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 yth 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 less tightly binding operators less tightly, but there are no syntactic constraints.)

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> ~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 NaNs 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:

v(x) = (println(x); x)

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.

Rounding functions

Function Description Return type
round(x) round x to the nearest integer FloatingPoint
iround(x) round x to the nearest integer Integer
floor(x) round x towards -Inf FloatingPoint
ifloor(x) round x towards -Inf Integer
ceil(x) round x towards +Inf FloatingPoint
iceil(x) round x towards +Inf Integer
trunc(x) round x towards zero FloatingPoint
itrunc(x) round x towards zero Integer

Division functions

Function Description
div(x,y) truncated division; quotient rounded towards zero
fld(x,y) floored division; quotient rounded towards -Inf
rem(x,y) remainder; satisfies x == div(x,y)*y + rem(x,y); sign matches x
divrem(x,y) returns (div(x,y),rem(x,y))
mod(x,y) modulus; satisfies x == fld(x,y)*y + mod(x,y); sign matches y
mod2pi(x) modulus with respect to 2pi; 0 <= mod2pi(x)  < 2pi
gcd(x,y...) greatest common divisor of x, y,...; sign matches x
lcm(x,y...) least common multiple of x, y,...; sign matches x

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