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There are several ways to "program" the Emacs Calculator, depending on the nature of the problem you need to solve.
EvalRules, they will be applied automatically to all Calc
results in just the same way as an internal "rule" is applied to
evaluate `sqrt(9)' to 3 and so on. See section Automatic Rewrites.
Programming features are available through the z and Z
prefix keys. New commands that you define are two-key sequences
beginning with z. Commands for managing these definitions
use the shift-Z prefix. (The Z T (calc-timing)
command is described elsewhere; see section Troubleshooting Commands.
The Z C (calc-user-define-composition) command is also
described elsewhere; see section User-Defined Compositions.)
Any Calculator command may be bound to a key using the Z D
(calc-user-define) command. Actually, it is bound to a two-key
sequence beginning with the lower-case z prefix.
The Z D command first prompts for the key to define. For example,
press Z D a to define the new key sequence z a. You are then
prompted for the name of the Calculator command that this key should
run. For example, the calc-sincos command is not normally
available on a key. Typing Z D s sincos RET programs the
z s key sequence to run calc-sincos. This definition will remain
in effect for the rest of this Emacs session, or until you redefine
z s to be something else.
You can actually bind any Emacs command to a z key sequence by backspacing over the `calc-' when you are prompted for the command name.
As with any other prefix key, you can type z ? to see a list of all the two-key sequences you have defined that start with z. Initially, no z sequences (except z ? itself) are defined.
User keys are typically letters, but may in fact be any key. (META-keys are not permitted, nor are a terminal's special function keys which generate multi-character sequences when pressed.) You can define different commands on the shifted and unshifted versions of a letter if you wish.
The Z U (calc-user-undefine) command unbinds a user key.
For example, the key sequence Z U s will undefine the sincos
key we defined above.
The Z P (calc-user-define-permanent) command makes a key
binding permanent so that it will remain in effect even in future Emacs
sessions. (It does this by adding a suitable bit of Lisp code into
your `.emacs' file.) For example, Z P s would register
our sincos command permanently. If you later wish to unregister
this command you must edit your `.emacs' file by hand.
(See section General Mode Commands, for a way to tell Calc to use a
different file instead of `.emacs'.)
The Z P command also saves the user definition, if any, for the command bound to the key. After Z F and Z C, a given user key could invoke a command, which in turn calls an algebraic function, which might have one or more special display formats. A single Z P command will save all of these definitions.
To save a command or function without its key binding (or if there is no key binding for the command or function), type ' (the apostrophe) when prompted for a key. Then, type the function name, or backspace to change the `calcFunc-' prefix to `calc-' and enter a command name. (If the command you give implies a function, the function will be saved, and if the function has any display formats, those will be saved, but not the other way around: Saving a function will not save any commands or key bindings associated with the function.)
The Z E (calc-user-define-edit) command edits the definition
of a user key. This works for keys that have been defined by either
keyboard macros or formulas; further details are contained in the relevant
following sections.
The easiest way to "program" the Emacs Calculator is to use standard keyboard macros. Press C-x ( to begin recording a macro. From this point on, keystrokes you type will be saved away as well as performing their usual functions. Press C-x ) to end recording. Press shift-X (or the standard Emacs key sequence C-x e) to execute your keyboard macro by replaying the recorded keystrokes. See section 'Keyboard Macros' in the Emacs Manual, for further information.
When you use X to invoke a keyboard macro, the entire macro is
treated as a single command by the undo and trail features. The stack
display buffer is not updated during macro execution, but is instead
fixed up once the macro completes. Thus, commands defined with keyboard
macros are convenient and efficient. The C-x e command, on the
other hand, invokes the keyboard macro with no special treatment: Each
command in the macro will record its own undo information and trail entry,
and update the stack buffer accordingly. If your macro uses features
outside of Calc's control to operate on the contents of the Calc stack
buffer, or if it includes Undo, Redo, or last-arguments commands, you
must use C-x e to make sure the buffer and undo list are up-to-date
at all times. You could also consider using K (calc-keep-args)
instead of M-RET (calc-last-args).
