expipiplus1/exact-real

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exact-real

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 :+ (-8.742278e-8)```

... 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

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