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README
exactreal
Exact real arithmetic implemented by fast binary Cauchy sequences.
Motivating Example
Compare evaluating Euler's identity with a Float
:
Note that you'll need the DataKinds
extension turned on to evaluate the
examples in this readme.
λ> let i = 0 :+ 1
λ> exp (i * pi) + 1 :: Complex Float
0.0 :+ (8.742278e8)
... and with a CReal
:
λ> import Data.CReal
λ> let i = 0 :+ 1
λ> exp (i * pi) + 1 :: Complex (CReal 0)
0 :+ 0
Implementation
CReal
's phantom type parameter n :: Nat
represents the precision at which
values should be evaluated at when converting to a less precise representation.
For instance the definition of x == y
in the instance for Eq
evaluates x 
y
at precision n
and compares the resulting Integer
to zero. I think that
this is the most reasonable solution to the fact that lots of of operations
(such as equality) are not computable on the reals but we want to pretend that
they are for the sake of writing useful programs. Please see the
Caveats section for more information.
The CReal
type is an instance of Num
, Fractional
, Floating
, Real
,
RealFrac
, RealFloat
, Eq
, Ord
, Show
and Read
. The only functions not
implemented are a handful from RealFloat
which assume the number is
implemented with a mantissa and exponent.
There is a comprehensive test suite to test the properties of these classes.
The performance isn't terrible on most operations but it's obviously not nearly
as speedy as performing the operations on Float
or Double
. The only two
super slow functions are asinh
and atanh
at the moment.
Caveats
The implementation is not without its caveats however. The big gotcha is that
although internally the CReal n
s are represented exactly, whenever a value is
extracted to another type such as a Rational
or Float
it is evaluated to
within 2^p
of the true value.
For example when using the CReal 0
type (numbers within 1 of the true value)
one can produce the following:
λ> 0.5 == (1 :: CReal 0)
True
λ> 0.5 * 2 == (1 :: CReal 0) * 2
False
Contributing
Contributions and bug reports are welcome!
Please feel free to contact me on GitHub or as "jophish" on freenode.
Joe