Calc extends the standard Emacs keyboard macros in several ways. Keyboard macros can be used to create user-defined commands. Keyboard macros can include conditional and iteration structures, somewhat analogous to those provided by a traditional programmable calculator.
Once you have defined a keyboard macro, you can bind it to a z
key sequence with the Z K (calc-user-define-kbd-macro) command.
This command prompts first for a key, then for a command name. For
example, if you type C-x ( n TAB n TAB C-x ) you will
define a keyboard macro which negates the top two numbers on the stack
(TAB swaps the top two stack elements). Now you can type
Z K n RET to define this keyboard macro onto the z n key
sequence. The default command name (if you answer the second prompt with
just the RET key as in this example) will be something like
`calc-User-n'. The keyboard macro will now be available as both
z n and M-x calc-User-n. You can backspace and enter a more
descriptive command name if you wish.
Macros defined by Z K act like single commands; they are executed in the same way as by the X key. If you wish to define the macro as a standard no-frills Emacs macro (to be executed as if by C-x e), give a negative prefix argument to Z K.
Once you have bound your keyboard macro to a key, you can use Z P to register it permanently with Emacs. See section Creating User Keys.
The Z E (calc-user-define-edit) command on a key that has
been defined by a keyboard macro tries to use the edit-kbd-macro
command to edit the macro. This command may be found in the
`macedit' package, a copy of which comes with Calc. It decomposes
the macro definition into full Emacs command names, like calc-pop
and calc-add. Type M-# M-# to finish editing and update
the definition stored on the key, or, to cancel the edit, type
M-# x.
If you give a negative numeric prefix argument to Z E, the keyboard
macro is edited in spelled-out keystroke form. For example, the editing
buffer might contain the nine characters `1 RET 2 +'. When you press
M-# M-#, the read-kbd-macro feature of the `macedit'
package is used to reinterpret these key names. The
notations RET, LFD, TAB, SPC, DEL, and
NUL must be written in all uppercase, as must the prefixes C-
and M-. Spaces and line breaks are ignored. Other characters are
copied verbatim into the keyboard macro. Basically, the notation is the
same as is used in all of this manual's examples, except that the manual
takes some liberties with spaces: When we say ' [1 2 3] RET, we take
it for granted that it is clear we really mean ' [1 SPC 2 SPC 3] RET,
which is what read-kbd-macro wants to see.
If `macedit' is not available, Z E edits the keyboard macro in "raw" form; the editing buffer simply contains characters like `1^M2+' (here `^M' represents the carriage-return character). Editing in this mode, you will have to use C-q to enter new control characters into the buffer.
The M-# m (read-kbd-macro) command reads an Emacs "region"
of spelled-out keystrokes and defines it as the current keyboard macro.
It is a convenient way to define a keyboard macro that has been stored
in a file, or to define a macro without executing it at the same time.
The M-# m command works only if `macedit' is present.
The Z [ (calc-kbd-if) and Z ] (calc-kbd-end-if)
commands allow you to put simple tests in a keyboard macro. When Calc
sees the Z [, it pops an object from the stack and, if the object is
a non-zero value, continues executing keystrokes. But if the object is
zero, or if it is not provably nonzero, Calc skips ahead to the matching
Z ] keystroke. See section Logical Operations, for a set of commands for
performing tests which conveniently produce 1 for true and 0 for false.
For example, RET 0 a < Z [ n Z ] implements an absolute-value
function in the form of a keyboard macro. This macro duplicates the
number on the top of the stack, pushes zero and compares using a <
(calc-less-than), then, if the number was less than zero,
executes n (calc-change-sign). Otherwise, the change-sign
command is skipped.
To program this macro, type C-x (, type the above sequence of keystrokes, then type C-x ). Note that the keystrokes will be executed while you are making the definition as well as when you later re-execute the macro by typing X. Thus you should make sure a suitable number is on the stack before defining the macro so that you don't get a stack-underflow error during the definition process.
