Exact real arithmetic implemented by fast binary Cauchy sequences.
Compare evaluating Euler’s identity with a
λ> let i = 0 :+ 1 λ> exp (i * pi) + 1 :: Complex Float 0.0 :+ (-8.742278e-8)
… and with a
λ> import Data.CReal λ> let i = 0 :+ 1 λ> exp (i * pi) + 1 :: Complex (CReal 0) 0 :+ 0
The basic operations have explanations and proofs of correctness here.
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
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.
CReal type is an instance of
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
Double. The only two
super slow functions are
atanh at the moment.
The implementation is not without its caveats however. The big gotcha is that
although internally the
CReal ns are represented exactly, whenever a value is
extracted to another type such as a
Float it is evaluated to
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
Contributions and bug reports are welcome!
Please feel free to contact me on GitHub or as “jophish” on freenode.