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#### 6.8.10.1 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)
    mod    400
    +/-    300
    !!     210    (as in n!!)
    !      210    (as in n!)
    ^      200
    -      197    (as in -n)
    *      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 (see 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.

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