Introduction

“You see, Vergon 6 was once filled with the super-dense substance known as dark matter, each pound of which weighs over 10,000 pounds.” — Futurama, S1E4

Numbat is a statically typed programming language for scientific computations with first class support for physical dimensions and units.

You can use it for simple mathematical computations:

>>> 1920/16*9

    = 1080

>>> 2^32

    = 4294967296

>>> sqrt(1.4^2 + 1.5^2) * cos(pi/3)^2

    = 0.512957

The real strength of Numbat, however, is to perform calculations with physical units:

>>> 8 km / (1 h + 25 min)

  8 kilometer / (1 hour + 25 minute)

    = 5.64706 km/h    [Velocity]

>>> 140 € -> GBP

  140 euro ➞ british_pound

    = 120.768 £    [Money]

>>> atan2(30 cm, 1 m) -> deg

  atan2(30 centimeter, 1 meter) ➞ degree

    = 16.6992°

>>> let ω = 2π c / 660 nm

  let ω: Frequency = 2 π × c / 660 nanometer

>>> ℏ ω -> eV

  ℏ × ω ➞ electronvolt

    = 1.87855 eV    [Energy]

Read the tutorial to learn more about the language or look at some example programs. You can also jump directly to the syntax reference.

Tutorial

In this tutorial, you will use Numbat to calculate how many bananas you would need to power a house. This is based on an article in the great what if? series by the author of the xkcd comics.

Bananas contain potassium. In its natural form, potassium contains a tiny fraction (0.0117%) of the isotope ⁴⁰K, which is radioactive. The idea is to use the radioactive decay energy as a power source. Open an interactive Numbat session by typing numbat in your favorite terminal emulator. We start by entering a few facts about potassium-40:

let halflife = 1.25 billion years
let occurrence = 0.0117%
let molar_mass = 40 g / mol

New constants are introduced with the let keyword. We define these physical quantities with their respective physical units (years, percent, g / mol) in order to profit from Numbat’s unit-safety and unit-conversion features later on.

Our first goal is to compute the radioactivity of natural potassium. Instead of dealing with the half-life, we want to know the decay rate. When entering the following computation, you can try Numbat’s auto-completion functionality. Instead of typing out halflife, just type half and press Tab.

let decay_rate = ln(2) / halflife

As you can see, we can use typical mathematical functions such as the natural logarithm ln. Next, we are interested how much radioactivity comes from a certain mass of potassium:

let radioactivity =
    N_A * occurrence * decay_rate / molar_mass -> Bq / g

The -> Bq / g part at the end converts the expression to Becquerel per gram. If you type in radioactivity, you should see a result of roughly 31 Bq / g, i.e. 31 radioactive decays per second, per gram of potassium.

The unit conversion also serves another purpose. If anything would be wrong with our calculation at the units-level, Numbat would detect that and show an error. Unit safety is a powerful concept not just because you can eliminate an entire category of errors, but also because it makes your computations more readable.

We are interested in the radioactivity of bananas, so we first introduce a new (base) unit:

unit banana

This lets us write readable code like

let potassium_per_banana = 451 mg / banana

let radioactivity_banana = potassium_per_banana * radioactivity -> Bq / banana

and should give you a result of roughly 14 Bq / banana. Adding unit conversions at the end of unit definitions is one way to enforce unit safety. An even more powerful way to do this is to add type annotations: For example, to define the decay energy for a single potassium-40 atom, you can optionally add a : Energy annotation that will be enforced by Numbat:

let energy_per_decay: Energy = 11% × 1.5 MeV + 89% × 1.3 MeV

This also works with custom units since Numbat adds new physical dimensions (types) implicitly:

let power_per_banana: Power / Banana = radioactivity_banana * energy_per_decay

You’ll also notice that types can be combined via mathematical operators such as / in this example.

How many bananas we need to power a household is going to depend on the average power consumption of that household. So we are defining a simple function

fn household_power(annual_consumption: Energy) -> Power = annual_consumption / year

This allows us to finally answer the original question (for a typical US household in 2021)

household_power(10000 kWh) / power_per_banana

This should give you a result of roughly 4×10¹⁴ bananas¹.

Attribution

The images in this tutorial are from https://what-if.xkcd.com/158/. They are licensed under the Creative Commons Attribution-NonCommercial 2.5 License. Details and usage notes can be found at https://xkcd.com/license.html.

Examples

This chapter shows some exemplary Numbat programs from various disciplines and sciences.

Acidity

Run this example

# Compute the pH (acidity) of a solution based
# on the activity of hydrogen ions
#
# https://en.wikipedia.org/wiki/PH

fn pH_acidity(activity_hplus: Molarity) -> Scalar =
    - log10(activity_hplus / (mol / L))

print(pH_acidity(5e-6 mol / L))

Barometric formula

Run this example

# This script calculates the air pressure at a specified
# height above sea level using the barometric formula.

let p0: Pressure = 1 atm
let t0: Temperature = 288.15 K

dimension TemperatureGradient = Temperature / Length
let lapse_rate: TemperatureGradient = 0.65 K / 100 m

fn air_pressure(height: Length) -> Pressure =
    p0 · (1 - lapse_rate · height / t0)^5.255

print("Air pressure 1500 m above sea level: {air_pressure(1500 m) -> hPa}")

Body mass index

Run this example

# This script calculates the Body Mass Index (BMI) based on
# the provided mass and height values.

unit BMI: Mass / Length^2 = kg / m^2

fn body_mass_index(mass: Mass, height: Length) =
    mass / height² -> BMI

print(body_mass_index(70 kg, 1.75 m))

Factorial

Run this example

# Naive factorial implementation to showcase recursive
# functions and conditionals.

fn factorial(n) =
  if n < 1
    then 1
    else n × factorial(n - 1)

# Compare result with the builtin factorial operator
assert_eq(factorial(10), 10!)

Flow rate in a pipe

Run this example

# This script calculates and prints the flow rate in a pipe
# using the Hagen-Poiseuille equation. It assumes the dynamic
# viscosity of water and allows for inputs of pipe radius,
# pipe length, and pressure difference.

let μ_water: DynamicViscosity = 1 mPa·s

fn flow_rate(radius: Length, length: Length, Δp: Pressure) -> FlowRate =
    π × radius^4 × Δp / (8 μ_water × length)

let pipe_radius = 1 cm
let pipe_length = 10 m
let Δp = 0.1 bar

let Q = flow_rate(pipe_radius, pipe_length, Δp)
print("Flow rate: {Q -> L/s}")

Medication dosage

Run this example

# This script calculates the total daily dose and per intake
# dose of a medication based on a person's body weight.

@aliases(takings)
unit taking

let body_weight = 75 kg
let dosage = (60 mg / kg) / day
let frequency = 3 takings / day

let total_daily_dose = dosage * body_weight -> mg / day
print("Total daily dose: {total_daily_dose}")

let single_dose = total_daily_dose / frequency
print("Single dose:      {single_dose}")

Molarity

Run this example

# This script calculates and prints the molarity of a salt
# water solution, given a fixed mass of salt (NaCl) and a
# volume of water.

let molar_mass_NaCl = 58.44 g / mol

fn molarity(mass: Mass, volume: Volume) -> Molarity =
    (mass / molar_mass_NaCl) / volume

let salt_mass = 9 g
let water_volume = 1 L

print(molarity(salt_mass, water_volume) -> mmol / l)

Musical note frequency

Run this example

# Musical note frequencies in the 12 equal temperament system

let frequency_A4: Frequency = 440 Hz  # the A above middle C, A4

fn note_frequency(n: Scalar) -> Frequency = frequency_A4 * 2^(n / 12)

print("A5: {note_frequency(12)}")  # one octave higher up, 880 Hz
print("E4: {note_frequency(7)}")
print("C4: {note_frequency(-3)}")

Paper sizes

Run this example

# Compute ISO 216 paper sizes for the A series
#
# https://en.wikipedia.org/wiki/ISO_216

struct PaperSize {
    width: Length,
    height: Length,
}

fn paper_size_A(n: Scalar) -> PaperSize =
  if n == 0
    then
      PaperSize {
        width: 841 mm,
        height: 1189 mm
      }
    else
      PaperSize {
        width: floor_in(mm, paper_size_A(n - 1).height / 2),
        height: paper_size_A(n - 1).width,
      }


fn paper_area(size: PaperSize) -> Area =
    size.width * size.height


fn size_as_string(size: PaperSize) = "{size.width:>4} × {size.height:>5}   {paper_area(size) -> cm²:>6.1f}"
fn row(n) = "A{n:<3}   {size_as_string(paper_size_A(n))}"

print("Name    Width     Height        Area  ")
print("----   -------   --------   ----------")
print(join(map(row, range(0, 10)), "\n"))

Population growth

Run this example

# Exponential model for population growth

let initial_population = 50_000 people
let growth_rate = 2% per year

fn predict_population(t) =
    initial_population × e^(growth_rate·t) |> round_in(people)

print("Population in  20 years: {predict_population(20 years)}")
print("Population in 100 years: {predict_population(1 century)}")

Recipe

Run this example

# Scale ingredient quantities based on desired servings.

@aliases(servings)
unit serving

let original_recipe_servings = 2 servings
let desired_servings = 3 servings

fn scale(quantity) =
    quantity × desired_servings / original_recipe_servings

print("Milk:          {scale(500 ml)}")
print("Flour:         {scale(250 g)}")
print("Sugar:         {scale(2 cups)}")
print("Baking powder: {scale(4 tablespoons)}")

Voyager

Run this example

# How many photons are received per bit transmitted from Voyager 1?
#
# This calculation is adapted from a Physics Stack Exchange answer [1].
#
# [1] https://physics.stackexchange.com/a/816710

# Voyager radio transmission:
let datarate = 160 bps
let f = 8.3 GHz
let P_transmit = 23 W

let ω = 2π f
let λ = c / f

@aliases(photon)
unit photons

let energy_per_photon = ℏ ω / photon

let photon_rate = P_transmit / energy_per_photon -> photons/s

print("Voyager sends data at a rate of {datarate} with {P_transmit}.")
print("At a frequency of {f}, this amounts to {photon_rate:.0e}.")

