Steffen Müller

# NAME

Math::SymbolicX::Inline - Inlined Math::Symbolic functions

# SYNOPSIS

``````  use Math::SymbolicX::Inline <<'END';
foo = x * bar
bar = partial_derivative(x^2, x)
x (:=) arg0 + 1
END

print bar(3);
# prints '8' which is 2*(3+1)...

print foo(3);
# prints '32' which is 2*(3+1)*(3+1)

print x(3);
# Throws an error because the parenthesis around the operator make
# the declaration of x private.``````

# DESCRIPTION

This module is an extension to the Math::Symbolic module. A basic familiarity with that module is required.

Math::SymbolicX::Inline allows easy creation of Perl functions from symbolic expressions in the context of Math::Symbolic. That means you can define arbitrary Math::Symbolic trees (including derivatives) and let this module compile them to package subroutines.

There are relatively few syntax elements that aren't standard in Math::Symbolic expressions, but those that exist are easier to explain using examples. Thus, please refer to the discussion of a simple example below.

## EXPORT

This module does not export any functions, but its intended usage is to create functions in the current namespace for you.

## A SIMPLE EXAMPLE

A contrived sample usage would be to create a function that computes the derivative of the square of the sine. You could do the math yourself and find that the x-derivative of `sin(x)*sin(x)` is `2*sin(x)*cos(x)`. On the other hand, you might want to change the source function later or the derivative is very complicated or you are just too lazy to do the math. Then you can write the following code to do allow of this for you:

``````  use Math::SymbolicX::Inline <<'HERE';
myfunction = partial_derivative( sin(arg0) * sin(arg0), arg0 )
HERE``````

After that, you can use your appropriately named function from Perl. This has almost no performance penalty compared to the version you would write by hand since Math::Symbolic can compile trees to Perl code. (You could, if you were crazy enough, compile it to C using Math::Symbolic::Custom::CCompiler.)

``  print myfunction(2);``

That would print `-0.756802495307928`.

## EXTENDED USAGE

You will have noticed the usage of the `arg0` variable in the above example. Rather unspectacularily, `argX` refers to the X+1th argument to the function. Thus, `arg19` refers to the twentieth argument.

But it is atypical to use `arg0` as a variable in a mathematical expression. We want to use the names `x` and `y` to compute the x-derivative of `sin(x*y)*sin(x*y)`. Furthermore, we want the sine to be exchangeable with a cosine with as little effort as possible. That is rather simple to implement:

``````  my \$function = 'sin';

use Math::SymbolicX::Inline <<HERE;

# Our function:
myfunction = partial_derivative(inner, x)

# Supportive declarations:
inner (=) \$function(x*y)^2
x (:=) arg0
y (:=) arg1
HERE``````

This short piece of code adds three symbolic declarations. All of these new declarations have their assignment operators enclosed in parenthesis to signify that they are not to be exported. That means you will not be able to call `inner(2, 3)` afterwards. But you will be able to call `myfunction(2, 3)`. The variable \$function is interpolated into the HERE document. The manual pages that come with Perl will tell you all the details about this kind of quoted string.

The declarations are relatively whitespace insensitive. All you need to do is put a new declaration with the assignment operator on a new line. It does not matter how man lines a single equation takes. This is valid:

``````  myfunction =
partial_derivative(
inner, x
)
inner (=) \$function(x*y)^2
...``````

Whereas this is not:

``````  myfunction
= partial_derivative(inner, x)
...``````

It is relevant to note that the order of the declarations is irrelevant. You could have written

``````  x (:=) arg0
...
myfunction = partial_derivative(inner, x)``````

instead and you would have gotten the same result.

You can also remove any of the parenthesis around the assignment operators to make the declared function accessible from your Perl code.

You may have wondered about the `:=` operator used in the declaration of `x` and `y`. This operator is interesting in the context of derivatives only. Say, you want to compute the partial x-derivative of a function `inner`. If you want to be really correct about it, that derivative is `0`! That's because The term you are deriving (`inner`) is - strictly speaking - not dependent on `x`. You have to put the function definition of `inner` into place before deriving to get a sensible result.

Therefore, in general, you want to replace any usage of a function with its definition in order to be able to derive it.

Now, this brings up another problem. If we do the same for `x`, we will have `arg0` in its place and can't derive either. That's where the `:=` operator comes in. It replaces the function after the applying all derivatives.

The consequence of this is that you cannot reference a normal function like `inner` in the definitions for late-replace functions like `x`.

## THE LAST BITS

All calls to functions that don't exist in the block of declarations passed to Math::SymbolicX::Inline will be resolved to subroutine calls in the current package. If the subroutines don't exist, the module will throw an error with a stack backtrace.

New versions of this module can be found on http://steffen-mueller.net or CPAN.

Math::Symbolic

This module does not use the Inline module an thus is not in the Inline:: hierarchy of modules. Nonetheless, similar modules usually can be found in that hierarchy: Inline

# AUTHOR

Steffen Müller, <symbolic-module at steffen-mueller dot net<gt>