Spire

• I learn new languages by lots of reading
• “Spire is a numeric library for Scala which is intended to be generic, fast, and precise.”
• https://github.com/non/spire

About me

• 5 years experience w/ Java, 2 years w/ Scala
• Expertise is computational statistics applied to evolutionary biology
• taking time off from PhD to focus on new venture

Abstracting over numerical types

• Good code uses abstraction: prefer List<T> over ArrayList<T>, etc.
• “Correct” numeric type not always obvious
• int or long; float, double, or BigDecimal?

OO solution: inheritance

trait Multiplicable[T] {
  def *(a: T): T
}

class MyInt(val i: Int) extends Multiplicable[MyInt] { ... }
class MyLong(val i: Long) extends Multiplicable[MyLong] { ... }

def square[T <: Multiplicable[T]](x: T): T = x * x

square(new MyInt(2))

Moving towards typeclasses

trait IsMultiplicable[T] {
  def times(a: T, b: T): T
}

object IntIsMultiplicable extends IsMultiplicable[Int] {
  def times(a: Int, b: Int) = a * b
}
object LongIsMultiplicable extends IsMultiplicable[Long] { ... }

def square[T](x: T, impl: IsMultiplicable[T]): T =
  impl.times(x, x)

square(2, IntIsMultiplicable)

Implicits in Scala

1. implicit conversions

implicit def intWrapper(x: Int) = new RichInt(x)
class RichInt(val self: Int) {
  def until(end: Int): Range = Range(self, end)
}
for (i <- 0 until 10) println(i)

2. implicit parameters

class Config(option: Boolean)
def f(x: String)(implicit cfg: Config) = { ... }

implicit config = new Config(false)
f("hello world")

Typeclasses

trait IsMultiplicable[T] {
  def times(a: T, b: T): T
}

implicit object IntIsMultiplicable extends IsMultiplicable[Int] {
  def times(a: Int, b: Int) = a * b
}

def square[T](x: T)(implicit impl: IsMultiplicable[T]): T =
  impl.times(x, x)

square(2)

Boxing

• Primitives vs references: balloons vs. strings to balloons
• Java: int vs Integer; more transparent in Scala (Int)
• Generics force boxing of primitives as objects

Boxed implementation

trait IsMultiplicable[T] {
  def times(a: T, b: T): T
}

implicit object IntIsMultiplicable extends IsMultiplicable[Int] {
  def times(a: Int, b: Int) = a * b
}

def square[T](x: T)(implicit impl: IsMultiplicable[T]): T =
  impl.times(x, x)

square(2)

Boxed implementation

public final class Main$
{

    public Object square(Object x, Main.IsMultiplicable impl)
    {
        return impl.times(x, x);
    }

    private Main$()
    {
        MODULE$ = this;
        square(BoxesRunTime.boxToInteger(2), Main.IntIsMultiplicable..MODULE$);
    }

    public static Main$ MODULE$;

    static
    {
        new Main$();
    }
}

Boxed implementation

public static class Main$IntIsMultiplicable$
    implements Main.IsMultiplicable
{

    public int times(int a, int b)
    {
        return a * b;
    }

    public volatile Object times(Object a, Object b)
    {
        return BoxesRunTime.boxToInteger(times(BoxesRunTime.unboxToInt(a), BoxesRunTime.unboxToInt(b)));
    }

}

Specialization

trait IsMultiplicable[@specialized T] {
  def times(a: T, b: T): T
}

implicit object IntIsMultiplicable extends IsMultiplicable[Int] {
  def times(a: Int, b: Int) = a * b
}

def square[@specialized T](x: T)(implicit impl: IsMultiplicable[T]): T =
  impl.times(x, x)

square(2)

Specialized implementation

public final class Main$
{
    public int square$mIc$sp(int x, Main.IsMultiplicable impl)
    {
        return impl.times$mcI$sp(x, x);
    }

    private Main$()
    {
        MODULE$ = this;
        square$mIc$sp(2, Main.IntIsMultiplicable..MODULE$);
    }

    public static Main$ MODULE$;

    static
    {
        new Main$();
    }
}

Specialized implementation

public static class Main$IntIsMultiplicable$
    implements sp
{

    public int times(int a, int b)
    {
        return times$mcI$sp(a, b);
    }

    public int times$mcI$sp(int a, int b)
    {
        return a * b;
    }

}

Nicer syntax

def uglySquare[@specialized T](x: T)(implicit impl: IsMultiplicable[T]): T =
  impl.times(x, x)

class MultiplyOps[@specialized T](self: T)(implicit impl: IsMultiplicable[T]) {
  def * (other: T): T = impl.times(self, other)
}

implicit def toMultiplyOps[@specialized T: IsMultiplicable](t: T): MultiplyOps[T] =
  new MultiplyOps(t)

def square[@specialized T: IsMultiplicable](x: T) = x * x

Decompiling again

public final class Main$
{

    public int square$mIc$sp(int x, Main.IsMultiplicable evidence$2)
    {
        return toMultiplyOps$mIc$sp(x, evidence$2).$times$mcI$sp(x);
    }

