rad(x)

Convert x from degrees to radians.

See also: rad2deg, sind.

Examples

julia> deg2rad(90)
1.5707963267948966

rad2deg(x)

Convert x from radians to degrees.

Examples

julia> rad2deg(pi)
180.0

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 https://arxiv.org/abs/1904.09481

hypot(x...)

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

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

julia> hypot(-5.7)
5.7

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

log(x)

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

See also [log1p], [log2], [log10].

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
[...]

log(b,x)

Compute the base b logarithm of x. Throws DomainError for negative Real 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
[...]

julia> log(100,1000000)
2.9999999999999996

julia> log10(1000000)/2
3.0

log2(x)

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

See also: exp2, ldexp, ispow2.

Examples

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362

julia> log2(-2)
ERROR: DomainError with -2.0:
log2 will only return a complex result if called with a complex argument. Try log2(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
[...]

log10(x)

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

Examples

julia> log10(100)
2.0

julia> log10(2)
0.3010299956639812

julia> log10(-2)
ERROR: DomainError with -2.0:
log10 will only return a complex result if called with a complex argument. Try log10(Complex(x)).
Stacktrace:
 [1] throw_complex_domainerror(f::Symbol, x::Float64) at ./math.jl:31
[...]

log1p(x)

Accurate natural logarithm of 1+x. Throws DomainError 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
[...]

frexp(val)

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

Examples

julia> frexp(12.8)
(0.8, 4)

exp(x)

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

See also exp2, exp10 and cis.

Examples

julia> exp(1.0)
2.718281828459045

julia> exp(im * pi) == cis(pi)
true

exp2(x)

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

See also ldexp, <<.

Examples

julia> exp2(5)
32.0

julia> 2^5
32

julia> exp2(63) > typemax(Int)
true

exp10(x)

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

Examples

julia> exp10(2)
100.0

julia> 10^2
100

ldexp(x, n)

Compute $x \times 2^n$.

Examples

julia> ldexp(5., 2)
20.0

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)

expm1(x)

Accurately compute $e^x-1$. It avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small values of x.

Examples

julia> expm1(1e-16)
1.0e-16

julia> exp(1e-16) - 1
0.0

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 convert.

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 RoundNearest, 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

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).

RoundingMode

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

Currently supported rounding modes are:

• RoundNearest (default)
• RoundNearestTiesAway
• RoundNearestTiesUp
• RoundToZero
• RoundFromZero (BigFloat only)
• RoundUp
• RoundDown

RoundNearest

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

RoundNearestTiesAway

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

RoundNearestTiesUp

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

RoundToZero

round using this rounding mode is an alias for trunc.

RoundFromZero

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

Examples

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

RoundUp

round using this rounding mode is an alias for ceil.

RoundDown

round using this rounding mode is an alias for floor.

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 RoundingMode 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

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.

Keywords digits, sigdigits and base work as for round.

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.

Keywords digits, sigdigits and base work as for round.

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 the absolute value of x.

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

Keywords digits, sigdigits and base work as for round.

See also: %, floor, unsigned, unsafe_trunc.

Examples

julia> trunc(2.22)
2.0

julia> trunc(-2.22, digits=1)
-2.2

julia> trunc(Int, -2.22)
-2

unsafe_trunc(T, x)

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

Examples

julia> unsafe_trunc(Int, -2.2)
-2

julia> unsafe_trunc(Int, NaN)
-9223372036854775808

min(x, y, ...)

Return the minimum of the arguments (with respect to isless). See also the minimum function to take the minimum element from a collection.

Examples

julia> min(2, 5, 1)
1

max(x, y, ...)

Return the maximum of the arguments (with respect to isless). See also the maximum function to take the maximum element from a collection.

Examples

julia> max(2, 5, 1)
5

minmax(x, y)

Return (min(x,y), max(x,y)).

See also extrema that returns (minimum(x), maximum(x)).

Examples

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

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.

See also clamp!, min, max.

