The Scala API for Quantities, Units of Measure and Dimensional Analysis
Squants is a framework of data types and a domain specific language (DSL) for representing Quantities, their Units of Measure, and their Dimensional relationships. The API supports typesafe dimensional analysis, improved domain models and more. All types are immutable and thread-safe.
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Current Release: 0.6.2 (API Docs)
Development Build: 0.7.1-SNAPSHOT (API Docs)
Build services provided by Travis CI
NOTE - This README reflects the feature set in the branch it can be found.
For more information on feature availability of a specific version see the Release History or the README for a that version
Repository hosting for Squants is provided by Sonatype. To use Squants in your SBT project add the following dependency to your build.
"com.squants" %% "squants" % "0.6.2"
or
"com.squants" %% "squants" % "0.7.1-SNAPSHOT"
To use Squants in your Maven project add the following dependency
<dependency>
<groupId>com.squants</groupId>
<artifactId>squants_2.11</artifactId>
<version>0.6.2</version>
</dependency>Beginning with Squants 0.4.x series, both Scala 2.10 and 2.11 builds are available.
To use Squants interactively in the Scala REPL, clone the git repo and run sbt squantsJVM/console
git clone https://github.com/garyKeorkunian/squants
cd squants
sbt squantsJVM/console
The Trouble with Doubles
When building programs that perform dimensional analysis, developers are quick to declare quantities using a basic numeric type, usually Double. While this may be satisfactory in some situations, it can often lead to semantic and other logic issues.
For example, when using a Double to describe quantities of Energy (kWh) and Power (kW), it is possible to compile a program that adds these two values together. This is not appropriate as kW and kWh measure quantities of two different dimensions. The unit kWh is used to measure an amount of Energy used or produced. The unit kW is used to measure Power/Load, the rate at which Energy is being used or produced, that is, Power is the first time derivative of Energy.
Power = Energy / Time
Consider the following code
val loadKw = 1.2 // Double: 1.2
val energyMwh = 24.2 // Double: 24.2
val sumKw = loadKw + energyMwh // Double: 25.4which not only adds quantities of different dimensions (Power vs Energy), it also fails to convert the scales implied in the val names (Mega vs Kilo). Because this code compiles, detection of these errors is pushed further into the development cycle.
Only quantities with the same dimensions may be compared, equated, added, or subtracted.
Squants helps prevent errors like these by type checking operations at compile time and automatically applying scale and type conversions at run-time. For example,
val load1: Power = Kilowatts(12) // returns Power(12, Kilowatts) or 12 kW
val load2: Power = Megawatts(0.023) // Power: 0.023 MW
val sum = load1 + load2 // Power: 35 kW - unit on left side is preserved
sum should be(Kilowatts(35))
sum should be(Megawatts(0.035)) // comparisons automatically convert scale works because Kilowatts and Megawatts are both units of Power. Only the scale is different and the library applies an appropriate conversion. Also, notice that keeping track of the scale within the value name is no longer needed.
val load: Power = Kilowatts(1.2) // Power: 1.2 kW
val energy: Energy = KilowattHours(23.0) // Energy: 23 kWH
val sum = load + energy // Invalid operation - does not compileThe unsupported operation in this expression prevents the code from compiling, catching the error made when using Double in the example above.
One may take quantities with different dimensions, and multiply or divide them.
Dimensionally correct type conversions are a key feature of Squants. Conversions are implemented by defining relationships between Quantity types using the * and / operators.
The following code demonstrates creating ratio between two quantities of the same dimension, resulting in a dimensionless value:
val ratio = Days(1) / Hours(3) // Double: 8.0This code demonstrates use of the Power.* method that takes a Time and returns an Energy:
val load = Kilowatts(1.2) // Power: 1.2 kW
val time = Hours(2) // Time: 2 h
val energyUsed = load * time // Energy: 2.4 kWhThis code demonstrates use of the Energy./ method that takes a Time and returns a Power:
val aveLoad: Power = energyUsed / time // Power: 1.2 kWQuantity values are based in the units used to create them.
val loadA: Power = Kilowatts(1200) // Power: 1200.0 kW
val loadB: Power = Megawatts(1200) // Power: 1200.0 MWSince Squants properly equates values of a like dimension, regardless of the unit, there is usually no reason to explicitly convert from one to the other. This is especially true if the user code is primarily performing dimensional analysis.
