This is Info file calc.info, produced by Makeinfo-1.55 from the input
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   This file documents Calc, the GNU Emacs calculator.
   Copyright (C) 1990, 1991 Free Software Foundation, Inc.
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   Permission is granted to copy and distribute translations of this
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File: calc.info,  Node: Mathematica Language Mode,  Next: Maple Language Mode,  Prev: Eqn Language Mode,  Up: Language Modes
Mathematica Language Mode
-------------------------
The `d M' (`calc-mathematica-language') command selects the conventions
of Mathematica, a powerful and popular mathematical tool from Wolfram
Research, Inc.  Notable differences in Mathematica mode are that the
names of built-in functions are capitalized, and function calls use
square brackets instead of parentheses.  Thus the Calc formula `sin(2
x)' is entered and displayed `Sin[2 x]' in Mathematica mode.
   Vectors and matrices use curly braces in Mathematica.  Complex
numbers are written `3 + 4 I'.  The standard special constants in Calc
are written `Pi', `E', `I', `GoldenRatio', `EulerGamma', `Infinity',
`ComplexInfinity', and `Indeterminate' in Mathematica mode.
Non-decimal numbers are written, e.g., `16^^7fff'.  Floating-point
numbers in scientific notation are written `1.23*10.^3'.  Subscripts
use double square brackets: `a[[i]]'.
File: calc.info,  Node: Maple Language Mode,  Next: Compositions,  Prev: Mathematica Language Mode,  Up: Language Modes
Maple Language Mode
-------------------
The `d W' (`calc-maple-language') command selects the conventions of
Maple, another mathematical tool from the University of Waterloo.
   Maple's language is much like C.  Underscores are allowed in symbol
names; square brackets are used for subscripts; explicit `*'s for
multiplications are required.  Use either `^' or `**' to denote powers.
   Maple uses square brackets for lists and curly braces for sets.  Calc
interprets both notations as vectors, and displays vectors with square
brackets.  This means Maple sets will be converted to lists when they
pass through Calc.  As a special case, matrices are written as calls to
the function `matrix', given a list of lists as the argument, and can
be read in this form or with all-capitals `MATRIX'.
   The Maple interval notation `2 .. 3' has no surrounding brackets;
Calc reads `2 .. 3' as the closed interval `[2 .. 3]', and writes any
kind of interval as `2 .. 3'.  This means you cannot see the difference
between an open and a closed interval while in Maple display mode.
   Maple writes complex numbers as `3 + 4*I'.  Its special constants
are `Pi', `E', `I', and `infinity' (all three of `inf', `uinf', and
`nan' display as `infinity').  Floating-point numbers are written
`1.23*10.^3'.
   Among things not currently handled by Calc's Maple mode are the
various quote symbols, procedures and functional operators, and inert
(`&') operators.
File: calc.info,  Node: Compositions,  Next: Syntax Tables,  Prev: Maple Language Mode,  Up: Language Modes
Compositions
------------
There are several "composition functions" which allow you to get
displays in a variety of formats similar to those in Big language mode.
Most of these functions do not evaluate to anything; they are
placeholders which are left in symbolic form by Calc's evaluator but
are recognized by Calc's display formatting routines.
   Two of these, `string' and `bstring', are described elsewhere.
*Note Strings::.  For example, `string("ABC")' is displayed as `ABC'.
When viewed on the stack it will be indistinguishable from the variable
`ABC', but internally it will be stored as `string([65, 66, 67])' and
can still be manipulated this way; for example, the selection and
vector commands `j 1 v v j u' would select the vector portion of this
object and reverse the elements, then deselect to reveal a string whose
characters had been reversed.
   The composition functions do the same thing in all language modes
(although their components will of course be formatted in the current
language mode).  The one exception is Unformatted mode (`d U'), which
does not give the composition functions any special treatment.  The
functions are discussed here because of their relationship to the
language modes.
* Menu:
* Composition Basics::
* Horizontal Compositions::
* Vertical Compositions::
* Other Compositions::
* Information about Compositions::
* User-Defined Compositions::
File: calc.info,  Node: Composition Basics,  Next: Horizontal Compositions,  Prev: Compositions,  Up: Compositions
Composition Basics
..................
Compositions are generally formed by stacking formulas together
horizontally or vertically in various ways.  Those formulas are
themselves compositions.  TeX users will find this analogous to TeX's
"boxes."  Each multi-line composition has a "baseline"; horizontal
compositions use the baselines to decide how formulas should be
positioned relative to one another.  For example, in the Big mode
formula
               2
          a + b
     17 + ------
            c
the second term of the sum is four lines tall and has line three as its
baseline.  Thus when the term is combined with 17, line three is placed
on the same level as the baseline of 17.
   Another important composition concept is "precedence".  This is an
integer that represents the binding strength of various operators.  For
example, `*' has higher precedence (195) than `+' (180), which means
that `(a * b) + c' will be formatted without the parentheses, but `a *
(b + c)' will keep the parentheses.
   The operator table used by normal and Big language modes has the
following precedences:
     _     1200   (subscripts)
     %     1100   (as in n%)
     -     1000   (as in -n)
     !     1000   (as in !n)
     mod    400
     +/-    300
     !!     210    (as in n!!)
     !      210    (as in n!)
