•OASIS Open Document Format for Office Applications (OpenDocument) Version 1.2. 29 September 2011. OASIS Standard. http://docs.oasis-open.org/office/v1.2/os/OpenDocument-v1.2-os.html.

Abstract:

This document is Part 4 of the Open Document Format for Office Applications (OpenDocument) Version 1.3 specification.

Status:

This document was last revised or approved by the OASIS Open Document Format for Office Applications (OpenDocument) TC on the above date. The level of approval is also listed above. Check the “Latest stage” location noted above for possible later revisions of this document. Any other numbered Versions and other technical work produced by the Technical Committee (TC) are listed at https://www.oasis-open.org/committees/tc_home.php?wg_abbrev=office#technical.

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Note that any machine-readable content (Computer Language Definitions) declared Normative for this Work Product is provided in separate plain text files. In the event of a discrepancy between any such plain text file and display content in the Work Product's prose narrative document(s), the content in the separate plain text file prevails.

Citation format:

When referencing this specification the following citation format should be used:

[OpenDocument-v1.3-part4]

Open Document Format for Office Applications (OpenDocument) Version 1.3. Part 4: Recalculated Formula (OpenFormula) Format. Edited by Patrick Durusau. 04 November 2019. OASIS Committee Specification Draft 02 / Public Review Draft 02. https://docs.oasis-open.org/office/OpenDocument/v1.3/csprd02/part4-formula/OpenDocument-v1.3-csprd02-part4-formula.html. Latest stage: https://docs.oasis-open.org/office/OpenDocument/v1.3/OpenDocument-v1.3-part4-formula.html.

Notices

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1.4 Normative References

1.5 Non-Normative References

2 Expressions and Evaluators

2.1 Introduction

2.2 OpenDocument Formula Expression

2.3 Evaluator Conformance

2.3.1 OpenDocument Formula Evaluator

2.3.2 OpenDocument Formula Small Group Evaluator

2.3.3 OpenDocument Formula Medium Group Evaluator

2.3.4 OpenDocument Formula Large Group Evaluator

2.4 Variances (Implementation-defined, Unspecified, and Behavioral Changes)

3 Formula Processing Model

3.1 General

3.2 Expression Evaluation

3.2.1 General

3.2.2 Expression Calculation

3.2.3 Operator and Function Evaluation

3.3 Non-Scalar Evaluation (aka 'Array expressions')

3.4 Host-Defined Behaviors

3.5 When recalculation occurs

4.3.7 Logical (Number)

4.4 Complex Number

4.5 Logical (Boolean)

4.11.12 Sequences (NumberSequence, NumberSequenceList, DateSequence, LogicalSequence, and ComplexSequence)

4.11.13 Any

5 Expression Syntax

5.1 General

5.2 Basic Expressions

5.3 Constant Numbers

5.4 Constant Strings

5.5 Operators

5.6 Functions and Function Parameters

5.7 Nonstandard Function Names

5.10.2 Lookup of Defined Labels

5.10.3 Automatic Lookup of Labels

5.10.4 Implicit Intersection

5.10.5 Automatic Range

5.10.6 Automatic Intersection

5.11 Named Expressions

5.12 Constant Errors

5.13 Inline Arrays

5.14 Whitespace

6 Standard Operators and Functions

6.1 General

6.2 Common Template for Functions and Operators

6.3 Implicit Conversion Operators

6.3.1 General

6.3.2 Conversion to Scalar

6.3.3 Implied intersection

6.3.4 Force to array context (ForceArray)

6.3.5 Conversion to Number

6.3.6 Conversion to Integer

6.3.7 Conversion to NumberSequence

6.3.8 Conversion to NumberSequenceList

6.3.9 Conversion to DateSequence

6.3.10 Conversion to Complex Number

6.3.11 Conversion to ComplexSequence

6.3.12 Conversion to Logical

6.3.13 Conversion to LogicalSequence

6.3.14 Conversion to Text

6.3.15 Conversion to DateParam

6.3.16 Conversion to TimeParam

6.4 Standard Operators

6.4.1 General

6.4.2 Infix Operator "+"

6.4.3 Infix Operator "-"

6.4.4 Infix Operator "*"

6.4.5 Infix Operator "/"

6.4.6 Infix Operator "^"

6.4.7 Infix Operator "="

6.4.8 Infix Operator "<>"

6.4.9 Infix Operator Ordered Comparison ("<", "<=", ">", ">=")

6.4.10 Infix Operator "&"

6.4.11 Infix Operator Reference Range (":")

6.4.12 Infix Operator Reference Intersection ("!")

6.4.13 Infix Operator Reference Concatenation ("~") (aka Union)

6.4.14 Postfix Operator "%"

6.4.15 Prefix Operator "+"

6.4.16 Prefix Operator "-"

6.6 Bit operation functions

6.7 Byte-position text functions

6.8 Complex Number Functions

6.9 Database Functions

6.10 Date and Time Functions

6.11 External Access Functions

6.11.1 General

6.11.2 DDE

6.11.3 HYPERLINK

6.12 Financial Functions

6.13 Information Functions

6.14 Lookup Functions

6.14.10 MULTIPLE.OPERATIONS

6.14.11 OFFSET

6.14.12 VLOOKUP

6.15 Logical Functions

6.16 Mathematical Functions

6.17 Rounding Functions

6.18 Statistical Functions

6.18.9 BINOM.DIST.RANGE

6.18.10 BINOMDIST

6.18.11 LEGACY.CHIDIST

6.18.12 CHISQDIST

6.18.13 LEGACY.CHIINV

6.18.14 CHISQINV

6.18.15 LEGACY.CHITEST

6.18.54 LEGACY.NORMSDIST

6.18.55 LEGACY.NORMSINV

6.19 Number Representation Conversion Functions

7.2 Inline constant arrays

7.3 Inline non-constant arrays

7.4 Year 1583

8 Non-portable Features

8.1 General

8.2 Distinct Logical

Appendix A. Changes From Previous Specification Versions (Non Normative)

A.1. Changes from “Open Document Format for Office Applications (OpenDocument) v1.2”

1 Introduction

1.1 Introduction

This document is part of the Open Document Format for Office Applications (OpenDocument) Version 1.3 specification. It defines a formula language for OpenDocument documents, which is also called OpenFormula.