Conditionals can be nested arbitrarily. However, there should be exactly one Z ] for each Z [ in a keyboard macro.
The Z : (calc-kbd-else) command allows you to choose between
two keystroke sequences. The general format is cond Z [
then-part Z : else-part Z ]. If cond is true
(i.e., if the top of stack contains a non-zero number after cond
has been executed), the then-part will be executed and the
else-part will be skipped. Otherwise, the then-part will
be skipped and the else-part will be executed.
The Z | (calc-kbd-else-if) command allows you to choose
between any number of alternatives. For example,
cond1 Z [ part1 Z : cond2 Z | part2 Z :
part3 Z ] will execute part1 if cond1 is true,
otherwise it will execute part2 if cond2 is true, otherwise
it will execute part3.
More precisely, Z [ pops a number and conditionally skips to the next matching Z : or Z ] key. Z ] has no effect when actually executed. Z : skips to the next matching Z ]. Z | pops a number and conditionally skips to the next matching Z : or Z ]; thus, Z [ and Z | are functionally equivalent except that Z [ participates in nesting but Z | does not.
Calc's conditional and looping constructs work by scanning the keyboard macro for occurrences of character sequences like `Z:' and `Z]'. One side-effect of this is that if you use these constructs you must be careful that these character pairs do not occur by accident in other parts of the macros. Since Calc rarely uses shift-Z for any purpose except as a prefix character, this is not likely to be a problem. Another side-effect is that it will not work to define your own custom key bindings for these commands. Only the standard shift-Z bindings will work correctly.
If Calc gets stuck while skipping characters during the definition of a macro, type Z C-g to cancel the definition. (Typing plain C-g actually adds a C-g keystroke to the macro.)
The Z < (calc-kbd-repeat) and Z >
(calc-kbd-end-repeat) commands pop a number from the stack,
which must be an integer, then repeat the keystrokes between the brackets
the specified number of times. If the integer is zero or negative, the
body is skipped altogether. For example, 1 TAB Z < 2 * Z >
computes two to a nonnegative integer power. First, we push 1 on the
stack and then swap the integer argument back to the top. The Z <
pops that argument leaving the 1 back on top of the stack. Then, we
repeat a multiply-by-two step however many times.
Once again, the keyboard macro is executed as it is being entered.
In this case it is especially important to set up reasonable initial
conditions before making the definition: Suppose the integer 1000 just
happened to be sitting on the stack before we typed the above definition!
Another approach is to enter a harmless dummy definition for the macro,
then go back and edit in the real one with a Z E command. Yet
another approach is to type the macro as written-out keystroke names
in a buffer, then use M-# m (read-kbd-macro) to read the
macro.
The Z / (calc-kbd-break) command allows you to break out
of a keyboard macro loop prematurely. It pops an object from the stack;
if that object is true (a non-zero number), control jumps out of the
innermost enclosing Z < ... Z > loop and continues
after the Z >. If the object is false, the Z / has no
effect. Thus cond Z / is similar to `if (cond) break;'
in the C language.
The Z ( (calc-kbd-for) and Z ) (calc-kbd-end-for)
commands are similar to Z < and Z >, except that they make the
value of the counter available inside the loop. The general layout is
init final Z ( body step Z ). The Z (
command pops initial and final values from the stack. It then creates
a temporary internal counter and initializes it with the value init.
The Z ( command then repeatedly pushes the counter value onto the
stack and executes body and step, adding step to the
counter each time until the loop finishes.
By default, the loop finishes when the counter becomes greater than (or less than) final, assuming initial is less than (greater than) final. If initial is equal to final, the body executes exactly once. The body of the loop always executes at least once. For example, 0 1 10 Z ( 2 ^ + 1 Z ) computes the sum of the squares of the integers from 1 to 10, in steps of 1.
