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#### 18.5.1 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)"

A good place to put your `defmath` commands is your Calc init file (the file given by `calc-settings-file`, typically `~/.emacs.d/calc.el`), which will not be loaded until Calc starts. 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 reasonable to put your favorite `defmath` commands there, 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! If you want to put the `defmath` commands there (for example, if you redefine `calc-settings-file` to be `.emacs`), 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`.