OpenFormula is a specification of an open format for exchanging recalculated formulas between office applications, in particular, formulas in spreadsheet documents. OpenFormula defines data types, syntax, and semantics for recalculated formulas, including predefined functions and operations.

OpenFormula is intended to be a supporting document to the Open Document Format for Office Applications (OpenDocument) format, particularly for defining its attributes table:formula and text:formula. It can also be used in other circumstances where a simple, easy-to-read infix text notation is desired for exchanging recalculated formulas.

Note: Using OpenFormula allows document creators to change the office application they use, exchange formulas with others (who may use a different application), and access formulas far in the future, with confidence that the recalculated formulas in their documents will produce equivalent results if given equivalent inputs.

1.2 Terminology

All text is normative unless otherwise labeled.

Within the normative text of this specification, the terms "shall", "shall not", "should", "should not", "may" and “need not” are to be interpreted as described in Annex H of [ISO/IEC Directives].

1.3 Purpose

OpenFormula defines:

1.data types
2.syntax
3.semantics

for recalculated formulas. 3.5

OpenFormula also defines functions.

OpenFormula does not define:

1.a user interface
2.a general notation for mathematical expressions

1.4 Normative References

[CharModel] Martin J. Dürst, et. al., Character Model for the World Wide Web 1.0: Fundamentals, http://www.w3.org/TR/2005/REC-charmod-20050215/, W3C, 2005.

[ISO/IEC Directives] ISO/IEC Directives, Part 2 (Fifth Edition) Rules for the structure and drafting of International Standards, International Organization for Standardization and International Electrotechnical Commission, 2004.

[ISO4217] ISO 4217:2008 Codes for the representation of currencies and funds, International Organization for Standardization and International Electrotechnical Commission, 2008.

[ISO8601] ISO 8601:2004 Data elements and interchange formats -- Information interchange -- Representation of dates and times, International Organization for Standardization and International Electrotechnical Commission, 2004.

[RFC3986] T. Berners-Lee, R. Fielding, L. Masinter, Uniform Resource Identifier (URI): Generic Syntax, http://www.ietf.org/rfc/rfc3986.txt, IETF, 2005.

[RFC3987] M. Duerst, M. Suignard, Internationalized Resource Identifiers (IRIs), http://www.ietf.org/rfc/rfc3987.txt, IETF, 2005.

[UNICODE] The Unicode Consortium. The Unicode Standard, Version 5.2.0, defined by: The Unicode Standard, Version 5.2 (Mountain View, CA, The Unicode Consortium, 2009. ISBN 978-1-936213-00-9). (http://www.unicode.org/versions/Unicode5.2.0/).

[UTR15] Mark Davis, Martin Dürst, Unicode Normalization Forms, Unicode Technical Report #15, http://www.unicode.org/reports/tr15/tr15-25.html, 2005.

[XML1.0] Tim Bray, Jean Paoli, C. M. Sperberg-McQueen, Eve Maler, François Yergeau , Extensible Markup Language (XML) 1.0 (Fourth Edition), http://www.w3.org/TR/2006/REC-xml-20060816/, W3C, 2006.

1.5 Non-Normative References

[JISX0201] The Unicode Consortium., JIS X 0201 (1976) to Unicode 1.1 Table, 1994, http://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/JIS/JIS0201.TXT.

[JISX0208] The Unicode Consortium., JIS X 0208 (1990) to Unicode, 1994, http://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/JIS/JIS0208.TXT.

[UAX11] Asmus Freytag, East Asian Width, Unicode Standard Annex #11, http://www.unicode.org/reports/tr11/tr11-19.html, 2009.

2 Expressions and Evaluators

2.1 Introduction

The OpenDocument specification defines conformance for formula expressions and evaluators. For evaluators, there are three groups of features that an evaluator may support. This chapter defines the basic requirements for the individual conformance targets.

2.2 OpenDocument Formula Expression

An OpenDocument formula expression shall adhere to the expression syntax defined in chapter 4. It may use subsets or supersets of OpenFormula.

2.3 Evaluator Conformance

2.3.1 OpenDocument Formula Evaluator

An OpenDocument Formula Evaluator is a program that can parse and recalculate OpenDocument formula expressions, and that meets the following additional requirements:

A)It may implement subsets or supersets of this specification.
B)It shall conform to one of: (C16) OpenDocument Formula Small Group Evaluator, (C17) OpenDocument Formula Medium Group Evaluator or (C18) OpenDocument Formula Large Group Evaluator
C)It may implement additional functions beyond those defined in this specification. It may further implement additional formula syntax, additional operations, additional optional parameters for functions, or may consider function parameters to be optional when they are required by this specification.
D)Applications should clearly document their extensions in their user documentation, both online and paper, in a manner so users would be likely to be aware when they are using a non-standard extension.

Note 1: An expression may reference a function not defined by this specification by name, or depend on implementation-defined behavior, or on semantics not guaranteed by this specification. Reference to or dependence upon functions or behavior not defined by this standard may impair the interoperability of the resulting expression(s).

Note 2: This specification defines formulas in terms of a canonical text representation used for exchange. If formulas are contained in XML attributes some characters shall be escaped as required by the XML specification (e.g., the character & shall be escaped in XML attributes using notations such as &). All string and character literals references by this specification are in the value space defined by [UNICODE] thus, “A” is U+0041, “Z” is U+005A, and the range of characters “A-Z” is the range U+0041 through U+005A inclusive.

2.3.2 OpenDocument Formula Small Group Evaluator

An OpenDocument Formula Small Group Evaluator is an OpenDocument Formula Evaluator that meets the following additional requirements:

