Programming with Formulas
=========================

Another way to create a new Calculator command uses algebraic
formulas.  The `Z F' (`calc-user-define-formula') command stores the
formula at the top of the stack as the definition for a key.  This
command prompts for five things: The key, the command name, the
function name, the argument list, and the behavior of the command when
given non-numeric arguments.

For example, suppose we type `' a+2b RET' to push the formula `a +
2*b' onto the stack.  We now type `Z F m' to define this formula on
the `z m' key sequence.  The next prompt is for a command name,
beginning with `calc-', which should be the long (`M-x') form for the
new command.  If you simply press RET, a default name like
`calc-User-m' will be constructed.  In our example, suppose we enter
`spam RET' to define the new command as `calc-spam'.

If you want to give the formula a long-style name only, you can press
SPC or RET when asked which single key to use.  For example `Z F RET
spam RET' defines the new command as `M-x calc-spam', with no keyboard
equivalent.

The third prompt is for a function name.  The default is to use the
same name as the command name but with `calcFunc-' in place of
`calc-'.  This is the name you will use if you want to enter your new
function in an algebraic formula.  Suppose we enter `yow RET'.  Then
the new function can be invoked by pushing two numbers on the stack
and typing `z m' or `x spam', or by entering the algebraic formula
`yow(x,y)'.

The fourth prompt is for the function's argument list.  This is used
to associate values on the stack with the variables that appear in the
formula.  The default is a list of all variables which appear in the
formula, sorted into alphabetical order.  In our case, the default
would be `(a b)'.  This means that, when the user types `z m', the
Calculator will remove two numbers from the stack, substitute these
numbers for `a' and `b' (respectively) in the formula, then simplify
the formula and push the result on the stack.  In other words, `10 RET
100 z m' would replace the 10 and 100 on the stack with the number
210, which is `a + 2 b' with `a=10' and `b=100'.  Likewise, the
formula `yow(10, 100)' will be evaluated by substituting `a=10' and
`b=100' in the definition.

You can rearrange the order of the names before pressing RET to
control which stack positions go to which variables in the formula.
If you remove a variable from the argument list, that variable will be
left in symbolic form by the command.  Thus using an argument list of
`(b)' for our function would cause `10 z m' to replace the 10 on the
stack with the formula `a + 20'.  If we had used an argument list of
`(b a)', the result with inputs 10 and 100 would have been 120.

You can also put a nameless function on the stack instead of just a
formula, as in `<a, b : a + 2 b>'.  See Specifying Operators.  In
this example, the command will be defined by the formula `a + 2 b'
using the argument list `(a b)'.

The final prompt is a y-or-n question concerning what to do if
symbolic arguments are given to your function.  If you answer `y',
then executing `z m' (using the original argument list `(a b)') with
arguments `10' and `x' will leave the function in symbolic form, i.e.,
`yow(10,x)'.  On the other hand, if you answer `n', then the formula
will always be expanded, even for non-constant arguments: `10 + 2 x'.
If you never plan to feed algebraic formulas to your new function, it
doesn't matter how you answer this question.

If you answered `y' to this question you can still cause a function
call to be expanded by typing `a "' (`calc-expand-formula').  Also,
Calc will expand the function if necessary when you take a derivative
or integral or solve an equation involving the function.

Once you have defined a formula on a key, you can retrieve this
formula with the `Z G' (`calc-user-define-get-defn') command.  Press a
key, and this command pushes the formula that was used to define that
key onto the stack.  Actually, it pushes a nameless function that
specifies both the argument list and the defining formula.  You will
get an error message if the key is undefined, or if the key was not
defined by a `Z F' command.

The `Z E' (`calc-user-define-edit') command on a key that has been
defined by a formula uses a variant of the `calc-edit' command to edit
the defining formula.  Press `M-# M-#' to finish editing and store the
new formula back in the definition, or `M-# x' to cancel the edit.
(The argument list and other properties of the definition are
unchanged; to adjust the argument list, you can use `Z G' to grab the
function onto the stack, edit with ``', and then re-execute the `Z F'
command.)

As usual, the `Z P' command records your definition permanently.  In
this case it will permanently record all three of the relevant
definitions: the key, the command, and the function.

You may find it useful to turn off the default simplifications with `m
O' (`calc-no-simplify-mode') when entering a formula to be used as a
function definition.  For example, the formula `deriv(a^2,v)' which
might be used to define a new function `dsqr(a,v)' will be
"simplified" to 0 immediately upon entry since `deriv' considers `a'
to be constant with respect to `v'.  Turning off default
simplifications cures this problem: The definition will be stored in
symbolic form without ever activating the `deriv' function.  Press `m
D' to turn the default simplifications back on afterwards.