# Voyager dish antenna:
let d_voyager = 3.7 m

# Voyagers distance to Earth:
let R = 23.5 billion kilometers  # as of 2024

# Diameter of receiver dish:
let d_receiver = 70 m

let irradiance = P_transmit / (4π R²)
let P_received: Power = irradiance × (π d_voyager / λ)² × (π d_receiver² / 4)

print("A {d_receiver} dish on Earth will receive {P_received -> aW:.1f} of power.")

let photon_rate_receiver = P_received / energy_per_photon -> photons/s
let photons_per_bit = photon_rate_receiver / datarate -> photons/bit

print()
print("This corresponds to {photon_rate_receiver}.")
print("Which means {photons_per_bit:.0}.")

XKCD 681

Run this example

# Gravity wells
#
# https://xkcd.com/681/

use extra::astronomy

fn well_depth(mass: Mass, radius: Length) -> Length =
    G × mass / (g0 × radius) -> km

print("Gravity well depths:")
print("Sun     {well_depth(solar_mass, solar_radius):8.0f}")
print("Earth   {well_depth(earth_mass, earth_radius):8.0f}")
print("Moon    {well_depth(lunar_mass, lunar_radius):8.0f}")
print("Mars    {well_depth(mars_mass,  mars_radius):8.0f}")
print("Jupiter {well_depth(jupiter_mass, jupiter_radius):8.0f}")

Source: https://xkcd.com/681/

XKCD 687

Run this example

# Dimensional analysis
#
# https://xkcd.com/687/

let core_pressure = 3.5 million atmospheres
let prius_milage = 50 miles per gallon
let min_width_channel = 21 miles

# Make sure that the result is dimensionless:
let r: Scalar =
    planck_energy / core_pressure × prius_milage / min_width_channel

print("{r} ≈ π ?")

Source: https://xkcd.com/687/

XKCD 2585

Run this example

# Rounding
#
# https://xkcd.com/2585/

let speed = 17 mph |>
  round_in(meters/sec) |>
  round_in(knots) |>
  round_in(fathoms/sec) |>
  round_in(furlongs/min) |>
  round_in(fathoms/sec) |>
  round_in(kph) |>
  round_in(knots) |>
  round_in(kph) |>
  round_in(furlongs/hour) |>
  round_in(mi/h) |>
  round_in(m/s) |>
  round_in(furlongs/min) |>
  round_in(yards/sec) |>
  round_in(fathoms/sec) |>
  round_in(m/s) |>
  round_in(mph) |>
  round_in(furlongs/min) |>
  round_in(knots) |>
  round_in(yards/sec) |>
  round_in(fathoms/sec) |>
  round_in(knots) |>
  round_in(furlongs/min) |>
  round_in(mph)

print("I can ride my bike at {speed}.")
print("If you round.")

Source: https://xkcd.com/2585/

XKCD 2812

Run this example

# Solar panel placement
#
# Solar energy tip: To maximize sun exposure, always
# orient your panels downward and install them on the
# surface of the sun.
#
# https://xkcd.com/2812/
#
# [1] https://en.wikipedia.org/wiki/Solar_luminosity
# [2] https://en.wikipedia.org/wiki/Sun

let net_metering_rate = $ 0.20 / kWh
let panel_area = 1 m²
let panel_efficiency = 20 %

fn savings(i: Irradiance) -> Money / Time =
    net_metering_rate × i × panel_area × panel_efficiency -> $/year

print("Option A: On the roof, south facing")

let savings_a = savings(4 kWh/m²/day)

print(savings_a |> round_in($/year))

print()
print("Option B: On the sun, downward facing")

dimension Luminosity = Power

let sun_luminosity: Luminosity = 3.828e26 W  # [1]
let sun_area: Area = 6.09e12 km^2            # [2]

let savings_b = savings(sun_luminosity / sun_area)

print(savings_b |> round_in($/year))

Source: https://xkcd.com/2812/

Basics

This chapter introduces language features that are required to perform basic computations in Numbat or to write small programs.

Number notation

Numbers in Numbat can be written in the following forms:

• Integer notation
  • 12345
  • 12_345 — with decimal separators
• Floating point notation
  • 0.234
  • .234 — without the leading zero
• Scientific notation
  • 1.234e15
  • 1.234e+15
  • 1e-9
  • 1.0e-9
• Non-decimal bases notation
  • 0x2A — Hexadecimal
  • 0o52 — Octal
  • 0b101010 — Binary
• Non-finite numbers
  • NaN — Not a number
  • inf — Infinity

Convert numbers to other bases

You can use the bin, oct, dec and hex functions to convert numbers to binary, octal, decimal and hexadecimal bases, respectively. You can call those using hex(2^16 - 1), but they are also available as targets of the conversion operator ->/to, so you can write expressions like:

Examples:

0xffee -> bin
42 -> oct
2^16 - 1 -> hex

# using 'to':
0xffee to bin

You can also use base(b, n) to convert a number n to base b. Using the reverse function application operator |> you can write this in a similar style to the previous examples:

273 |> base(3)
144 |> base(12)

Unit notation

Most units can be entered in the same way that they would appear in textbook calculations. They usually have a long form (meter, degrees, byte, …), a plural form (meters, degrees, bytes), and a short alias (m, °, B). For a full list of supported units, see this page.

All SI-accepted units support metric prefixes (mm, cm, km, … or millimeter, centimeter, kilometer, …) and — where sensible — units allow for binary prefixes (MiB, GiB, … or mebibyte, gibibyte, …). Note that the short-form prefixes can only be used with the short version of the unit, and vice versa (that is: kmeter and kilom are not allowed, only km and kilometer).

Units can be combined using mathematical operations such as multiplication, division and exponentiation: kg * m/s^2, km/h, m², meter per second.

The following snippet shows various styles of entering units:

2 min + 1 s
150 cm
sin(30°)
50 mph
6 MiB

2 minutes + 1 second
150 centimeters
sin(30 degrees)
50 miles per hour
6 mebibyte

Note that Numbat also allows you to define new units.

Operations and precedence

Numbat operators and other language constructs, ordered by precedence form high to low:

Note that implicit multiplication has a higher precedence than division, i.e. 50 cm / 2 m will be parsed as 50 cm / (2 m).

Also, note that per-division has a higher precedence than /-division. This means 1 / meter per second will be parsed as 1 / (meter per second).

If in doubt, you can always look at the pretty-printing output (second line in the snippet below) to make sure that your input was parsed correctly:

>>> 1 / meter per second

  1 / (meter / second)

    = 1 s/m

Constants

New constants can be introduced with the let keyword:

let pipe_radius = 1 cm
let pipe_length = 10 m
let Δp = 0.1 bar

Definitions may contain a type annotation after the identifier (let Δp: Pressure = 0.1 bar). This annotation will be verified by the type checker. For more complex definitions it can be desirable to add type annotations, as it often improves readability and allows you to catch potential errors early:

let μ_water: DynamicViscosity = 1 mPa·s
let Q: FlowRate = π × pipe_radius^4 × Δp / (8 μ_water × pipe_length)

Unit conversions

The conversion operator -> attempts to convert the physical quantity on its left hand side to the unit of the expression on its right hand side. This means that you can write an arbitrary expression on the right hand side — but only the unit part will be extracted. For example:

# simple unit conversion:
> 120 km/h -> mph

  = 74.5645 mi/h

# expression on the right hand side:
> 120 m^3 -> km * m^2

  = 0.12 m²·km

# convert x1 to the same unit as x2:
> let x1 = 50 km / h
> let x2 = 3 m/s -> x1

  x2 = 10.8 km/h

Conversion functions

The conversion operator -> (or to) can not just be used for unit conversions, but also for other types of conversions. The way this is set up in Numbat is that you can call x -> f for any function f that takes a single argument of the same type as x.

The following functions are available for this purpose:

# Convert a date and time to a Unix timestamp
now() -> unixtime

# Convert a date and time to a different timezone
now() -> tz("Asia/Kathmandu")

# Convert a duration to years, months, days, hours, minutes, seconds
10 million seconds -> human

# Convert an angle to degrees, minutes, seconds (48° 46′ 32″)
48.7756° -> DMS

# Convert an angle to degrees, decimal minutes (48° 46.536′)
48.7756° -> DM

# Convert a number to its binary representation
42 -> bin

# Convert a number to its octal representation
42 -> oct

# Convert a number to its hexadecimal representation
2^31-1 -> hex

# Convert a code point number to a character
0x2764 -> chr

# Convert a character to a code point number
"❤" -> ord

# Convert a string to upper/lower case
"numbat is awesome" -> uppercase
"vier bis elf weiße Querbänder" -> lowercase

Note that the tz(…) call above returns a function, i.e. the right hand side of the conversion operator is still a function.

Function definitions

Numbat comes with a large number of predefined functions, but it is also possible to add new functions. A function definition is introduced with the fn keyword:

fn max_distance(v: Velocity, θ: Angle) -> Length = v² · sin(2 θ) / g0

This exemplary function computes the maximum distance of a projectile under the influence of Earths gravity. It takes two parameters (the initial velocity v and the launch angle θ), which are both annotated with their corresponding physical dimension (their type). The function returns a distance, and so the return type is specified as Length.

Type inference

Numbat has a powerful type inference system, which is able to infer missing types when they are not explicitly specified. For example, consider the following function definition for the braking distance of a car, given its velocity v:

fn braking_distance(v) = v t_reaction + v² / 2 µ g0
  where t_reaction = 1 s # driver reaction time
    and µ = 0.7          # coefficient of friction

If you enter this function into the Numbat REPL, you will see that all types are filled in automatically:

fn braking_distance(v: Velocity) -> Length = v × t_reaction + (v² / (2 µ × g0))
  where t_reaction: Time = 1 second
    and µ: Scalar = 0.7

In particular, note that the type of the function argument v is correctly inferred as Velocity, and the return type is Length.

Note: This is possible because the types of t_reaction, µ, and g0 (gravitational acceleration) are known. The + operator imposes a constraint on the types: two quantities can only be added if their physical dimension is the same. The type inference algorithm records constraints like this, and then tries to find a solution that satisfies all of them. In this case, only a single equation needs to be solved:

type(v) × type(t_reaction) = type(v)² / (type(µ) × type(g0)      )
type(v) × Time             = type(v)² / (      1 × Length / Time²)

which has the solution type(v) = Length / Time = Velocity. Note that this also works if there are multiple constraints on the types. In fact, type inference is always decidable.

