Mathematical functions
Basics · Transcendental functions · Trigonometry · Statistics · Random sampling, distributions · Number theory · Numerical methods · Geometry · Algebra · Trigonometry (extra)
Basics
Defined in: core::functions
id
(Identity function)
Return the input value.
fn id<A>(x: A) -> A
abs
(Absolute value)
Return the absolute value \( |x| \) of the input. This works for quantities, too: abs(-5 m) = 5 m
.
More information here.
fn abs<T: Dim>(x: T) -> T
sqrt
(Square root)
Return the square root \( \sqrt{x} \) of the input: sqrt(121 m^2) = 11 m
.
More information here.
fn sqrt<D: Dim>(x: D^2) -> D
cbrt
(Cube root)
Return the cube root \( \sqrt[3]{x} \) of the input: cbrt(8 m^3) = 2 m
.
More information here.
fn cbrt<D: Dim>(x: D^3) -> D
sqr
(Square function)
Return the square of the input, \( x^2 \): sqr(5 m) = 25 m^2
.
fn sqr<D: Dim>(x: D) -> D^2
round
(Rounding)
Round to the nearest integer. If the value is half-way between two integers, round away from \( 0 \). See also: round_in
.
More information here.
fn round(x: Scalar) -> Scalar
round_in
(Rounding)
Round to the nearest multiple of base
. For example: round_in(m, 5.3 m) == 5 m
.
fn round_in<D: Dim>(base: D, value: D) -> D
floor
(Floor function)
Returns the largest integer less than or equal to \( x \). See also: floor_in
.
More information here.
fn floor(x: Scalar) -> Scalar
floor_in
(Floor function)
Returns the largest integer multiple of base
less than or equal to value
. For example: floor_in(m, 5.7 m) == 5 m
.
fn floor_in<D: Dim>(base: D, value: D) -> D
ceil
(Ceil function)
Returns the smallest integer greater than or equal to \( x \). See also: ceil_in
.
More information here.
fn ceil(x: Scalar) -> Scalar
ceil_in
(Ceil function)
Returns the smallest integer multuple of base
greater than or equal to value
. For example: ceil_in(m, 5.3 m) == 6 m
.
fn ceil_in<D: Dim>(base: D, value: D) -> D
trunc
(Truncation)
Returns the integer part of \( x \). Non-integer numbers are always truncated towards zero. See also: trunc_in
.
More information here.
fn trunc(x: Scalar) -> Scalar
trunc_in
(Truncation)
Truncates to an integer multiple of base
(towards zero). For example: trunc_in(m, -5.7 m) == -5 m
.
fn trunc_in<D: Dim>(base: D, value: D) -> D
mod
(Modulo)
Calculates the least nonnegative remainder of \( a (\mod b) \). More information here.
fn mod<T: Dim>(a: T, b: T) -> T
Transcendental functions
Defined in: math::transcendental
exp
(Exponential function)
The exponential function, \( e^x \). More information here.
fn exp(x: Scalar) -> Scalar
ln
(Natural logarithm)
The natural logarithm with base \( e \). More information here.
fn ln(x: Scalar) -> Scalar
log
(Natural logarithm)
The natural logarithm with base \( e \). More information here.
fn log(x: Scalar) -> Scalar
log10
(Common logarithm)
The common logarithm with base \( 10 \). More information here.
fn log10(x: Scalar) -> Scalar
log2
(Binary logarithm)
The binary logarithm with base \( 2 \). More information here.
fn log2(x: Scalar) -> Scalar
gamma
(Gamma function)
The gamma function, \( \Gamma(x) \). More information here.
fn gamma(x: Scalar) -> Scalar
Trigonometry
Defined in: math::trigonometry
sin
(Sine)
More information here.
fn sin(x: Scalar) -> Scalar
cos
(Cosine)
More information here.
fn cos(x: Scalar) -> Scalar
tan
(Tangent)
More information here.
fn tan(x: Scalar) -> Scalar
asin
(Arc sine)
More information here.
fn asin(x: Scalar) -> Scalar
acos
(Arc cosine)
More information here.
fn acos(x: Scalar) -> Scalar
atan
(Arc tangent)
More information here.
fn atan(x: Scalar) -> Scalar
atan2
More information here.
fn atan2<T: Dim>(y: T, x: T) -> Scalar
sinh
(Hyperbolic sine)
More information here.