A)It shall implement at least the limits defined in the “Basic Limits” section. 3.7
B)It shall implement the syntax defined in these sections on syntax: Criteria 4.11.11; Basic Expressions 5.2; Constant Numbers 5.3; Constant Strings 5.4; Operators 5.5; Functions and Function Parameters 5.6; Nonstandard Function Names 5.7; References 5.8; Simple Named Expressions ; Errors 5.12; Whitespace 5.14.
C)It shall implement all implicit conversions for the types it implements, at least Text 6.3.14, Conversion to Number 6.3.5, Reference , Conversion to Logical 6.3.12, and when an expression returns an Error.
D)It shall implement the following operators (which are all the operators except reference union (~)): Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.3.5; Infix Operator "&” 6.4.10; Infix Operator "+” 6.4.2; Infix Operator "-” 6.4.3; Infix Operator "*” 6.4.4; Infix Operator "/” 6.4.5; Infix Operator "^” 6.4.6; Infix Operator "=” 6.4.7; Infix Operator "<>” 6.4.8; Postfix Operator “%” 6.4.14; Prefix Operator “+” 6.4.15; Prefix Operator “-” 6.4.16; Infix Operator Reference Intersection ("!") 6.4.12; Infix Operator Range (":") 6.4.11.
E)It shall implement at least the following functions as defined in this specification: ABS 6.16.2 ; ACOS 6.16.3 ; AND 6.15.2 ; ASIN 6.16.7 ; ATAN 6.16.9 ; ATAN2 6.16.10 ; AVERAGE 6.18.3 ; AVERAGEIF 6.18.5 ; CHOOSE 6.14.3 ; COLUMNS 6.13.5 ; COS 6.16.19 ; COUNT 6.13.6 ; COUNTA 6.13.7 ; COUNTBLANK 6.13.8 ; COUNTIF 6.13.9 ; DATE 6.10.2 ; DAVERAGE 6.9.2 ; DAY 6.10.5 ; DCOUNT 6.9.3 ; DCOUNTA 6.9.4 ; DDB 6.12.14 ; DEGREES 6.16.25 ; DGET 6.9.5 ; DMAX 6.9.6 ; DMIN 6.9.7 ; DPRODUCT 6.9.8 ; DSTDEV 6.9.9 ; DSTDEVP 6.9.10 ; DSUM 6.9.11 ; DVAR 6.9.12 ; DVARP 6.9.13 ; EVEN 6.16.30 ; EXACT 6.20.8 ; EXP 6.16.31 ; FACT 6.16.32 ; FALSE 6.15.3 ; FIND 6.20.9 ; FV 6.12.20 ; HLOOKUP 6.14.5 ; HOUR 6.10.10 ; IF 6.15.4 ; INDEX 6.14.6 ; INT 6.17.2 ; IRR 6.12.24 ; ISBLANK 6.13.14 ; ISERR 6.13.15 ; ISERROR 6.13.16 ; ISLOGICAL 6.13.19 ; ISNA 6.13.20 ; ISNONTEXT 6.13.21 ; ISNUMBER 6.13.22 ; ISTEXT 6.13.25 ; LEFT 6.20.12 ; LEN 6.20.13 ; LN 6.16.39 ; LOG 6.16.40 ; LOG10 6.16.41 ; LOWER 6.20.14 ; MATCH 6.14.9 ; MAX 6.18.45 ; MID 6.20.15 ; MIN 6.18.48 ; MINUTE 6.10.12 ; MOD 6.16.42 ; MONTH 6.10.13 ; N 6.13.26 ; NA 6.13.27 ; NOT 6.15.7 ; NOW 6.10.15 ; NPER 6.12.29 ; NPV 6.12.30 ; ODD 6.16.44 ; OR 6.15.8 ; PI 6.16.45 ; PMT 6.12.36 ; POWER 6.16.46 ; PRODUCT 6.16.47 ; PROPER 6.20.16 ; PV 6.12.41 ; RADIANS 6.16.49 ; RATE 6.12.42 ; REPLACE 6.20.17 ; REPT 6.20.18 ; RIGHT 6.20.19 ; ROUND 6.17.5 ; ROWS 6.13.30 ; SECOND 6.10.16 ; SIN 6.16.55 ; SLN 6.12.45 ; SQRT 6.16.58 ; STDEV 6.18.72 ; STDEVP 6.18.74 ; SUBSTITUTE 6.20.21 ; SUM 6.16.61 ; SUMIF 6.16.62 ; SYD 6.12.46 ; T 6.20.22 ; TAN 6.16.69 ; TIME 6.10.17 ; TODAY 6.10.19 ; TRIM 6.20.24 ; TRUE 6.15.9 ; TRUNC 6.17.8 ; UPPER 6.20.27 ; VALUE 6.13.34 ; VAR 6.18.82 ; VARP 6.18.84 ; VLOOKUP 6.14.12 ; WEEKDAY 6.10.20 ; YEAR 6.10.23
F)It need not evaluate references that contain more than one area.
G)It need not implement inline arrays 5.13, complex numbers 4.4, and the reference union operator 6.4.13.

Note: This specification does not mandate a user interface for international characters, so a resource-constrained application may choose to not show the traditional glyph (e.g., it may show the [UNICODE] numeric code instead).

2.3.3 OpenDocument Formula Medium Group Evaluator

An OpenDocument Formula Medium Group Evaluator is an OpenDocument Small Group Formula Evaluator that meets the following additional requirements:

2.3.4 OpenDocument Formula Large Group Evaluator

An OpenDocument Formula Large Group Evaluator is an OpenDocument Medium Group Formula Evaluator that meets the following additional requirements:

A)It shall implement the syntax defined in these sections on syntax: Inline Arrays 5.13; Automatic Intersection 5.10.6; External Named Expressions 5.11.
B)It shall implement the complex number type as discussed in the section on Complex Number 4.4, array formulas, and Sheet-local Named Expressions.