     ^      200
     *      195    (or implicit multiplication)
     / % \  190
     + -    180    (as in a+b)
     |      170
     < =    160    (and other relations)
     &&     110
     ||     100
     ? :     90
     !!!     85
     &&&     80
     |||     75
     :=      50
     ::      45
     =>      40
   The general rule is that if an operator with precedence `n' occurs
as an argument to an operator with precedence `m', then the argument is
enclosed in parentheses if `n < m'.  Top-level expressions and
expressions which are function arguments, vector components, etc., are
formatted with precedence zero (so that they normally never get
additional parentheses).
   For binary left-associative operators like `+', the righthand
argument is actually formatted with one-higher precedence than shown in
the table.  This makes sure `(a + b) + c' omits the parentheses, but
the unnatural form `a + (b + c)' keeps its parentheses.
Right-associative operators like `^' format the lefthand argument with
one-higher precedence.
   The `cprec' function formats an expression with an arbitrary
precedence.  For example, `cprec(abc, 185)' will combine into sums and
products as follows:  `7 + abc', `7 (abc)' (because this `cprec' form
has higher precedence than addition, but lower precedence than
multiplication).
   A final composition issue is "line breaking".  Calc uses two
different strategies for "flat" and "non-flat" compositions.  A
non-flat composition is anything that appears on multiple lines (not
counting line breaking).  Examples would be matrices and Big mode
powers and quotients.  Non-flat compositions are displayed exactly as
specified.  If they come out wider than the current window, you must
use horizontal scrolling (`<' and `>') to view them.
   Flat compositions, on the other hand, will be broken across several
lines if they are too wide to fit the window.  Certain points in a
composition are noted internally as "break points".  Calc's general
strategy is to fill each line as much as possible, then to move down to
the next line starting at the first break point that didn't fit.
However, the line breaker understands the hierarchical structure of
formulas.  It will not break an "inner" formula if it can use an
earlier break point from an "outer" formula instead.  For example, a
vector of sums might be formatted as:
     [ a + b + c, d + e + f,
       g + h + i, j + k + l, m ]
If the `m' can fit, then so, it seems, could the `g'.  But Calc prefers
to break at the comma since the comma is part of a "more outer"
formula.  Calc would break at a plus sign only if it had to, say, if
the very first sum in the vector had itself been too large to fit.
   Of the composition functions described below, only `choriz'
generates break points.  The `bstring' function (*note Strings::.) also
generates breakable items:  A break point is added after every space
(or group of spaces) except for spaces at the very beginning or end of
the string.
   Composition functions themselves count as levels in the formula
hierarchy, so a `choriz' that is a component of a larger `choriz' will
be less likely to be broken.  As a special case, if a `bstring' occurs
as a component of a `choriz' or `choriz'-like object (such as a vector
or a list of arguments in a function call), then the break points in
that `bstring' will be on the same level as the break points of the
surrounding object.
File: calc.info,  Node: Horizontal Compositions,  Next: Vertical Compositions,  Prev: Composition Basics,  Up: Compositions
Horizontal Compositions
.......................
The `choriz' function takes a vector of objects and composes them
horizontally.  For example, `choriz([17, a b/c, d])' formats as
`17a b / cd' in normal language mode, or as
       a b
     17---d
        c
in Big language mode.  This is actually one case of the general
function `choriz(VEC, SEP, PREC)', where either or both of SEP and PREC
may be omitted.  PREC gives the "precedence" to use when formatting
each of the components of VEC.  The default precedence is the
precedence from the surrounding environment.
   SEP is a string (i.e., a vector of character codes as might be
entered with `" "' notation) which should separate components of the
composition.  Also, if SEP is given, the line breaker will allow lines
to be broken after each occurrence of SEP.  If SEP is omitted, the
composition will not be breakable (unless any of its component
compositions are breakable).
   For example, `2 choriz([a, b c, d = e], " + ", 180)' is formatted as
`2 a + b c + (d = e)'.  To get the `choriz' to have precedence 180
"outwards" as well as "inwards," enclose it in a `cprec' form:  `2
cprec(choriz(...), 180)' formats as `2 (a + b c + (d = e))'.
   The baseline of a horizontal composition is the same as the
baselines of the component compositions, which are all aligned.
File: calc.info,  Node: Vertical Compositions,  Next: Other Compositions,  Prev: Horizontal Compositions,  Up: Compositions
Vertical Compositions
.....................
The `cvert' function makes a vertical composition.  Each component of
the vector is centered in a column.  The baseline of the result is by
default the top line of the resulting composition.  For example,
`f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))' formats in Big mode as
     f( a ,  2    )
       bb   a  + 1
       ccc     2
              b
   There are several special composition functions that work only as
components of a vertical composition.  The `cbase' function controls
the baseline of the vertical composition; the baseline will be the same
as the baseline of whatever component is enclosed in `cbase'.  Thus
`f(cvert([a, cbase(bb), ccc]), cvert([a^2 + 1, cbase(b^2)]))' displays
             2
            a  + 1
        a      2
     f(bb ,   b   )
       ccc
   There are also `ctbase' and `cbbase' functions which make the
baseline of the vertical composition equal to the top or bottom line
(rather than the baseline) of that component.  Thus `cvert([cbase(a /
b)]) + cvert([ctbase(a / b)]) + cvert([cbbase(a / b)])' gives
             a
     a       -
     - + a + b
     b   -
         b
   There should be only one `cbase', `ctbase', or `cbbase' function in
a given vertical composition.  These functions can also be written with
no arguments:  `ctbase()' is a zero-height object which means the
baseline is the top line of the following item, and `cbbase()' means
the baseline is the bottom line of the preceding item.