However, there are times when you may need to set a Quantity value to a specific unit (eg, for proper JSON encoding).
When necessary, a quantity can be converted to another unit using the in method.
val loadA = Kilowatts(1200) // Power: 1200.0 kW
val loadB = loadA in Megawatts // Power: 1.2 MW
val loadC = loadA in Gigawatts // Power: 0.0012 GWSometimes you need to get the numeric value of the quantity in a specific unit (eg, for submission to an external service that requires a numeric in a specified unit or to perform analysis beyond Squant's domain)
When necessary, the value can be extracted in the desired unit with the to method.
val load: Power = Kilowatts(1200)
val kw: Double = load to Kilowatts // Double: 1200.0
val mw: Double = load to Megawatts // Double: 1.2
val gw: Double = load to Gigawatts // Double: 0.0012Most types include methods with convenient aliases for the to methods.
val kw: Double = load toKilowatts // Double: 1200.0
val mw: Double = load toMegawatts // Double: 1.20
val gw: Double = load toGigawatts // Double: 0.0012NOTE - It is important to use the to method for extracting the numeric value,
as this ensures you will be getting the numeric value for the desired unit.
Quantity.value should not be accessed directly.
To prevent improper usage, direct access to the Quantity.value field may be deprecated in a future version.
Creating strings formatted in the desired unit:
val kw: String = load toString Kilowatts // String: “1200.0 kW”
val mw: String = load toString Megawatts // String: “1.2 MW”
val gw: String = load toString Gigawatts // String: “0.0012 GW”Creating Tuple2(Double, String) that includes a numeric value and unit symbol:
val load: Power = Kilowatts(1200)
val kw: Tuple2 = load toTuple // Tuple2: (1200, "kW")
val mw: Tuple2 = load toTuple Megawatts // Tuple2: (1.2, "MW)
val gw: Tuple2 = load toTuple Gigawatts // Tuple2: (0.0012, "GW")This can be useful for passing properly scaled quantities to other processes that do not use Squants, or require use of more basic types (Double, String)
Simple console based conversions (using DSL described below)
1.kilograms to Pounds // Double: 2.2046226218487757
kilogram / pound // Double: 2.2046226218487757
2.1.pounds to Kilograms // Double: 0.952543977
2.1.pounds / kilogram // Double: 0.952543977
100.C to Fahrenheit // Double: 212.0Apply a Double => Double operation to the underlying value of a quantity, while preserving its type and unit.
val load = Kilowatts(2.0) // 2.0 kW
val newLoad = load.map(v => v * 2 + 10) // Power: 14.0 kWThe q.map(f) method effectively expands to q.unit(f(q.to(q.unit))
Create an implicit Quantity value to be used as a tolerance in approximations.
Then use the approx method (or =~, ~=, ≈ operators) like you would use the equals method (== operator).
implicit val tolerance = Watts(.1) // implicit Power: 0.1 W
val load = Kilowatts(2.0) // Power: 2.0 kW
val reading = Kilowatts(1.9999) // Power: 1.9999 kW
// uses implicit tolerance
load =~ reading should be(true)
load ≈ reading should be(true)
load approx reading should be(true)The =~ and ≈ are the preferred operators as they have the correct precedence for equality operations.
The ~= is provided for those who wish to use a more natural looking approx operator using standard characters.
However, because of its lower precedence, user code may require parenthesis around these comparisons.
All Quantity types in Squants represent the scalar value of a quantity.
That is, there is no direction information encoded in any of the Quantity types.
This is true even for Quantities which are normally vector quantities (ie. Velocity, Acceleration, etc).
Vector quantities in Squants are implemented as case classes that takes a variable parameter list of like quantities
representing a set of point coordinates in Cartesian space.
The SVector object is a factory for creating DoubleVectors and QuantityVectors.
The dimensionality of the vector is determined by the number of arguments.