The fact that it is possible to omit type annotations does not mean that it is always a good idea to do so. Type annotations can help to make the code more readable and can also help to catch errors earlier.

In some cases, type inference will also lead to function types that are overly generic. For example, consider the following function to compute the kinetic energy of a massive object in motion:

fn kinetic_energy(mass, speed) = 1/2 * mass * speed^2

In the absence of any type annotations, this function has an overly generic type where mass and speed can have arbitrary dimensions (but the return type is constrained accordingly):

fn kinetic_energy<A: Dim, B: Dim>(mass: A, speed: B) -> A × B² = …

In this example, it would be better to specify the types of mass and speed explicitly (Mass, Velocity). The return type can then be inferred (Energy). It is still valuable to specify it explicitly, in order to ensure there are no mistakes in the function implementation.

Generic functions

Sometimes it is useful to write generic functions. For example, consider max(a, b) — a function that returns the larger of the two arguments. We might want to use that function with dimensionful arguments such as max(1 m, 1 yd). To define such a generic function, you can introduce type parameters in angle brackets:

fn max<D: Dim>(a: D, b: D) -> D =
  if a > b then a else b

This function signature tells us that max takes two arguments of arbitrary dimension type D (but they need to match!), and returns a quantity of the same type D. The D: Dim syntax is a type constraint (or bound) that ensures that D is a dimension type (Scalar, Length, Velocity, etc), and not something like Bool or DateTime.

Note that you can perform the usual operations with (dimension) type parameters, such as multiplying / dividing them with other types, or raising to rational powers. For example, consider this cube-root function

fn cube_root<T>(x: T^3) -> T = x^(1/3)

that can be called with a scalar (cube_root(8) == 2) or a dimensionful argument (cube_root(1 liter) == 10 cm).

Note: cube_root can also be defined as fn cube_root<T>(x: T) -> T^(1/3), which is equivalent to the definition above.

Functions can also be generic over all types, not just dimension types. In this case, no type constraints are needed. For example:

fn second_element<A>(xs: List<A>) -> A =
  head(tail(xs))

second_element([10 cm, 2 m, 3 inch]) # returns 2 m
second_element(["a", "b", "c"])      # returns "b"

Note that the type annotations for all examples in this section are optional and can also be inferred.

Recursive functions

It is also possible to define recursive functions. For example, a naive recursive implementation to compute Fibonacci numbers in Numbat looks like this:

fn fib(n) =
  if n ≤ 2
    then 1
    else fib(n - 2) + fib(n - 1)

Conditionals

Numbat has if-then-else conditional expressions with the following syntax

if <cond> then <expr1> else <expr2>

where <cond> is a condition that evaluates to a Boolean value, like 3 ft < 3 m. The types of <expr1> and <expr2> need to match.

For example, you can defined a simple step function using

fn step(x) = if x < 0 then 0 else 1

Lists

Numbat has a built-in data type for lists. The elements can be of any type, including other lists. Lists can be created using the […] syntax. For example:

[30 cm, 110 cm, 2 m]
["a", "b", "c"]
[[1, 2], [3, 4]]

The type of a list is written as List<T>, where T is the type of the elements. The types of the lists above are List<Length>, List<String>, and List<List<Scalar>>, respectively.

The standard library provides a number of functions to work with lists. Some useful things to do with lists are:

# Get the length of a list
len([1, 2, 3])  # returns 3

# Sum all elements of a list:
sum([30 cm, 130 cm, 2 m])  # returns 360 cm

# Get the average of a list:
mean([30 cm, 130 cm, 2 m])  # returns 120 cm

# Filter a list:
filter(is_finite, [20 cm, inf, 1 m])  # returns [20 cm, 1 m]

# Map a function over a list:
map(sqr, [10 cm, 2 m])  # returns [100 cm², 4 m²]

# Generate a range of numbers:
range(1, 5)  # returns [1, 2, 3, 4, 5]

# Generate a list of evenly spaced quantities:
linspace(0 m, 1 m, 5)  # returns [0 m, 0.25 m, 0.5 m, 0.75 m, 1 m]

Structs

Numbat has compound data structures in the form of structs:

struct Vector {
  x: Length,
  y: Length,
}

let origin   = Vector { x: 0 m, y: 0 m }
let position = Vector { x: 6 m, y: 8 m }

# A function with a struct as a parameter
fn euclidean_distance(a: Vector, b: Vector) =
  sqrt((a.x - b.x)² + (a.y - b.y)²)

assert_eq(euclidean_distance(origin, position), 10 m)

# Struct fields can be accessed using `.field` notation
let x = position.x

Date and time

Numbat supports date and time handling based on the proleptic Gregorian calendar, which is the (usual) Gregorian calendar extended to dates before its introduction in 1582.

A few examples of useful operations that can be performed on dates and times:

# How many days are left until September 1st?
date("2024-11-01") - today() -> days

# What time is it in Nepal right now?
now() -> tz("Asia/Kathmandu")  # use tab completion to find time zone names

# What is the local time when it is 2024-11-01 12:30:00 in Australia?
datetime("2024-11-01 12:30:00 Australia/Sydney") -> local

# Which date was 1 million seconds ago?
now() - 1 million seconds

# Which date is 40 days from now?
calendar_add(now(), 40 days)

# Which weekday was the 1st day of this century?
date("2000-01-01") -> weekday

# What is the current UNIX timestamp?
now() -> unixtime

# What is the date corresponding to a given UNIX timestamp?
from_unixtime(1707568901)

# How long are one million seconds in years, months, days, hours, minutes, seconds?
1 million seconds -> human

Date and time arithmetic

The following operations are supported for DateTime objects:

Date, time, and duration functions

The following functions are available for date and time handling:

• now() -> DateTime: Returns the current date and time.
• today() -> DateTime: Returns the current date at midnight (in the local time).
• datetime(input: String) -> DateTime: Parses a string (date and time) into a DateTime object.
• date(input: String) -> DateTime: Parses a string (only date) into a DateTime object.
• time(input: String) -> DateTime: Parses a string (only time) into a DateTime object.
• format_datetime(format: String, dt: DateTime) -> String: Formats a DateTime object as a string. See this page for possible format specifiers.
• tz(tz: String) -> Fn[(DateTime) -> DateTime]: Returns a timezone conversion function, typically used with the conversion operator (datetime -> tz("Europe/Berlin"))
• local(dt: DateTime) -> DateTime: Timezone conversion function targeting the users local timezone (datetime -> local)
• get_local_timezone() -> String: Returns the users local timezone
• unixtime(dt: DateTime) -> Scalar: Converts a DateTime to a UNIX timestamp.
• from_unixtime(ut: Scalar) -> DateTime: Converts a UNIX timestamp to a DateTime object.
• calendar_add(dt: DateTime, span: Time): Add a span of time to a DateTime object, taking proper calendar arithmetic into accound.
• calendar_sub(dt: DateTime, span: Time): Subtract a span of time from a DateTime object, taking proper calendar arithmetic into accound.
• weekday(dt: DateTime) -> String: Returns the weekday of a DateTime object as a string.
• human(duration: Time) -> String: Converts a Time to a human-readable string in days, hours, minutes and seconds.
• julian_date(dt: DateTime) -> Scalar: Convert a DateTime to a Julian date.

Date time formats

The following formats are supported by datetime. UTC offsets are mandatory for the RFC 3339 and RFC 2822 formats. The other formats can optionally include a time zone name or UTC offset. If no time zone is specified, the local time zone is used.

The date function supports the following formats. It returns a DateTime object with the time set to midnight in the specified timezone (or the local timezone if no timezone is specified).

The time function supports the following formats. It returns a DateTime object with the date set to the current date. If no timezone is specified, the local timezone is used.

Printing, testing, debugging

Printing

Numbat has a builtin print procedure that can be used to print the value of an expression:

print(2 km/h)
print(3 ft < 1 m)

You can also print out simple messages as strings. This is particularly useful when combined with string interpolation to print results of a computation:

let radius: Length = sqrt(footballfield / 4 pi) -> meter
print("A football field would fit on a sphere of radius {radius}")

You can use almost every expression inside a string interpolation field. For example:

print("3² + 4² = {hypot2(3, 4)}²")

let speed = 25 km/h
print("Speed of the bicycle: {speed} ({speed -> mph})")

Format specifiers are also supported in interpolations. For instance:

print("{pi:0.2f}")  # Prints "3.14"

For more information on supported format specifiers, please see this page.

Testing

The assert_eq procedure can be used to test for (approximate) equality of two quantities. This is often useful to make sure that (intermediate) results in longer calculations have a certain value, e.g. when restructuring the code. The general syntax is

assert_eq(q1, q2)
assert_eq(q1, q2, ε)

where the first version tests for exact equality while the second version tests for approximate equality \( |q_1-q_2| <= \epsilon \) with a specified accuracy of \( \epsilon \). Note that the input quantities are converted to the units of \( \epsilon \) before comparison. For example:

assert_eq(2 + 3, 5)
assert_eq(1 ft × 77 in², 4 gal)

assert_eq(alpha, 1 / 137, 1e-4)
assert_eq(3.3 ft, 1 m, 1 cm)

There is also a plain assert procedure that can test any boolean condition. For example:

assert(1 yard < 1 meter)
assert(str_contains("bar", "foobar"))

A runtime error is thrown if an assertion fails. Otherwise, nothing happens.

Debugging

You can use the builtin type procedure to see the type (or physical dimension) of a quantity:

>>> type(g0)

  Length / Time²

>>> type(2 < 3)

  Bool

Advanced

This chapter covers more advanced topics, like defining custom physical units or new physical dimensions.

Dimension definitions

New (physical) dimensions can be introduced with the dimension keyword. Similar like for units, there are base dimensions (like length, time and mass) and dimensions that are derived from those base dimensions (like momentum, which is mass · length / time). Base dimensions are simply introduced by declaring their name:

dimension Length
dimension Time
dimension Mass

Derived dimensions need to specify their relation to base dimensions (or other derived dimensions). For example:

dimension Velocity = Length / Time
dimension Momentum = Mass * Velocity
dimension Force = Mass * Acceleration = Momentum / Time
dimension Energy = Momentum^2 / Mass = Mass * Velocity^2 = Force * Length

In the definition of Force and Energy, we can see that multiple alternative definitions can be specified. This is entirely optional. When given, the compiler will make sure that all definitions are equivalent.