It shall implement the following functions as defined in this specification: AMORLINC 6.12.4 ; ARABIC 6.19.2 ; AREAS 6.13.2 ; ASC 6.20.2 ; AVERAGEA 6.18.4 ; AVERAGEIFS 6.18.6 ; BASE 6.19.3 ; BIN2DEC 6.19.4 ; BIN2HEX 6.19.5 ; BIN2OCT 6.19.6 ; BINOM.DIST.RANGE 6.18.9 ; BITAND 6.6.2 ; BITLSHIFT 6.6.3 ; BITOR 6.6.4 ; BITRSHIFT 6.6.5 ; BITXOR 6.6.6 ; CHISQDIST 6.18.12 ; CHISQINV 6.18.14 ; COMBINA 6.16.17 ; COMPLEX 6.8.2 ; COUNTIFS 6.13.10 ; CSC 6.16.23 ; 6.16.23CSCH 6.16.24 ; DATEDIF 6.10.3 ; DAYS 6.10.6 ; DDE 6.11.2 ; DEC2BIN 6.19.7 ; DEC2HEX 6.19.8 ; DEC2OCT 6.19.9 ; DECIMAL 6.19.10 ; DELTA 6.16.26 ; EDATE 6.10.8 ; ERROR.TYPE 6.13.11; EUROCONVERT 6.16.29 ; FACTDOUBLE 6.16.33 ; FDIST 6.18.22 ; FINDB 6.7.2 ; FINV 6.18.24 ; FORMULA 6.13.12 ; FREQUENCY 6.18.29 ; FVSCHEDULE 6.12.21 ; GAMMA 6.16.34 ; GAUSS 6.18.33 ; GESTEP 6.16.37 ; GETPIVOTDATA 6.14.4 ; GROWTH 6.18.35 ; HEX2BIN 6.19.11 ; HEX2DEC 6.19.12 ; HEX2OCT 6.19.13 ; HYPERLINK 6.11.3 ; IFERROR 6.15.5 ; IFNA 6.15.6 ; IMABS 6.8.3 ; IMAGINARY 6.8.4 ; IMARGUMENT 6.8.5 ; IMCONJUGATE 6.8.6 ; IMCOS 6.8.7 ; IMCOT 6.8.9 ; IMCSC 6.8.10 ; IMCSCH 6.8.11 ; IMDIV 6.8.12 ; IMEXP 6.8.13 ; IMLN 6.8.14 ; IMLOG10 6.8.15 ; IMLOG2 6.8.16 ; IMPOWER 6.8.17 ; IMPRODUCT 6.8.18 ; IMREAL 6.8.19 ; IMSEC 6.8.22 ; IMSECH 6.8.23 ; IMSIN 6.8.20 ; IMSQRT 6.8.24 ; IMSUB 6.8.25 ; IMSUM 6.8.26 ; IMTAN 6.8.27; INDIRECT 6.14.7 ; INFO 6.13.13 ; IPMT 6.12.23 ; ISFORMULA 6.13.18 ; ISPMT 6.12.25 ; ISREF 6.13.24 ; JIS 6.20.11 ; LEFTB 6.7.3 ; LENB 6.7.4 ; MAXA 6.18.46 ; MDETERM 6.5.2 ; MULTIPLE.OPERATIONS 6.14.10 ; MUNIT 6.5.5 ; MIDB 6.7.5 ; MINA 6.18.49 ; NORMDIST 6.18.52 ; NORMINV 6.18.53 ; NUMBERVALUE 6.13.28 ; OCT2BIN 6.19.14 ; OCT2DEC 6.19.15 ; OCT2HEX 6.19.16 ; PDURATION 6.12.35 ; PERMUTATIONA 6.18.60 ; PHI 6.18.61 ; PPMT 6.12.37 ; PRICEDISC 6.12.39 ; REPLACEB 6.7.6 ; RIGHTB 6.7.7 ; RRI 6.12.44 ; SEARCH 6.20.20 ; SEARCHB 6.7.8 ; SEC 6.16.52 ; SECH 6.16.57 ; SHEET 6.13.31 ; SHEETS 6.13.32 ; SUMIFS 6.16.63 ; TEXT 6.20.23 ; UNICHAR 6.20.25 ; UNICODE 6.20.26 ; VARPA 6.18.85 ; XOR 6.15.10

Note: The following functions are documented by this specification, but not included even in the Large group:CELL 6.13.3 ; DOLLAR 6.20.7

2.4 Variances (Implementation-defined, Unspecified, and Behavioral Changes)

Applications should document all implementation-defined variances from this standard in a manner that enables application users to obtain the information.

In a few cases a specific approach is required (e.g., string indexes begin at one), which may vary in the user interfaces of different implementations.

Note 1: In practice, for nearly all documents the differences are irrelevant. The primary variances and differences from OpenFormula and some existing applications are:

•Some conversions between types are not required to be automatic. In particular, applications may, but need not, perform automatic conversion of text in a cell when it is to be used as a number.
•There need not be a distinguishable Logical type. Applications may have a Logical type distinct from Number and Text (see Distinct Logical 8.2), but Logical values may also be represented by the Number type using the values 1 (TRUE) and 0 (FALSE). This means that functions that take number sequences (such as SUM) may but need not include true and false values in the sequence.
•Applications vary on the set of Errors they support. In this specification the only distinguished Error is #N/A; all other errors are simply errors, allowing applications to choose the Error set that best meets their needs.
•In this specification, string index positions start from 1. Users of applications with string index positions starting from 0 shall add and subtract 1 on import/export of this format, as appropriate.
•Database criteria match patterns (such as the pattern matching language for text) have historically varied: Some support glob syntax (e.g., a*b is a, followed by 0 or more characters, followed by b), while others support traditional regular expression syntax (e.g., a*b is zero or more a’s, followed by b). This specification supports both pattern languages.

Note 2: Interoperability is improved by the use of the DATE 6.10.2 and TIME 6.10.17 functions or the textual [ISO8601] date representation because dates in that format do not rely upon epoch or locale-specific settings.

In an OpenDocument file, calculation settings impact formula recalculation, which can be the same or different from a particular application's defaults. These include whether or not text comparisons are case-sensitive, or if search criteria apply to the whole cell.

3 Formula Processing Model

3.1 General

This section describes the basic formula processing model: how expressions are calculated, when recalculation occurs, and limits on formulas.

3.2 Expression Evaluation

3.2.1 General

OpenFormula defines rules for the evaluation of expressions as well as the functions and operators that appear in expressions.

3.2.2 Expression Calculation

Expressions in OpenFormula shall be evaluated by application of the following rules:

1)If an expression consists of a constant Number (5.3), a constant String (5.4), a Reference (5.8), constant Error (per section 5.12), the value of that type is returned.
2)If an expression consists of one or more operations, apply the operators in order of precedence and associativity as defined by Table 1 in 5.5 (Operators). Precedence of operators may be altered by the use of "(" (LEFT PARENTHESES, U+0028) and ")" (RIGHT PARENTHESES, U+0029) to group operators. Evaluate the operator as described in Operator and Function Evaluation, 3.2.3.
3)If an expression consists of a function call (5.6, 5.7), evaluate the function as described in Operator and Function Evaluation, 3.2.3.
4)If an expression consists of a named expression (5.11), the result of evaluating the named expression is returned.
5)If an expression consists of a QuotedLabel (5.10), AutomaticIntersection (5.10.6), or Array (5.13), its value is returned. Expression Syntax 5

Once evaluation has completed:

1.If the result is a Reference and a single non-reference value is needed, it is converted to the referenced value, using the rules of Non-Scalar Evaluation, 3.3, 1.2.
2.If the result an Array, for the display area, apply the rules of Non-Scalar Evaluation, 3.3, 1.1.