Most basic vector operations are currently supported (addition, subtraction, scaling, cross and dot products)
val vector: QuantityVector[Length] = SVector(Kilometers(1.2), Kilometers(4.3), Kilometers(2.3))
val magnitude: Length = vector.magnitude // returns the scalar value of the vector
val normalized = vector.normalize(Kilometers) // returns a corresponding vector scaled to 1 of the given unit
val vector2: QuantityVector[Length] = SVector(Kilometers(1.2), Kilometers(4.3), Kilometers(2.3))
val vectorSum = vector + vector2 // returns the sum of two vectors
val vectorDiff = vector - vector2 // return the difference of two vectors
val vectorScaled = vector * 5 // returns vector scaled 5 times
val vectorReduced = vector / 5 // returns vector reduced 5 time
val vectorDouble = vector / 5.meters // returns vector reduced and converted to DoubleVector
val dotProduct = vector * vectorDouble // returns the Dot Product of vector and vectorDouble
val crossProduct = vector crossProduct vectorDouble // currently only supported for 3-dimensional vectorsSimple non-quantity (Double based) vectors are also supported.
val vector: DoubleVector = SVector(1.2, 4.3, 2.3, 5.4) // a Four-dimensional vectorNOTE - This feature is currently under development and the final implementation being evaluated. The following type of operation is the goal.
val vectorLength = QuantityVector(Kilometers(1.2), Kilometers(4.3), Kilometers(2.3))
val vectorArea = vectorLength * Kilometers(2) // QuantityVector(2.4 km², 8.6 km², 4.6 km²)
val vectorVelocity = vectorLength / Seconds(1) // QuantityVector(1200.0 m/s, 4300.0 m/s, 2300.0 m/s)
val vectorDouble = DoubleVector(1.2, 4.3, 2.3)
val vectorLength = vectorDouble.to(Kilometers) // QuantityVector(1.2 km, 4.3 km, 2.3 km)Currently dimensional conversions are supported by using the slightly verbose, but flexible map method.
val vectorLength = QuantityVector(Kilometers(1.2), Kilometers(4.3), Kilometers(2.3))
val vectorArea = vectorLength.map[Area](_ * Kilometers(2)) // QuantityVector(2.4 km², 8.6 km², 4.6 km²)
val vectorVelocity = vectorLength.map[Velocity](_ / Seconds(1)) // QuantityVector(1200.0 m/s, 4300.0 m/s, 2300.0 m/s)
val vectorDouble = DoubleVector(1.2, 4.3, 2.3)
val vectorLength = vectorDouble.map[Length](Kilometers(_)) // QuantityVector(1.2 km, 4.3 km, 2.3 km)Convert QuantityVectors to specific units using the to or in method - much like Quantities.
val vectorLength = QuantityVector(Kilometers(1.2), Kilometers(4.3), Kilometers(2.3))
val vectorMetersNum = vectorLength.to(Meters) // DoubleVector(1200.0, 4300.0, 2300.0)
val vectorMeters = vectorLength.in(Meters) // QuantityVector(1200.0 m, 4300.0 m, 2300.0 m)Market Types are similar but not quite the same as other quantities in the library. The primary type, Money, is a Dimensional Quantity, and its Units of Measure are Currencies. However, because the conversion multipliers between currency units can not be predefined, many of the behaviors have been overridden and augmented to realize correct behavior.
A Quantity of purchasing power measured in Currency units. Like other quantities, the Unit of Measures are used to create Money values.
val tenBucks = USD(10) // Money: 10 USD
val someYen = JPY(1200) // Money: 1200 JPY
val goldStash = XAU(50) // Money: 50 XAU
val digitalCache = BTC(50) // Money: 50 BTCA Ratio between Money and another Quantity. A Price value is typed on a Quantity and can be denominated in any defined Currency.
Price = Money / Quantity
val threeForADollar = USD(1) / Each(3) // Price[Dimensionless]: 1 USD / 3 ea
val energyPrice = USD(102.20) / MegawattHours(1) // Price[Energy]: 102.20 USD / megawattHour
val milkPrice = USD(4) / UsGallons(1) // Price[Volume]: 4 USD / gallon
val costForABunch = threeForADollar * Dozen(10) // Money: 40 USD
val energyCost = energyPrice * MegawattHours(4) // Money: 408.80 USD
val milkQuota = milkPrice * USD(20) // Volume: 5 galCurrency Exchange Rates are used to define the conversion factors between currencies
// create an exchange rate
val rate = CurrencyExchangeRate(USD(1), JPY(100))
// OR
val rate = USD / JPY(100)
// OR
val rate = JPY(100) -> USD(1)
// OR
val rate = JPY(100) toThe USD(1)
val someYen: Money = JPY(350)
val someBucks: Money = USD(23.50)
// Use the convert method which automatically converts the money to the 'other' currency
val dollarAmount: Money = rate.convert(someYen) // Money: 3.5 USD
val yenAmount: Money = rate.convert(someBucks) // Money: 2360 JPY
// or just use the * operator in either direction (money * rate, or rate * money)
val dollarAmount2: Money = rate * someYen // Money: 3.5 USD
val yenAmount2: Money = someBucks * rate // Money: 2360 JPYA MoneyContext can be implicitly declared to define default settings and applicable exchange rates within its scope. This allows your application to work with a default currency based on an application configuration or other dynamic source. It also provides support for updating exchange rates and using those rates for automatic conversions between currencies. The technique and frequency chosen for exchange rate updates is completely in control of the application.
