Numeric Literals

Meaning of Numeric Literals

The meaning of a numeric literal is determined as follows:

• If the literal ends with l or L, it is a Long integer (and must fit in its legal range).
• If the literal ends with f or F, it is a single precision floating point number of type Float.
• If the literal ends with d or D, it is a double precision floating point number of type Double.

In each of these cases the conversion to a number is exactly as in Scala 2 or in Java. If a numeric literal does not end in one of these suffixes, its meaning is determined by the expected type:

1. If the expected type is Int, Long, Float, or Double, the literal is treated as a standard literal of that type.
2. If the expected type is a fully defined type T that has a given instance of type scala.util.FromDigits[T], the literal is converted to a value of type T by passing it as an argument to the fromDigits method of that instance (more details below).
3. Otherwise, the literal is treated as a Double literal (if it has a decimal point or an exponent), or as an Int literal (if not). (This last possibility is again as in Scala 2 or Java.)

With these rules, the definition

val x: Long = -10_000_000_000

is legal by rule (1), since the expected type is Long. The definitions

val y: BigInt = 0x123_abc_789_def_345_678_901
val z: BigDecimal = 111222333444.55

are legal by rule (2), since both BigInt and BigDecimal have FromDigits instances (which implement the FromDigits subclasses FromDigits.WithRadix and FromDigits.Decimal, respectively). On the other hand,

val x = -10_000_000_000

gives a type error, since without an expected type -10_000_000_000 is treated by rule (3) as an Int literal, but it is too large for that type.

The FromDigits Trait

To allow numeric literals, a type simply has to define a given instance of the scala.util.FromDigits type class, or one of its subclasses. FromDigits is defined as follows:

trait FromDigits[T]:
  def fromDigits(digits: String): T

Implementations of fromDigits convert strings of digits to the values of the implementation type T. The digits string consists of digits between 0 and 9, possibly preceded by a sign ("+" or "-"). Number separator characters _ are filtered out before the string is passed to fromDigits.

The companion object FromDigits also defines subclasses of FromDigits for whole numbers with a given radix, for numbers with a decimal point, and for numbers that can have both a decimal point and an exponent:

object FromDigits:

  /** A subclass of `FromDigits` that also allows to convert whole
   *  number literals with a radix other than 10
   */
  trait WithRadix[T] extends FromDigits[T]:
    def fromDigits(digits: String): T = fromDigits(digits, 10)
    def fromDigits(digits: String, radix: Int): T

  /** A subclass of `FromDigits` that also allows to convert number
   *  literals containing a decimal point ".".
   */
  trait Decimal[T] extends FromDigits[T]

  /** A subclass of `FromDigits`that allows also to convert number
   *  literals containing a decimal point "." or an
   *  exponent `('e' | 'E')['+' | '-']digit digit*`.
   */
  trait Floating[T] extends Decimal[T]

A user-defined number type can implement one of those, which signals to the compiler that hexadecimal numbers, decimal points, or exponents are also accepted in literals for this type.

Error Handling

FromDigits implementations can signal errors by throwing exceptions of some subtype of FromDigitsException. FromDigitsException is defined with three subclasses in the FromDigits object as follows:

abstract class FromDigitsException(msg: String) extends NumberFormatException(msg)

class NumberTooLarge (msg: String = "number too large")         extends FromDigitsException(msg)
class NumberTooSmall (msg: String = "number too small")         extends FromDigitsException(msg)
class MalformedNumber(msg: String = "malformed number literal") extends FromDigitsException(msg)

Example

As a fully worked out example, here is an implementation of a new numeric class, BigFloat, that accepts numeric literals. BigFloat is defined in terms of a BigInt mantissa and an Int exponent:

case class BigFloat(mantissa: BigInt, exponent: Int):
  override def toString = s"${mantissa}e${exponent}"

BigFloat literals can have a decimal point as well as an exponent. E.g. the following expression should produce the BigFloat number BigFloat(-123, 997):