3.2.3 Operator and Function Evaluation

Operators and functions in OpenFormula shall be evaluated according to their definitions by applying the following rules:

1.The value of all expression arguments are computed. Exceptions to computation of all arguments are noted in a function's specification.
Note: The practice of computing all argument expressions is known as "eager" evaluation. The IF function is an example of a function that does not require computation of all arguments.
2.If an argument expression evaluates to Error, calculation of the operator or function may short-circuit and return the Error if the function does not suppress error propagation as noted in the function's specification.
3.If an operator or function is passed a value of incorrect type, call the appropriate implicit conversion function to convert the value to the correct type. If conversion is not possible, generate an Error.
4.The function or operation is called with its argument expressions' results, and the result of the function or operation is the evaluation of the expression.

3.3 Non-Scalar Evaluation (aka 'Array expressions')

Non-scalar values passed as arguments to functions are evaluated by intersection or iteration.

1)Evaluation as an implicit intersection of the argument with the expression's evaluation position.
1. 1.1)Inline Arrays
  Element (0;0) of the array is used in place of the array.
  Note 1:
  =ABS({-3;-4}) => ABS(-3) // row vector
  =ABS({-3|-4}) => ABS(-3) // column vector
  =ABS({-3;-4|-6;-8}) => ABS(-3) // matrix
  ={1;2;3|4;5;6} => 1 // simple display
2. 1.2)References
  1. 1.2.1)If the target reference is a row-vector (Nx1), use the value at the intersection of the evaluation position's column and the reference's row.
    Note 2:
    in cell B2 : =ABS(A1:C1) => ABS(B1)
    If there is no intersection the result is #N/A or a more specific Error value.
    Note 3: in cell D4 : =ABS(A1:C1) => #N/A or a more specific Error value.
  2. 1.2.2)If the target reference is a column-vector (1xM), the value at the intersection of the evaluation position's row and the reference's column.
    Note 4:
    in cell B2 : =ABS(A1:A3) => ABS(A2)
    in cell D4 : =ABS(A1:A3) => #N/A or a more specific Error value.
2)Matrix evaluation.
If an expression is being evaluated in a cell flagged as a being part of a 'Matrix' (OpenDocument Part 3, 19.683 table:number-matrix-columns-spanned):
1. 2.1)The portion of a non-scalar result to be displayed need not be co-extensive with a specified display area. The portion of the non-scalar result to be displayed is determined by:
2. 2.2)Calculations with non-scalar inputs are a generalization of (2.1).
  
  When evaluating a formula in 'matrix' mode, and a non-scalar value is passed to a function argument that expects a scalar, the function is evaluated multiple times, iterating over the non-scalar input(s) and putting the function result into a matrix at the position corresponding to the input. Unary/Binary operators, other than range and union, follow the rules for scalar functions when passed non-scalar values.
  Inline arrays and references are interchangeable.

3.4 Host-Defined Behaviors

A Formula Evaluator operates in an execution environment (a "host"). The behavior of the Formula Evaluator is parametrized by host-defined properties and functions.

The following properties are host-defined:

1.HOST-CASE-SENSITIVE: if true, text comparisons are case-sensitive. This influences the operators =, <>, <, <=, >, and >=, as well as database query functions that use them. Note that the EXACT function is always case-sensitive, regardless of this calculation setting.
2.HOST-PRECISION-AS-SHOWN: If true, calculations are performed using rounded values of those displayed; otherwise, calculations are performed using the precision of the underlying numeric representation.
Note: This does not impose a particular numeric model. Since implementations may use binary representations, this rounding may be inexact for decimal values.
3.HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL If true, the specified search criteria shall apply to the entire cell contents if it is a text match using = or <>; if not, only a subpart of the cell content needs to match the text.
4.HOST-AUTOMATIC-FIND-LABELS: if true, row and column labels are automatically found.
5.HOST-USE-REGULAR-EXPRESSIONS: If true, regular expressions are used for character string comparisons and when searching.
6.HOST-USE-WILDCARDS: If true, wildcards question mark '?' and asterisk '*' are used for character-string comparisons and when searching. Wildcards may be escaped with a tilde '~' character.
7.HOST-NULL-YEAR: This defines how to convert a two-digit year into a four-digit year. Each two-digit year value is interpreted as a year that equals or follows this year.
8.HOST-NULL-DATE: Defines the beginning of the epoch; a numeric date of 0 equals this date.
9.HOST-LOCALE: The locale to be used for locale-dependent operations, such as conversion of text to dates, or text to numbers.
10. HOST-ITERATION-STATUS: If enabled, iterative calculations of cyclic references are performed.
11. HOST-ITERATION-MAXIMUM-DIFFERENCE: If iterative calculations of cyclic references are enabled, the maximum absolute difference between calculation steps that all involved formula cells must yield for the iteration to end and yield a result.
12. HOST-ITERATION-STEPS: If iterative calculations of cyclic references are enabled, the maximum number of steps iterations that are performed if the results are not within HOST-ITERATION-MAXIMUM-DIFFERENCE.

The function HOST-REFERENCE-RESOLVER(Reference) is implementation defined. This function takes as input a Unicode string containing a Reference according to section 4.8 and returns a resolved value.

3.5 When recalculation occurs

Implementations of OpenFormula typically recalculate formulas when its information is needed. Typical implementations will note what values a formula depends on, and when those dependent values are changed and the formula's results are displayed, it will re-execute the formulas that depend on them to produce the new results (choosing the formulas in the right order based on their dependencies). Implementations may recalculate when a value changes (this is termed automatic recalculation) or on user command (this is termed manual recalculation).

Some functions' dependencies are difficult to determine and/or should be recalculated more frequently. These include functions that return today's date or time, random number generator functions (such as RAND 6.16.50), or ones that indirectly determine the cells to act on. Many implementations always recalculate formulas including such functions whenever a recalculation occurs. Functions that are always recalculated whenever a recalculation occurs are termed volatile functions. Functions that are often volatile functions include CELL 6.13.3, HYPERLINK 6.11.3, INDIRECT 6.14.7, INFO 6.13.13, NOW 6.10.15, OFFSET 6.14.11, RAND 6.16.50 and TODAY 6.10.19. Functions that depend on the cell position of the formula they are contained in or the position of a cell they reference need to be recalculated whenever that cell is moved, such functions are COLUMN 6.13.4, ROW 6.13.29 and SHEET 6.13.31. In addition, formulas may indicate that they should always be recalculated during a recalculation process by including a forced recalculation marker, as described in the syntax below.