val exchangeRates = List(USD / CAD(1.05), USD / MXN(12.50), USD / JPY(100))
implicit val moneyContext = defaultMoneyContext withExchangeRates exchangeRates
val someMoney = Money(350) // 350 in the default Cur
val usdMoney: Money = someMoney in USD
val usdBigDecimal: BigDecimal = someMoney to USD
val yenCost: Money = (energyPrice * MegawattHours(5)) in JPY
val northAmericanSales: Money = (CAD(275) + USD(350) + MXN(290)) in USDUsed to represent a range of Quantity values between an upper and lower bound
val load1: Power = Kilowatts(1000)
val load2: Power = Kilowatts(5000)
val range: QuantityRange[Power] = QuantityRange(load1, load2)Use multiplication and division to create a Seq of ranges from the original
// Create a Seq of 10 sequential ranges starting with the original and each the same size as the original
val rs1 = range * 10
// Create a Seq of 10 sequential ranges each 1/10th of the original size
val rs2 = range / 10
// Create a Seq of 10 sequential ranges each with a size of 400 kilowatts
val rs3 = range / Kilowatts(400)Apply foreach, map and foldLeft/foldRight directly to QuantityRanges with a divisor
// foreach over each of the 400 kilometer ranges within the range
range.foreach(Kilometers(400)) {r => ???}
// map over each of 10 even parts of the range
range.map(10) {r => ???}
// fold over each 10 even parts of the range
range.foldLeft(10)(0) {(z, r) => ???}NOTE - Because these implementations of foreach, map and fold* take a parameter (the divisor), these methods are not directly compatible with Scala's for comprehensions. To use in a for comprehension, apply the * or / operators as described above to create a Seq from the Range.
for {
interval <- (0.seconds to 1.seconds) * 60 // 60 time ranges, 0s to 1s, 1s to 2s, ...., 59s to 60s
...
} yield ...Implicit conversions give the DSL some features that allows user code to express quantities in a more naturally expressive and readable way.
Create Quantities using Unit Of Measure Factory objects (no implicits required)
val load = Kilowatts(100)
val time = Hours(3.75)
val money = USD(112.50)
val price = Price(money, MegawattHours(1))Create Quantities using Unit of Measure names and/or symbols (uses implicits)
val load1 = 100 kW // Simple expressions don’t need dots
val load2 = 100 megaWatts
val time = 3.hours + 45.minutes // Compound expressions may need dotsCreate Quantities using operations between other Quantities
val energyUsed = 100.kilowatts * (3.hours + 45.minutes)
val price = 112.50.USD / 1.megawattHours
val speed = 55.miles / 1.hoursCreate Quantities using formatted Strings
val load = Power("40 MW") // 40 MWCreate Quantities using Tuples
val load = Power((40, "MW")) // 40 MWUse single unit values to simplify expressions
// Hours(1) == 1.hours == hour
val ramp = 100.kilowatts / hour
val speed = 100.kilometers / hour
// MegawattHours(1) == 1.megawattHours == megawattHour == MWh
val hi = 100.dollars / MWh
val low = 40.dollars / megawattHourImplicit conversion support for using Double on the left side of operations
val price = 10 / dollar // 1 USD / 10 ea
val freq = 60 / second // 60 Hz
val load = 10 * 4.MW // 40 MWCreate Quantity Ranges using to or plusOrMinus (+-) operators
val range1 = 1000.kW to 5000.kW // 1000.kW to 5000.kW
val range2 = 5000.kW plusOrMinus 1000.kW // 4000.kW to 6000.kW
val range2 = 5000.kW +- 1000.kW // 4000.kW to 6000.kWMost Quantities that support implicit conversions also include an implicit Numeric object that can be imported to your code where Numeric support is required. These follow the following pattern:
import squants.mass.MassConversions.MassNumeric
val sum = List(Kilograms(100), Grams(34510)).sumNOTE - Because a quantity can not be multiplied by a like quantity and return a like quantity, the Numeric.times
operation of numeric is implemented to throw an UnsupportedOperationException for all types except Dimensionless.
