The units package provides a mechanism for compile-time dimensional analysis
in Haskell programs. It defines an embedded type system based on
units-of-measure. The units and dimensions defined are fully extensible, and
need not relate to physical properties. This package exports definitions
Number. The set of units and dimensions from
the International System (SI) are exported from the companion package
This package supports independent notions of dimension and unit. Examples of dimensions include length and mass. Examples of unit include meter and gram. Every unit measures a particular dimension, but a given dimension may be measured by many different units. For example, both meters and feet measure length.
The package supports defining multiple inter-convertible units of the same
dimension, such as
Foot. When extracting a numerical value from
a quantity, the desired unit must be specified, and the value is converted
into that unit.
The laws of nature have dimensions, and they hold true regardless of the units
used. For example, the gravitational force between two bodies is
(gravitational constant) * (mass 1) * (mass 2) / (distance between body 1 and 2)^2, regardless of whether the distance is given in meters or feet
or centimeters. In other words, every law of nature is unit-polymorphic.
The units package supports unit-polymorphic programs through the coherent
system of units (CSU) mechanism. A CSU is essentially a mapping from
dimensions to the units. All dimensioned quantities (generally just called
quantities) are expressed using the
Qu type. The
Qu type constructor takes
a (perhaps compound) dimension, a CSU and a numerical value type as arguments.
Internally, the quantity is stored as a number in the units as specified in
the CSU – this may matter if you are worried about rounding errors. In the
sequence of computations that works within one CSU, there is no unit
conversion. Unit conversions are needed only when putting values in and out of
quantities, or converting between two different CSUs.
Checking out units
Units has a git submodule, so you’ll want to use
git clone --recursive. Example:
git clone --recursive https://github.com/goldfirere/units
It is easy to imagine any number of built-in facilities that would go well with this package (sets of definitions of units for various systems, vector operations, a suite of polymorphic functions that are commonly needed but hard to define, etc.). Yet, I (Richard) don’t have the time to imagine or write all of these. If you write code that is sufficiently general and might want to be included with this package (but you don’t necessarily want to create your own new package), please write me!
The units package exports several key modules. Note that you will generally
import only one of
This is the main exported module. It exports all the necessary functionality for you to build your own set of units and operate with them.
This re-exports most of the definitions from
Data.Metrology.Poly, but restricts a few operators to work only with the default LCSU, as this is simpler for new users to
This also re-exports a similar set of definitions as
Data.Metrology.Poly, but provides numerical operations based on
vector-spaceinstead of the standard numerical classes.
This exports a set of definitions for interoperability with the
linearpackage. This is /not/ a top-level import; generally, import this with
This module contains mostly-internal definitions that may appear in GHC’s error messages. Users will generally not need to use these definitions in their code. However, by exporting this module from within
Data.Metrology.Poly, we can reduce the module-prefix clutter in error messages.
This module exports the constructor for the central datatype that stores quantities. With this constructor, you can arbitrarily change units! Use at your peril.
This module defines a
Showinstance for quantities, printing out the number stored along with its canonical dimension. This behavior may not be the best for every setting, so it is exported separately. Importing this module reduces the guaranteed unit-safety of your code, because it allows you to inspect (in a round-about way) how your quantities are stored.
This module allows users to create custom unit parsers. The user specifies a set of prefixes and a set of units to parse, and then a quasi-quoting parser is generated. See the module documentation for details.
This module exports several functions, written with Template Haskell, that make programming with
unitssomewhat easier. In particular, see
declareMonoUnit, which gets rid of a lot of the boilerplate if you don’t want unit polymorphism.
This module defines a
Quantityclass to enable easy, safe conversions with non-
unitstypes. See the module for more documentation.
We will build up a full working example in several sections. It is awkward to explain the details of the pieces until the whole example is built, so please read on to see how it all works. For more complete(-ish) examples, see this test case (for examples of how to use units) and units-defs (for examples of how to define units).
When setting up your well-typed units-of-measure program, the first step is
to define the dimensions you will be working in. (If your application involves
physical quantities, you may want to check
Data.Dimensions.SI in the
units-defs package first.)
data LengthDim = LengthDim -- each dimension is a datatype that acts as its own proxy instance Dimension LengthDim data TimeDim = TimeDim instance Dimension TimeDim
We can now build up dimensions from these base dimensions:
type VelocityDim = LengthDim :/ TimeDim
We then define units to work with these dimensions. Here, we define two different
inter-convertible units for length. (Note that just about all of this boilerplate
can be generated by functions in the
data Meter = Meter instance Unit Meter where -- declare Meter as a Unit type BaseUnit Meter = Canonical -- Meters are "canonical" type DimOfUnit Meter = LengthDim -- Meters measure Lengths instance Show Meter where -- Show instances are optional but useful show _ = "m" -- do *not* examine the argument! data Foot = Foot instance Unit Foot where type BaseUnit Foot = Meter -- Foot is defined in terms of Meter conversionRatio _ = 0.3048 -- do *not* examine the argument! -- We don't need to specify the `DimOfUnit`; -- it's implied by the `BaseUnit`. instance Show Foot where show _ = "ft" data Second = Second instance Unit Second where type BaseUnit Second = Canonical type DimOfUnit Second = TimeDim instance Show Second where show _ = "s"
A unit assignment
To perform computations with units, we must define a so-called local coherent set of units, or LCSU. This is a mapping from dimensions to units, and it informs exactly how the quantities are stored. For example:
type LCSU = MkLCSU '[(LengthDim, Meter), (TimeDim, Second)]
This definition says that we wish to store lengths in meters and times in seconds.
