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
necessarily support transfinite values. In particular,
Rational do not have portable representations.
For the Glasgow compiler (GHC 6.8.2), GHC.Real defines
0%0 as representations for
but most operations on them will raise exceptions. If
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.
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
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
Eq for some types.
In particular, this class extends the ordered projection to have
a maximum value
infinity and a minimum value
as well as an exceptional value
notANumber. All the natural
negativeInfinity should pertain.
(Some of these are discussed below.)
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
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.
An exceptional transfinite value for dealing with undefined
results when manipulating infinite values. The following
operations must return
inf is any value
Additionally, any mathematical operations on
must also return
notANumber, and any equality or ordering
notANumber must return
the law of the excluded middle, often assumed but not required
ne are preferred over (
/=)). Since it returns false for equality, there may be
more than one machine representation of this
Return true for both
false for all other values.
Return true only for
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 propegate
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
Float) are supported. If you define your
own instance of
Transfinite, verify the above equation holds
negativeInfinity. If it doesn't, then you
should avoid importing our
log and will probably want converters
to handle the discrepancy.