3.6 Numerical Models

This specification does not, by itself, specify a numerical implementation model, though it does imply some minimal levels of accuracy for most functions. For example, an application cannot say that it implements the infix operator “/” as specified in this document if it implements integer-only arithmetic.

3.7 Basic Limits

Evaluators which claim to support “basic limits” shall support at least the following limits:

1.formulas up to at least 1024 characters long, as measured when in OpenDocument interchange format not counting the square brackets around cell addresses, or the “.” in a cell address when the sheet name is omitted.
2.at least 30 parameters per function when the function prototype permits a list of parameters.
3.permit strings of ASCII characters of up to 32,767 (2^15-1) characters.
4.support at least 7 nesting levels of functions.

4 Types

4.1 General

All values defined by OpenFormula have a type. OpenFormula defines Text 4.2, Number 4.3, Complex Number 4.4, Logical 4.5, Error 4.6, Reference 4.8, ReferenceList 4.9 and Array 4.10 types.

4.2 Text (String)

A Text value (also called a string value) is a Character string as specified in [CharModel].

A text value of length zero is termed the empty string.

Index positions in a text value begin at 1.

Whether or not Unicode Normalization [UTR15] is performed on formulas, formula results or user inputs is implementation-defined. Some functions defined in OpenFormula are labeled as "normalization-sensitive", meaning that the results of the formula evaluation may differ depending on whether normalization occurs, and which normalization form is used. Mixing operands of different normalization forms in the same calculation is undefined.

4.3 Number

4.3.1 General

A number is a numeric value.

Numbers shall be able to represent fractional values (they shall not be limited to only integers). Evaluators may implement Number with a fixed or with a variable bit length. A cell with a constant numeric value has the Number type.

Implementations typically support many subtypes of Number, including Date, Time, DateTime, Percentage, fixed-point arithmetic, and arithmetic supporting arbitrarily long integers, and determine the display format from this. All such Number subtypes shall yield TRUE for the ISNUMBER 6.13.22 function.

Note: This specification does not require that specific subtypes be distinguishable from each other, or that the subtype be tracked, but in practice most implementations do such tracking because requiring users to manually format every cell appropriately becomes tedious very quickly. Automatically determining the most likely subtype is especially important for a good user interface when generating OpenDocument format, since some subtypes (such as date, time, and currency) are stored in a different manner depending on their subtype. Thus, this specification identifies some common subtypes and identifies those subtypes where relevant in some function definitions, as an aid to implementing good user interfaces. Many applications vary in the subtype produced when combining subtypes (e.g., what is the result when percentages are multiplied together), so unless otherwise noted these are unspecified. Typical implementations try to heuristically determine the "right" format for a cell when a formula is first created, based on the operations in the formula. Users can then override this format, so as a result the heuristics are not important for data exchange (and thus outside the scope of this specification).

All Number subtypes shall yield TRUE for the ISNUMBER function.

4.3.7 Logical (Number)

Applications may have a Logical type distinct from both Number and Text (see 4.5 Logical (Boolean)), but Logical values may also be represented by the Number type using the values 1 (True) and 0 (False). (see 8.2 Distinct Logical for details)

4.4 Complex Number

A complex number (sometimes also called an imaginary number) is a pair of real numbers including a real part and an imaginary part. In mathematics, complex numbers are often written as x + iy, where x (the real part) and y (the imaginary part) are real numbers and i is

\sqrt{- 1}

. A complex number can also be written as reiθ = rcos(θ) + irsin(θ), where r is the modulus of the complex number (a real number) and θ is the argument or phase (a real number representing an angle in radians).

A complex number may, but need not be, represented as a Number or Text. The results of the functions ISNUMBER() 6.13.22 and ISTEXT() 6.13.25 are implementation-defined when applied to a complex number.

Functions and operators that accept complex numbers shall accept Text values as complex numbers (Conversion to Complex Number 6.3.10), as well as Numbers that are not complex numbers.

Note 1: IMSUM("3i";4) will produce the same result as COMPLEX(4;3).

Note 2: Expression authors should be aware that use of functions that are not defined by OpenFormula as accepting complex numbers as input may impair interoperability.

Equality can be tested using IMSUB to compute the difference, use IMABS to find the absolute difference, and then ensure the absolute difference is smaller than or equal to some nonnegative value (for exact equality, compare for equality with 0).

4.5 Logical (Boolean)

Applications may have a Logical type distinct from both Number and Text, but Logical values may also be represented by the Number type using the values 1 (True) and 0 (False) (see 4.3.7 Logical (Number)). (see 8.2 Distinct Logical for details)

4.6 Error

An Error is one of a set of possible error values. Implementations may have many different error values (see 5.12), but one error value in particular is distinct: #N/A, the result of the NA() function. Users may choose to enter some data values as #N/A, so that this error value propagates to any other formula that uses it, and may test for this using the function ISNA().

Functions and operators that receive one or more error values as an input shall produce one of those input error values as their result, except when the formula or operator is specifically defined to do otherwise.

In an OpenDocument document, if an error value is the result of a cell computation it shall be stored as if it was a string. That is, the office:value-type (OpenDocument Part 3, 19.389) of an error value is string; if the computed value is stored, it is stored in the attribute office:string-value (OpenDocument Part 3, 19.383).

Note: This does not change an Error into a string type (since the Error will be restored on recalculation); this enables applications which cannot recalculate values to display the error information.

4.7 Empty Cell

An empty cell is neither zero nor the empty string, and an empty cell can be distinguished from cells containing values (including zero and the empty string). An empty cell is not the same as an Error, in particular, it is distinguishable from the Error #N/A (not available).

4.8 Reference

A cell position is the location of a single cell at the intersection of a column and a row.

A cell strip consists of cell positions in the same row and in one or more contiguous columns.

A cell rectangle consists of cell positions in the same cell strips of one or more contiguous rows.

A cell cuboid consists of cell positions in the same cell rectangles of one or more contiguous sheets.

A reference is the smallest cuboid that (1) contains specifically-identified cell positions and/or specifically-identified complete columns/rows such that (2) removal of any cell positions either violates condition (1) or does not leave a cuboid.

