# Mathematical Operations and Elementary Functions¶

Julia provides a complete collection of basic arithmetic and bitwise operators across all of its numeric primitive types, as well as providing portable, efficient implementations of a comprehensive collection of standard mathematical functions.

## Arithmetic Operators¶

The following arithmetic operators are supported on all primitive numeric types:

Expression | Name | Description |
---|---|---|

`+x` |
unary plus | the identity operation |

`-x` |
unary minus | maps values to their additive inverses |

`x + y` |
binary plus | performs addition |

`x - y` |
binary minus | performs subtraction |

`x * y` |
times | performs multiplication |

`x / y` |
divide | performs division |

`x \ y` |
inverse divide | equivalent to `y / x` |

`x ^ y` |
power | raises `x` to the `y` th power |

`x % y` |
remainder | equivalent to `rem(x,y)` |

as well as the negation on `Bool`

types:

Expression | Name | Description |
---|---|---|

`!x` |
negation | changes `true` to `false` and vice versa |

Julia’s promotion system makes arithmetic operations on mixtures of argument types “just work” naturally and automatically. See Conversion and Promotion for details of the promotion system.

Here are some simple examples using arithmetic operators:

```
julia> 1 + 2 + 3
6
julia> 1 - 2
-1
julia> 3*2/12
0.5
```

(By convention, we tend to space operators more tightly if they get applied before
other nearby operators. For instance, we would generally write `-x + 2`

to reflect
that first `x`

gets negated, and then `2`

is added to that result.)

## Bitwise Operators¶

The following bitwise operators are supported on all primitive integer types:

Expression | Name |
---|---|

`~x` |
bitwise not |

`x & y` |
bitwise and |

`x | y` |
bitwise or |

`x ⊻ y` |
bitwise xor (exclusive or) |

`x >>> y` |
logical shift right |

`x >> y` |
arithmetic shift right |

`x << y` |
logical/arithmetic shift left |

Here are some examples with bitwise operators:

```
julia> ~123
-124
julia> 123 & 234
106
julia> 123 | 234
251
julia> 123 ⊻ 234
145
julia> xor(123, 234)
145
julia> ~UInt32(123)
0xffffff84
julia> ~UInt8(123)
0x84
```

## Updating operators¶

Every binary arithmetic and bitwise operator also has an updating
version that assigns the result of the operation back into its left
operand. The updating version of the binary operator is formed by placing a
`=`

immediately after the operator. For example, writing `x += 3`

is
equivalent to writing `x = x + 3`

:

```
julia> x = 1
1
julia> x += 3
4
julia> x
4
```

The updating versions of all the binary arithmetic and bitwise operators are:

```
+= -= *= /= \= ÷= %= ^= &= |= ⊻= >>>= >>= <<=
```

Note

An updating operator rebinds the variable on the left-hand side. As a result, the type of the variable may change.

```
julia> x = 0x01; typeof(x)
UInt8
julia> x *= 2 #Same as x = x * 2
2
julia> isa(x, Int)
true
```

## Numeric Comparisons¶

Standard comparison operations are defined for all the primitive numeric types:

Operator | Name |
---|---|

`==` |
equality |

`!=` `≠` |
inequality |

`<` |
less than |

`<=` `≤` |
less than or equal to |

`>` |
greater than |

`>=` `≥` |
greater than or equal to |

Here are some simple examples:

```
julia> 1 == 1
true
julia> 1 == 2
false
julia> 1 != 2
true
julia> 1 == 1.0
true
julia> 1 < 2
true
julia> 1.0 > 3
false
julia> 1 >= 1.0
true
julia> -1 <= 1
true
julia> -1 <= -1
true
julia> -1 <= -2
false
julia> 3 < -0.5
false
```

Integers are compared in the standard manner — by comparison of bits. Floating-point numbers are compared according to the IEEE 754 standard:

- Finite numbers are ordered in the usual manner.
- Positive zero is equal but not greater than negative zero.
`Inf`

is equal to itself and greater than everything else except`NaN`

.`-Inf`

is equal to itself and less then everything else except`NaN`

.`NaN`

is not equal to, not less than, and not greater than anything, including itself.

