Copyright | Copyright (c) 2007--2015 wren gayle romano |
---|---|

License | BSD3 |

Maintainer | wren@community.haskell.org |

Stability | stable |

Portability | portable |

Safe Haskell | None |

Language | Haskell98 |

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:

- http://www.haskell.org/pipermail/haskell-prime/2006-February/000791.html
- http://www.haskell.org/pipermail/haskell-prime/2006-February/000821.html

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

# Documentation

class PartialOrd a => Transfinite a where Source

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.)

Hugs (September 2006) has buggy Prelude definitions for
`isNaN`

and `isInfinite`

on Float and Double.
This module provides correct definitions, so long as Hugs.RealFloat
is compiled correctly.

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 :: a Source

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 :: a Source

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`

:

infinity + negativeInfinity

negativeInfinity + infinity

infinity - infinity

negativeInfinity - negativeInfinity

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`

(violating
the law of the excluded middle, often assumed but not required
for `Eq`

; thus, `eq`

and `ne`

are preferred over (`==`

) and
(`/=`

)). Since it returns false for equality, there may be
more than one machine representation of this `value`

.

isInfinite :: a -> Bool Source

Return true for both `infinity`

and `negativeInfinity`

,
false for all other values.

Return true only for `notANumber`

.

log :: (Floating a, Transfinite a) => a -> a Source

Since the normal `log`

throws an error on zero, we
have to redefine it in order for things to work right. Arguing
from limits we can see that `log 0 == negativeInfinity`

. Newer
versions of GHC have this behavior already, but older versions
and Hugs do not.

This function will raise an error when taking the log of negative
numbers, rather than returning `notANumber`

as the newer GHC
implementation does. The reason being that typically this is a
logical error, and `notANumber`

allows the error to propagate
silently.

In order to improve portability, the `Transfinite`

class is
required to indicate that the `Floating`

type does in fact have
a representation for negative infinity. Both native floating
types (`Double`

and `Float`

) are supported. If you define your
own instance of `Transfinite`

, verify the above equation holds
for your `0`

and `negativeInfinity`

. If it doesn't, then you
should avoid importing our `log`

and will probably want converters
to handle the discrepancy.

For GHC, this version of `log`

has rules for fusion with `exp`

.
These can give different behavior by preventing overflow to
`infinity`

and preventing errors for taking the logarithm of
negative values. For `Double`

and `Float`

they can also give
different answers due to eliminating floating point fuzz. The
rules strictly improve mathematical accuracy, however they should
be noted in case your code depends on the implementation details.