If you give a numeric prefix argument of 1 to Z (, the loop is forced to use upward-counting conventions. In this case, if initial is greater than final the body will not be executed at all. Note that step may still be negative in this loop; the prefix argument merely constrains the loop-finished test. Likewise, a prefix argument of -1 forces downward-counting conventions.
The Z { (calc-kbd-loop) and Z }
(calc-kbd-end-loop) commands are similar to Z < and
Z >, except that they do not pop a count from the stack--they
effectively create an infinite loop. Every Z { ... Z }
loop ought to include at least one Z / to make sure the loop
doesn't run forever. (If any error message occurs which causes Emacs
to beep, the keyboard macro will also be halted; this is a standard
feature of Emacs. You can also generally press C-g to halt a
running keyboard macro, although not all versions of Unix support
this feature.)
The conditional and looping constructs are not actually tied to keyboard macros, but they are most often used in that context. For example, the keystrokes 10 Z < 23 RET Z > push ten copies of 23 onto the stack. This can be typed "live" just as easily as in a macro definition.
See section Conditionals in Keyboard Macros, for some additional notes about conditional and looping commands.
Keyboard macros sometimes want to operate under known conditions without affecting surrounding conditions. For example, a keyboard macro may wish to turn on Fraction Mode, or set a particular precision, independent of the user's normal setting for those modes.
Macros also sometimes need to use local variables. Assignments to
local variables inside the macro should not affect any variables
outside the macro. The Z ` (calc-kbd-push) and Z '
(calc-kbd-pop) commands give you both of these capabilities.
When you type Z ` (with a backquote or accent grave character),
the values of various mode settings are saved away. The ten "quick"
variables q0 through q9 are also saved. When
you type Z ' (with an apostrophe), these values are restored.
Pairs of Z ` and Z ' commands may be nested.
If a keyboard macro halts due to an error in between a Z ` and a Z ', the saved values will be restored correctly even though the macro never reaches the Z ' command. Thus you can use Z ` and Z ' without having to worry about what happens in exceptional conditions.
If you type Z ` "live" (not in a keyboard macro), Calc puts you into a "recursive edit." You can tell you are in a recursive edit because there will be extra square brackets in the mode line, as in `[(Calculator)]'. These brackets will go away when you type the matching Z ' command. The modes and quick variables will be saved and restored in just the same way as if actual keyboard macros were involved.
The modes saved by Z ` and Z ' are the current precision and binary word size, the angular mode (Deg, Rad, or HMS), the simplification mode, Algebraic mode, Symbolic mode, Infinite mode, Matrix or Scalar mode, Fraction mode, and the current complex mode (Polar or Rectangular). The ten "quick" variables' values (or lack thereof) are also saved.
Most mode-setting commands act as toggles, but with a numeric prefix they force the mode either on (positive prefix) or off (negative or zero prefix). Since you don't know what the environment might be when you invoke your macro, it's best to use prefix arguments for all mode-setting commands inside the macro.
In fact, C-u Z ` is like Z ` except that it sets the modes listed above to their default values. As usual, the matching Z ' will restore the modes to their settings from before the C-u Z `. Also, Z ` with a negative prefix argument resets algebraic mode to its default (off) but leaves the other modes the same as they were outside the construct.
The contents of the stack and trail, values of non-quick variables, and other settings such as the language mode and the various display modes, are not affected by Z ` and Z '.
The Z = (calc-kbd-report) command displays an informative
message including the value on the top of the stack. You are prompted
to enter a string. That string, along with the top-of-stack value,
is displayed unless m w (calc-working) has been used
to turn such messages off.
The Z # (calc-kbd-query) command displays a prompt message
(which you enter during macro definition), then does an algebraic entry
which takes its input from the keyboard, even during macro execution.
This command allows your keyboard macros to accept numbers or formulas
as interactive input. All the normal conventions of algebraic input,
including the use of $ characters, are supported.
See section 'Kbd Macro Query' in the Emacs Manual, for a description of
C-x q (kbd-macro-query), the standard Emacs way to accept
keyboard input during a keyboard macro. In particular, you can use
C-x q to enter a recursive edit, which allows the user to perform
any Calculator operations interactively before pressing C-M-c to
return control to the keyboard macro.
