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