Examples

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

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

clamp(x, T)::T

Clamp x between typemin(T) and typemax(T) and convert the result to type T.

See also trunc.

Examples

julia> clamp(200, Int8)
127

julia> clamp(-200, Int8)
-128

julia> trunc(Int, 4pi^2)
39

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

Clamp x to lie within range r.

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

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

Examples

julia> row = collect(-4:4)';

julia> clamp!(row, 0, Inf)
1×9 adjoint(::Vector{Int64}) with eltype Int64:
 0  0  0  0  0  1  2  3  4

julia> clamp.((-4:4)', 0, Inf)
1×9 Matrix{Float64}:
 0.0  0.0  0.0  0.0  0.0  1.0  2.0  3.0  4.0

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.

See also: abs2, unsigned, sign.

Examples

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808

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.

The overflow protection may impose a perceptible performance penalty.

Base.checked_neg(x)

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

The overflow protection may impose a perceptible performance penalty.

Base.checked_add(x, y)

Calculates x+y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_sub(x, y)

Calculates x-y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_mul(x, y)

Calculates x*y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_div(x, y)

Calculates div(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_rem(x, y)

Calculates x%y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_fld(x, y)

Calculates fld(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_mod(x, y)

Calculates mod(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.checked_cld(x, y)

Calculates cld(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

Base.add_with_overflow(x, y) -> (r, f)

Calculates r = x+y, with the flag f indicating whether overflow has occurred.

Base.sub_with_overflow(x, y) -> (r, f)

Calculates r = x-y, with the flag f indicating whether overflow has occurred.

Base.mul_with_overflow(x, y) -> (r, f)

Calculates r = x*y, with the flag f indicating whether overflow has occurred.

abs2(x)

Squared absolute value of x.

Examples

julia> abs2(-3)
9

copysign(x, y) -> z

Return z which has the magnitude of x and the same sign as y.

Examples

julia> copysign(1, -2)
-1

julia> copysign(-1, 2)
1

sign(x)

Return zero if x==0 and $x/|x|$ otherwise (i.e., ±1 for real x).

See also signbit, zero, copysign, flipsign.

Examples

julia> sign(-4.0)
-1.0

julia> sign(99)
1

julia> sign(-0.0)
-0.0

julia> sign(0 + im)
0.0 + 1.0im

signbit(x)

Returns true if the value of the sign of x is negative, otherwise false.

See also sign and copysign.

Examples

julia> signbit(-4)
true

julia> signbit(5)
false

julia> signbit(5.5)
false

julia> signbit(-4.1)
true

flipsign(x, y)

Return x with its sign flipped if y is negative. For example abs(x) = flipsign(x,x).

Examples

julia> flipsign(5, 3)
5

julia> flipsign(5, -3)
-5

sqrt(x)

Return $\sqrt{x}$. Throws DomainError for negative Real arguments. Use complex negative arguments instead. The prefix operator √ is equivalent to sqrt.

See also: hypot.

Examples

julia> sqrt(big(81))
9.0

julia> sqrt(big(-81))
ERROR: DomainError with -81.0:
NaN result for non-NaN input.
Stacktrace:
 [1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]

julia> sqrt(big(complex(-81)))
0.0 + 9.0im

julia> .√(1:4)
4-element Vector{Float64}:
 1.0
 1.4142135623730951
 1.7320508075688772
 2.0

isqrt(n::Integer)

Integer square root: the largest integer m such that m*m <= n.

julia> isqrt(5)
2

cbrt(x::Real)

Return the cube root of x, i.e. $x^{1/3}$. Negative values are accepted (returning the negative real root when $x < 0$).

The prefix operator ∛ is equivalent to cbrt.

Examples

julia> cbrt(big(27))
3.0

julia> cbrt(big(-27))
-3.0

real(z)

Return the real part of the complex number z.

See also: imag, reim, complex, isreal, Real.

Examples

julia> real(1 + 3im)
1

imag(z)

Return the imaginary part of the complex number z.

See also: conj, reim, adjoint, angle.