Calc provides one special keyboard macro, called up by M-# z
(calc-user-invocation), that is intended to allow you to define
your own special way of starting Calc. To define this "invocation
macro," create the macro in the usual way with C-x ( and
C-x ), then type Z I (calc-user-define-invocation).
There is only one invocation macro, so you don't need to type any
additional letters after Z I. From now on, you can type
M-# z at any time to execute your invocation macro.
For example, suppose you find yourself often grabbing rectangles of numbers into Calc and multiplying their columns. You can do this by typing M-# r to grab, and V R : * to multiply columns. To make this into an invocation macro, just type C-x ( M-# r V R : * C-x ), then Z I. Then, to multiply a rectangle of data, just mark the data in its buffer in the usual way and type M-# z.
Invocation macros are treated like regular Emacs keyboard macros; all the special features described above for Z K-style macros do not apply. M-# z is just like C-x e, except that it uses the macro that was last stored by Z I. (In fact, the macro does not even have to have anything to do with Calc!)
The m m command saves the last invocation macro defined by Z I along with all the other Calc mode settings. See section General Mode Commands.
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 section 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.
The Calculator can be programmed quite extensively in Lisp. All you
do is write a normal Lisp function definition, but with defmath
in place of defun. This has the same form as defun, but it
automagically replaces calls to standard Lisp functions like + and
zerop with calls to the corresponding functions in Calc's own library.
Thus you can write natural-looking Lisp code which operates on all of the
standard Calculator data types. You can then use Z D if you wish to
bind your new command to a z-prefix key sequence. The Z E command
will not edit a Lisp-based definition.
Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section assumes a familiarity with Lisp programming concepts; if you do not know Lisp, you may find keyboard macros or rewrite rules to be an easier way to program the Calculator.
This section first discusses ways to write commands, functions, or small programs to be executed inside of Calc. Then it discusses how your own separate programs are able to call Calc from the outside. Finally, there is a list of internal Calc functions and data structures for the true Lisp enthusiast.
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:
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.
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.
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.
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.
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.)
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 defuns need not be enclosed with calc-define.
If a defmath form contains an interactive clause, it defines
a Calculator command. Actually such a defmath results in two
function definitions: One, a `calcFunc-' function as was just described,
with the interactive clause removed. Two, a `calc-' function
with a suitable interactive clause and some sort of wrapper to make
the command work in the Calc environment.
In the simple case, the interactive clause has the same form as
for normal Emacs Lisp commands:
(defmath increase-precision (delta) "Increase precision by DELTA." ; This is the "documentation string" (interactive "p") ; Register this as a M-x-able command (setq calc-internal-prec (+ calc-internal-prec delta)))
This expands to the pair of definitions,
(defun calc-increase-precision (delta) "Increase precision by DELTA." (interactive "p") (calc-wrapper (setq calc-internal-prec (math-add calc-internal-prec delta)))) (defun calcFunc-increase-precision (delta) "Increase precision by DELTA." (setq calc-internal-prec (math-add calc-internal-prec delta)))
where in this case the latter function would never really be used! Note
that since the Calculator stores small integers as plain Lisp integers,
the math-add function will work just as well as the native
+ even when the intent is to operate on native Lisp integers.
The `calc-wrapper' call invokes a macro which surrounds the body of the function with code that looks roughly like this:
(let ((calc-command-flags nil))
(unwind-protect
(save-excursion
(calc-select-buffer)
body of function
renumber stack
clear Working message)
realign cursor and window
clear Inverse, Hyperbolic, and Keep Args flags
update Emacs mode line))
The calc-select-buffer function selects the `*Calculator*'
buffer if necessary, say, because the command was invoked from inside
the `*Calc Trail*' window.
