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 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 percent × 1.5 MeV + 89 percent × 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

# 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

# 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

# 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

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

fn factorial(n: Scalar) -> Scalar =
  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

# 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

# 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

# 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

# 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)}")

Population growth

# Exponential model for population growth

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

fn predict_population(t: Time) =
    initial_population × e^(growth_rate·t) // round

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

Recipe

# Scale ingredient quantities based on desired servings.

@aliases(servings)
unit serving

let original_recipe_servings = 2 servings
let desired_servings = 3 servings

fn scale<D>(quantity: D) -> D =
    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)}")

XKCD 687

# 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 ?")

XKCD 2585

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

17 mph

ans -> meters/sec    // round
ans -> knots         // round
ans -> fathoms/sec   // round
ans -> furlongs/min  // round
ans -> fathoms/sec   // round
ans -> kph           // round
ans -> knots         // round
ans -> kph           // round
ans -> furlongs/hour // round
ans -> mph           // round
ans -> m/s           // round
ans -> furlongs/min  // round
ans -> yards/sec     // round
ans -> fathoms/sec   // round
ans -> m/s           // round
ans -> mph           // round
ans -> furlongs/min  // round
ans -> knots         // round
ans -> yards/sec     // round
ans -> fathoms/sec   // round
ans -> knots         // round
ans -> furlongs/min  // round
ans -> mph           // round

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

XKCD 2812

# 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

unit $: Money

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)

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)

IDE / editor integration

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

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

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

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

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

The return type annotation may be omitted, but it is often desirable to add it for better readability of the code and in order to catch potential errors.

The parameter types can also (sometimes) be omitted, in which case Numbat tries to infer their type. However, this often leads to overly generic function signatures. 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

Without any type annotations, this function has an overly generic type where mass and speed can have arbitrary dimensions (and the return type is type(mass) * type(speed)^2). So for this case, it is probably better to add parameter and return types.

Generic functions

Sometimes however, 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<T>(a: T, b: T) -> T = if a > b then a else b

This function signature tells us that max takes two arguments of arbitrary type T (but they need to match!), and returns a quantity of the same type T.

Note that you can perform the usual operations with 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.

Recursive functions

It is also possible to define recursive functions. In order to do so, you currently need to specify the return type — as the type signature can not (yet) be inferred otherwise.

For example, a naive recursive implementation to compute Fibonacci numbers in Numbat looks like this:

fn fib(n: Scalar) -> Scalar =
  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: Scalar) -> Scalar = if x < 0 then 0 else 1

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})")

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 \( \epsilon \). 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(π != 3)

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.

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
...

Dimension definitions

New (physical) dimensions can be introduced with the dimension keyword. Similar to 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 alternative definitions can be given. This is entirely optional. If specified, the compiler will make sure that all definitions are equivalent.

Custom dimensions

It is often useful to introduce ‘fictional’ physical dimensions. For example, we might want to do calculations with screen resolutions and ‘dot densities’. Introducing a new dimension for dots then allows us to define units like dpi without sacrificing unit safety:

dimension Dot

@aliases(dots)
unit dot: Dot

unit dpi = dots / inch

fn inter_dot_spacing(resolution: Dot / Length) -> Length = 1 dot / resolution

inter_dot_spacing(72 dpi) -> µm  # 353 µm

There is also a shorthand notation for creating a new dimension and a corresponding unit:

unit book

@aliases(pages)
unit page

@aliases(words)
unit word

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

Here, the base unit definitions will implicitly create new dimensions which are capitalized versions of the unit names (Book, Page, Word). This allows you 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.

