Debug.SimpleExpr.Tutorial

Tutorial, Quick start or Demo for 'simple-expr' package.

Quick start

>>> import Prelude (String)
>>> import Debug.SimpleExpr (variable, unaryFunc, binaryFunc)
>>> import NumHask (sin, (**))

Let us build an example symbolic expression for

\[ f(x) := \sin x^2 \]

It can be done as follows

>>> x = variable "x"
>>> sin (x ** 2)
sin(x^2)

where terms x and sin (x ** 2) have type SimpleExpr. It is just a syntactic tree where the role of leaves is played by variables and numbers. We used variable :: String -> SimpleExpr to build the expression for variable x. For the sine function we attracted a predefined term sin :: SimpleExpr -> SimpleExpr.

As well we can define a custom function using unaryFunc and binary functoins using binaryFunc as follows

>>> f = unaryFunc "f"
>>> (-*-) = binaryFunc "-*-"
>>> f x -*- f x
f(x)-*-f(x)

There is also a typeclass Expr that includes SimpleExpr as well as it's tuples and lists.

Expression simplification

>>> import Prelude (($))
>>> import Debug.SimpleExpr (variable, simplify)
>>> import NumHask ((+), (-), (*))

We can try to simplify an expressions with the aid of quite a primitive simplify method

>>> x = variable "x"
>>> simplify $ (x + 0) * 1 - x * (3 - 2)
0

Visualisation

>>> import Debug.SimpleExpr (variable, unaryFunc)
>>> import Debug.SimpleExpr.GraphUtils (plotExpr, plotDGraphPng, exprToGraph)
>>> import NumHask (exp, (*), (+), (-))

There is a built-in tool to visualize expression that attracts graphite package to transform expressions to graphs and graphviz to render the images.

Consider first a simple composition for two functions f and g

>>> x = variable "x"
>>> f = unaryFunc "f"
>>> g = unaryFunc "g"
>>> expr = g (f x)
>>> expr
g(f(x))

This symbolic expression can be plotted by plotExpr :: Expr d => d -> IO ThreadId like

 plotExpr expr

To save the image as a file use, for example,

plotDGraphPng (exprToGraph expr) pathToFile ,

where

exprToGraph :: Expr d => d -> DGraph String ()

transforms an expression to a graph and

plotDGraphPng :: DGraph v e -> FilePath -> IO FilePath.

plats the graph.

Consider now a more representative example

\[ e^{i k x} + e^{ - i k x} \]

>>> :{
  x, k, i, expr :: SimpleExpr
  x = variable "x"
  k = variable "k"
  i = variable "i"
  expr = exp (i * k * x) + exp (-(i * k * x))
:}