You can call, for example, (calc-set-command-flag 'no-align) to set
the above-mentioned command flags. The following command flags are
recognized by Calc routines:
renum-stack
calc-push.
clear-message
no-align
position-point
calc-position-point-line and
calc-position-point-column to position the cursor after
this command finishes.
keep-flags
calc-inverse-flag, calc-hyperbolic-flag,
and calc-keep-args-flag at the end of this command.
do-edit
hold-trail
Calc reserves a special prefix key, shift-Y, for user-written
extensions to Calc. There are no built-in commands that work with
this prefix key; you must call define-key from Lisp (probably
from inside a calc-define property) to add to it. Initially only
Y ? is defined; it takes help messages from a list of strings
(initially nil) in the variable calc-Y-help-msgs. All
other undefined keys except for Y are reserved for use by
future versions of Calc.
If you are writing a Calc enhancement which you expect to give to others, it is best to minimize the number of Y-key sequences you use. In fact, if you have more than one key sequence you should consider defining three-key sequences with a Y, then a key that stands for your package, then a third key for the particular command within your package.
Users may wish to install several Calc enhancements, and it is possible
that several enhancements will choose to use the same key. In the
example below, a variable inc-prec-base-key has been defined
to contain the key that identifies the inc-prec package. Its
value is initially "P", but a user can change this variable
if necessary without having to modify the file.
Here is a complete file, `inc-prec.el', which makes a Y P I command that increases the precision, and a Y P D command that decreases the precision.
;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
;;; (Include copyright or copyleft stuff here.)
(defvar inc-prec-base-key "P"
"Base key for inc-prec.el commands.")
(put 'calc-define 'inc-prec '(progn
(define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
'increase-precision)
(define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
'decrease-precision)
(setq calc-Y-help-msgs
(cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
calc-Y-help-msgs))
(defmath increase-precision (delta)
"Increase precision by DELTA."
(interactive "p")
(setq calc-internal-prec (+ calc-internal-prec delta)))
(defmath decrease-precision (delta)
"Decrease precision by DELTA."
(interactive "p")
(setq calc-internal-prec (- calc-internal-prec delta)))
)) ; end of calc-define property
(run-hooks 'calc-check-defines)
To define a new computational command which takes and/or leaves arguments
on the stack, a special form of interactive clause is used.
(interactive num tag)
where num is an integer, and tag is a string. The effect is
to pop num values off the stack, resimplify them by calling
calc-normalize, and hand them to your function according to the
function's argument list. Your function may include &optional and
&rest parameters, so long as calling the function with num
parameters is legal.
Your function must return either a number or a formula in a form
acceptable to Calc, or a list of such numbers or formulas. These value(s)
are pushed onto the stack when the function completes. They are also
recorded in the Calc Trail buffer on a line beginning with tag,
a string of (normally) four characters or less. If you omit tag
or use nil as a tag, the result is not recorded in the trail.
As an example, the definition
(defmath myfact (n)
"Compute the factorial of the integer at the top of the stack."
(interactive 1 "fact")
(if (> n 0)
(* n (myfact (1- n)))
(and (= n 0) 1)))
is a version of the factorial function shown previously which can be used as a command as well as an algebraic function. It expands to
(defun calc-myfact ()
"Compute the factorial of the integer at the top of the stack."
(interactive)
(calc-slow-wrapper
(calc-enter-result 1 "fact"
(cons 'calcFunc-myfact (calc-top-list-n 1)))))
(defun calcFunc-myfact (n)
"Compute the factorial of the integer at the top of the stack."
(if (math-posp n)
(math-mul n (calcFunc-myfact (math-add n -1)))
(and (math-zerop n) 1)))
The calc-slow-wrapper function is a version of calc-wrapper
that automatically puts up a `Working...' message before the
computation begins. (This message can be turned off by the user
with an m w (calc-working) command.)
The calc-top-list-n function returns a list of the specified number
of values from the top of the stack. It resimplifies each value by
calling calc-normalize. If its argument is zero it returns an
empty list. It does not actually remove these values from the stack.
