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 defuns need not be enclosed with calc-define.