   The `crule' function builds a "rule," or horizontal line, across a
vertical composition.  By itself `crule()' uses `-' characters to build
the rule.  You can specify any other character, e.g., `crule("=")'.
The argument must be a character code or vector of exactly one
character code.  It is repeated to match the width of the widest item
in the stack.  For example, a quotient with a thick line is `cvert([a +
1, cbase(crule("=")), b^2])':
     a + 1
     =====
       2
      b
   Finally, the functions `clvert' and `crvert' act exactly like
`cvert' except that the items are left- or right-justified in the
stack.  Thus `clvert([a, bb, ccc]) + crvert([a, bb, ccc])' gives:
     a   +   a
     bb     bb
     ccc   ccc
   Like `choriz', the vertical compositions accept a second argument
which gives the precedence to use when formatting the components.
Vertical compositions do not support separator strings.
File: calc.info,  Node: Other Compositions,  Next: Information about Compositions,  Prev: Vertical Compositions,  Up: Compositions
Other Compositions
..................
The `csup' function builds a superscripted expression.  For example,
`csup(a, b)' looks the same as `a^b' does in Big language mode.  This
is essentially a horizontal composition of `a' and `b', where `b' is
shifted up so that its bottom line is one above the baseline.
   Likewise, the `csub' function builds a subscripted expression.  This
shifts `b' down so that its top line is one below the bottom line of
`a' (note that this is not quite analogous to `csup').  Other
arrangements can be obtained by using `choriz' and `cvert' directly.
   The `cflat' function formats its argument in "flat" mode, as
obtained by `d O', if the current language mode is normal or Big.  It
has no effect in other language modes.  For example, `a^(b/c)' is
formatted by Big mode like `csup(a, cflat(b/c))' to improve its
readability.
   The `cspace' function creates horizontal space.  For example,
`cspace(4)' is effectively the same as `string("    ")'.  A second
string (i.e., vector of characters) argument is repeated instead of the
space character.  For example, `cspace(4, "ab")' looks like `abababab'.
If the second argument is not a string, it is formatted in the normal
way and then several copies of that are composed together:  `cspace(4,
a^2)' yields
      2 2 2 2
     a a a a
If the number argument is zero, this is a zero-width object.
   The `cvspace' function creates vertical space, or a vertical stack
of copies of a certain string or formatted object.  The baseline is the
center line of the resulting stack.  A numerical argument of zero will
produce an object which contributes zero height if used in a vertical
composition.
   There are also `ctspace' and `cbspace' functions which create
vertical space with the baseline the same as the baseline of the top or
bottom copy, respectively, of the second argument.  Thus `cvspace(2,
a/b) + ctspace(2, a/b) + cbspace(2, a/b)' displays as:
             a
             -
     a       b
     -   a   a
     b + - + -
     a   b   b
     -   a
     b   -
         b
File: calc.info,  Node: Information about Compositions,  Next: User-Defined Compositions,  Prev: Other Compositions,  Up: Compositions
Information about Compositions
..............................
The functions in this section are actual functions; they compose their
arguments according to the current language and other display modes,
then return a certain measurement of the composition as an integer.
   The `cwidth' function measures the width, in characters, of a
composition.  For example, `cwidth(a + b)' is 5, and `cwidth(a / b)' is
5 in normal mode, 1 in Big mode, and 11 in TeX mode (for `{a \over
b}').  The argument may involve the composition functions described in
this section.
   The `cheight' function measures the height of a composition.  This
is the total number of lines in the argument's printed form.
   The functions `cascent' and `cdescent' measure the amount of the
height that is above (and including) the baseline, or below the
baseline, respectively.  Thus `cascent(X) + cdescent(X)' always equals
`cheight(X)'.  For a one-line formula like `a + b', `cascent' returns 1
and `cdescent' returns 0.  For `a / b' in Big mode, `cascent' returns 2
and `cdescent' returns 1.  The only formula for which `cascent' will
return zero is `cvspace(0)' or equivalents.
File: calc.info,  Node: User-Defined Compositions,  Prev: Information about Compositions,  Up: Compositions
User-Defined Compositions
.........................
The `Z C' (`calc-user-define-composition') command lets you define the
display format for any algebraic function.  You provide a formula
containing a certain number of argument variables on the stack.  Any
time Calc formats a call to the specified function in the current
language mode and with that number of arguments, Calc effectively
replaces the function call with that formula with the arguments
replaced.
   Calc builds the default argument list by sorting all the variable
names that appear in the formula into alphabetical order.  You can edit
this argument list before pressing RET if you wish.  Any variables in
the formula that do not appear in the argument list will be displayed
literally; any arguments that do not appear in the formula will not
affect the display at all.
   You can define formats for built-in functions, for functions you have
defined with `Z F' (*note Algebraic Definitions::.), or for functions
which have no definitions but are being used as purely syntactic
objects.  You can define different formats for each language mode, and
for each number of arguments, using a succession of `Z C' commands.
When Calc formats a function call, it first searches for a format
defined for the current language mode (and number of arguments); if
there is none, it uses the format defined for the Normal language mode.
If neither format exists, Calc uses its built-in standard format for
that function (usually just `FUNC(ARGS)').
   If you execute `Z C' with the number 0 on the stack instead of a
formula, any defined formats for the function in the current language
mode will be removed.  The function will revert to its standard format.