The calc-enter-result function takes an integer num and string
tag as described above, plus a third argument which is either a
Calculator data object or a list of such objects. These objects are
resimplified and pushed onto the stack after popping the specified number
of values from the stack. If tag is non-nil, the values
being pushed are also recorded in the trail.
Note that if calcFunc-myfact returns nil this represents
"leave the function in symbolic form." To return an actual empty list,
in the sense that calc-enter-result will push zero elements back
onto the stack, you should return the special value `'(nil)', a list
containing the single symbol nil.
The interactive declaration can actually contain a limited
Emacs-style code string as well which comes just before num and
tag. Currently the only Emacs code supported is `"p"', as in
(defmath foo (a b &optional c) (interactive "p" 2 "foo") body)
In this example, the command calc-foo will evaluate the expression
`foo(a,b)' if executed with no argument, or `foo(a,b,n)' if
executed with a numeric prefix argument of n.
The other code string allowed is `"m"' (unrelated to the usual `"m"'
code as used with defun). It uses the numeric prefix argument as the
number of objects to remove from the stack and pass to the function.
In this case, the integer num serves as a default number of
arguments to be used when no prefix is supplied.
Anywhere a parameter name can appear in the parameter list you can also use an argument qualifier. Thus the general form of a definition is:
(defmath name (param param...
&optional param param...
&rest param)
body)
where each param is either a symbol or a list of the form
(qual param)
The following qualifiers are recognized:
For example,
(defmath foo (a (constp (not-matrixp b)) &optional (float c)
&rest (integer d))
body)
expands to
(defun calcFunc-foo (a b &optional c &rest d)
(and (math-matrixp b)
(math-reject-arg b 'not-matrixp))
(or (math-constp b)
(math-reject-arg b 'constp))
(and c (setq c (math-check-float c)))
(setq d (mapcar 'math-check-integer d))
body)
which performs the necessary checks and conversions before executing the body of the function.
This section includes some Lisp programming examples on a larger scale. These programs make use of some of the Calculator's internal functions; see section Calculator Internals.
Calc does not include a built-in function for counting the number of "one" bits in a binary integer. It's easy to invent one using b u to convert the integer to a set, and V # to count the elements of that set; let's write a function that counts the bits without having to create an intermediate set.
(defmath bcount ((natnum n))
(interactive 1 "bcnt")
(let ((count 0))
(while (> n 0)
(if (oddp n)
(setq count (1+ count)))
(setq n (lsh n -1)))
count))
When this is expanded by defmath, it will become the following
Emacs Lisp function:
(defun calcFunc-bcount (n)
(setq n (math-check-natnum n))
(let ((count 0))
(while (math-posp n)
(if (math-oddp n)
(setq count (math-add count 1)))
(setq n (calcFunc-lsh n -1)))
count))
If the input numbers are large, this function involves a fair amount of arithmetic. A binary right shift is essentially a division by two; recall that Calc stores integers in decimal form so bit shifts must involve actual division.
To gain a bit more efficiency, we could divide the integer into n-bit chunks, each of which can be handled quickly because they fit into Lisp integers. It turns out that Calc's arithmetic routines are especially fast when dividing by an integer less than 1000, so we can set n = 9 bits and use repeated division by 512:
(defmath bcount ((natnum n))
(interactive 1 "bcnt")
(let ((count 0))
(while (not (fixnump n))
(let ((qr (idivmod n 512)))
(setq count (+ count (bcount-fixnum (cdr qr)))
n (car qr))))
(+ count (bcount-fixnum n))))
(defun bcount-fixnum (n)
(let ((count 0))
(while (> n 0)
(setq count (+ count (logand n 1))
n (lsh n -1)))
count))
Note that the second function uses defun, not defmath.
Because this function deals only with native Lisp integers ("fixnums"),
it can use the actual Emacs + and related functions rather
than the slower but more general Calc equivalents which defmath
uses.