Unit definitions

New units of measurement can be introduced with the unit keyword. There are two types of units: base units and derived units.

A new base unit can be defined by specifying the physical dimension it represents. For example, in the International System of Units (SI), the second is the base unit for measuring times:

unit second: Time

Here, Time denotes the physical dimension. To learn more, you can read the corresponding chapter. But for now, we can just assume that they are already given.

Derived units are also introduced with the unit keyword. But unlike base units, they are defined through their relation to other units. For example, a minute can be defined as

unit minute: Time = 60 second

Here, the : Time annotation is optional. If a dimension is specified, it will be used to verify that the right hand side expression (60 second) is indeed of physical dimension Time. This is apparent in this simple example, but can be useful for more complicated unit definitions like

unit farad: Capacitance = ampere^2 second^4 / (kilogram meter^2)

Prefixes

If a unit may be used with metric prefixes such as milli/m, kilo/k or mega/M, we can prepend the unit definition with the @metric_prefixes decorator:

@metric_prefixes
unit second: Time

This allows identifiers such as millisecond to be used in calculations. See the section below how prefixes interact with aliases.

Similarly, if a unit should be prependable with binary (IEC) prefixes such as kibi/Ki, mebi/Mi or gibi/Gi, you can add the @binary_prefixes decorator. A unit might also allow for both metric and binary prefixes, for example:

@binary_prefixes
@metric_prefixes
unit byte = 8 bit

This allows the usage of both mebibyte (1024² byte) as well as megabyte (1000² byte).

Aliases

It is often useful to define alternative names for a unit. For example, we might want to use the plural form seconds or the commonly used short version s. We can use the @aliases decorator to specify them:

@metric_prefixes
@aliases(meters, metre, metres, m: short)
unit meter: Length

In addition to the name, we can also specify how aliases interact with prefixes using : long (the default), : short, : both or : none. The actual unit name (meter) and all long aliases will accept the long version of prefixes (…, milli, kilo, mega, giga, …). All short aliases (m in the example above) will only accept the respective short versions of the prefixes (…, m, k, M, G, …). Aliases annotated with : both or : none accept either both long and short prefixes, or none of them. The unit definition above allows all of following expressions:

millimeter
kilometer

millimeters
kilometers

millimetre
kilometre

millimetres
kilometres

mm
km
...

Ad-hoc units

It is often useful to introduce ‘fictional’ physical units (and dimensions). This comes up frequently when you want to count things. For example:

unit book

@aliases(pages)
unit page

@aliases(words)
unit word

let words_per_book = 500 words/page × 300 pages/book

Note that those base unit definitions will implicitly create new dimensions which are capitalized versions of the unit names (Book, Page, Word). A definition like unit book is a shorthand for dimension Book; unit book: Book. Those units now allow us to count books, pages and words independently without any risk of mixing them. The words_per_book constant in this examples has a type of Word / Book.

Another example shows how we introduce a dot unit to do calculations with screen resolutions:

@aliases(dots)
unit dot

unit dpi = dots / inch

# Note: a `Dot` dimension was implicitly created for us
fn inter_dot_spacing(resolution: Dot / Length) -> Length = 1 dot / resolution

inter_dot_spacing(72 dpi) -> µm  # 353 µm

Syntax overview

# This is a line comment. It can span over
# multiple lines

# 1. Imports

use prelude        # This is not necessary. The 'prelude'
                   # module will always be loaded upon startup

use units::stoney  # Load a specific module

# 2. Numbers

12345       # integer notation
12_345      # optional decimal separators

0.234       # floating point notation
.234        # without the leading zero

1.234e15    # scientific notation
1.234e+15
1e-9
1.0e-9

0x2A        # hexadecimal
0o52        # octal
0b101010    # binary

NaN         # Not a number
inf         # Infinity

# 3. Simple expressions

3 + (4 - 3)       # Addition and subtraction

1920 / 16 * 9     # Multiplication, division
1920 ÷ 16 × 9     # Unicode-style, '·' or '⋅' works as well
2 pi              # Whitespace is implicit multiplication
meter per second  # 'per' keyword can be used for division

2^3               # Exponentiation
2**3              # Python-style
2³                # Unicode exponents
2^-3              # Negative exponents

mod(17, 4)        # Modulo

3 in -> cm        # Unit conversion, can also be → or ➞
3 in to cm        # Unit conversion with the 'to' keyword

cos(pi/3 + pi)    # Call mathematical functions
pi/3 + pi |> cos  # Same, 'arg |> f' is equivalent to 'f(arg)'
                  # The '|>' operator has the lowest precedence
                  # which makes it very useful for interactive
                  # terminals (press up-arrow, and add '|> f')

# 4. Constants

let n = 4                          # Simple numerical constant
let q1 = 2 m/s                     # Right hand side can be any expression
let q2: Velocity = 2 m/s           # With optional type annotation
let q3: Length / Time = 2 m/s      # more complex type annotation

# 5. Function definitions

fn foo(z: Scalar) -> Scalar = 2 * z + 3                   # A simple function
fn speed(len: Length, dur: Time) -> Velocity = len / dur  # Two parameters
fn my_sqrt<T: Dim>(q: T^2) -> T = q^(1/2)                 # A generic function
fn is_non_negative(x: Scalar) -> Bool = x ≥ 0             # Returns a bool
fn power_4(x: Scalar) = z                                 # A function with local variables
  where y = x * x
    and z = y * y

# 6. Dimension definitions

dimension Fame                            # A new base dimension
dimension Deceleration = Length / Time^2  # A new derived dimension

# 7. Unit definitions

@aliases(quorks)                 # Optional aliases-decorator
unit quork = 0.35 meter          # A new derived unit

@metric_prefixes                 # Optional decorator to allow 'milliclonk', etc.
@aliases(ck: short)              # short aliases can be used with short prefixes (mck)
unit clonk: Time = 0.2 seconds   # Optional type annotation

@metric_prefixes
@aliases(wh: short)
unit warhol: Fame                # New base unit for the "Fame" dimension

unit thing                       # New base unit with automatically generated
                                 # base dimension "Thing"

# 8. Conditionals

fn bump(x: Scalar) -> Scalar =   # The construct 'if <cond> then <expr> else <expr>'
  if x >= 0 && x <= 1            # is an expression, not a statement. It can span
    then 1                       # multiple lines.
    else 0

# 9. Procedures

print(2 kilowarhol)              # Print the value of an expression
print("hello world")             # Print a message
print("value of pi = {pi}")      # String interpolation
print("sqrt(10) = {sqrt(10)}")   # Expressions in string interpolation
print("value of π ≈ {π:.3}")     # Format specifiers

assert(1 yard < 1 meter)         # Assertion

assert_eq(1 ft, 12 in)           # Assert that two quantities are equal
assert_eq(1 yd, 1 m, 10 cm)      # Assert that two quantities are equal, up to
                                 # the given precision
type(2 m/s)                      # Print the type of an expression


# 10. Structs

struct Element {                 # Define a struct
    name: String,
    atomic_number: Scalar,
    density: MassDensity,
}

let hydrogen = Element {         # Instantiate it
    name: "Hydrogen",
    atomic_number: 1,
    density: 0.08988 g/L,
}

hydrogen.density                 # Access the field of a struct

The prelude

Numbat comes with a special module called prelude that is always loaded on startup (unless --no-prelude is specified on the command line). This module is split into multiple submodules and sets up a useful default environment with mathematical functions, constants but also dimension definitions, unit definitions and physical constants.

You can find the full source code of the standard library on GitHub.

This chapter is a reference to the prelude module.

Predefined functions

See sub-chapters for a list of predefined functions in the standard library.

Mathematical functions

Basics · Transcendental functions · Trigonometry · Statistics · Combinatorics · Random sampling, distributions · Number theory · Numerical methods · Percentage calculations · Geometry · Algebra · Trigonometry (extra)

Basics

Defined in: core::functions

id (Identity function)

Return the input value.

fn id<A>(x: A) -> A

id(8 kg)

    = 8 kg    [Mass]

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

abs(-22.2 m)

    = 22.2 m    [Length]

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

sqrt(4 are) -> m

    = 20 m    [Length]

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

cbrt(8 L) -> cm

    = 20.0 cm    [Length]

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

sqr(7)

    = 49

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(5.5)

    = 6

round(-5.5)

    = -6

round_in (Rounding)

Round to the nearest multiple of base.

fn round_in<D: Dim>(base: D, value: D) -> D

round_in(m, 5.3 m)

    = 5 m    [Length]

round_in(cm, 5.3 m)

    = 530 cm    [Length]

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(5.5)

    = 5

floor_in (Floor function)

Returns the largest integer multiple of base less than or equal to value.

fn floor_in<D: Dim>(base: D, value: D) -> D

floor_in(m, 5.7 m)

    = 5 m    [Length]

floor_in(cm, 5.7 m)

    = 570 cm    [Length]

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(5.5)

    = 6

ceil_in (Ceil function)

Returns the smallest integer multiple of base greater than or equal to value.

fn ceil_in<D: Dim>(base: D, value: D) -> D

ceil_in(m, 5.3 m)

    = 6 m    [Length]

ceil_in(cm, 5.3 m)

    = 530 cm    [Length]

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(5.5)

    = 5

trunc(-5.5)

    = -5

trunc_in (Truncation)

Truncates to an integer multiple of base (towards zero).

fn trunc_in<D: Dim>(base: D, value: D) -> D

trunc_in(m, 5.7 m)

    = 5 m    [Length]

trunc_in(cm, 5.7 m)

    = 570 cm    [Length]

fract (Fractional part)

Returns the fractional part of \( x \), i.e. the remainder when divided by 1. If \( x < 0 \), then so will be fract(x). Note that due to floating point error, a number’s fractional part can be slightly “off”; for instance, fract(1.2) == 0.1999...996 != 0.2. More information here.

fn fract(x: Scalar) -> Scalar

fract(0.0)

    = 0

fract(5.5)