The last point is potentially surprising and thus worth noting:

```
julia> NaN == NaN
false
julia> NaN != NaN
true
julia> NaN < NaN
false
julia> NaN > NaN
false
```

and can cause especial headaches with Arrays:

```
julia> [1 NaN] == [1 NaN]
false
```

Julia provides additional functions to test numbers for special values, which can be useful in situations like hash key comparisons:

Function | Tests if |
---|---|

`isequal(x, y)` |
`x` and `y` are identical |

`isfinite(x)` |
`x` is a finite number |

`isinf(x)` |
`x` is infinite |

`isnan(x)` |
`x` is not a number |

`isequal()`

considers `NaN`

s equal to each other:

```
julia> isequal(NaN,NaN)
true
julia> isequal([1 NaN], [1 NaN])
true
julia> isequal(NaN,NaN32)
true
```

`isequal()`

can also be used to distinguish signed zeros:

```
julia> -0.0 == 0.0
true
julia> isequal(-0.0, 0.0)
false
```

Mixed-type comparisons between signed integers, unsigned integers, and floats can be tricky. A great deal of care has been taken to ensure that Julia does them correctly.

For other types, `isequal()`

defaults to calling `==()`

, so if you want to
define equality for your own types then you only need to add a `==()`

method. If you define your own equality function, you should probably
define a corresponding `hash()`

method to ensure that `isequal(x,y)`

implies `hash(x) == hash(y)`

.

### Chaining comparisons¶

Unlike most languages, with the notable exception of Python, comparisons can be arbitrarily chained:

```
julia> 1 < 2 <= 2 < 3 == 3 > 2 >= 1 == 1 < 3 != 5
true
```

Chaining comparisons is often quite convenient in numerical code.
Chained comparisons use the `&&`

operator for scalar comparisons,
and the `&`

operator for elementwise comparisons, which allows them to
work on arrays. For example, `0 .< A .< 1`

gives a boolean array whose
entries are true where the corresponding elements of `A`

are between 0
and 1.

The operator `.<`

is intended for array objects; the operation
`A .< B`

is valid only if `A`

and `B`

have the same dimensions. The
operator returns an array with boolean entries and with the same dimensions
as `A`

and `B`

. Such operators are called *elementwise*; Julia offers a
suite of elementwise operators: `.*`

, `.+`

, etc. Some of the elementwise
operators can take a scalar operand such as the example `0 .< A .< 1`

in
the preceding paragraph.
This notation means that the scalar operand should be replicated for each entry of
the array.

Note the evaluation behavior of chained comparisons:

```
julia> v(x) = (println(x); x)
v (generic function with 1 method)
julia> v(1) < v(2) <= v(3)
2
1
3
true
julia> v(1) > v(2) <= v(3)
2
1
false
```

The middle expression is only evaluated once, rather than twice as it
would be if the expression were written as
`v(1) < v(2) && v(2) <= v(3)`

. However, the order of evaluations in a
chained comparison is undefined. It is strongly recommended not to use
expressions with side effects (such as printing) in chained comparisons.
If side effects are required, the short-circuit `&&`

operator should
be used explicitly (see Short-Circuit Evaluation).

### Operator Precedence¶

Julia applies the following order of operations, from highest precedence to lowest:

Category | Operators |
---|---|

Syntax | `.` followed by `::` |

Exponentiation | `^` and its elementwise equivalent `.^` |

Fractions | `//` and `.//` |

Multiplication | `* / % & \` and `.* ./ .% .\` |

Bitshifts | `<< >> >>>` and `.<< .>> .>>>` |

Addition | `+ - | ⊻` and `.+ .-` |

Syntax | `: ..` followed by `|>` |

Comparisons | `> < >= <= == === != !== <:` and `.> .< .>= .<= .== .!=` |

Control flow | `&&` followed by `||` followed by `?` |

Assignments | `= += -= *= /= //= \= ^= ÷= %= |= &= ⊻= <<= >>= >>>=` and `.+= .-= .*= ./= .//= .\= .^= .÷= .%=` |

### Elementary Functions¶

Julia provides a comprehensive collection of mathematical functions and operators. These mathematical operations are defined over as broad a class of numerical values as permit sensible definitions, including integers, floating-point numbers, rationals, and complexes, wherever such definitions make sense.

Moreover, these functions (like any Julia function) can be applied
in “vectorized” fashion to arrays and other collections with the
syntax `f.(A)`

, e.g. `sin.(A)`

will compute the elementwise
sine of each element of an array `A`

. See Dot Syntax for Vectorizing Functions.

## Numerical Conversions¶

Julia supports three forms of numerical conversion, which differ in their handling of inexact conversions.

- The notation
`T(x)`

or`convert(T,x)`

converts`x`

to a value of type`T`

.- If
`T`

is a floating-point type, the result is the nearest representable value, which could be positive or negative infinity. - If
`T`

is an integer type, an`InexactError`

is raised if`x`

is not representable by`T`

.

