Defining New Functions
----------------------

The `defmath' function (actually a Lisp macro) is like `defun' except
that code in the body of the definition can make use of the full range
of Calculator data types.  The prefix `calcFunc-' is added to the
specified name to get the actual Lisp function name.  As a simple
example,

     (defmath myfact (n)
       (if (> n 0)
           (* n (myfact (1- n)))
         1))

This actually expands to the code,

     (defun calcFunc-myfact (n)
       (if (math-posp n)
           (math-mul n (calcFunc-myfact (math-add n -1)))
         1))

This function can be used in algebraic expressions, e.g., `myfact(5)'.

The `myfact' function as it is defined above has the bug that an
expression `myfact(a+b)' will be simplified to 1 because the formula
`a+b' is not considered to be `posp'.  A robust factorial function
would be written along the following lines:

     (defmath myfact (n)
       (if (> n 0)
           (* n (myfact (1- n)))
         (if (= n 0)
             1
           nil)))    ; this could be simplified as: (and (= n 0) 1)

If a function returns `nil', it is left unsimplified by the Calculator
(except that its arguments will be simplified).  Thus, `myfact(a+1+2)'
will be simplified to `myfact(a+3)' but no further.  Beware that every
time the Calculator reexamines this formula it will attempt to
resimplify it, so your function ought to detect the returning-`nil'
case as efficiently as possible.

The following standard Lisp functions are treated by `defmath': `+',
`-', `*', `/', `%', `^' or `expt', `=', `<', `>', `<=', `>=', `/=',
`1+', `1-', `logand', `logior', `logxor', `logandc2', `lognot'.  Also,
`~=' is an abbreviation for `math-nearly-equal', which is useful in
implementing Taylor series.

For other functions FUNC, if a function by the name `calcFunc-FUNC'
exists it is used, otherwise if a function by the name `math-FUNC'
exists it is used, otherwise if FUNC itself is defined as a function
it is used, otherwise `calcFunc-FUNC' is used on the assumption that
this is a to-be-defined math function.  Also, if the function name is
quoted as in `('integerp a)' the function name is always used exactly
as written (but not quoted).

Variable names have `var-' prepended to them unless they appear in the
function's argument list or in an enclosing `let', `let*', `for', or
`foreach' form, or their names already contain a `-' character.  Thus
a reference to `foo' is the same as a reference to `var-foo'.

A few other Lisp extensions are available in `defmath' definitions:

   * The `elt' function accepts any number of index variables.  Note
     that Calc vectors are stored as Lisp lists whose first element is
     the symbol `vec'; thus, `(elt v 2)' yields the second element of
     vector `v', and `(elt m i j)' yields one element of a Calc
     matrix.

   * The `setq' function has been extended to act like the Common Lisp
     `setf' function.  (The name `setf' is recognized as a synonym of
     `setq'.)  Specifically, the first argument of `setq' can be an
     `nth', `elt', `car', or `cdr' form, in which case the effect is
     to store into the specified element of a list.  Thus, `(setq (elt
     m i j) x)' stores `x' into one element of a matrix.

   * A `for' looping construct is available.  For example, `(for ((i 0
     10)) body)' executes `body' once for each binding of `i' from
     zero to 10.  This is like a `let' form in that `i' is temporarily
     bound to the loop count without disturbing its value outside the
     `for' construct.  Nested loops, as in `(for ((i 0 10) (j 0 (1- i)
     2)) body)', are also available.  For each value of `i' from zero
     to 10, `j' counts from 0 to `i-1' in steps of two.  Note that
     `for' has the same general outline as `let*', except that each
     element of the header is a list of three or four things, not just
     two.

   * The `foreach' construct loops over elements of a list.  For
     example, `(foreach ((x (cdr v))) body)' executes `body' with `x'
     bound to each element of Calc vector `v' in turn.  The purpose of
     `cdr' here is to skip over the initial `vec' symbol in the
     vector.

   * The `break' function breaks out of the innermost enclosing
     `while', `for', or `foreach' loop.  If given a value, as in
     `(break x)', this value is returned by the loop.  (Lisp loops
     otherwise always return `nil'.)

   * The `return' function prematurely returns from the enclosing
     function.  For example, `(return (+ x y))' returns `x+y' as the
     value of a function.  You can use `return' anywhere inside the
     body of the function.

Non-integer numbers (and extremely large integers) cannot be included
directly into a `defmath' definition.  This is because the Lisp reader
will fail to parse them long before `defmath' ever gets control.
Instead, use the notation, `:"3.1415"'.  In fact, any algebraic
formula can go between the quotes.  For example,

     (defmath sqexp (x)     ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
       (and (numberp x)
            (exp :"x * 0.5")))

expands to

     (defun calcFunc-sqexp (x)
       (and (math-numberp x)
            (calcFunc-exp (math-mul x '(float 5 -1)))))

Note the use of `numberp' as a guard to ensure that the argument is a
number first, returning `nil' if not.  The exponential function could
itself have been included in the expression, if we had preferred:
`:"exp(x * 0.5)"'.  As another example, the
multiplication-and-recursion step of `myfact' could have been written

     :"n * myfact(n-1)"

If a file named `.emacs' exists in your home directory, Emacs reads
and executes the Lisp forms in this file as it starts up.  While it
may seem like a good idea to put your favorite `defmath' commands
here, this has the unfortunate side-effect that parts of the
Calculator must be loaded in to process the `defmath' commands whether
or not you will actually use the Calculator!  A better effect can be
had by writing

     (put 'calc-define 'thing '(progn
      (defmath ... )
      (defmath ... )
     ))

The `put' function adds a "property" to a symbol.  Each Lisp symbol
has a list of properties associated with it.  Here we add a property
with a name of `thing' and a `(progn ...)' form as its value.  When
Calc starts up, and at the start of every Calc command, the property
list for the symbol `calc-define' is checked and the values of any
properties found are evaluated as Lisp forms.  The properties are
removed as they are evaluated.  The property names (like `thing') are
not used; you should choose something like the name of your project so
as not to conflict with other properties.

The net effect is that you can put the above code in your `.emacs'
file and it will not be executed until Calc is loaded.  Or, you can
put that same code in another file which you load by hand either
before or after Calc itself is loaded.

The properties of `calc-define' are evaluated in the same order that
they were added.  They can assume that the Calc modules `calc.el',
`calc-ext.el', and `calc-macs.el' have been fully loaded, and that the
`*Calculator*' buffer will be the current buffer.

If your `calc-define' property only defines algebraic functions, you
can be sure that it will have been evaluated before Calc tries to call
your function, even if the file defining the property is loaded after
Calc is loaded.  But if the property defines commands or key
sequences, it may not be evaluated soon enough.  (Suppose it defines
the new command `tweak-calc'; the user can load your file, then type
`M-x tweak-calc' before Calc has had chance to do anything.)  To
protect against this situation, you can put

     (run-hooks 'calc-check-defines)

at the end of your file.  The `calc-check-defines' function is what
looks for and evaluates properties on `calc-define'; `run-hooks' has
the advantage that it is quietly ignored if `calc-check-defines' is
not yet defined because Calc has not yet been loaded.

Examples of things that ought to be enclosed in a `calc-define'
property are `defmath' calls, `define-key' calls that modify the Calc
key map, and any calls that redefine things defined inside Calc.
Ordinary `defun's need not be enclosed with `calc-define'.