The general syntax of a formula is as follows:

formula:
expression

expression:
( expression )
constant
prefix-operator expression
expression infix-operator expression
expression postfix-operator
cell-reference
function-call
name

where expression is an arbitrarily complex expression involving constants (§3.17.2.1), operators (§0), cell references (§3.17.2.3), calls to functions (§3.17.2.4), and names (§3.17.2.5).

A token is the minimal lexical element of a formula. The categories of tokens are: constants (except for array-constant), operators, cell references, function names, names, and punctuators. The punctuators are:

• Left parenthesis (() and right parenthesis ()) used for expression grouping and in a function call.

• Comma (,) used in a function call and an array-constant.

• Left brace ({), right brace (}), and semicolon (;) used in an array-constant.

In a formula, an arbitrary number of space characters (U+0020) can precede the first token or follow the final token. An arbitrary number of space characters can separate two adjacent tokens, except that no space characters shall separate a function-name from the left parenthesis (() that follows it. Such space characters have no effect on the semantics of a formula; however, such spaces shall be distinguished from the space operator (§0).

All arithmetic terms in an expression are real numbers. [Example: In the expression 1/3, although the operands appear to be integers, they are, in fact, real numbers, and the result is 0.33…, not 0, as would result from integer division. end example]

As ranges of data are fundamental to spreadsheet calculations, many SpreadsheetML functions are able to take arrays as inputs and to return arrays as outputs. The way in which formula return values are returned into the worksheet differs based on whether the formula in a given cell was array entered or not, but intermediate calculations are always done for the full arrays when they are used as inputs. The differences with an array-entered formula are:

• No implicit intersection is performed on cell range inputs.

• The results of the formula’s calculation can be returned across multiple cells in the sheet.

When a range of cell references is used in a formula that is array-entered in an area larger than that range, the excess cells take on a value of #N/A.

[Example: Here are some formulas taking array constants and ranges:

• (B2:B4*C2:C4)+10.5 performs three calculations: (B2*C2)+10.5, (B3*C3)+10.5, and (B4*C4)+10.5.

• SQRT({1,2,3,4}) returns 1 when entered normally.

• SQRT({1,2,3,4}) returns 1 when array-entered into a single cell, but if it’s array-entered in four or more cells in a contiguous row, it will return 1, 1.41, 1.73, and 2 in the first four cells, respectively, and #N/A in any additional cells in the horizontal range for which it was array-entered. (For display purposes, the values returned have been truncated to two decimal places.)

• SUM(SQRT({1,2,3,4})) returns 6.14 when entered normally, since array calculations are always performed by the SQRT function, and the array output is understood as a valid input by the SUM function.

With A1:A4 holding the values 1, 2, 3, and 4, respectively:

• SQRT(A1:A4) entered normally will do implicit intersection if it is in any of the rows 1–4, and return the SQRT of the number in the same row.

• SQRT(A1:A4) returns 1 when array-entered into a single cell, since it does not do implicit intersection in this case. If it’s array-entered in multiple cells in a contiguous column, it will return 1, 1.41, 1.73, 2, #N/A, …, respectively, in the cells in its vertical output range. end example]