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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 an algebraic function name. The default is to
use the same name as the command name but without the ‘`calc-`’
prefix. (If this is of the form ‘`User-m`’, the hyphen is removed so
it won't be taken for a minus sign in algebraic formulas.)
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 `C-c C-c` to finish editing and
store the new formula back in the definition, or kill the buffer with
`C-x k` 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.