The exact-real package

[Tags:benchmark, library, mit, test]

A type to represent exact real numbers using fast binary Cauchy sequences.

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Versions,,,,,,,, 0.9.0, 0.10.0, 0.11.0, 0.11.1, 0.11.2, 0.11.3, 0.12.0, 0.12.1, 0.12.2
Dependencies base (>=4.8 && <4.10), integer-gmp (<, memoize (>=0.7 && <0.9), random (>=1.0 && <1.2) [details]
License MIT
Copyright 2015 Joe Hermaszewski
Author Joe Hermaszewski
Maintainer Joe Hermaszewski <>
Category Math
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Source repository head: git clone
Uploaded Mon Jan 16 13:36:16 UTC 2017 by jophish
Distributions NixOS:0.12.2
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Readme for exact-real

Readme for exact-real-0.12.2


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Exact real arithmetic implemented by fast binary Cauchy sequences.

Motivating Example

Compare evaluating Euler's identity with a Float:

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


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 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.


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 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)
λ> 0.5 * 2 == (1 :: CReal 0) * 2


Contributions and bug reports are welcome!

Please feel free to contact me on GitHub or as "jophish" on freenode.