- If
`x % T`

converts an integer`x`

to a value of integer type`T`

congruent to`x`

modulo`2^n`

, where`n`

is the number of bits in`T`

. In other words, the binary representation is truncated to fit.- The Rounding functions take a type
`T`

as an optional argument. For example,`round(Int,x)`

is a shorthand for`Int(round(x))`

.

The following examples show the different forms.

```
julia> Int8(127)
127
julia> Int8(128)
ERROR: InexactError()
in Int8(::Int64) at ./sysimg.jl:66
...
julia> Int8(127.0)
127
julia> Int8(3.14)
ERROR: InexactError()
in convert(::Type{Int8}, ::Float64) at ./float.jl:635
in Int8(::Float64) at ./sysimg.jl:66
...
julia> Int8(128.0)
ERROR: InexactError()
in convert(::Type{Int8}, ::Float64) at ./float.jl:635
in Int8(::Float64) at ./sysimg.jl:66
...
julia> 127 % Int8
127
julia> 128 % Int8
-128
julia> round(Int8,127.4)
127
julia> round(Int8,127.6)
ERROR: InexactError()
in trunc(::Type{Int8}, ::Float64) at ./float.jl:628
in round(::Type{Int8}, ::Float64) at ./float.jl:332
...
```

See Conversion and Promotion for how to define your own conversions and promotions.

### Rounding functions¶

Function | Description | Return type |
---|---|---|

`round(x)` |
round `x` to the nearest integer |
`typeof(x)` |

`round(T, x)` |
round `x` to the nearest integer |
`T` |

`floor(x)` |
round `x` towards `-Inf` |
`typeof(x)` |

`floor(T, x)` |
round `x` towards `-Inf` |
`T` |

`ceil(x)` |
round `x` towards `+Inf` |
`typeof(x)` |

`ceil(T, x)` |
round `x` towards `+Inf` |
`T` |

`trunc(x)` |
round `x` towards zero |
`typeof(x)` |

`trunc(T, x)` |
round `x` towards zero |
`T` |

### Division functions¶

Function | Description |
---|---|

`div(x,y)` |
truncated division; quotient rounded towards zero |

`fld(x,y)` |
floored division; quotient rounded towards `-Inf` |

`cld(x,y)` |
ceiling division; quotient rounded towards `+Inf` |

`rem(x,y)` |
remainder; satisfies `x == div(x,y)*y + rem(x,y)` ; sign matches `x` |

`mod(x,y)` |
modulus; satisfies `x == fld(x,y)*y + mod(x,y)` ; sign matches `y` |

`mod1(x,y)` |
`mod()` with offset 1; returns `r∈(0,y]` for `y>0` or `r∈[y,0)` for `y<0` , where `mod(r, y) == mod(x, y)` |

`mod2pi(x)` |
modulus with respect to 2pi; `0 <= mod2pi(x) < 2pi` |

`divrem(x,y)` |
returns `(div(x,y),rem(x,y))` |

`fldmod(x,y)` |
returns `(fld(x,y),mod(x,y))` |

`gcd(x,y...)` |
greatest positive common divisor of `x` , `y` ,... |

`lcm(x,y...)` |
least positive common multiple of `x` , `y` ,... |

### Sign and absolute value functions¶

Function | Description |
---|---|

`abs(x)` |
a positive value with the magnitude of `x` |

`abs2(x)` |
the squared magnitude of `x` |

`sign(x)` |
indicates the sign of `x` , returning -1, 0, or +1 |

`signbit(x)` |
indicates whether the sign bit is on (true) or off (false) |

`copysign(x,y)` |
a value with the magnitude of `x` and the sign of `y` |

`flipsign(x,y)` |
a value with the magnitude of `x` and the sign of `x*y` |

### Powers, logs and roots¶

Function | Description |
---|---|

`sqrt(x)` `√x` |
square root of `x` |

`cbrt(x)` `∛x` |
cube root of `x` |

`hypot(x,y)` |
hypotenuse of right-angled triangle with other sides of length `x` and `y` |

`exp(x)` |
natural exponential function at `x` |

`expm1(x)` |
accurate `exp(x)-1` for `x` near zero |

`ldexp(x,n)` |
`x*2^n` computed efficiently for integer values of `n` |

`log(x)` |
natural logarithm of `x` |

`log(b,x)` |
base `b` logarithm of `x` |

`log2(x)` |
base 2 logarithm of `x` |

`log10(x)` |
base 10 logarithm of `x` |

`log1p(x)` |
accurate `log(1+x)` for `x` near zero |

`exponent(x)` |
binary exponent of `x` |

`significand(x)` |
binary significand (a.k.a. mantissa) of a floating-point number `x` |

For an overview of why functions like `hypot()`

, `expm1()`

, and `log1p()`

are necessary and useful, see John D. Cook’s excellent pair
of blog posts on the subject: expm1, log1p,
erfc,
and
hypot.