    = 0.5

fract(-5.5)

    = -0.5

mod (Modulo)

Calculates the least nonnegative remainder of \( a (\mod b) \). More information here.

fn mod<T: Dim>(a: T, b: T) -> T

mod(27, 5)

    = 2

Transcendental functions

Defined in: math::transcendental

exp (Exponential function)

The exponential function, \( e^x \). More information here.

fn exp(x: Scalar) -> Scalar

exp(4)

    = 54.5982

ln (Natural logarithm)

The natural logarithm with base \( e \). More information here.

fn ln(x: Scalar) -> Scalar

ln(20)

    = 2.99573

log (Natural logarithm)

The natural logarithm with base \( e \). More information here.

fn log(x: Scalar) -> Scalar

log(20)

    = 2.99573

log10 (Common logarithm)

The common logarithm with base \( 10 \). More information here.

fn log10(x: Scalar) -> Scalar

log10(100)

    = 2

log2 (Binary logarithm)

The binary logarithm with base \( 2 \). More information here.

fn log2(x: Scalar) -> Scalar

log2(256)

    = 8

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.

fn maximum<D: Dim>(xs: List<D>) -> D

maximum([30 cm, 2 m])

    = 2 m    [Length]

minimum (Minimum)

Get the smallest element of a list.

fn minimum<D: Dim>(xs: List<D>) -> D

minimum([30 cm, 2 m])

    = 30 cm    [Length]

mean (Arithmetic mean)

Calculate the arithmetic mean of a list of quantities. More information here.

fn mean<D: Dim>(xs: List<D>) -> D

mean([1 m, 2 m, 300 cm])

    = 2 m    [Length]

variance (Variance)

Calculate the population variance of a list of quantities. More information here.

fn variance<D: Dim>(xs: List<D>) -> D^2

variance([1 m, 2 m, 300 cm])

    = 0.666667 m²    [Area]

stdev (Standard deviation)

Calculate the population standard deviation of a list of quantities. More information here.

fn stdev<D: Dim>(xs: List<D>) -> D

stdev([1 m, 2 m, 300 cm])

    = 0.816497 m    [Length]

median (Median)

Calculate the median of a list of quantities. More information here.

fn median<D: Dim>(xs: List<D>) -> D

median([1 m, 2 m, 400 cm])

    = 2 m    [Length]

Combinatorics

Defined in: math::combinatorics

factorial (Factorial)

The product of the integers 1 through n. Numbat also supports calling this via the postfix operator n!. More information here.

fn factorial(n: Scalar) -> Scalar

factorial(4)

    = 24

4!

    = 24

falling_factorial (Falling factorial)

Equal to \( n⋅(n-1)⋅…⋅(n-k+2)⋅(n-k+1) \) (k terms total). If n is an integer, this is the number of k-element permutations from a set of size n. k must always be an integer. More information here.

fn falling_factorial(n: Scalar, k: Scalar) -> Scalar

falling_factorial(4, 2)

    = 12

binom (Binomial coefficient)

Equal to falling_factorial(n, k)/k!, this is the coefficient of \( x^k \) in the series expansion of \( (1+x)^n \) (see “binomial series”). If n is an integer, then this this is the number of k-element subsets of a set of size n, often read “n choose k”. k must always be an integer. More information here.

fn binom(n: Scalar, k: Scalar) -> Scalar

binom(5, 2)

    = 10

fibonacci (Fibonacci numbers)

The nth Fibonacci number, where n is a nonnegative integer. The Fibonacci sequence is given by \( F_0=0 \), \( F_1=1 \), and \( F_n=F_{n-1}+F_{n-2} \) for \( n≥2 \). The first several elements, starting with \( n=0 \), are \( 0, 1, 1, 2, 3, 5, 8, 13 \). More information here.

fn fibonacci(n: Scalar) -> Scalar

fibonacci(5)

    = 5

lucas (Lucas numbers)

The nth Lucas number, where n is a nonnegative integer. The Lucas sequence is given by \( L_0=2 \), \( L_1=1 \), and \( L_n=L_{n-1}+L_{n-2} \) for \( n≥2 \). The first several elements, starting with \( n=0 \), are \( 2, 1, 3, 4, 7, 11, 18, 29 \). More information here.

fn lucas(n: Scalar) -> Scalar

lucas(5)

    = 11

catalan (Catalan numbers)

The nth Catalan number, where n is a nonnegative integer. The Catalan sequence is given by \( C_n=\frac{1}{n+1}\binom{2n}{n}=\binom{2n}{n}-\binom{2n}{n+1} \). The first several elements, starting with \( n=0 \), are \( 1, 1, 2, 5, 14, 42, 132, 429 \). More information here.

fn catalan(n: Scalar) -> Scalar

catalan(5)

    = 42

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

gcd(60, 42)

    = 6

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

lcm(14, 4)

    = 28

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, Δx: X) -> Y / X

use numerics::diff
fn polynomial(x) = x² - x - 1
diff(polynomial, 1, 1e-10)

    = 1.0

use numerics::diff
fn distance(t) = 0.5 g0 t²
fn velocity(t) = diff(distance, t, 1e-10 s)
velocity(2 s)

    = 19.6133 m/s    [Velocity]

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

use numerics::solve
fn f(x) = x² +x -2
root_bisect(f, 0, 100, 0.01, 0.01)

    = 1.00098

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

use numerics::solve
fn f(x) = x² -3x +2
fn f_prime(x) = 2x -3
root_newton(f, f_prime, 0 , 0.01)

    = 0.996078

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

use numerics::fixed_point
fn function(x) = x/2 - 1
fixed_point(function, 0, 0.01)

    = -1.99219

Percentage calculations

Defined in: math::percentage_calculations

increase_by

Increase a quantity by the given percentage. More information here.

fn increase_by<D: Dim>(percentage: Scalar, quantity: D) -> D

72 € |> increase_by(15%)

    = 82.8 €    [Money]

decrease_by

Decrease a quantity by the given percentage. More information here.

fn decrease_by<D: Dim>(percentage: Scalar, quantity: D) -> D

210 cm |> decrease_by(10%)

    = 189 cm    [Length]

percentage_change

By how many percent has a given quantity increased or decreased?. More information here.

fn percentage_change<D: Dim>(old: D, new: D) -> Scalar

percentage_change(35 kg, 42 kg)

    = 20 %

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

hypot2(3 m, 4 m)

    = 5 m    [Length]

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

hypot3(8, 9, 12)

    = 17

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>

use extra::algebra
quadratic_equation(2, -1, -1)

    = [1, -0.5]    [List]

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

List-related functions

Defined in: core::lists

len

Get the length of a list.

fn len<A>(xs: List<A>) -> Scalar

len([3, 2, 1])

    = 3

head

Get the first element of a list. Yields a runtime error if the list is empty.

fn head<A>(xs: List<A>) -> A

head([3, 2, 1])

    = 3

tail

Get everything but the first element of a list. Yields a runtime error if the list is empty.

fn tail<A>(xs: List<A>) -> List<A>

tail([3, 2, 1])

    = [2, 1]    [List]

cons

Prepend an element to a list.

fn cons<A>(x: A, xs: List<A>) -> List<A>

cons(77, [3, 2, 1])

    = [77, 3, 2, 1]    [List]

cons_end

Append an element to the end of a list.

fn cons_end<A>(x: A, xs: List<A>) -> List<A>

cons_end(77, [3, 2, 1])

    = [3, 2, 1, 77]    [List]

is_empty

Check if a list is empty.

fn is_empty<A>(xs: List<A>) -> Bool

is_empty([3, 2, 1])

    = false    [Bool]

is_empty([])

    = true    [Bool]

concat

Concatenate two lists.

fn concat<A>(xs1: List<A>, xs2: List<A>) -> List<A>

concat([3, 2, 1], [10, 11])

    = [3, 2, 1, 10, 11]    [List]

take

Get the first n elements of a list.

fn take<A>(n: Scalar, xs: List<A>) -> List<A>

take(2, [3, 2, 1, 0])

    = [3, 2]    [List]

drop

Get everything but the first n elements of a list.

fn drop<A>(n: Scalar, xs: List<A>) -> List<A>

drop(2, [3, 2, 1, 0])

    = [1, 0]    [List]

element_at

Get the element at index i in a list.

fn element_at<A>(i: Scalar, xs: List<A>) -> A

element_at(2, [3, 2, 1, 0])

    = 1

range

Generate a range of integer numbers from start to end (inclusive).

fn range(start: Scalar, end: Scalar) -> List<Scalar>

range(2, 12)

    = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]    [List]

reverse

Reverse the order of a list.

fn reverse<A>(xs: List<A>) -> List<A>

reverse([3, 2, 1])

    = [1, 2, 3]    [List]

map

Generate a new list by applying a function to each element of the input list.

fn map<A, B>(f: Fn[(A) -> B], xs: List<A>) -> List<B>

map(sqr, [3, 2, 1])

    = [9, 4, 1]    [List]

filter

Filter a list by a predicate.

fn filter<A>(p: Fn[(A) -> Bool], xs: List<A>) -> List<A>

filter(is_finite, [0, 1e10, NaN, -inf])

    = [0, 10_000_000_000]    [List]

foldl

Fold a function over a list.

fn foldl<A, B>(f: Fn[(A, B) -> A], acc: A, xs: List<B>) -> A

foldl(str_append, "", ["Num", "bat", "!"])