Note that, even though
Meter is defined as the
Canonical length, we could have
Foot in our LCSU. Canonical units are used only in conversion between
units, not the choice of how to store a quantity.
To use all these pieces to build the actual type that will store quantities, we
use one of the
MkQu_xxx type synonyms, as follows:
type Length = MkQu_DLN LengthDim LCSU Double -- Length stores lengths in our defined LCSU, using `Double` as the numerical type type Length' = MkQu_ULN Foot LCSU Double -- same as Length. Note the `U` in `MkQu_ULN`, allowing it to take a unit type Time = MkQu_DLN TimeDim LCSU Double
We now show some example computations on the defined types:
extend :: Length -> Length -- a function over lengths extend x = redim $ x |+| (1 % Meter) inMeters :: Length -> Double -- extract the # of meters inMeters = (# Meter) -- more on this later conversion :: Length -- mixing units conversion = (4 % Meter) |+| (10 % Foot) vel :: Length %/ Time -- The `%*` and `%/` operators allow -- you to combine types vel = (3 % Meter) |/| (2 % Second)
Let’s pick this apart. The
data LengthDim = LengthDim declaration creates both the
LengthDim and a term-level proxy for it. It would be possible to get away
without the proxies and use lots of type annotations, but who would want to?
We must define an instance of
Dimension to declare that
LengthDim is a dimension.
Why suffix with
Dim? To distinguish the length dimension from the length type.
Generally, the type is mentioned more often and should be the shorter name.
We then create a
TimeDim to operate alongside the
LengthDim. Using the
:/ combinator, we can create a
VelocityDim out of the two dimensions defined
so far. See below for more information on unit combinators.
Then, we make some units, using similar
data definitions. We define an
Unit to make
Meter into a proper unit. The
Unit class is
primarily responsible for handling unit conversions. In the case of
we define that as the canonical unit of length, meaning that all lengths
will internally be stored in meters. It also means that we don’t need to
define a conversion ratio for meters. You will also see that we say that
Meters measure the dimension
LengthDim, through the
We also include a
Show instance for
Meter so that lengths can be printed
easily. If you don’t need to
show your lengths, there is no need for this
Foot, we say that its
Meter, meaning that
Foot is inter-convertible with
Meter. This declaration also says that
the dimension measured by a
Foot must be the same as the dimension for
Meter. We must then define the conversion
ratio, which is the number of meters in a foot. Note that the
conversionRatio method must take a parameter to fix its type parameter, but
it must not inspect that parameter. Internally, it will be passed
undefined quite often.
The definition for
Second is quite similar to that for
The next section of code constructs an “LCSU” – a local coherent set of units. The idea is that we wish to be able to choose a set of units which are to be used in the internal, concrete representation. An LCSU is just an association list giving a concrete unit for each dimension in your domain. The particular LCSU here says that length is stored in meters and time is stored in seconds. It would be invalid to specify an LCSU with repeats for either dimension or unit.
With all this laid out, we can make the types that store values. units
MkQu_xxx type synonyms that vary in the arguments they
MkQu_DLN, for example, takes a dimension, an LCSU, and a
numerical type. With the definition above,
Length is now a type suitable
for storing lengths.
Length' are the same type. The
MkQu machinery notices
that these two are inter-convertible and will produce the same dimensioned
Note that, as you can see in the function examples at the end, it is necessary
to specify the choice of unit when creating a quantity or
extracting from a quantity. Thus, other than thinking about the
vagaries of floating point wibbles and the
Show instance, it is completely
irrelevant which unit the concrete unit in the LCSU.
Length defined here could be
used equally well in a program that deals exclusively in feet as it could in a
program with meters.
As a tangential note: I have experimented both with definitions like
data Meter = Meter and
data Meter = Meters (note the
s at the end). The second
often flows more nicely in code, but the annoyance of having to remember
whether I was at the type level or the term level led me to use the former in
Here is how to define the “kilo” prefix:
data Kilo = Kilo instance UnitPrefix Kilo where multiplier _ = 1000 kilo :: unit -> Kilo :@ unit kilo = (Kilo :@)
We define a prefix in much the same way as an ordinary unit, with a datatype
and a constructor to serve as a proxy. Instead of the
Unit class, though,
we use the
UnitPrefix class, which contains a
multiplier method. As with
other methods, this may not inspect its argument.
Due to the way units are encoded, it is necessary to explicitly apply prefixes
:@ combinator (available at both the type and term level). It is often
convenient to then define a function like
kilo to make the code flow more
longWayAway :: Length longWayAway = 150 % kilo Meter longWayAwayInMeters :: Double longWayAwayInMeters = longWayAway # Meter -- 150000.0
There are several ways of combining units to create other units.