### Trigonometric and hyperbolic functions¶

All the standard trigonometric and hyperbolic functions are also defined:

```
sin cos tan cot sec csc
sinh cosh tanh coth sech csch
asin acos atan acot asec acsc
asinh acosh atanh acoth asech acsch
sinc cosc atan2
```

These are all single-argument functions, with the exception of
atan2, which gives the angle
in radians between the *x*-axis
and the point specified by its arguments, interpreted as *x* and *y*
coordinates.

Additionally, `sinpi(x)`

and `cospi(x)`

are provided for more accurate computations
of `sin(pi*x)`

and `cos(pi*x)`

respectively.

In order to compute trigonometric functions with degrees
instead of radians, suffix the function with `d`

. For example, `sind(x)`

computes the sine of `x`

where `x`

is specified in degrees.
The complete list of trigonometric functions with degree variants is:

```
sind cosd tand cotd secd cscd
asind acosd atand acotd asecd acscd
```

### Special functions¶

Function | Description |
---|---|

`erf(x)` |
error function at `x` |

`erfc(x)` |
complementary error function, i.e. the accurate version of `1-erf(x)` for large `x` |

`erfinv(x)` |
inverse function to `erf()` |

`erfcinv(x)` |
inverse function to `erfc()` |

`erfi(x)` |
imaginary error function defined as `-im * erf(x * im)` , where `im` is the imaginary unit |

`erfcx(x)` |
scaled complementary error function, i.e. accurate `exp(x^2) * erfc(x)` for large `x` |

`dawson(x)` |
scaled imaginary error function, a.k.a. Dawson function, i.e. accurate `exp(-x^2) * erfi(x) * sqrt(pi) / 2` for large `x` |

`gamma(x)` |
gamma function at `x` |

`lgamma(x)` |
accurate `log(gamma(x))` for large `x` |

`lfact(x)` |
accurate `log(factorial(x))` for large `x` ; same as `lgamma(x+1)` for `x > 1` , zero otherwise |

`digamma(x)` |
digamma function (i.e. the derivative of `lgamma()` ) at `x` |

`beta(x,y)` |
beta function at `x,y` |

`lbeta(x,y)` |
accurate `log(beta(x,y))` for large `x` or `y` |

`eta(x)` |
Dirichlet eta function at `x` |

`zeta(x)` |
Riemann zeta function at `x` |

`airy(z)` , `airyai(z)` , `airy(0,z)` |
Airy Ai function at `z` |

`airyprime(z)` , `airyaiprime(z)` , `airy(1,z)` |
derivative of the Airy Ai function at `z` |

`airybi(z)` , `airy(2,z)` |
Airy Bi function at `z` |

`airybiprime(z)` , `airy(3,z)` |
derivative of the Airy Bi function at `z` |

`airyx(z)` , `airyx(k,z)` |
scaled Airy AI function and `k` th derivatives at `z` |

`besselj(nu,z)` |
Bessel function of the first kind of order `nu` at `z` |

`besselj0(z)` |
`besselj(0,z)` |

`besselj1(z)` |
`besselj(1,z)` |

`besseljx(nu,z)` |
scaled Bessel function of the first kind of order `nu` at `z` |

`bessely(nu,z)` |
Bessel function of the second kind of order `nu` at `z` |

`bessely0(z)` |
`bessely(0,z)` |

`bessely1(z)` |
`bessely(1,z)` |

`besselyx(nu,z)` |
scaled Bessel function of the second kind of order `nu` at `z` |

`besselh(nu,k,z)` |
Bessel function of the third kind (a.k.a. Hankel function) of order `nu` at `z` ; `k` must be either `1` or `2` |

`hankelh1(nu,z)` |
`besselh(nu, 1, z)` |

`hankelh1x(nu,z)` |
scaled `besselh(nu, 1, z)` |

`hankelh2(nu,z)` |
`besselh(nu, 2, z)` |

`hankelh2x(nu,z)` |
scaled `besselh(nu, 2, z)` |

`besseli(nu,z)` |
modified Bessel function of the first kind of order `nu` at `z` |

`besselix(nu,z)` |
scaled modified Bessel function of the first kind of order `nu` at `z` |

`besselk(nu,z)` |
modified Bessel function of the second kind of order `nu` at `z` |

`besselkx(nu,z)` |
scaled modified Bessel function of the second kind of order `nu` at `z` |