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I need to manipulate expressions like 1 + sqrt(3) and do basic arithmetic like addition, subtraction, and division. I d like the result to be in some sort of canonical form so that it can be used as a key in a map. Turning 1 + sqrt(3) into a float is not feasible due to roundoff problems.

I used SymPy for this task in Python. Is there an equivalent native library for Haskell?

Answer 1

Please check out the numbers package. If all you need is to store exact numbers like "1 + √3", you may want to use Data.Number.CReal instead of symbolic arithmetics. It stores the expressions and can be computed to arbitrary number of digits when needed.

Prelude Data.Number.CReal> let cx = 1 + sqrt (3 :: CReal)
Prelude Data.Number.CReal> showCReal 400 cx 
"2.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756756261414154067030299699450949989524788116555120943736485280932319023055820679748201010846749232650153123432669033228866506722546689218379712270471316603678615880190499865373798593894676503475065760507566183481296061009476021871903250831458295239598329977898245082887144638329173472241639845878553977"

There is also a Data.Number.Symbolic module in the package but the description says "It s mainly useful for debugging".

Answer 2

It seems you are looking for Computer Algebra System (CAS) in Haskell. Inspite of so many references to algebraic objects in the names of Haskell packages/modules, I ve never heard of a general purpose and well-maintained CA system in Haskell (like SymPy or Sage in Python).

However in the list of Computer Algebra Systems on Wikipedia I ve found a reference to

DoCon. The Algebraic Domain Constructor

It uses a non-standard license, but I dare say it is still Open Source (though with rename and attribution requirements). As of July 2010 docon-2.11 still builds with GHC 6.12.1 and runs demos/tests (I only had to insert a LANGUAGE FlexibleContexts pragma in one file of the demo).

DoCon is well documented (362 pages of the Manual). Its Manual is packed inside of the zip with sources, so I put it online separately for convenience:

DoCon 2.11 Manual.ps

Please look through to check if it suits your needs.

Answer 3

Check out the cyclotomic package, which implements exact arithmetic on the cyclotomic numbers. These include all algebraic numbers (hence in particular 1+sqrt(3)) and the key operations (like equality) are decidable.

They do not provide an Ord instance (for the same reason the complex numbers do not), but one can implement a non-semantic instance if all one needs is to use them as keys in a lookup table. You may want to contact the author about how to do this correctly, as there may be some invariants that are not obvious (e.g. one may need to be careful about zeros in the coeffs map).

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