Units can be multiplied and divided with the operators
:/, at either
the term or type level. For example:
type MetersPerSecond = Meter :/ Second type Velocity1 = MkQu_ULN MetersPerSecond LCSU Double speed :: Velocity1 speed = 20 % (Meter :/ Second)
The units package also provides combinators “%*” and “%/” to combine the types of quantities.
type Velocity2 = Length %/ Time -- same type as Velocity1
There are also exponentiation combinators
:^ (for units) and
quantities) to raise to a power. To represent the power, the
units package exports
Zero, positive numbers
MFive. At the term level, precede the number
p (mnemonic: “power”). For example:
type MetersSquared = Meter :^ Two type Area1 = MkQu_ULN MetersSquared LCSU Double type Area2 = Length %^ Two -- same type as Area1 roomSize :: Area1 roomSize = 100 % (Meter :^ pTwo) roomSize' :: Area1 roomSize' = 100 % (Meter :* Meter)
Note that addition and subtraction on units does not make physical sense, so those operations are not provided.
The haddock documentation shows the term-level quantity
combinators. The only one deserving special mention is
dimension-safe cast operator. Expressions written with the units package can
have their types inferred. This works just fine in practice, but the types are
terrible, unfortunately. Much better is to use top-level annotations (using
Time) for your functions. However, it may
happen that the inferred type of your expression and the given type of your
function may not exactly match up. This is because quantities have
a looser notion of type equality than Haskell does. For example, “meter *
second” should be the same as “second * meter”, even though these are in
different order. The
redim function checks (at compile time) to make sure its
input type and output type represent the same underlying dimension and then
performs a cast from one to the other. This cast is completely free at
runtime. When providing type annotations, it is good practice to start your
function with a
redim $ to prevent the possibility of type errors. For
example, say we redefine velocity a different way:
type Velocity3 = (MkQu_ULN Number LCSU Double) %/ Time %* Length addVels :: Velocity1 -> Velocity1 -> Velocity3 addVels v1 v2 = redim $ v1 |+| v2
This is a bit contrived, but it demonstrates the point. Without the
addVels function would not type-check. Because
redim needs to know its
result type to type-check, it should only be used at the top level, such as
here, where there is a type annotation to guide it.
redim is always dimension-safe – it will not convert a time to a
units provides a facility for ignoring LCSUs, if your application does not
need to worry about numerical precision. The facility is through the type
DefaultUnitOfDim. For example, with the definitions above, we could
type instance DefaultUnitOfDim LengthDim = Meter type instance DefaultUnitOfDim TimeDim = Second
and then use the
DefaultLCSU for our LCSU. To make the use of the default
LCSU even easier, the
MkQu_xxx operators that don’t mention an LCSU all
use the default one. So, we can say
type Length = MkQu_D LengthDim
and get to work. (This uses
Double as the underlying numerical representation.)
Data.Metrology.SI from the units-defs package exports type
DefaultUnitOfDim for the SI types, meaning that you can use
definitions like this right away.
Check out some of the test examples we have written to get more of a feel for how this all works, here.
Release notes for
New interface with the
Readinstances for dimensionless quantities.
NFDatainstances for quantities, thanks to @rimmington.
GHC 8 compatibility.
Data.Metrology.TH.evalTypenow works in GHC 7.10 on dimensions like
Volumeinstead of just base dimensions.
Break out the units parser into its own package:
Data.Metrology.Unsafe.UnsafeQu, which has
Functorand other instances.
Fix the fixity of
%. It was way too high! Now it’s 5.
Add the ability to convert in and out of point quantities. See
quOfPointand friends in
Compatibility with GHC 7.10.
Some types of arithmetic operations are different to aid in type inference. For example,
*|does not normalize its dimension list.
The types of the
derive...TH functions now allow for deriving units based on composite dimensions/units.
New TH function to help declare constants, called
Includes a decently comprehensive test suite.
Add support for unit parsing within expressions:
g = 9.8 % [si| m/s^2 |]
Data.Metrologyexports operators that work with the default LCSU. Use
Data.Metrology.Polyto get the old, more flexible operators.
Data.Metrology.Poly. This allows users to import
Showinstance for quantities.
Numeric operations are available based on vector spaces, as implemented in the
Some documentation cleanup.
evalTypethat evaluates a type, using Template Haskell. This allows for easier instance declarations for quantities.
Quantitythat allows for easy conversions with non-
A few bugfixes.
Ordinstances now work over any quantity, not just dimensionless ones.
New functions in
Unitinstances for you.
This is a major update.
units now supports the notion of a local unit set
and of separable dimensions and units. See the description in the
draft paper for
This update will very likely break any code that used
The update was written in partnership with Takayuki Muranushi.
Added dependency on the singletons library
Brought up to date with changes for GHC 7.8
Generalized numerical representation
Improved Haddock headers
- Fixed dependency on base to force compilation with GHC >= 7.7
- First release