The idivmod function does an integer division, returning both
the quotient and the remainder at once. Again, note that while it
might seem that `(logand n 511)' and `(lsh n -9)' are
more efficient ways to split off the bottom nine bits of n,
actually they are less efficient because each operation is really
a division by 512 in disguise; idivmod allows us to do the
same thing with a single division by 512.
A somewhat limited sine function could be defined as follows, using the well-known Taylor series expansion for @c{$\sin x$} `sin(x)':
(defmath mysin ((float (anglep x)))
(interactive 1 "mysn")
(setq x (to-radians x)) ; Convert from current angular mode.
(let ((sum x) ; Initial term of Taylor expansion of sin.
newsum
(nfact 1) ; "nfact" equals "n" factorial at all times.
(xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
(for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
(working "mysin" sum) ; Display "Working" message, if enabled.
(setq nfact (* nfact (1- n) n)
x (* x xnegsqr)
newsum (+ sum (/ x nfact)))
(if (~= newsum sum) ; If newsum is "nearly equal to" sum,
(break)) ; then we are done.
(setq sum newsum))
sum))
The actual sin function in Calc works by first reducing the problem
to a sine or cosine of a nonnegative number less than @c{$\pi \over 4$}
pi/4. This
ensures that the Taylor series will converge quickly. Also, the calculation
is carried out with two extra digits of precision to guard against cumulative
round-off in `sum'. Finally, complex arguments are allowed and handled
by a separate algorithm.
(defmath mysin ((float (scalarp x)))
(interactive 1 "mysn")
(setq x (to-radians x)) ; Convert from current angular mode.
(with-extra-prec 2 ; Evaluate with extra precision.
(cond ((complexp x)
(mysin-complex x))
((< x 0)
(- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
(t (mysin-raw x))))))
(defmath mysin-raw (x)
(cond ((>= x 7)
(mysin-raw (% x (two-pi)))) ; Now x < 7.
((> x (pi-over-2))
(- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
((> x (pi-over-4))
(mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
((< x (- (pi-over-4)))
(- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
(t (mysin-series x)))) ; so the series will be efficient.
where mysin-complex is an appropriate function to handle complex
numbers, mysin-series is the routine to compute the sine Taylor
series as before, and mycos-raw is a function analogous to
mysin-raw for cosines.
The strategy is to ensure that x is nonnegative before calling
mysin-raw. This function then recursively reduces its argument
to a suitable range, namely, plus-or-minus @c{$\pi \over 4$}
pi/4. Note that each
test, and particularly the first comparison against 7, is designed so
that small roundoff errors cannnot produce an infinite loop. (Suppose
we compared with `(two-pi)' instead; if due to roundoff problems
the modulo operator ever returned `(two-pi)' exactly, an infinite
recursion could result!) We use modulo only for arguments that will
clearly get reduced, knowing that the next rule will catch any reductions
that this rule misses.
If a program is being written for general use, it is important to code it carefully as shown in this second example. For quick-and-dirty programs, when you know that your own use of the sine function will never encounter a large argument, a simpler program like the first one shown is fine.
A later section (see section Calculator Internals) gives a full description of
Calc's internal Lisp functions. It's not hard to call Calc from
inside your programs, but the number of these functions can be daunting.
So Calc provides one special "programmer-friendly" function called
calc-eval that can be made to do just about everything you
need. It's not as fast as the low-level Calc functions, but it's
much simpler to use!
It may seem that calc-eval itself has a daunting number of
options, but they all stem from one simple operation.
In its simplest manifestation, `(calc-eval "1+2")' parses the
string "1+2" as if it were a Calc algebraic entry and returns
the result formatted as a string: "3".
Since calc-eval is on the list of recommended autoload
functions, you don't need to make any -level Calc functions, but it's
much simpler to use!
It may seem that calc-eval itself has a daunting number of
options, but they all stem from one simple operation.
In its simplest manifestation, `(calc-eval "1+2")' parses the
string "1+2" as if it were a Calc algebraic entry and returns
the result formatted as a string: "3".
Since calc-eval is on the list of recommended autoload
functions, you don't need to make any