   For example, the default format for the binomial coefficient function
`choose(n, m)' in the Big language mode is
      n
     ( )
      m
You might prefer the notation,
      C
     n m
To define this notation, first make sure you are in Big mode, then put
the formula
     choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
on the stack and type `Z C'.  Answer the first prompt with `choose'.
The second prompt will be the default argument list of `(C m n)'.  Edit
this list to be `(n m)' and press RET.  Now, try it out:  For example,
turn simplification off with `m O' and enter `choose(a,b) + choose(7,3)'
as an algebraic entry.
      C  +  C
     a b   7 3
   As another example, let's define the usual notation for Stirling
numbers of the first kind, `stir1(n, m)'.  This is just like the
regular format for binomial coefficients but with square brackets
instead of parentheses.
     choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
   Now type `Z C stir1 RET', edit the argument list to `(n m)', and
type RET.
   The formula provided to `Z C' usually will involve composition
functions, but it doesn't have to.  Putting the formula `a + b + c'
onto the stack and typing `Z C foo RET RET' would define the function
`foo(x,y,z)' to display like `x + y + z'.  This "sum" will act exactly
like a real sum for all formatting purposes (it will be parenthesized
the same, and so on).  However it will be computationally unrelated to
a sum.  For example, the formula `2 * foo(1, 2, 3)' will display as `2
(1 + 2 + 3)'.  Operator precedences have caused the "sum" to be written
in parentheses, but the arguments have not actually been summed.
(Generally a display format like this would be undesirable, since it
can easily be confused with a real sum.)
   The special function `eval' can be used inside a `Z C' composition
formula to cause all or part of the formula to be evaluated at display
time.  For example, if the formula is `a + eval(b + c)', then `foo(1,
2, 3)' will be displayed as `1 + 5'.  Evaluation will use the default
simplifications, regardless of the current simplification mode.  There
are also `evalsimp' and `evalextsimp' which simplify as if by `a s' and
`a e' (respectively).  Note that these "functions" operate only in the
context of composition formulas (and also in rewrite rules, where they
serve a similar purpose; *note Rewrite Rules::.).  On the stack, a call
to `eval' will be left in symbolic form.
   It is not a good idea to use `eval' except as a last resort.  It can
cause the display of formulas to be extremely slow.  For example, while
`eval(a + b)' might seem quite fast and simple, there are several
situations where it could be slow.  For example, `a' and/or `b' could
be polar complex numbers, in which case doing the sum requires
trigonometry.  Or, `a' could be the factorial `fact(100)' which is
unevaluated because you have typed `m O'; `eval' will evaluate it
anyway to produce a large, unwieldy integer.
   You can save your display formats permanently using the `Z P'
command (*note Creating User Keys::.).
File: calc.info,  Node: Syntax Tables,  Prev: Compositions,  Up: Language Modes
Syntax Tables
-------------
Syntax tables do for input what compositions do for output:  They allow
you to teach custom notations to Calc's formula parser.  Calc keeps a
separate syntax table for each language mode.
   (Note that the Calc "syntax tables" discussed here are completely
unrelated to the syntax tables described in the Emacs manual.)
   The `Z S' (`calc-edit-user-syntax') command edits the syntax table
for the current language mode.  If you want your syntax to work in any
language, define it in the normal language mode.  Type `M-# M-#' to
finish editing the syntax table, or `M-# x' to cancel the edit.  The `m
m' command saves all the syntax tables along with the other mode
settings; *note General Mode Commands::..
* Menu:
* Syntax Table Basics::
* Precedence in Syntax Tables::
* Advanced Syntax Patterns::
* Conditional Syntax Rules::
File: calc.info,  Node: Syntax Table Basics,  Next: Precedence in Syntax Tables,  Prev: Syntax Tables,  Up: Syntax Tables
Syntax Table Basics
...................
"Parsing" is the process of converting a raw string of characters, such
as you would type in during algebraic entry, into a Calc formula.
Calc's parser works in two stages.  First, the input is broken down
into "tokens", such as words, numbers, and punctuation symbols like
`+', `:=', and `+/-'.  Space between tokens is ignored (except when it
serves to separate adjacent words).  Next, the parser matches this
string of tokens against various built-in syntactic patterns, such as
"an expression followed by `+' followed by another expression" or "a
name followed by `(', zero or more expressions separated by commas, and
`)'."
   A "syntax table" is a list of user-defined "syntax rules", which
allow you to specify new patterns to define your own favorite input
notations.  Calc's parser always checks the syntax table for the
current language mode, then the table for the normal language mode,
before it uses its built-in rules to parse an algebraic formula you
have entered.  Each syntax rule should go on its own line; it consists
of a "pattern", a `:=' symbol, and a Calc formula with an optional
"condition".  (Syntax rules resemble algebraic rewrite rules, but the
notation for patterns is completely different.)
   A syntax pattern is a list of tokens, separated by spaces.  Except
for a few special symbols, tokens in syntax patterns are matched
literally, from left to right.  For example, the rule,
     foo ( ) := 2+3
would cause Calc to parse the formula `4+foo()*5' as if it were
`4+(2+3)*5'.  Notice that the parentheses were written as two separate
tokens in the rule.  As a result, the rule works for both `foo()' and
`foo (  )'.  If we had written the rule as `foo () := 2+3', then Calc
would treat `()' as a single, indivisible token, so that `foo( )' would
not be recognized by the rule.  (It would be parsed as a regular
zero-argument function call instead.)  In fact, this rule would also
make trouble for the rest of Calc's parser:  An unrelated formula like
`bar()' would now be tokenized into `bar ()' instead of `bar ( )', so
that the standard parser for function calls would no longer recognize
   While it is possible to make a token with a mixture of letters and
punctuation symbols, this is not recommended.  It is better to break it
into several tokens, as we did with `foo()' above.