    = "Numbat!"    [String]

sort_by_key

Sort a list of elements, using the given key function that maps the element to a quantity.

fn sort_by_key<A, D: Dim>(key: Fn[(A) -> D], xs: List<A>) -> List<A>

fn last_digit(x) = mod(x, 10)
sort_by_key(last_digit, [701, 313, 9999, 4])

    = [701, 313, 4, 9999]    [List]

sort

Sort a list of quantities in ascending order.

fn sort<D: Dim>(xs: List<D>) -> List<D>

sort([3, 2, 7, 8, -4, 0, -5])

    = [-5, -4, 0, 2, 3, 7, 8]    [List]

contains

Returns true if the element x is in the list xs.

fn contains<A>(x: A, xs: List<A>) -> Bool

[3, 2, 7, 8, -4, 0, -5] |> contains(0)

    = true    [Bool]

[3, 2, 7, 8, -4, 0, -5] |> contains(1)

    = false    [Bool]

unique

Remove duplicates from a given list.

fn unique<A>(xs: List<A>) -> List<A>

unique([1, 2, 2, 3, 3, 3])

    = [1, 2, 3]    [List]

intersperse

Add an element between each pair of elements in a list.

fn intersperse<A>(sep: A, xs: List<A>) -> List<A>

intersperse(0, [1, 1, 1, 1])

    = [1, 0, 1, 0, 1, 0, 1]    [List]

sum

Sum all elements of a list.

fn sum<D: Dim>(xs: List<D>) -> D

sum([3 m, 200 cm, 1000 mm])

    = 6 m    [Length]

linspace

Generate a list of n_steps evenly spaced numbers from start to end (inclusive).

fn linspace<D: Dim>(start: D, end: D, n_steps: Scalar) -> List<D>

linspace(-5 m, 5 m, 11)

    = [-5 m, -4 m, -3 m, -2 m, -1 m, 0 m, 1 m, 2 m, 3 m, 4 m, 5 m]    [List]

join

Convert a list of strings into a single string by concatenating them with a separator.

fn join(xs: List<String>, sep: String) -> String

join(["snake", "case"], "_")

    = "snake_case"    [String]

split

Split a string into a list of strings using a separator.

fn split(input: String, separator: String) -> List<String>

split("Numbat is a statically typed programming language.", " ")

    = ["Numbat", "is", "a", "statically", "typed", "programming", "language."]    [List]

String-related functions

Defined in: core::strings

str_length

The length of a string.

fn str_length(s: String) -> Scalar

str_length("Numbat")

    = 6

str_slice

Subslice of a string.

fn str_slice(start: Scalar, end: Scalar, s: String) -> String

str_slice(3, 6, "Numbat")

    = "bat"    [String]

chr

Get a single-character string from a Unicode code point.

fn chr(n: Scalar) -> String

0x2764 -> chr

    = "❤"    [String]

ord

Get the Unicode code point of the first character in a string.

fn ord(s: String) -> Scalar

"❤" -> ord

    = 10084

lowercase

Convert a string to lowercase.

fn lowercase(s: String) -> String

lowercase("Numbat")

    = "numbat"    [String]

uppercase

Convert a string to uppercase.

fn uppercase(s: String) -> String

uppercase("Numbat")

    = "NUMBAT"    [String]

str_append

Concatenate two strings.

fn str_append(a: String, b: String) -> String

"Numbat" |> str_append("!")

    = "!Numbat"    [String]

str_prepend

Concatenate two strings.

fn str_prepend(a: String, b: String) -> String

"!" |> str_prepend("Numbat")

    = "!Numbat"    [String]

str_find

Find the first occurrence of a substring in a string.

fn str_find(needle: String, haystack: String) -> Scalar

str_find("typed", "Numbat is a statically typed programming language.")

    = 23

str_contains

Check if a string contains a substring.

fn str_contains(needle: String, haystack: String) -> Bool

str_contains("typed", "Numbat is a statically typed programming language.")

    = true    [Bool]

str_replace

Replace all occurrences of a substring in a string.

fn str_replace(pattern: String, replacement: String, s: String) -> String

str_replace("statically typed programming language", "scientific calculator", "Numbat is a statically typed programming language.")

    = "Numbat is a scientific calculator."    [String]

str_repeat

Repeat the input string n times.

fn str_repeat(n: Scalar, a: String) -> String

str_repeat(4, "abc")

    = "abcabcabcabc"    [String]

base

Convert a number to the given base.

fn base(b: Scalar, x: Scalar) -> String

42 |> base(16)

    = "2a"    [String]

bin

Get a binary representation of a number.

fn bin(x: Scalar) -> String

42 -> bin

    = "0b101010"    [String]

oct

Get an octal representation of a number.

fn oct(x: Scalar) -> String

42 -> oct

    = "0o52"    [String]

dec

Get a decimal representation of a number.

fn dec(x: Scalar) -> String

0b111 -> dec

    = "7"    [String]

hex

Get a hexadecimal representation of a number.

fn hex(x: Scalar) -> String

2^31-1 -> hex

    = "0x7fffffff"    [String]

Date and time

See this page for a general introduction to date and time handling in Numbat.

Defined in: datetime::functions, datetime::human

now

Returns the current date and time.

fn now() -> DateTime

datetime

Parses a string (date and time) into a DateTime object. See here for an overview of the supported formats.

fn datetime(input: String) -> DateTime

datetime("2022-07-20T21:52+0200")

    = 2022-07-20 19:52:00 UTC    [DateTime]

datetime("2022-07-20 21:52 Europe/Berlin")

    = 2022-07-20 21:52:00 CEST (UTC +02), Europe/Berlin    [DateTime]

datetime("2022/07/20 09:52 PM +0200")

    = 2022-07-20 21:52:00 (UTC +02)    [DateTime]

format_datetime

Formats a DateTime object as a string.

fn format_datetime(format: String, input: DateTime) -> String

format_datetime("This is a date in %B in the year %Y.", datetime("2022-07-20 21:52 +0200"))

    = "This is a date in July in the year 2022."    [String]

get_local_timezone

Returns the users local timezone.

fn get_local_timezone() -> String

get_local_timezone()

    = "UTC"    [String]

tz

Returns a timezone conversion function, typically used with the conversion operator.

fn tz(tz: String) -> Fn[(DateTime) -> DateTime]

datetime("2022-07-20 21:52 +0200") -> tz("Europe/Amsterdam")

    = 2022-07-20 21:52:00 CEST (UTC +02), Europe/Amsterdam    [DateTime]

datetime("2022-07-20 21:52 +0200") -> tz("Asia/Taipei")

    = 2022-07-21 03:52:00 CST (UTC +08), Asia/Taipei    [DateTime]

unixtime

Converts a DateTime to a UNIX timestamp. Can be used on the right hand side of a conversion operator: now() -> unixtime.

fn unixtime(input: DateTime) -> Scalar

datetime("2022-07-20 21:52 +0200") -> unixtime

    = 1_658_346_720

from_unixtime

Converts a UNIX timestamp to a DateTime object.

fn from_unixtime(input: Scalar) -> DateTime

from_unixtime(2^31)

    = 2038-01-19 03:14:08 UTC    [DateTime]

today

Returns the current date at midnight (in the local time).

fn today() -> DateTime

date

Parses a string (only date) into a DateTime object.

fn date(input: String) -> DateTime

date("2022-07-20")

    = 2022-07-20 00:00:00 UTC    [DateTime]

time

Parses a string (time only) into a DateTime object.

fn time(input: String) -> DateTime

calendar_add

Adds the given time span to a DateTime. This uses leap-year and DST-aware calendar arithmetic with variable-length days, months, and years.

fn calendar_add(dt: DateTime, span: Time) -> DateTime

calendar_add(datetime("2022-07-20 21:52 +0200"), 2 years)

    = 2024-07-20 21:52:00 (UTC +02)    [DateTime]

calendar_sub

Subtract the given time span from a DateTime. This uses leap-year and DST-aware calendar arithmetic with variable-length days, months, and years.

fn calendar_sub(dt: DateTime, span: Time) -> DateTime

calendar_sub(datetime("2022-07-20 21:52 +0200"), 3 years)

    = 2019-07-20 21:52:00 (UTC +02)    [DateTime]

weekday

Get the day of the week from a given DateTime.

fn weekday(dt: DateTime) -> String

weekday(datetime("2022-07-20 21:52 +0200"))

    = "Wednesday"    [String]

julian_date (Julian date)

Convert a DateTime to a Julian date, the number of days since the origin of the Julian date system (noon on November 24, 4714 BC in the proleptic Gregorian calendar). More information here.

fn julian_date(dt: DateTime) -> Time

julian_date(datetime("2022-07-20 21:52 +0200"))

    = 2.45978e+6 day    [Time]

human (Human-readable time duration)

Converts a time duration to a human-readable string in days, hours, minutes and seconds. More information here.

fn human(time: Time) -> String

century/1e6 -> human

    = "52 minutes + 35.693 seconds"    [String]

Other functions

Error handling · Floating point · Quantities · Chemical elements · Mixed unit conversion · Temperature conversion · Color format conversion

Error handling

Defined in: core::error

error

Throw an error with the specified message. Stops the execution of the program.

fn error<T>(message: String) -> T

Floating point

Defined in: core::numbers

is_nan

Returns true if the input is NaN. More information here.

fn is_nan<T: Dim>(n: T) -> Bool

is_nan(37)

    = false    [Bool]

is_nan(NaN)

    = true    [Bool]

is_infinite

Returns true if the input is positive infinity or negative infinity. More information here.

fn is_infinite<T: Dim>(n: T) -> Bool

is_infinite(37)

    = false    [Bool]

is_infinite(-inf)

    = true    [Bool]

is_finite

Returns true if the input is neither infinite nor NaN.

fn is_finite<T: Dim>(n: T) -> Bool

is_finite(37)

    = true    [Bool]

is_finite(-inf)

    = false    [Bool]

is_zero

Returns true if the input is 0 (zero).

fn is_zero<D: Dim>(value: D) -> Bool

is_zero(37)

    = false    [Bool]

is_zero(0)

    = true    [Bool]

is_nonzero

Returns true unless the input is 0 (zero).

fn is_nonzero<D: Dim>(value: D) -> Bool

is_nonzero(37)

    = true    [Bool]

is_nonzero(0)

    = false    [Bool]

is_integer

Returns true if the input is an integer.

fn is_integer(x: Scalar) -> Bool

is_integer(3)

    = true    [Bool]

is_integer(pi)

    = false    [Bool]

Quantities

Defined in: core::quantities

value_of

Extract the plain value of a quantity (the 20 in 20 km/h). This can be useful in generic code, but should generally be avoided otherwise.

fn value_of<T: Dim>(x: T) -> Scalar

value_of(20 km/h)

    = 20

unit_of

Extract the unit of a quantity (the km/h in 20 km/h). This can be useful in generic code, but should generally be avoided otherwise. Returns an error if the quantity is zero.

fn unit_of<T: Dim>(x: T) -> T

unit_of(20 km/h)