Examples

julia> imag(1 + 3im)
3

reim(z)

Return both the real and imaginary parts of the complex number z.

Examples

julia> reim(1 + 3im)
(1, 3)

conj(z)

Compute the complex conjugate of a complex number z.

See also: angle, adjoint.

Examples

julia> conj(1 + 3im)
1 - 3im

angle(z)

Compute the phase angle in radians of a complex number z.

See also: atan, cis.

Examples

julia> rad2deg(angle(1 + im))
45.0

julia> rad2deg(angle(1 - im))
-45.0

julia> rad2deg(angle(-1 - im))
-135.0

cis(A::AbstractMatrix)

Compute $\exp(i A)$ for a square matrix $A$.

Examples

julia> cis([π 0; 0 π]) ≈ -I
true

cis(z)

Return $\exp(iz)$.

See also cispi, angle.

Examples

julia> cis(π) ≈ -1
true

cispi(z)

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

Examples

julia> cispi(1)
-1.0 + 0.0im

julia> cispi(0.25 + 1im)
0.030556854645952924 + 0.030556854645952924im

binomial(n::Integer, k::Integer)

The binomial coefficient $\binom{n}{k}$, being the coefficient of the $k$th term in the polynomial expansion of $(1+x)^n$.

If $n$ is non-negative, then it is the number of ways to choose k out of n items:

\[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]

where $n!$ is the factorial function.

If $n$ is negative, then it is defined in terms of the identity

\[\binom{n}{k} = (-1)^k \binom{k-n-1}{k}\]

See also factorial.

Examples

julia> binomial(5, 3)
10

julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
10

julia> binomial(-5, 3)
-35

External links

• Binomial coefficient on Wikipedia.

factorial(n::Integer)

Factorial of n. If n is an Integer, the factorial is computed as an integer (promoted to at least 64 bits). Note that this may overflow if n is not small, but you can use factorial(big(n)) to compute the result exactly in arbitrary precision.

See also binomial.

Examples

julia> factorial(6)
720

julia> factorial(21)
ERROR: OverflowError: 21 is too large to look up in the table; consider using `factorial(big(21))` instead
Stacktrace:
[...]

julia> factorial(big(21))
51090942171709440000

External links

• Factorial on Wikipedia.

gcd(x, y...)

Greatest common (positive) divisor (or zero if all arguments are zero). The arguments may be integer and rational numbers.

Examples

julia> gcd(6,9)
3

julia> gcd(6,-9)
3

julia> gcd(6,0)
6

julia> gcd(0,0)
0

julia> gcd(1//3,2//3)
1//3

julia> gcd(1//3,-2//3)
1//3

julia> gcd(1//3,2)
1//3

julia> gcd(0, 0, 10, 15)
5

lcm(x, y...)

Least common (positive) multiple (or zero if any argument is zero). The arguments may be integer and rational numbers.

Examples

julia> lcm(2,3)
6

julia> lcm(-2,3)
6

julia> lcm(0,3)
0

julia> lcm(0,0)
0

julia> lcm(1//3,2//3)
2//3

julia> lcm(1//3,-2//3)
2//3

julia> lcm(1//3,2)
2//1

julia> lcm(1,3,5,7)
105

gcdx(a, b)

Computes the greatest common (positive) divisor of a and b and their Bézout coefficients, i.e. the integer coefficients u and v that satisfy $ua+vb = d = gcd(a, b)$. $gcdx(a, b)$ returns $(d, u, v)$.

The arguments may be integer and rational numbers.

Examples

julia> gcdx(12, 42)
(6, -3, 1)

julia> gcdx(240, 46)
(2, -9, 47)

ispow2(n::Number) -> Bool

Test whether n is an integer power of two.

See also count_ones, prevpow, nextpow.

Examples

julia> ispow2(4)
true

julia> ispow2(5)
false

julia> ispow2(4.5)
false

julia> ispow2(0.25)
true

julia> ispow2(1//8)
true

nextpow(a, x)

The smallest a^n not less than x, where n is a non-negative integer. a must be greater than 1, and x must be greater than 0.