   The symbol `#' in a syntax pattern matches any Calc expression.  On
the righthand side, the things that matched the `#'s can be referred to
as `#1', `#2', and so on (where `#1' matches the leftmost `#' in the
pattern).  For example, these rules match a user-defined function,
prefix operator, infix operator, and postfix operator, respectively:
     foo ( # ) := myfunc(#1)
     foo # := myprefix(#1)
     # foo # := myinfix(#1,#2)
     # foo := mypostfix(#1)
   Thus `foo(3)' will parse as `myfunc(3)', and `2+3 foo' will parse as
`mypostfix(2+3)'.
   It is important to write the first two rules in the order shown,
because Calc tries rules in order from first to last.  If the pattern
`foo #' came first, it would match anything that could match the `foo (
# )' rule, since an expression in parentheses is itself a valid
expression.  Thus the `foo ( # )' rule would never get to match
anything.  Likewise, the last two rules must be written in the order
shown or else `3 foo 4' will be parsed as `mypostfix(3) * 4'.  (Of
course, the best way to avoid these ambiguities is not to use the same
symbol in more than one way at the same time!  In case you're not
convinced, try the following exercise:  How will the above rules parse
the input `foo(3,4)', if at all?  Work it out for yourself, then try it
in Calc and see.)
   Calc is quite flexible about what sorts of patterns are allowed.
The only rule is that every pattern must begin with a literal token
(like `foo' in the first two patterns above), or with a `#' followed by
a literal token (as in the last two patterns).  After that, any mixture
is allowed, although putting two `#'s in a row will not be very useful
since two expressions with nothing between them will be parsed as one
expression that uses implicit multiplication.
   As a more practical example, Maple uses the notation `sum(a(i),
i=1..10)' for sums, which Calc's Maple mode doesn't recognize at
present.  To handle this syntax, we simply add the rule,
     sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
to the Maple mode syntax table.  As another example, C mode can't read
assignment operators like `++' and `*='.  We can define these operators
quite easily:
     # *= # := muleq(#1,#2)
     # ++ := postinc(#1)
     ++ # := preinc(#1)
To complete the job, we would use corresponding composition functions
and `Z C' to cause these functions to display in their respective Maple
and C notations.  (Note that the C example ignores issues of operator
precedence, which are discussed in the next section.)
   You can enclose any token in quotes to prevent its usual
interpretation in syntax patterns:
     # ":=" # := becomes(#1,#2)
   Quotes also allow you to include spaces in a token, although once
again it is generally better to use two tokens than one token with an
embedded space.  To include an actual quotation mark in a quoted token,
precede it with a backslash.  (This also works to include backslashes
in tokens.)
     # "bad token" # "/\"\\" # := silly(#1,#2,#3)
This will parse `3 bad token 4 /"\ 5' to `silly(3,4,5)'.
   The token `#' has a predefined meaning in Calc's formula parser; it
is not legal to use `"#"' in a syntax rule.  However, longer tokens
that include the `#' character are allowed.  Also, while `"$"' and
`"\""' are allowed as tokens, their presence in the syntax table will
prevent those characters from working in their usual ways (referring to
stack entries and quoting strings, respectively).
   Finally, the notation `%%' anywhere in a syntax table causes the
rest of the line to be ignored as a comment.
File: calc.info,  Node: Precedence in Syntax Tables,  Next: Advanced Syntax Patterns,  Prev: Syntax Table Basics,  Up: Syntax Tables
Precedence
..........
Different operators are generally assigned different "precedences".  By
default, an operator defined by a rule like
     # foo # := foo(#1,#2)
will have an extremely low precedence, so that `2*3+4 foo 5 == 6' will
be parsed as `(2*3+4) foo (5 == 6)'.  To change the precedence of an
operator, use the notation `#/P' in place of `#', where P is an integer
precedence level.  For example, 185 lies between the precedences for
`+' and `*', so if we change this rule to
     #/185 foo #/186 := foo(#1,#2)
then `2+3 foo 4*5' will be parsed as `2+(3 foo (4*5))'.  Also, because
we've given the righthand expression slightly higher precedence, our
new operator will be left-associative: `1 foo 2 foo 3' will be parsed
as `(1 foo 2) foo 3'.  By raising the precedence of the lefthand
expression instead, we can create a right-associative operator.
   *Note Composition Basics::, for a table of precedences of the
standard Calc operators.  For the precedences of operators in other
language modes, look in the Calc source file `calc-lang.el'.
File: calc.info,  Node: Advanced Syntax Patterns,  Next: Conditional Syntax Rules,  Prev: Precedence in Syntax Tables,  Up: Syntax Tables
Advanced Syntax Patterns
........................