The MoneyNumeric implementation is a bit different than the implementations for other quantity types in a few important ways.
- MoneyNumeric is a class, not an object like the others.
- To create a MoneyNumeric value there must be an implicit MoneyContext in scope.
- The MoneyContext must contain applicable exchange rates if you will be applying cross-currency Numeric ops.
The following code provides a basic example for creating a MoneyNumeric:
import MoneyConversions._
implicit val moneyContext = defaultMoneyContext
implicit val moneyNum = new MoneyNumeric()
val sum = List(USD(100), USD(10)).sumThe type hierarchy includes the following core types: Quantity, Dimension, and UnitOfMeasure
A Dimension represents a type of Quantity. For example: Mass, Length, Time, etc.
A Quantity represents a dimensional value or measurement. A Quantity is a combination of a numeric value and a unit. For example: 2 lb, 10 km, 3.4 hr.
Squants has built in support for 54 quantity dimensions.
UnitOfMeasure is the scale or multiplier in which the Quantity is being measured. Squants has built in support for over 257 units of measure
For each Dimension a set of UOM objects implement a primary UOM trait typed to that Quantity. The UOM objects define the unit symbols, conversion factors, and factory methods for creating Quantities in that unit.
The code for specific implementations include
- A class representing the Quantity including cross-dimensional operations
- A companion object representing the Dimension and set of available units
- A base trait for its Units
- A set of objects defining specific units, their symbols and conversion factors
This is an abbreviated example of how a Quantity type is constructed:
class Length(val value: Double, val unit: LengthUnit) extends Quantity[Length] { ... }
object Length extends Dimension[Length] { ... }
trait LengthUnit extends UnitOfMeasure[Length] { ... }
object Meters extends LengthUnit { ... }
object Yards extends LengthUnit { ... }The apply method of the UOM objects are implemented as factories for creating Quantity values.
val len1: Length = Meters(4.3)
val len2: Length = Yards(5)Squants currently supports 257 units of measure
Special traits are used to establish a time derivative relationship between quantities.
For example Velocity is the 1st Time Derivative of Length (Distance), Acceleration is the 2nd Time Derivative.
class Length( ... ) extends Quantity[Length] with TimeIntegral[Velocity]
...
class Velocity( ... ) extends Quantity[Velocity] with TimeDerivative[Length] with TimeIntegral[Acceleration]
...
class Acceleration( ... ) extends Quantity[Acceleration] with TimeDerivative[Velocity]These traits provide operations with time operands which result in correct dimensional transformations.
val distance: Length = Kilometers(100)
val time: Time = Hours(2)
val velocity: Velocity = distance / time
val acc: Acceleration = velocity / Seconds(1)
val gravity = 32.feet / second.squared// Power is the 1st Time Derivative of Energy, PowerRamp is the 2nd
val power = Kilowatts(100)
val time: Time = Hours(2)
val energy = power * time
val ramp = Kilowatt(50) / Hours(1)The primary use case for Squants, as described above, is to produce code that is typesafe within domains that perform dimensional analysis.
val energyPrice: Price[Energy] = 45.25.money / megawattHour
val energyUsage: Energy = 345.kilowatts * 5.4.hours
val energyCost: Money = energyPrice * energyUsage
val dodgeViper: Acceleration = 60.miles / hour / 3.9.seconds
val speedAfter5Seconds: Velocity = dodgeViper * 5.seconds
val timeTo100MPH: Time = 100.miles / hour / dodgeViper
val density: Density = 1200.kilograms / cubicMeter
val volFlowRate: VolumeFlowRate = 10.gallons / minute
val flowTime: Time = 30.minutes
val totalMassFlow: Mass = volFlowRate * flowTime * densityAnother excellent use case for Squants is stronger typing for fields in your domain model. This is OK ...