}

Back to Spire

• (Noting that Scala has a Numeric[T] typeclass)
• Spire implements a typeclass hiearchy based on mathematical definitions (Rings, Fields, etc.)
• Lots of good macros, lots of @inline-ing
• Definitions for vectorized operations on arrays, etc.
• Caveats: importing right implicits and syntax

The cfor macro

cfor(0)(_ < 10, _ + 1) { i =>
  println(i)
}

  def cforMacro[A](c: Context)(init: c.Expr[A])
     (test: c.Expr[A => Boolean], next: c.Expr[A => A])
     (body: c.Expr[A => Unit]): c.Expr[Unit] = {

    import c.universe._
    val util = SyntaxUtil[c.type](c)
    val index = util.name("index")

    val tree = if (util.isClean(test, next, body)) {
      q"""
      var $index = $init
      while ($test($index)) {
        $body($index)
        $index = $next($index)
      }
      """

    } else {
      ...
    }

    new InlineUtil[c.type](c).inlineAndReset[Unit](tree)
  }

Extending to your own types

• Define appropriate typeclasses, Spire gives syntax etc. for free
• Example: Define a Ring and VectorSpace for a square matrix

implicit def matrixIsRing[N <: Int with Singleton : Witness.Aux, R : Field]: Ring[Matrix[N, R]] = new Ring[Matrix[N, R]] {

  override def negate(x: Matrix[N, R]): Matrix[N, R] =
    Matrix((i, j) => -x(i, j))

  override def zero: Matrix[N, R] =
    Matrix((i, j) => Field[R].zero)

  override def plus(x: Matrix[N, R], y: Matrix[N, R]): Matrix[N, R] =
    Matrix((i, j) => x(i, j) + y(i, j))

  override def minus(x: Matrix[N, R], y: Matrix[N, R]): Matrix[N, R] =
    Matrix((i, j) => x(i, j) - y(i, j))

  override def one: Matrix[N, R] =
    Matrix((i, j) => if (i == j) Field[R].one else Field[R].zero)

  override def times(x: Matrix[N, R], y: Matrix[N, R]): Matrix[N, R] =
    Matrix((i, j) => x.rows(i) dot y.columns(j))

}

implicit def matrixIsVectorSpace[N <: Int with Singleton : Witness.Aux, R](implicit f: Field[R]): VectorSpace[Matrix[N, R], R] = new VectorSpace[Matrix[N, R], R] {

  override def scalar: Field[R] = f

  override def timesl(r: R, v: Matrix[N, R]): Matrix[N, R] =
    Matrix[N, R]((i: Int, j: Int) => r * v(i, j))

  override def divr(v: Matrix[N, R], r: R): Matrix[N, R] =
    Matrix[N, R]((i: Int, j: Int) => v(i, j) / r)

  override def negate(x: Matrix[N, R]): Matrix[N, R] =
    Ring[Matrix[N, R]].negate(x)

  override def zero: Matrix[N, R] =
    Ring[Matrix[N, R]].zero

  override def plus(x: Matrix[N, R], y: Matrix[N, R]): Matrix[N, R] =
    Ring[Matrix[N, R]].plus(x, y)

  override def minus(x: Matrix[N, R], y: Matrix[N, R]): Matrix[N, R] =
    Ring[Matrix[N, R]].minus(x, y)

}

Use case: automatic differentiation

Dual numbers

• Define \(\epsilon\) such that \(\epsilon^2 = 0\)
• (Similar to imaginary number \(i^2 = -1\))
• \(z = a + b\epsilon\) is a dual number
• Special property that \[f(x + \epsilon) = f(x) + f'(x)\epsilon\]
• Spire implements these as Jet

AD in Spire

import spire.implicits._
import spire.algebra.Ring
import spire.math.{Jet, JetDim}
implicit val jd = JetDim(1)
def square[T : Ring](x: T) = x * x
square(Jet(2.0, 0)) // (4.0 + [4.0]h)

Concluding thoughts

• If you use numbers or love math Spire is fun to work with
• I think it makes great reading material
• Use more typeclasses
• Spire demonstrates typeclasses that are performant and practical
• Questions?