To match a function with a variable number of arguments, you could write
     foo ( # ) := myfunc(#1)
     foo ( # , # ) := myfunc(#1,#2)
     foo ( # , # , # ) := myfunc(#1,#2,#3)
but this isn't very elegant.  To match variable numbers of items, Calc
uses some notations inspired regular expressions and the "extended BNF"
style used by some language designers.
     foo ( { # }*, ) := apply(myfunc,#1)
   The token `{' introduces a repeated or optional portion.  One of the
three tokens `}*', `}+', or `}?' ends the portion.  These will match
zero or more, one or more, or zero or one copies of the enclosed
pattern, respectively.  In addition, `}*' and `}+' can be followed by a
separator token (with no space in between, as shown above).  Thus `{ #
}*,' matches nothing, or one expression, or several expressions
separated by commas.
   A complete `{ ... }' item matches as a vector of the items that
matched inside it.  For example, the above rule will match `foo(1,2,3)'
to get `apply(myfunc,[1,2,3])'.  The Calc `apply' function takes a
function name and a vector of arguments and builds a call to the
function with those arguments, so the net result is the formula
`myfunc(1,2,3)'.
   If the body of a `{ ... }' contains several `#'s (or nested `{ ...
}' constructs), then the items will be strung together into the
resulting vector.  If the body does not contain anything but literal
tokens, the result will always be an empty vector.
     foo ( { # , # }+, ) := bar(#1)
     foo ( { { # }*, }*; ) := matrix(#1)
will parse `foo(1,2,3,4)' as `bar([1,2,3,4])', and `foo(1,2;3,4)' as
`matrix([[1,2],[3,4]])'.  Also, after some thought it's easy to see how
this pair of rules will parse `foo(1,2,3)' as `matrix([[1,2,3]])',
since the first rule will only match an even number of arguments.  The
     foo ( # { , # , # }? ) := bar(#1,#2)
will parse `foo(2,3,4)' as `bar(2,[3,4])', and `foo(2)' as `bar(2,[])'.
   The notation `{ ... }?.' (note the trailing period) works just the
same as regular `{ ... }?', except that it does not count as an
argument; the following two rules are equivalent:
     foo ( # , { also }? # ) := bar(#1,#3)
     foo ( # , { also }?. # ) := bar(#1,#2)
Note that in the first case the optional text counts as `#2', which
will always be an empty vector, but in the second case no empty vector
is produced.
   Another variant is `{ ... }?$', which means the body is optional
only at the end of the input formula.  All built-in syntax rules in
Calc use this for closing delimiters, so that during algebraic entry
you can type `[sqrt(2), sqrt(3 RET', omitting the closing parenthesis
and bracket.  Calc does this automatically for trailing `)', `]', and
`>' tokens in syntax rules, but you can use `{ ... }?$' explicitly to
get this effect with any token (such as `"}"' or `end').  Like `{ ...
}?.', this notation does not count as an argument.  Conversely, you can
use quotes, as in `")"', to prevent a closing-delimiter token from
being automatically treated as optional.
   Calc's parser does not have full backtracking, which means some
patterns will not work as you might expect:
     foo ( { # , }? # , # ) := bar(#1,#2,#3)
Here we are trying to make the first argument optional, so that
`foo(2,3)' parses as `bar([],2,3)'.  Unfortunately, Calc first tries to
match `2,' against the optional part of the pattern, finds a match, and
so goes ahead to match the rest of the pattern.  Later on it will fail
to match the second comma, but it doesn't know how to go back and try
the other alternative at that point.  One way to get around this would
be to use two rules:
     foo ( # , # , # ) := bar([#1],#2,#3)
     foo ( # , # ) := bar([],#1,#2)
   More precisely, when Calc wants to match an optional or repeated
part of a pattern, it scans forward attempting to match that part.  If
it reaches the end of the optional part without failing, it "finalizes"
its choice and proceeds.  If it fails, though, it backs up and tries
the other alternative.  Thus Calc has "partial" backtracking.  A fully
backtracking parser would go on to make sure the rest of the pattern
matched before finalizing the choice.
File: calc.info,  Node: Conditional Syntax Rules,  Prev: Advanced Syntax Patterns,  Up: Syntax Tables
Conditional Syntax Rules
........................
It is possible to attach a "condition" to a syntax rule.  For example,
the rules
     foo ( # ) := ifoo(#1) :: integer(#1)
     foo ( # ) := gfoo(#1)
will parse `foo(3)' as `ifoo(3)', but will parse `foo(3.5)' and
`foo(x)' as calls to `gfoo'.  Any number of conditions may be attached;
all must be true for the rule to succeed.  A condition is "true" if it
evaluates to a nonzero number.  *Note Logical Operations::, for a list
of Calc functions like `integer' that perform logical tests.
   The exact sequence of events is as follows:  When Calc tries a rule,
it first matches the pattern as usual.  It then substitutes `#1', `#2',
etc., in the conditions, if any.  Next, the conditions are simplified
and evaluated in order from left to right, as if by the `a s' algebra
command (*note Simplifying Formulas::.).  Each result is true if it is
a nonzero number, or an expression that can be proven to be nonzero
(*note Declarations::.).  If the results of all conditions are true,
the expression (such as `ifoo(#1)') has its `#'s substituted, and that
is the result of the parse.  If the result of any condition is false,
Calc goes on to try the next rule in the syntax table.
   Syntax rules also support `let' conditions, which operate in exactly
the same way as they do in algebraic rewrite rules.  *Note Other
Features of Rewrite Rules::, for details.  A `let' condition is always
true, but as a side effect it defines a variable which can be used in
later conditions, and also in the expression after the `:=' sign:
     foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
The `dnumint' function tests if a value is numerically an integer,
i.e., either a true integer or an integer-valued float.  This rule will
parse `foo' with a half-integer argument, like `foo(3.5)', to a call
like `hifoo(4.)'.
