logfloat-0.9.1: Log-domain floating point numbersSource codeContentsIndex
Portabilitynon-portable (CPP, MPTC, OverlappingInstances)

This module presents a type class for numbers which have representations for transfinite values. The idea originated from the IEEE-754 floating-point special values, used by Data.Number.LogFloat. However not all Fractional types necessarily support transfinite values. In particular, Ratio types including Rational do not have portable representations.

For the Glasgow compiler (GHC 6.8.2), GHC.Real defines 1%0 and 0%0 as representations for infinity and notANumber, but most operations on them will raise exceptions. If toRational is used on an infinite floating value, the result is a rational with a numerator sufficiently large that it will overflow when converted back to a Double. If used on NaN, the result would buggily convert back as negativeInfinity. For more discussion on why this approach is problematic, see:

Hugs (September 2006) stays closer to the haskell98 spec and offers no way of constructing those values, raising arithmetic overflow errors if attempted.

class PartialOrd a => Transfinite a where
infinity :: a
negativeInfinity :: a
notANumber :: a
isInfinite :: a -> Bool
isNaN :: a -> Bool
log :: (Floating a, Transfinite a) => a -> a
class (Real a, Fractional b) => RealToFrac a b where
realToFrac :: a -> b
class PartialOrd a => Transfinite a whereSource

Many numbers are not Bounded yet, even though they can represent arbitrarily large values, they are not necessarily able to represent transfinite values such as infinity itself. This class is for types which are capable of representing such values. Notably, this class does not require the type to be Fractional nor Floating since integral types could also have representations for transfinite values. By popular demand the Num restriction has been lifted as well, due to complications of defining Show or Eq for some types.

In particular, this class extends the ordered projection to have a maximum value infinity and a minimum value negativeInfinity, as well as an exceptional value notANumber. All the natural laws regarding infinity and negativeInfinity should pertain. (Some of these are discussed below.)

infinity :: aSource
A transfinite value which is greater than all finite values. Adding or subtracting any finite value is a no-op. As is multiplying by any non-zero positive value (including infinity), and dividing by any positive finite value. Also obeys the law negate infinity = negativeInfinity with all appropriate ramifications.
negativeInfinity :: aSource
A transfinite value which is less than all finite values. Obeys all the same laws as infinity with the appropriate changes for the sign difference.
notANumber :: aSource

An exceptional transfinite value for dealing with undefined results when manipulating infinite values. The following operations must return notANumber, where inf is any value which isInfinite:

  • inf + inf
  • inf - inf
  • inf * 0
  • 0 * inf
  • inf / inf
  • inf div inf
  • 0 / 0
  • 0 div 0

Additionally, any mathematical operations on notANumber must also return notANumber, and any equality or ordering comparison on notANumber must return False. Since it returns false for equality, there may be more than one machine representation of this value.

isInfinite :: a -> BoolSource
Return true for both infinity and negativeInfinity, false for all other values.
isNaN :: a -> BoolSource
Return true only for notANumber.
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log :: (Floating a, Transfinite a) => a -> aSource
class (Real a, Fractional b) => RealToFrac a b whereSource

The realToFrac function is defined to pivot through a Rational according to the haskell98 spec. This is non-portable and problematic as discussed above. Since there is some resistance to breaking from the spec, this class defines a reasonable variant which deals with transfinite values appropriately.

N.B. The generic instance for transfinite types uses expensive checks to ensure correctness. On GHC there are specialized versions which use primitive converters instead. These instances are hidden from other compilers by the CPP. Be warned that the instances are overlapped, so you'll need to give type signatures if the arguments to realToFrac are polymorphic.

If any of these restrictions (CPP, GHC-only, OverlappingInstances) are onerous to you, contact the maintainer (we like patches :)

realToFrac :: a -> bSource
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