fn sinh(x: Scalar) -> Scalar
cosh
(Hyperbolic cosine)
More information here.
fn cosh(x: Scalar) -> Scalar
tanh
(Hyperbolic tangent)
More information here.
fn tanh(x: Scalar) -> Scalar
asinh
(Area hyperbolic sine)
More information here.
fn asinh(x: Scalar) -> Scalar
acosh
(Area hyperbolic cosine)
More information here.
fn acosh(x: Scalar) -> Scalar
atanh
(Area hyperbolic tangent )
More information here.
fn atanh(x: Scalar) -> Scalar
Statistics
Defined in: math::statistics
maximum
(Maxmimum)
Get the largest element of a list: maximum([30 cm, 2 m]) = 2 m
.
fn maximum<D: Dim>(xs: List<D>) -> D
minimum
(Minimum)
Get the smallest element of a list: minimum([30 cm, 2 m]) = 30 cm
.
fn minimum<D: Dim>(xs: List<D>) -> D
mean
(Arithmetic mean)
Calculate the arithmetic mean of a list of quantities: mean([1 m, 2 m, 300 cm]) = 2 m
.
More information here.
fn mean<D: Dim>(xs: List<D>) -> D
variance
(Variance)
Calculate the population variance of a list of quantities. More information here.
fn variance<D: Dim>(xs: List<D>) -> D^2
stdev
(Standard deviation)
Calculate the population standard deviation of a list of quantities. More information here.
fn stdev<D: Dim>(xs: List<D>) -> D
median
(Median)
Calculate the median of a list of quantities. More information here.
fn median<D: Dim>(xs: List<D>) -> D
Random sampling, distributions
Defined in: core::random
, math::distributions
random
(Standard uniform distribution sampling)
Uniformly samples the interval \( [0,1) \).
fn random() -> Scalar
rand_uniform
(Continuous uniform distribution sampling)
Uniformly samples the interval \( [a,b) \) if \( a \le b \) or \( [b,a) \) if \( b<a \) using inversion sampling. More information here.
fn rand_uniform<T: Dim>(a: T, b: T) -> T
rand_int
(Discrete uniform distribution sampling)
Uniformly samples integers from the interval \( [a, b] \). More information here.
fn rand_int(a: Scalar, b: Scalar) -> Scalar
rand_bernoulli
(Bernoulli distribution sampling)
Samples a Bernoulli random variable. That is, \( 1 \) with probability \( p \) and \( 0 \) with probability \( 1-p \). The parameter \( p \) must be a probability (\( 0 \le p \le 1 \)). More information here.
fn rand_bernoulli(p: Scalar) -> Scalar
rand_binom
(Binomial distribution sampling)
Samples a binomial distribution by doing \( n \) Bernoulli trials with probability \( p \). The parameter \( n \) must be a positive integer, the parameter \( p \) must be a probability (\( 0 \le p \le 1 \)). More information here.
fn rand_binom(n: Scalar, p: Scalar) -> Scalar
rand_norm
(Normal distribution sampling)
Samples a normal distribution with mean \( \mu \) and standard deviation \( \sigma \) using the Box-Muller transform. More information here.
fn rand_norm<T: Dim>(μ: T, σ: T) -> T
rand_geom
(Geometric distribution sampling)
Samples a geometric distribution (the distribution of the number of Bernoulli trials with probability \( p \) needed to get one success) by inversion sampling. The parameter \( p \) must be a probability (\( 0 \le p \le 1 \)). More information here.
fn rand_geom(p: Scalar) -> Scalar
rand_poisson
(Poisson distribution sampling)
Sampling a poisson distribution with rate \( \lambda \), that is, the distribution of the number of events occurring in a fixed interval if these events occur with mean rate \( \lambda \). The rate parameter \( \lambda \) must be non-negative. More information here.
fn rand_poisson(λ: Scalar) -> Scalar
rand_expon
(Exponential distribution sampling)
Sampling an exponential distribution (the distribution of the distance between events in a Poisson process with rate \( \lambda \)) using inversion sampling. The rate parameter \( \lambda \) must be positive. More information here.
fn rand_expon<T: Dim>(λ: T) -> 1 / T
rand_lognorm
(Log-normal distribution sampling)
Sampling a log-normal distribution, that is, a distribution whose logarithm is a normal distribution with mean \( \mu \) and standard deviation \( \sigma \). More information here.
fn rand_lognorm(μ: Scalar, σ: Scalar) -> Scalar
rand_pareto
(Pareto distribution sampling)
Sampling a Pareto distribution with minimum value min
and shape parameter \( \alpha \) using inversion sampling. Both parameters must be positive.
