# aern2-mp

Variable-precision interval arithmetic

## Numeric data types

This package provides the following two data types:

* `Dyadic`:  variable-precision floats with exact ring operations
  
* `MPBall`: variable-precision interval (float centre ± error bound) with field & elementary interval operations
  
The type `MPBall` has instances of both [mixed-types-num](https://hackage.haskell.org/package/mixed-types-num) type classes such as `CanAdd`, `CanSqrt` as well as with traditional Prelude type classes such as `Ord`, `Num` and `Floating`.
The type `Dyadic` also has an appropriate subset of such instances.

Package [aern2-real](../aern2-real/README.md) provides an arithmetic of exact real numbers as converging lazy sequences of `MPBall`s of increasing precision.  Exact real numbers offer additional convenience and readability to validated numeric programming.

### Examples

First, let us test interval arithmetic with **Prelude** operations:

    $ stack ghci aern2-mp:lib --no-load --ghci-options AERN2.MP
    *AERN2.MP> import Prelude
    *AERN2.MP Prelude>

    ...> pi100 = piBallP (prec 100)
    ...> pi10000 = piBallP (prec 10000)

    ...> pi100 ^ 2
    [9.8696044010893586188344909998725639610631902560... ± ~8.1120e-30 ~2^(-96)]

    ...> pi100 ^ pi100
    <interactive>:18:1: error:
        • No instance for (Integral MPBall) arising from a use of ‘^’

    ...> sin pi100
    [0.0000000000000000000000000000001694818351060767... ± ~7.8925e-31 ~2^(-99)]

    ...> sin pi10000
    [0.0000000000000000000000000000000000000000000000... ± ~0.0000 ~2^(-9999)]
    (0.07 secs, 64,466,432 bytes)

    ...> pi100 > 0
    True

    ...> pi100 == pi100
    *** Exception: Failed to decide equality of MPBalls.  If you switch to MixedTypesNumPrelude instead of Prelude, comparison of MPBalls returns Kleenean instead of Bool.

Some things do not work with Prelude. Let us try using **MixedTypesNumPrelude** operations:

    $ stack ghci aern2-mp:lib --no-load --ghci-options AERN2.MP
    *AERN2.MP> import MixedTypesNumPrelude
    *AERN2.MP MixedTypesNumPrelude>

    ...> pi100 = piBallP (prec 100)
    ...> pi10000 = piBallP (prec 10000)

    ...> pi100 ^ pi100
    [36.4621596072079117709908260226198218149834948802... ± ~1.8696e-28 ~2^(-92)]

    ...> pi10000 ^ pi10000
    [36.4621596072079117709908260226921236663655084022... ± ~0.0000 ~2^(-9992)]
    (0.27 secs, 204,657,248 bytes)

    ...> pi100 > 0
    CertainTrue

    ...> pi100 == pi100
    TrueOrFalse

### Internal types and backends

The type `MPBall` internally uses the type:

* `MPFloat`: arbitrary-precision floats with both upwards and downwards-rounded arithmetic operations such as `*^` and `*.`

The package uses [cdar-mBound](https://hackage.haskell.org/package/cdar-mBound), a fork of [cdar](https://github.com/jensblanck/cdar) as its backend for `Dyadic` and `MPFloat`.

In previous versions, there was an MPFR backend via [rounded](https://hackage.haskell.org/package/rounded).  This may be added again in future.

## Specifications and tests

This package also provides a fairly complete hspec/QuickCheck specification of algebraic properties for the above types.  

For `MPFloat`, the properties are given mostly as approximate versions of algebraic equalities with a small rounding error tolerance.  

For `MPBall`, the properties are given mostly as (interval) set over-approximations of the usual algebraic equalities.