    = 1 km/h    [Velocity]

has_unit

Returns true if quantity has the same unit as unit_query, or if quantity evaluates to zero.

fn has_unit<T: Dim>(quantity: T, unit_query: T) -> Bool

has_unit(20 km/h, km/h)

    = true    [Bool]

has_unit(20 km/h, m/s)

    = false    [Bool]

is_dimensionless

Returns true if quantity is dimensionless, or if quantity is zero.

fn is_dimensionless<T: Dim>(quantity: T) -> Bool

is_dimensionless(10)

    = true    [Bool]

is_dimensionless(10 km/h)

    = false    [Bool]

unit_name

Returns a string representation of the unit of quantity. Returns an empty string if quantity is dimensionless.

fn unit_name<T: Dim>(quantity: T) -> String

unit_name(20)

    = ""    [String]

unit_name(20 m^2)

    = "m²"    [String]

unit_name(20 km/h)

    = "km/h"    [String]

quantity_cast

Unsafe function that returns the quantity from unmodified with the target dimension To. This can be useful in generic code, but should generally be avoided otherwise.

fn quantity_cast<From: Dim, To: Dim>(f: From, t: To) -> To

quantity_cast(1 nm, m)

    = 1 nm    [Length]

Chemical elements

Defined in: chemistry::elements

element (Chemical element)

Get properties of a chemical element by its symbol or name (case-insensitive).

fn element(pattern: String) -> ChemicalElement

element("H")

    = ChemicalElement { symbol: "H", name: "Hydrogen", atomic_number: 1, group: 1, group_name: "Alkali metals", period: 1, melting_point: 13.99 K, boiling_point: 20.271 K, density: 0.00008988 g/cm³, electron_affinity: 0.754 eV, ionization_energy: 13.598 eV, vaporization_heat: 0.904 kJ/mol }    [ChemicalElement]

element("hydrogen").ionization_energy

    = 13.598 eV    [Energy or Torque]

Mixed unit conversion

Defined in: units::mixed

unit_list (Unit list)

Convert a value to a mixed representation using the provided units.

fn unit_list<D: Dim>(units: List<D>, value: D) -> List<D>

5500 m |> unit_list([miles, yards, feet, inches])

    = [3 mi, 734 yd, 2 ft, 7.43307 in]    [List]

DMS (Degrees, minutes, seconds)

Convert an angle to a mixed degrees, (arc)minutes, and (arc)seconds representation. Also called sexagesimal degree notation. More information here.

fn DMS(alpha: Angle) -> List<Angle>

46.5858° -> DMS

    = [46°, 35′, 8.88″]    [List]

DM (Degrees, decimal minutes)

Convert an angle to a mixed degrees and decimal minutes representation. More information here.

fn DM(alpha: Angle) -> List<Angle>

46.5858° -> DM

    = [46°, 35.148′]    [List]

feet_and_inches (Feet and inches)

Convert a length to a mixed feet and inches representation. More information here.

fn feet_and_inches(length: Length) -> List<Length>

180 cm -> feet_and_inches

    = [5 ft, 10.8661 in]    [List]

pounds_and_ounces (Pounds and ounces)

Convert a mass to a mixed pounds and ounces representation. More information here.

fn pounds_and_ounces(mass: Mass) -> List<Mass>

1 kg -> pounds_and_ounces

    = [2 lb, 3.27396 oz]    [List]

Temperature conversion

Defined in: physics::temperature_conversion

from_celsius

Converts from degree Celsius (°C) to Kelvin. More information here.

fn from_celsius(t_celsius: Scalar) -> Temperature

from_celsius(300)

    = 573.15 K    [Temperature]

celsius

Converts from Kelvin to degree Celcius (°C). This can be used on the right hand side of a conversion operator: 200 K -> celsius. More information here.

fn celsius(t_kelvin: Temperature) -> Scalar

300K -> celsius

    = 26.85

from_fahrenheit

Converts from degree Fahrenheit (°F) to Kelvin. More information here.

fn from_fahrenheit(t_fahrenheit: Scalar) -> Temperature

from_fahrenheit(300)

    = 422.039 K    [Temperature]

fahrenheit

Converts from Kelvin to degree Fahrenheit (°F). This can be used on the right hand side of a conversion operator: 200 K -> fahrenheit. More information here.

fn fahrenheit(t_kelvin: Temperature) -> Scalar

300K -> fahrenheit

    = 80.33

Color format conversion

Defined in: extra::color

rgb

Create a Color from RGB (red, green, blue) values in the range \( [0, 256) \).

fn rgb(red: Scalar, green: Scalar, blue: Scalar) -> Color

use extra::color
rgb(125, 128, 218)

    = Color { red: 125, green: 128, blue: 218 }    [Color]

color

Create a Color from a (hexadecimal) value.

fn color(rgb_hex: Scalar) -> Color

use extra::color
color(0xff7700)

    = Color { red: 255, green: 119, blue: 0 }    [Color]

color_rgb

Convert a color to its RGB representation.

fn color_rgb(color: Color) -> String

use extra::color
cyan -> color_rgb

    = "rgb(0, 255, 255)"    [String]

color_rgb_float

Convert a color to its RGB floating point representation.

fn color_rgb_float(color: Color) -> String

use extra::color
cyan -> color_rgb_float

    = "rgb(0.000, 1.000, 1.000)"    [String]

color_hex

Convert a color to its hexadecimal representation.

fn color_hex(color: Color) -> String

use extra::color
rgb(225, 36, 143) -> color_hex

    = "#e1248f"    [String]

Constants

Mathematical

• pi, π
• τ
• e
• golden_ratio, φ

Named numbers

Large numbers

• hundred
• thousand
• million
• billion
• trillion
• quadrillion
• quintillion
• googol

Unicode fractions:

• ½, ⅓, ⅔, ¼, ¾, …

Colloquial:

• quarter
• half
• semi
• double
• triple
• dozen

Physics

List of supported units

See also: Unit notation.

All SI-accepted units support metric prefixes (mm, cm, km, … or millimeter, centimeter, kilometer, …) and — where sensible — units allow for binary prefixes (MiB, GiB, … or mebibyte, gibibyte, …).

Installation

Linux

Ubuntu

… and other Debian-based Linux distributions.

Download the latest .deb package from the release page and install it via dpkg. For example:

curl -LO https://github.com/sharkdp/numbat/releases/download/v1.16.0/numbat_1.16.0_amd64.deb
sudo dpkg -i numbat_1.16.0_amd64.deb

Alternatively, if you want automatic updates, you can use a community-maintained Numbat PPA. The PPA only hosts packages for the amd64/x86_64 architecture.

sudo add-apt-repository ppa:apandada1/numbat
sudo apt update
sudo apt install numbat

Arch Linux

In Arch Linux and Arch based distributions, you can install the prebuilt package of Numbat from the AUR for the x86_64 architecture:

yay -S numbat-bin

You can also install the numbat AUR package, which will download the source and compile it. It works on all architectures.

yay -S numbat

Void Linux

You can install the numbat package using

sudo xbps-install -S numbat

Chimera Linux

Chimera Linux has a numbat package in its contrib repo. Enable it if you haven’t already, then install numbat:

doas apk add numbat

macOS

Homebrew

You can install Numbat with Homebrew:

brew install numbat

Windows

Scoop

You can install the numbat package using scoop:

scoop install main/numbat

NixOS

… or any distribution where Nix is installed.

Install numbat to your profile:

nix-env -iA nixpkgs.numbat

Or add it to your NixOS Configuration:

environment.systemPackages = [
  pkgs.numbat
];

From pre-built binaries

Download the latest release for your system from this page. Unpack the archive and place the numbat/numbat.exe binary in a folder that is on your PATH.

Note that the modules folder that is included in the archives is not strictly required to run Numbat. It serves more as a reference for interested users. However, if you want to get the best possible experience or if you are a package maintainer, please follow these guidelines.

From source

Clone the Git repository, and build Numbat with cargo:

git clone https://github.com/sharkdp/numbat
cd numbat/
cargo install -f --path numbat-cli

Or install the latest release using

cargo install numbat-cli

Guidelines for package maintainers

Thank you for packaging Numbat! This section contains instructions that are not strictly necessary to create a Numbat package, but provide users with the best-possible experience on your target platform.

Numbat has a standard library that is written in Numbat itself. The sources for this so called “prelude” are available in the numbat/modules folder. We also include this modules folder in the pre-built GitHub releases. Installing this folder as part of the package installation is not necessary for Numbat to work, as the prelude is also stored inside the numbat binary. But ideally, this folder should be made available for users. There are three reasons for this:

• Users might want to look at the code in the standard library to get a better understanding of the language itself.
• For some error messages, Numbat refers to locations in the source code. For example, if you type let meter = 2, the compiler will let you know that this identifier is already in use, and has been previously defined at a certain location inside the standard library. If the corresponding module is available as a file on the users system, they will see the proper path and can read the corresponding file.
• Users might want to make changes to the prelude. Ideally, this should be done via a user module folder, but the system-wide folder can serve as a template.

In order for this to work, the modules folder should ideally be placed in the standard location for the target operating system. If this is not possible, package maintainers can customize numbat during compilation by setting the environment variable NUMBAT_SYSTEM_MODULE_PATH to the final locatiom. If this variable is set during compilation, the specified path will be compiled into the numbat binary.

In order to test that everything is working as intended, you can open numbat and type let meter = 2. The path in the error message should point to the specified location (and not to <builtin>/…).

If your OS uses .desktop files, you should probably also install:

• assets/numbat.desktop (typically to /usr/share/applications)
• assets/numbat.svg (typically to /usr/share/icons/hicolor/scalable/apps)
• assets/numbat-*x*.png (typically to e.g. /usr/share/icons/hicolor/32x32/apps, depending on each icon’s size)

This allows users to e.g. pin Numbat to GNOME’s Dash.

Usage

Modes

You can run the Numbat command-line application in three different modes:

Command-line options

See numbat --help for more information.

Interactive sessions

Interactive sessions allow you to perform a sequence of calculations. You can use the special identifiers ans or _ to refer to the result of the last calculation. For example:

>>> 60 kW h / 150 kW

    = 0.4 h

>>> ans -> minutes

    = 24 min

Commands

There is a set of special commands that only work in interactive mode:

Key bindings

In interactive command-line mode, you can use the following key bindings. Most importantly, Tab for auto-completion, arrow keys and Ctrl-R for browsing the command history, and Ctrl-D for exiting the interactive session.