Cell positions in a cell cuboid/rectangle/strip can resolve to empty cells (section 4.7).

The definitions of specific operations and functions that allow references as operands and parameters stipulate any particular limitations there are on forms of references and how empty cells, when permitted, are interpreted.

4.9 ReferenceList

A reference list contains 1 or more references, in order. A reference list can be passed as an argument to functions where passing one reference results in an identical computation as an arbitrary sequence of single references occupying the identical cell range.

Note 1: For example, SUM([.A1:.B2]) is identical to SUM([.A1]~[.B2]~[.A2]~[.B1]), but COLUMNS([.A1:.B2]), resulting in 2 columns, is not identical to COLUMNS([.A1]~[.B2]~[.A2]~[.B1]), where iterating over the reference list would result in 4 columns.

A reference list cannot be converted to an array.

Note 2: For example, in array context {ABS([.A1]~[.B2]~[.A2]~[.B1])} is an invalid expression, whereas {ABS([.A1:.B2])} is not.

Passing a reference list where a function does not expect one shall generate an Error. Passing a reference list in array iteration context to a function expecting a scalar value shall generate an Error.

4.10 Array

An array is a set of rows each with the same number of columns that contain one or more values. There is a maximum of one value per intersection of row and column. The intersection of a row and column may contain no value.

4.11 Pseudotypes

4.11.1 General

Many functions require a type or a set of types with special properties, and/or process them specially. For example, a "Database" requires headers that are the field names. These specialized types are called pseudotypes.

4.11.2 Scalar

A Scalar value is a value that has a single value. A reference to more than one cell is not a scalar (by itself), and is converted to one as described below. An array with more than one element is not a scalar. The types Number (including a complex number), Logical, and Text are scalars.

4.11.3 DateParam

A DateParam is a value that is either a Number (interpreted as a serial number; 4.3.3) or Text; text is automatically converted to a date value. 6.3.15

4.11.4 TimeParam

A TimeParam is a value that is either a Number (interpreted as a serial number; 4.3.2) or Text; text is automatically converted to a time value (fraction of a day). 6.3.16

4.11.5 Integer

An integer is a subtype of Number that has no fractional value. An integer X is equal to INT(X). Division of one integer by another integer may produce a non-integer.

4.11.6 TextOrNumber

TextOrNumber is a value that is either a Number or Text.

4.11.7 Basis

4.11.7.1 General

A basis is a subtype of Integer that specifies the day-count convention to be used in a calculation.

This standard defines five day-count conventions, corresponding to widely used current and historical accounting conventions. Each of these five bases defines two things:

1.How to calculate the number of days between two dates, date1 and date2.
2.How to calculate the number of days in each year between two dates, date1 and date2.

Historically day-count bases used the naming convention x/y, which indicated that the convention assumed x days per month and y days per year. These names are given for reference purposes.

Date Basis	Historical Name	Day Count	Days in Year
0	US (NASD) 30/360	Procedure A, 4.11.7.3	Procedure D, 4.11.7.6
1	Actual/Actual	Procedure B, 4.11.7.4	Procedure E, 4.11.7.7
2	Actual/360	Procedure B, 4.11.7.4	Procedure D, 4.11.7.6
3	Actual/365	Procedure B, 4.11.7.4	Procedure F, 4.11.7.8
4	European 30/360	Procedure C, 4.11.7.5	Procedure D, 4.11.7.6

4.11.7.2 Procedural Notation

The day-count procedures are expressed using notations defined as:

•day(date) returns the day of the month for the given date value, an integer from 1 to 31
•month(date) returns the month of a given date value, an integer from 1-12
•year(date) returns the year of the given date value
•truncate(date) truncates any fractional (hours, minutes, seconds) of a date value and returns the whole date portion.
•Binary comparison operators date1>date2 and date1 == date2
•is-leap-year(year) returns true if year is a leap year, otherwise false.

Note: Some of the day count procedures use intermediate results that contain counter-factual dates, such as February 30th. This is not an error. The above functions work on such dates as well, e.g., day(February 30th) == 30.

4.11.7.3 Procedure A

1.truncate(date1), truncate(date2)
2.If date1==date2 return 0
3.If date1> date2, then swap the values of date1 and date2.
4.If day(date1)==31 then subtract 1 day from date1
5.If day(date1)==30 and day(date2)==31 then subtract 1 day from date2
6.If both date1 and date2 are the last day of February, change date2 to the 30th of the month.
7.If date1 is the last day of February, change it to the 30th of the month.
8.Return (year(date2)*360 + month(date2)*30 + day(date2)) - (year(date1)*360 + month(date1)*30 + day(date1)).

4.11.7.4 Procedure B

1.truncate(date1), truncate(date2)
2.If date1> date2, then swap the values of date1 and date2.
3.Return the actual numbers of days between date1 and date2, inclusive of date1, but not inclusive of date2.

4.11.7.5 Procedure C

1.truncate(date1), truncate(date2)
2.If date1==date2 return 0
3.If date1> date2, then swap the values of date1 and date2.
4.If day(date1)==31 then subtract 1 from date1
5.If day(date2)==31 then subtract 1 from date 2
6.Return (year(date2)*360 + month(date2)*30 + day(date2)) - (year(date1)*360 + month(date1)*30 + day(date1)).

4.11.7.6 Procedure D

1.Return 360

4.11.7.7 Procedure E

1.Evaluate A: year(date1) != year(date2)
2.Evaluate B: year(date2)!=year(date1)+1
3.Evaluate C: month(date1) < month(date2)
4.Evaluate D: month(date1) == month(date2)
5.Evaluate E: day(date1) < day(date2)
6.Evaluate F: (A and B) or (A and C) or (A and D and E)
7.If F is true then return the average of the number of days in each year between date1 and date2, inclusive.
8.Otherwise, if A and is-leap-year(year(date1)) then return 366
9.Otherwise, if a February 29 occurs between date1 and date2 then return 366
10. Otherwise, if date2 is a February 29, then return 366
11. Otherwise return 365

4.11.7.8 Procedure F

1.Return 365

4.11.8 Criterion

A criterion is a single cell Reference, Number or Text. It is used in comparisons with cell contents.

A reference to an empty cell is interpreted as the numeric value 0.