   The lefthand side of a syntax rule `let' must be a simple variable,
not the arbitrary pattern that is allowed in rewrite rules.
   The `matches' function is also treated specially in syntax rule
conditions (again, in the same way as in rewrite rules).  *Note
Matching Commands::.  If the matching pattern contains meta-variables,
then those meta-variables may be used in later conditions and in the
result expression.  The arguments to `matches' are not evaluated in
this situation.
     sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
This is another way to implement the Maple mode `sum' notation.  In
this approach, we allow `#2' to equal the whole expression `i=1..10'.
Then, we use `matches' to break it apart into its components.  If the
expression turns out not to match the pattern, the syntax rule will
fail.  Note that `Z S' always uses Calc's normal language mode for
editing expressions in syntax rules, so we must use regular Calc
notation for the interval `[b..c]' that will correspond to the Maple
mode interval `1..10'.
File: calc.info,  Node: Modes Variable,  Next: Calc Mode Line,  Prev: Language Modes,  Up: Mode Settings
The `Modes' Variable
====================
The `m g' (`calc-get-modes') command pushes onto the stack a vector of
numbers that describes the various mode settings that are in effect.
With a numeric prefix argument, it pushes only the Nth mode, i.e., the
Nth element of this vector.  Keyboard macros can use the `m g' command
to modify their behavior based on the current mode settings.
   The modes vector is also available in the special variable `Modes'.
In other words, `m g' is like `s r Modes RET'.  It will not work to
store into this variable; in fact, if you do, `Modes' will cease to
track the current modes.  (The `m g' command will continue to work,
however.)
   In general, each number in this vector is suitable as a numeric
prefix argument to the associated mode-setting command.  (Recall that
the `~' key takes a number from the stack and gives it as a numeric
prefix to the next command.)
   The elements of the modes vector are as follows:
  1. Current precision.  Default is 12; associated command is `p'.
  2. Binary word size.  Default is 32; associated command is `b w'.
  3. Stack size (not counting the value about to be pushed by `m g').
     This is zero if `m g' is executed with an empty stack.
  4. Number radix.  Default is 10; command is `d r'.
  5. Floating-point format.  This is the number of digits, plus the
     constant 0 for normal notation, 10000 for scientific notation,
     20000 for engineering notation, or 30000 for fixed-point notation.
     These codes are acceptable as prefix arguments to the `d n'
     command, but note that this may lose information:  For example, `d
     s' and `C-u 12 d s' have similar (but not quite identical) effects
     if the current precision is 12, but they both produce a code of
     10012, which will be treated by `d n' as `C-u 12 d s'.  If the
     precision then changes, the float format will still be frozen at
     12 significant figures.
  6. Angular mode.  Default is 1 (degrees).  Other values are 2
     (radians) and 3 (HMS).  The `m d' command accepts these prefixes.
  7. Symbolic mode.  Value is 0 or 1; default is 0.  Command is `m s'.
  8. Fraction mode.  Value is 0 or 1; default is 0.  Command is `m f'.
  9. Polar mode.  Value is 0 (rectangular) or 1 (polar); default is 0.
     Command is `m p'.
 10. Matrix/scalar mode.  Default value is -1.  Value is 0 for scalar
     mode, -2 for matrix mode, or N for NxN matrix mode.  Command is `m
     v'.
 11. Simplification mode.  Default is 1.  Value is -1 for off (`m O'),
     0 for `m N', 2 for `m B', 3 for `m A', 4 for `m E', or 5 for
     `m U'.  The `m D' command accepts these prefixes.
 12. Infinite mode.  Default is -1 (off).  Value is 1 if the mode is on,
     or 0 if the mode is on with positive zeros.  Command is `m i'.
   For example, the sequence `M-1 m g RET 2 + ~ p' increases the
precision by two, leaving a copy of the old precision on the stack.
Later, `~ p' will restore the original precision using that stack
value.  (This sequence might be especially useful inside a keyboard
macro.)
   As another example, `M-3 m g 1 - ~ DEL' deletes all but the oldest
(bottommost) stack entry.
   Yet another example:  The HP-48 "round" command rounds a number to
the current displayed precision.  You could roughly emulate this in
Calc with the sequence `M-5 m g 10000 % ~ c c'.  (This would not work
for fixed-point mode, but it wouldn't be hard to do a full emulation
with the help of the `Z [' and `Z ]' programming commands.  *Note
Conditionals in Macros::.)
File: calc.info,  Node: Calc Mode Line,  Prev: Modes Variable,  Up: Mode Settings
The Calc Mode Line
==================
This section is a summary of all symbols that can appear on the Calc
mode line, the highlighted bar that appears under the Calc stack window
(or under an editing window in Embedded Mode).
   The basic mode line format is:
     --%%-Calc: 12 Deg OTHER MODES       (Calculator)
   The `%%' is the Emacs symbol for "read-only"; it shows that regular
Emacs commands are not allowed to edit the stack buffer as if it were
text.
   The word `Calc:' changes to `CalcEmbed:' if Embedded Mode is
enabled.  The words after this describe the various Calc modes that are
in effect.
   The first mode is always the current precision, an integer.  The
second mode is always the angular mode, either `Deg', `Rad', or `Hms'.
   Here is a complete list of the remaining symbols that can appear on
the mode line:
`Alg'
     Algebraic mode (`m a'; *note Algebraic Entry::.).