Customization

Startup

By default, Numbat will load the following modules/files during startup, in order:

• Numbat Prelude (a module called prelude, either from <module-path>/prelude.nbt if available, or the builtin version)
• The user initialization file, if available (a file called init.nbt from <config-path>/init.nbt)

Config path

Numbat’s configuration folder (<config-path> above) can be found under:

Module paths

Numbat will load modules from the following sources. Entries higher up in the list take precedence.

Note that the System-location might be different for some installation methods. Refer to your package manager for details.

Customization

Configuration

Numbat’s configuration file is called config.toml, and it needs to be placed in <config-path> described above (~/.config/numbat/config.toml on Linux). You can generate a default configuration by calling

numbat --generate-config

The most important fields are:

# Controls the welcome message. Can be "long", "short", or "off".
intro-banner = "long"

# Controls the prompt character(s) in the interactive terminal.
prompt = ">>> "

# Whether or not to pretty-print expressions before showing the result.
# Can be "always", "never" or "auto". The latter uses pretty-printing
# only in interactive mode.
pretty-print = "auto"

[exchange-rates]
# When and if to load exchange rates from the European Central Bank for
# currency conversions. Can be "on-startup" to always fetch exchange rates
# in the background when the application is started. With "on-first-use",
# Numbat only fetches exchange rates when they are needed. Exchange rate
# fetching can also be disabled using "never". The latter will lead to
# "unknown identifier" errors when a currency unit is being used.
fetching-policy = "on-startup"

Custom functions, constants, units

If you want to add custom functions, constants, or units to your default environment, create a init.nbt file in your config folder (~/.config/numbat/init.nbt on Linux).

Custom modules

You can also create your own modules that can be loaded on demand. To this end, create a new file, say <module-path>/user/finance.nbt in one of the module folders (e.g. ~/.config/numbat/modules/custom/finance.nbt on Linux). This module can then be loaded using

use custom::finance

in your Numbat scripts or in the REPL. You can also load custom modules from init.nbt if you want to have them available all the time.

You can also organize modules into subfolders (e.g. <module-path>/custom/finance/functions.nbt). In that case, you can load them using

use custom::finance::functions

In fact, the custom folder is just a convention to avoid name clashes with the standard library.

Usage

The browser-based version of Numbat is available at https://numbat.dev/.

Interactive terminal

The terminal allows you to perform a sequence of calculations. You can use the arrow keys to browse through the command history. The special identifiers ans and _ refer to the result of the last calculation. For example:

>>> 60 kW h / 150 kW

    = 0.4 h

>>> ans -> minutes

    = 24 min

Commands

There is a set of special commands that only work in the web version:

Key bindings

In interactive command-line mode, you can use the following key bindings. Most importantly, Tab for auto-completion, arrow keys and Ctrl-R for browsing the command history, and Ctrl-D for exiting the interactive session.

Sharing calculations

To share the result of a calculation with someone else, you can just copy the URL from your browers address bar. As you enter new lines in the terminal, your input will be appended to the URL to build up something like https://numbat.dev/?q=let+P0+%3D+50_000+people%0A… that you can just copy and share. To reset the state and clear the URL, use the reset command (see above).

Type system

Numbat is a language with a special type system that treats physical dimensions as types. A type checker infers types for every expression in the program and ensures that everything is correct in terms of physical dimensions, which implies correctness in terms of physical units. For example, the expression 2 meter has a type of Length. The expression 3 inch also has a type of Length. The combined expression 2 meter + 3 inch is therefore well-typed. On the other hand, 2 meter + 3 second is ill-typed, as 3 second is of type Time.

The type system is static which means that the correctness of a Numbat program is verified before the program starts executing. Note that certain runtime errors (like division-by-zero) can still occur.

Algebra of types

Types in Numbat can be combined in various ways to produce new types. In its most general form, a type can be thought of as a product of physical (base) dimensions \( D_k \) with exponents \( \alpha_k \in \mathbb{Q} \): \[ \prod_k D_k^{\alpha_k} \] For example, the type Energy can be represented as Mass¹ × Length² × Time⁻².

Multiplication

This naturally allows us to multiply types (by combining the factors of both products into a single product). We can use the * operator to construct types for physical dimensions that are products of two or more (base) dimensions. For example:

dimension Time
dimension Current
dimension ElectricCharge = Current * Time

Exponentiation

We can also raise units to arbitrary powers \( n \in \mathbb{Q} \), by simply multiplying each \( \alpha_k \) with \( n \). The syntax uses the ^ exponentiation operator:

dimension Length
dimension Volume = Length^3

dimension Time
dimension Frequency = Time^(-1)

Division

Once we have multiplication and exponentiation, we can define the division of two types as

TypeA / TypeB ≡ TypeA * TypeB^(-1)

This is mostly for convenience. It allows us to write definitions like

dimension Power = Energy / Time

Note: When we talk about products of types in this section, we mean actual, literal products. Type theory also has the notion of product types which denote something else: compound types — like tuples or structs — that are built by combining two or more types. If we think of types in terms of the sets of all possible values that they represent, then product types represent the Cartesian product of those.

Type inference and type annotations

The type checker can infer the types of all expressions without explicitly declaring them. For example, the following definition does not mention any types:

let E_pot = 80 kg × 9.8 m/s² × 5 m

However, it is often helpful to specify the type anyway. This way, we can make sure that no mistakes were made:

let E_pot: Energy = 80 kg × 9.8 m/s² × 5 m

The type checker will compare the inferred type with the specified type and raise an error in case of inconsistency.

Function definitions also allow for type annotations, both for the parameters as well as the return type. The following example shows a function that takes a quantity of type Length and returns a Pressure:

let p0: Pressure = 101325 Pa
let t0: Temperature = 288.15 K

let lapse_rate = 0.65 K / 100 m

fn air_pressure(height: Length) -> Pressure =
  p0 · (1 - lapse_rate · height / t0)^5.255

See this chapter for more details on the type inference algorithm.

Generic types

Numbat’s type system also supports generic types (parametric polymorphism). These can be used for functions that work regardless of the physical dimension of the argument(s). For example, the type signature of the absolute value function is given by

fn abs<D: Dim>(x: D) -> D

where the angle brackets after the function name introduce new type parameters (D). This can be read as: abs takes an arbitrary physical quantity of dimension type D and returns a quantity of the same physical dimension D.

The Dim constraint makes sure that this can not be used with non-dimensional types like Bool or String.

As a more interesting example, we can look at the sqrt function. Its type signature can be written as

fn sqrt<D>(x: D^2) -> D

Alternatively, it could also be specified as fn sqrt<D>(x: D) -> D^(1/2).

Limitations

The static type system also has some limitations. Let’s look at an exponentiation expression like

expr1 ^ expr2

where expr1 and expr2 are arbitrary expressions. In order for that expression to properly type check, the type of expr2 must be Scalar — something like 2^meter does not make any sense. If the type of expr1 is also Scalar, everything is well and the type of the total expression is also Scalar. An example for this trivial case is an expression like e^(-x²/σ²). As long as the type of x is the same as the type of σ, this is fine.

A more interesting case arises if expr1 is dimensionfull, as in meter^3. Here, things become difficult: in order to compute the type of the total expression expr1 ^ expr2, we need to know the value of expr2. For the meter^3 example, the answer is Length^3. This seems straightforward. However, the syntax of the language allows arbitrary expressions in the exponent. This is important to support use cases like the above e^(-x²/σ²). But it poses a problem for the type checker. In order to compute the type of expr1 ^ expr2, we need to fully evaluate expr2 at compile time. This is not going to work in general. Just think of a hypothetical expression like meter^f() where f() could do anything. Maybe even get some input from the user at runtime.

Numbat’s solution to this problem looks like this: If expr1 is not dimensionless, we restrict expr2 to a small subset of allowed operations that can be fully evaluated at compile time (similar to constexpr expressions in C++, const expressions in Rust, etc). Expressions like meter^(2 * (2 + 1) / 3) are completely fine and can be typechecked (Length^2), but things like function calls are not allowed and will lead to a compile time error.

To summarize: Given an exponentiation expression like expr1 ^ expr2, the type checker requires that:

• expr2 is of type Scalar
• One of the following:
  • expr1 is also of type Scalar
  • expr2 can be evaluated at compile time and yields a rational number.

Remark: We would probably need to enter the world of dependent types if we wanted to fully support exponentiation expressions without the limitations above. For example, consider the function f(x, n) = x^n. The return type of that function depends on the value of the parameter n.

IDE / editor integration

There is syntax highlighting support for the following IDEs / text editors:

• Vim
• VS Code
• Sublime Text
• Helix (tree-sitter)

Comparison with other tools

The following table provides a comparison of Numbat with other scientific calculators and programming languages. This comparison is certainly not objective, as we only list criteria that we consider important. If you think that a tool or language is missing or misrepresented, please let us know.

Detailed comparison

• Qalculate is a fantastic calculator with a strong support for units and conversions. If you don’t need the full power of a programming language, Qalculate is probably more feature-complete than Numbat.
• Frink is a special-purpose programming language with a focus on scientific calculations and units of measurement. The language is probably more powerful than Numbat, but lacks a static type system. It’s also a imperative/OOP language, while Numbat is a functional/declarative language. Frink is not open-source.
• GNU Units is probably the most comprehensive tool in terms of pre-defined units. Numbat makes it very easy to define custom units. If you think that a unit should be part of the standard library, please let us know.
• Wolfram Alpha is a very powerful tool, but it’s focused on single-line queries instead of longer computations. The query language lacks a strict syntax (which some might consider a feature). The tool is not open source and sometimes has limitations with respect to the number/size of queries you can make.

Other interesting tools / languages

• F# is the only programming language that we know of that comes close in terms of having an expressive type system that is based on units of measure. In fact, Numbats type system is heavily inspired by F#, except that it uses physical dimensions instead of physical units on the type level. Both languages have full type inference. F# is not listed above, as it’s not really suitable as a scientific calculator.

Contact us

To contact us, either open a GitHub issue or discussion, or pop into our Discord server.