-0.123E+1000: BigFloat

The companion object of BigFloat defines an apply constructor method to construct a BigFloat from a digits string. Here is a possible implementation:

object BigFloat:
  import scala.util.FromDigits

  def apply(digits: String): BigFloat =
    val (mantissaDigits, givenExponent) =
      digits.toUpperCase.split('E') match
        case Array(mantissaDigits, edigits) =>
          val expo =
            try FromDigits.intFromDigits(edigits)
            catch case ex: FromDigits.NumberTooLarge =>
              throw FromDigits.NumberTooLarge(s"exponent too large: $edigits")
          (mantissaDigits, expo)
        case Array(mantissaDigits) =>
          (mantissaDigits, 0)
    val (intPart, exponent) =
      mantissaDigits.split('.') match
        case Array(intPart, decimalPart) =>
          (intPart ++ decimalPart, givenExponent - decimalPart.length)
        case Array(intPart) =>
          (intPart, givenExponent)
    BigFloat(BigInt(intPart), exponent)

To accept BigFloat literals, all that's needed in addition is a given instance of type FromDigits.Floating[BigFloat]:

given FromDigits: FromDigits.Floating[BigFloat] with
    def fromDigits(digits: String) = apply(digits)
end BigFloat

Note that the apply method does not check the format of the digits argument. It is assumed that only valid arguments are passed. For calls coming from the compiler that assumption is valid, since the compiler will first check whether a numeric literal has the correct format before it gets passed on to a conversion method.

Compile-Time Errors

With the setup of the previous section, a literal like

1e10_0000_000_000: BigFloat

would be expanded by the compiler to

BigFloat.FromDigits.fromDigits("1e100000000000")

Evaluating this expression throws a NumberTooLarge exception at run time. We would like it to produce a compile-time error instead. We can achieve this by tweaking the BigFloat class with a small dose of metaprogramming. The idea is to turn the fromDigits method into a macro, i.e. make it an inline method with a splice as right-hand side. To do this, replace the FromDigits instance in the BigFloat object by the following two definitions:

object BigFloat:
  ...

  class FromDigits extends FromDigits.Floating[BigFloat]:
    def fromDigits(digits: String) = apply(digits)

  given FromDigits with
    override inline def fromDigits(digits: String) = ${
      fromDigitsImpl('digits)
    }

Note that an inline method cannot directly fill in for an abstract method, since it produces no code that can be executed at runtime. That is why we define an intermediary class FromDigits that contains a fallback implementation which is then overridden by the inline method in the FromDigits given instance. That method is defined in terms of a macro implementation method fromDigitsImpl. Here is its definition:

private def fromDigitsImpl(digits: Expr[String])(using ctx: Quotes): Expr[BigFloat] =
    digits.value match
      case Some(ds) =>
        try
          val BigFloat(m, e) = apply(ds)
          '{BigFloat(${Expr(m)}, ${Expr(e)})}
        catch case ex: FromDigits.FromDigitsException =>
          ctx.error(ex.getMessage)
          '{BigFloat(0, 0)}
      case None =>
        '{apply($digits)}
end BigFloat

The macro implementation takes an argument of type Expr[String] and yields a result of type Expr[BigFloat]. It tests whether its argument is a constant string. If that is the case, it converts the string using the apply method and lifts the resulting BigFloat back to Expr level. For non-constant strings fromDigitsImpl(digits) is simply apply(digits), i.e. everything is evaluated at runtime in this case.

The interesting part is the catch part of the case where digits is constant. If the apply method throws a FromDigitsException, the exception's message is issued as a compile time error in the ctx.error(ex.getMessage) call.

With this new implementation, a definition like

val x: BigFloat = 1234.45e3333333333

would give a compile time error message:

3 |  val x: BigFloat = 1234.45e3333333333
  |                    ^^^^^^^^^^^^^^^^^^
  |                    exponent too large: 3333333333