See also prevpow.

Examples

julia> nextpow(2, 7)
8

julia> nextpow(2, 9)
16

julia> nextpow(5, 20)
25

julia> nextpow(4, 16)
16

prevpow(a, x)

The largest a^n not greater than x, where n is a non-negative integer. a must be greater than 1, and x must not be less than 1.

See also nextpow, isqrt.

Examples

julia> prevpow(2, 7)
4

julia> prevpow(2, 9)
8

julia> prevpow(5, 20)
5

julia> prevpow(4, 16)
16

nextprod(factors::Union{Tuple,AbstractVector}, n)

Next integer greater than or equal to n that can be written as $\prod k_i^{p_i}$ for integers $p_1$, $p_2$, etcetera, for factors $k_i$ in factors.

Examples

julia> nextprod((2, 3), 105)
108

julia> 2^2 * 3^3
108

invmod(n, m)

Take the inverse of n modulo m: y such that $n y = 1 \pmod m$, and $div(y,m) = 0$. This will throw an error if $m = 0$, or if $gcd(n,m) \neq 1$.

Examples

julia> invmod(2,5)
3

julia> invmod(2,3)
2

julia> invmod(5,6)
5

powermod(x::Integer, p::Integer, m)

Compute $x^p \pmod m$.

Examples

julia> powermod(2, 6, 5)
4

julia> mod(2^6, 5)
4

julia> powermod(5, 2, 20)
5

julia> powermod(5, 2, 19)
6

julia> powermod(5, 3, 19)
11

ndigits(n::Integer; base::Integer=10, pad::Integer=1)

Compute the number of digits in integer n written in base base (base must not be in [-1, 0, 1]), optionally padded with zeros to a specified size (the result will never be less than pad).

See also digits, count_ones.

Examples

julia> ndigits(12345)
5

julia> ndigits(1022, base=16)
3

julia> string(1022, base=16)
"3fe"

julia> ndigits(123, pad=5)
5

julia> ndigits(-123)
3

Base.add_sum(x, y)

The reduction operator used in sum. The main difference from + is that small integers are promoted to Int/UInt.

widemul(x, y)

Multiply x and y, giving the result as a larger type.

See also promote, Base.add_sum.

Examples

julia> widemul(Float32(3.0), 4.0) isa BigFloat
true

julia> typemax(Int8) * typemax(Int8)
1

julia> widemul(typemax(Int8), typemax(Int8))  # == 127^2
16129

evalpoly(x, p)

Evaluate the polynomial $\sum_k x^{k-1} p[k]$ for the coefficients p[1], p[2], ...; that is, the coefficients are given in ascending order by power of x. Loops are unrolled at compile time if the number of coefficients is statically known, i.e. when p is a Tuple. This function generates efficient code using Horner's method if x is real, or using a Goertzel-like ^[DK62] algorithm if x is complex.

Example

julia> evalpoly(2, (1, 2, 3))
17

@evalpoly(z, c...)

Evaluate the polynomial $\sum_k z^{k-1} c[k]$ for the coefficients c[1], c[2], ...; that is, the coefficients are given in ascending order by power of z. This macro expands to efficient inline code that uses either Horner's method or, for complex z, a more efficient Goertzel-like algorithm.

See also evalpoly.

Examples

julia> @evalpoly(3, 1, 0, 1)
10

julia> @evalpoly(2, 1, 0, 1)
5

julia> @evalpoly(2, 1, 1, 1)
7

@fastmath expr

Execute a transformed version of the expression, which calls functions that may violate strict IEEE semantics. This allows the fastest possible operation, but results are undefined – be careful when doing this, as it may change numerical results.

This sets the LLVM Fast-Math flags, and corresponds to the -ffast-math option in clang. See the notes on performance annotations for more details.

Examples

julia> @fastmath 1+2
3

julia> @fastmath(sin(3))
0.1411200080598672