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

# 3. Simple expressions

3 + (4 - 3)       # Addition and subtraction

1920 / 16 * 9     # Multiplication, division
1920 ÷ 16 × 9     # Unicode-style, '·' is also multiplication
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>(q: T^2) -> T = q^(1/2)                      # A generic function
fn is_non_negative(x: Scalar) -> Bool = x ≥ 0             # Returns a bool

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

# 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

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

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

Utility

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

Math

Basics

fn abs<T>(x: T) -> T
fn round<T>(x: T) -> T
fn floor<T>(x: T) -> T
fn ceil<T>(x: T) -> T
fn mod<T>(a: T, b: T) -> T
fn sqrt<D>(x: D^2) -> D
fn sqr<D>(x: D) -> D^2

Exponential and logarithm

fn exp(x: Scalar) -> Scalar
fn ln(x: Scalar) -> Scalar
fn log(x: Scalar) -> Scalar
fn log10(x: Scalar) -> Scalar
fn log2(x: Scalar) -> Scalar

Trigonometry

Basic:

fn cos(x: Scalar) -> Scalar
fn sin(x: Scalar) -> Scalar
fn tan(x: Scalar) -> Scalar
fn asin(x: Scalar) -> Scalar
fn acos(x: Scalar) -> Scalar
fn atan(x: Scalar) -> Scalar
fn atan2<T>(y: T, x: T) -> Scalar

Hyperbolic:

fn sinh(x: Scalar) -> Scalar
fn cosh(x: Scalar) -> Scalar
fn tanh(x: Scalar) -> Scalar
fn asinh(x: Scalar) -> Scalar
fn acosh(x: Scalar) -> Scalar
fn atanh(x: Scalar) -> Scalar

Extra:

fn cot(x: Scalar) -> Scalar
fn acot(x: Scalar) -> Scalar
fn coth(x: Scalar) -> Scalar
fn acoth(x: Scalar) -> Scalar
fn secant(x: Scalar) -> Scalar
fn arcsecant(x: Scalar) -> Scalar
fn cosecant(x: Scalar) -> Scalar
fn csc(x: Scalar) -> Scalar
fn acsc(x: Scalar) -> Scalar
fn sech(x: Scalar) -> Scalar
fn asech(x: Scalar) -> Scalar
fn csch(x: Scalar) -> Scalar
fn acsch(x: Scalar) -> Scalar

Others

fn gamma(x: Scalar) -> Scalar

Statistics

fn mean<D>(xs: D…) -> D
fn maximum<D>(xs: D…) -> D
fn minimum<D>(xs: D…) -> D

Geometry

fn hypot2<T>(x: T, y: T) -> T
fn hypot3<T>(x: T, y: T, z: T) -> T
fn circle_area<L>(radius: L) -> L^2
fn circle_circumference<L>(radius: L) -> L
fn sphere_area<L>(radius: L) -> L^2
fn sphere_volume<L>(radius: L) -> L^3

Physics

Temperature conversion

fn from_celsius(t_celsius: Scalar) -> Temperature
fn to_celsius(t_kelvin: Temperature) -> Scalar
fn from_fahrenheit(t_fahrenheit: Scalar) -> Temperature
fn to_fahrenheit(t_kelvin: Temperature) -> Scalar

Strings

fn str_length(s: String) -> Scalar
fn str_slice(s: String, start: Scalar, end: Scalar) -> String
fn str_append(a: String, b: String) -> String
fn str_contains(haystack: String, needle: String) -> Bool
fn str_replace(s: String, pattern: String, replacement: String) -> String
fn str_repeat(a: String, n: Scalar) -> 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/v1.8.0/numbat_1.8.0_amd64.deb
sudo dpkg -i numbat_1.8.0_amd64.deb

Arch Linux

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

yay -S numbat-bin

You can also install the numbat AUR package, which will download the source and compile it.

yay -S numbat

Void Linux

You can install the numbat package using

sudo xbps-install -S 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
];

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

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>/…).

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 (most) 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 gradient = 0.65 K / 100 m

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

Generic types

Numbat’s type system also supports generic types (type 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>(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 D and returns a quantity of the same physical dimension D.

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.

Contact us

To contact us, either open a GitHub issue or discussion, or pop into #numbat on Libera.Chat (link to webchat).