case class Generator(
id: String,
maxLoadKW: Double,
rampRateKWph: Double,
operatingCostPerMWh: Double,
currency: String,
maintenanceTimeHours: Double)
...
val gen1 = Generator("Gen1", 5000, 7500, 75.4, "USD", 1.5)
val gen2 = Generator("Gen2", 100, 250, 2944.5, "JPY", 0.5)... but this is much better
case class Generator(
id: String,
maxLoad: Power,
rampRate: PowerRamp,
operatingCost: Price[Energy],
maintenanceTime: Time)
...
val gen1 = Generator("Gen1", 5 MW, 7.5.MW/hour, 75.4.USD/MWh, 1.5 hours)
val gen2 = Generator("Gen2", 100 kW, 250 kWph, 2944.5.JPY/MWh, 30 minutes)Create wrappers around external services that use basic types to represent quantities. Your application code then uses the ACL to communicate with that system thus eliminating the need to deal with type and scale conversions in multiple places throughout your application logic.
class ScadaServiceAnticorruption(val service: ScadaService) {
// ScadaService returns meter load as Double representing Megawatts
def getLoad: Power = Megawatts(service.getLoad(meterId))
}
// ScadaService.sendTempBias requires a Double representing Fahrenheit
def sendTempBias(temp: Temperature) =
service.sendTempBias(temp.to(Fahrenheit))
}Implement the ACL as a trait and mix in to the application's services where needed.
trait WeatherServiceAntiCorruption {
val service: WeatherService
def getTemperature: Temperature = Celsius(service.getTemperature)
def getIrradiance: Irradiance = WattsPerSquareMeter(service.getIrradiance)
}Extend the pattern to provide multi-currency support
class MarketServiceAnticorruption(val service: MarketService)
(implicit val moneyContext: = MoneyContext) {
// MarketService.getPrice returns a Double representing $/MegawattHour
def getPrice: Price[Energy] =
(USD(service.getPrice) in moneyContext.defaultCurrency) / megawattHour
// MarketService.sendBid requires a Double representing $/MegawattHour
// and another Double representing the max amount of energy in MegawattHours
def sendBid(bid: Price[Energy], limit: Energy) =
service.sendBid((bid * megawattHour) to USD, limit to MegawattHours)
}Build Anticorruption into Akka routers
// LoadReading message used within a Squants enabled application context
case class LoadReading(meterId: String, time: Long, load: Power)
class ScadaLoadListener(router: Router) extends Actor {
def receive = {
// ScadaLoadReading - from an external service - sends load as a string
// eg, “10.3 MW”, “345 kW”
case msg @ ScadaLoadReading(meterId, time, loadString) ⇒
// Parse the string and on success emit the Squants enabled event to routees
Power(loadString) match {
case Success(p) => router.route(LoadReading(meterId, time, p), sender())
case Failure(e) => // react to QuantityStringParseException
}
}
}... and REST API's with contracts that require basic types
trait LoadRoute extends HttpService {
def repo: LoadRepository
val loadRoute = {
path("meter-reading") {
// REST API contract requires load value and units in different fields
// Units are string values that may be 'kW' or 'MW'
post {
parameters(meterId, time, loadDouble, unit) { (meterId, time, loadDouble, unit) =>
complete {
val load = unit match {
case "kW" => Kilowatts(loadDouble)
case "MW" => Megawatts(loadDouble)
}
repo.saveLoad(meterId, time, load)
}
}
} ~
// REST API contract requires load returned as a number representing megawatts
get {
parameters(meterId, time) { (meterId, time) =>
complete {
repo.getLoad(meterId, time) to Megawatts
}
}
}
}
}
}- Gary Keorkunian (garyKeorkunian)
- Jeremy Apthorp (nornagon)
- Steve Barham (stevebarham)
- Derek Morr (derekmorr)
- rmihael (rmihael)
- Florian Nussberger (fnussber)
- Ajay Chandran (ajaychandran)
- Gia Bảo (giabao)
- Josh Lemer (joshlemer)
- Dave DeCarpio (DaveDeCaprio)
- Carlos Quiroz (cquiroz)
Squants is a Typelevel Incubator Project and, as such, supports the Typelevel Code of Conduct.
Code is offered as-is, with no implied warranty of any kind. Comments, criticisms, and/or praise are welcome, especially from scientists, engineers and the like.
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