`Alg[('
     Incomplete algebraic mode (`C-u m a').
`Alg*'
     Total algebraic mode (`m t').
`Symb'
     Symbolic mode (`m s'; *note Symbolic Mode::.).
`Matrix'
     Matrix mode (`m v'; *note Matrix Mode::.).
`MatrixN'
     Dimensioned matrix mode (`C-u N m v').
`Scalar'
     Scalar mode (`m v'; *note Matrix Mode::.).
`Polar'
     Polar complex mode (`m p'; *note Polar Mode::.).
`Frac'
     Fraction mode (`m f'; *note Fraction Mode::.).
`Inf'
     Infinite mode (`m i'; *note Infinite Mode::.).
`+Inf'
     Positive infinite mode (`C-u 0 m i').
`NoSimp'
     Default simplifications off (`m O'; *note Simplification Modes::.).
`NumSimp'
     Default simplifications for numeric arguments only (`m N').
`BinSimpW'
     Binary-integer simplification mode; word size W (`m B', `b w').
`AlgSimp'
     Algebraic simplification mode (`m A').
`ExtSimp'
     Extended algebraic simplification mode (`m E').
`UnitSimp'
     Units simplification mode (`m U').
`Bin'
     Current radix is 2 (`d 2'; *note Radix Modes::.).
`Oct'
     Current radix is 8 (`d 8').
`Hex'
     Current radix is 16 (`d 6').
`RadixN'
     Current radix is N (`d r').
`Zero'
     Leading zeros (`d z'; *note Radix Modes::.).
`Big'
     Big language mode (`d B'; *note Normal Language Modes::.).
`Flat'
     One-line normal language mode (`d O').
`Unform'
     Unformatted language mode (`d U').
     C language mode (`d C'; *note C FORTRAN Pascal::.).
`Pascal'
     Pascal language mode (`d P').
`Fortran'
     FORTRAN language mode (`d F').
`TeX'
     TeX language mode (`d T'; *note TeX Language Mode::.).
`Eqn'
     "Eqn" language mode (`d E'; *note Eqn Language Mode::.).
`Math'
     Mathematica language mode (`d M'; *note Mathematica Language
     Mode::.).
`Maple'
     Maple language mode (`d W'; *note Maple Language Mode::.).
`NormN'
     Normal float mode with N digits (`d n'; *note Float Formats::.).
`FixN'
     Fixed point mode with N digits after the point (`d f').
`Sci'
     Scientific notation mode (`d s').
`SciN'
     Scientific notation with N digits (`d s').
`Eng'
     Engineering notation mode (`d e').
`EngN'
     Engineering notation with N digits (`d e').
`LeftN'
     Left-justified display indented by N (`d <'; *note
     Justification::.).
`Right'
     Right-justified display (`d >').
`RightN'
     Right-justified display with width N (`d >').
`Center'
     Centered display (`d =').
`CenterN'
     Centered display with center column N (`d =').
`WidN'
     Line breaking with width N (`d b'; *note Normal Language Modes::.).
`Wide'
     No line breaking (`d b').
`Break'
     Selections show deep structure (`j b'; *note Making Selections::.).
`Save'
     Record modes in `~/.emacs' (`m R'; *note General Mode Commands::.).
`Local'
     Record modes in Embedded buffer (`m R').
`LocEdit'
     Record modes as editing-only in Embedded buffer (`m R').
`LocPerm'
     Record modes as permanent-only in Embedded buffer (`m R').
`Global'
     Record modes as global in Embedded buffer (`m R').
`Manual'
     Automatic recomputation turned off (`m C'; *note Automatic
     Recomputation::.).
`Graph'
     GNUPLOT process is alive in background (*note Graphics::.).
`Sel'
     Top-of-stack has a selection (Embedded only; *note Making
     Selections::.).
`Dirty'
     The stack display may not be up-to-date (*note Display Modes::.).
`Inv'
     "Inverse" prefix was pressed (`I'; *note Inverse and
     Hyperbolic::.).
`Hyp'
     "Hyperbolic" prefix was pressed (`H').
`Keep'
     "Keep-arguments" prefix was pressed (`K').
`Narrow'
     Stack is truncated (`d t'; *note Truncating the Stack::.).
   In addition, the symbols `Active' and `~Active' can appear as minor
modes on an Embedded buffer's mode line.  *Note Embedded Mode::.
File: calc.info,  Node: Arithmetic,  Next: Scientific Functions,  Prev: Mode Settings,  Up: Top
Arithmetic Functions
********************
This chapter describes the Calc commands for doing simple calculations
on numbers, such as addition, absolute value, and square roots.  These
commands work by removing the top one or two values from the stack,
performing the desired operation, and pushing the result back onto the
stack.  If the operation cannot be performed, the result pushed is a
formula instead of a number, such as `2/0' (because division by zero is
illegal) or `sqrt(x)' (because the argument `x' is a formula).
   Most of the commands described here can be invoked by a single
keystroke.  Some of the more obscure ones are two-letter sequences
beginning with the `f' ("functions") prefix key.
   *Note Prefix Arguments::, for a discussion of the effect of numeric
prefix arguments on commands in this chapter which do not otherwise
interpret a prefix argument.
* Menu:
* Basic Arithmetic::
* Integer Truncation::
* Complex Number Functions::
* Conversions::
* Date Arithmetic::
* Financial Functions::
* Binary Functions::