More information here.
fn rand_pareto<T: Dim>(α: Scalar, min: T) -> T
Number theory
Defined in: math::number_theory
gcd
(Greatest common divisor)
The largest positive integer that divides each of the integers \( a \) and \( b \). More information here.
fn gcd(a: Scalar, b: Scalar) -> Scalar
lcm
(Least common multiple)
The smallest positive integer that is divisible by both \( a \) and \( b \). More information here.
fn lcm(a: Scalar, b: Scalar) -> Scalar
Numerical methods
Defined in: numerics::diff
, numerics::solve
, numerics::fixed_point
diff
(Numerical differentiation)
Compute the numerical derivative of the function \( f \) at point \( x \) using the central difference method. More information here.
fn diff<X: Dim, Y: Dim>(f: Fn[(X) -> Y], x: X) -> Y / X
root_bisect
(Bisection method)
Find the root of the function \( f \) in the interval \( [x_1, x_2] \) using the bisection method. The function \( f \) must be continuous and \( f(x_1) \cdot f(x_2) < 0 \). More information here.
fn root_bisect<X: Dim, Y: Dim>(f: Fn[(X) -> Y], x1: X, x2: X, x_tol: X, y_tol: Y) -> X
root_newton
(Newton’s method)
Find the root of the function \( f(x) \) and its derivative \( f’(x) \) using Newton’s method. More information here.
fn root_newton<X: Dim, Y: Dim>(f: Fn[(X) -> Y], f_prime: Fn[(X) -> Y / X], x0: X, y_tol: Y) -> X
fixed_point
(Fixed-point iteration)
Compute the approximate fixed point of a function \( f: X \rightarrow X \) starting from \( x_0 \), until \( |f(x) - x| < ε \). More information here.
fn fixed_point<X: Dim>(f: Fn[(X) -> X], x0: X, ε: X) -> X
Geometry
Defined in: math::geometry
hypot2
The length of the hypotenuse of a right-angled triangle \( \sqrt{x^2+y^2} \).
fn hypot2<T: Dim>(x: T, y: T) -> T
hypot3
The Euclidean norm of a 3D vector \( \sqrt{x^2+y^2+z^2} \).
fn hypot3<T: Dim>(x: T, y: T, z: T) -> T
circle_area
The area of a circle, \( \pi r^2 \).
fn circle_area<L: Dim>(radius: L) -> L^2
circle_circumference
The circumference of a circle, \( 2\pi r \).
fn circle_circumference<L: Dim>(radius: L) -> L
sphere_area
The surface area of a sphere, \( 4\pi r^2 \).
fn sphere_area<L: Dim>(radius: L) -> L^2
sphere_volume
The volume of a sphere, \( \frac{4}{3}\pi r^3 \).
fn sphere_volume<L: Dim>(radius: L) -> L^3
Algebra
Defined in: extra::algebra
quadratic_equation
(Solve quadratic equations)
Returns the solutions of the equation a x² + b x + c = 0. More information here.
fn quadratic_equation<A: Dim, B: Dim>(a: A, b: B, c: B^2 / A) -> List<B / A>
Trigonometry (extra)
Defined in: math::trigonometry_extra
cot
fn cot(x: Scalar) -> Scalar
acot
fn acot(x: Scalar) -> Scalar
coth
fn coth(x: Scalar) -> Scalar
acoth
fn acoth(x: Scalar) -> Scalar
secant
fn secant(x: Scalar) -> Scalar
arcsecant
fn arcsecant(x: Scalar) -> Scalar
cosecant
fn cosecant(x: Scalar) -> Scalar
csc
fn csc(x: Scalar) -> Scalar
acsc
fn acsc(x: Scalar) -> Scalar
sech
fn sech(x: Scalar) -> Scalar
asech
fn asech(x: Scalar) -> Scalar
csch
fn csch(x: Scalar) -> Scalar
acsch
fn acsch(x: Scalar) -> Scalar