A matching expression can be:

•A Number or Logical value. A matching cell content equals the Number or Logical value.
•A value beginning with a comparator (<, <=, >, >=). 6.4.9
For =, if the value is empty it matches empty cells. Empty cell 4.7, = 6.4.7
For <>, if the value is empty it matches non-empty cells. <> 6.4.8
For <>, if the value is not empty it matches any cell content except the value, including empty cells.
Note: "=0" does not match empty cells.
For = and <>, if the value is not empty and can not be interpreted as a Number type or one of its subtypes and the host-defined property HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL is true, comparison is against the entire cell contents, if false, comparison is against any subpart of the field that matches the criteria. For = and <>, if the value is not empty and can not be interpreted as a Number type or one of its subtypes 3.4 applies.
•Other Text value. If the host-defined property HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL is true, the comparison is against the entire cell contents, if false, comparison is against any subpart of the field that matches the criteria.

4.11.9 Database

A database is a rectangular organized set of data. Any database has a set of one or more fields that determine the structure of the database. A database has a set of zero or more records with data, and each record contains data for every field (though that field may be empty).

Evaluators shall support the use of ranges as databases if they support any database functions. The first row of a range is interpreted as a set of field names.

Note: Field names of type Text and unique without regard to case enhance the interoperability of data. It is also a common expectation that rows following the first row of data are data records that correspond to field names in the first row.

A single cell containing text can be used as a database; if it is, it is a database with a single field and no data records.

4.11.10 Field

A field is a value that selects a field in a database; it is also called a field selector. If the field selector is Text, it selects the field in the database with the same name.

Evaluators should match the database field name case-insensitively.

If a field selector is a Number, it is a positive integer and used to select the fields. Fields are numbered from left to right beginning with the number 1.

All functions that accept a field parameter shall, when evaluated, return an Error if the selected field does not exist.

4.11.11 Criteria

A criteria is a rectangular set of values, with at least one column and two rows, that selects matching records from a database. The first row lists fields against which expressions will be matched. 4.11.10 Rows after the first row contain fields with expressions for matching against database records.

For a record to be selected from a database, all of the expressions in a criteria row shall match.

A reference to an empty cell is interpreted as the numeric value 0.

•Expressions are matched as per 4.11.8 Criterion.

4.11.12 Sequences (NumberSequence, NumberSequenceList, DateSequence, LogicalSequence, and ComplexSequence)

Some functions accept a sequence, i.e., a value that is to be treated as a sequential series of values. The following are sequences: NumberSequence, NumberSequenceList, DateSequence, LogicalSequence, and ComplexSequence.

When evaluating a function that accepts a sequence, the evaluator shall follow the rules for that sequence as defined in section 6.3. When processing a ReferenceList, the references are processed in order (first, second if any, and so on). In a cuboid, the first sheet is first processed, followed by later sheets (if any) in order. Inside a sheet, it is implementation-defined as to whether the values are processed row-at-a-time or column-at-a-time, but it shall be one of these two processing orders. If processing row-at-a-time, the sequence shall be produced by processing each row in turn, from smallest to largest column value (e.g., A1, B1, C1). If processing column-at-a-time, the sequence shall be produced by processing each column at a time, from the smallest to the largest row value (e.g., A1, A2, A3).

4.11.13 Any

Any represents a value of any type defined in this standard, including Error values.

5 Expression Syntax

5.1 General

The OpenFormula syntax is defined using the BNF notation of the XML specification, chapter 6 [XML1.0]. Each syntax rule is defined using "::=".

Note: Formulas are typically embedded inside an XML document. When this occurs, characters (such as "<", ">", '"', and "&") shall be escaped, as described in section 2.4 of the XML specification [XML1.0]. In particular, the less-than symbol "<" is typically represented as “<”, the double-quote symbol as “"”, and the ampersand symbol as “&” (alternatively, a numeric character reference can be used).

5.2 Basic Expressions

Formulas may start with a '=' (EQUALS SIGN, U+003D), which if present may be followed by a “forced recalculate” marker '=' (EQUALS SIGN, U+003D), followed by an expression. If the second '=' (EQUALS SIGN, U+003D) is present, this formula is a "forced recalculation" formula. If a formula is marked as a "forced recalculation" formula, then it should be recalculated whenever one of its predecessors it depends on changes.

Expressed as a grammar in BNF notation, a formula is specified:

Formula ::= Intro? Expression

Intro ::= '=' ForceRecalc?

ForceRecalc ::= '='

The primary component of a formula is an Expression. Formulas are composed of Expressions, which may in turn be composed from other Expressions.

PrefixOp Expression |

Expression PostfixOp |

Expression InfixOp Expression |

'(' Expression ')' |

FunctionName Whitespace* '(' ParameterList ')' |

Reference |

QuotedLabel |

AutomaticIntersection |

NamedExpression |

Error

) Whitespace*

SingleQuoted ::= "'" ([^'] | "''")+ "'"

5.3 Constant Numbers

Constant numbers are written using '.' (FULL STOP, U+002E) dot as the decimal separator. Optional "E" or "e" denotes scientific notation. Syntactically, negative numbers are positive numbers with a prefix "-" (HYPHEN-MINUS, U+002D) operator. A constant number is of type Number.

Number ::= StandardNumber |

'.' [0-9]+ ([eE] [-+]? [0-9]+)?

StandardNumber ::= [0-9]+ ('.' [0-9]+)? ([eE] [-+]? [0-9]+)?

Evaluators should be able to read the Number format, which accepts a decimal fraction that starts with decimal point '.' (FULL STOP, U+002E), without a leading zero. Evaluators shall write numbers only using the StandardNumber format, which requires a leading digit, and shall not write numbers with a leading '.' (FULL STOP, U+002E).

5.4 Constant Strings

Constant strings are surrounded by double-quote characters (QUOTATION MARK, U+0022); a literal double-quote character '"' (QUOTATION MARK, U+0022) as string content is escaped by duplicating it. A constant string is of type Text.

String ::= '"' ([^"#x00] | '""')* '"'

5.5 Operators

Operators are functions with one or more parameters.

PrefixOp ::= '+' | '-'

PostfixOp ::= '%'

InfixOp ::= ArithmeticOp | ComparisonOp | StringOp | ReferenceOp

ArithmeticOp ::= '+' | '-' | '*' | '/' | '^'

ComparisonOp ::= '=' | '<>' | '<' | '>' | '<=' | '>='

StringOp ::= '&'

There are three predefined reference operators: reference intersection, reference concatenation, and range. The result of these operators may be a 3-dimensional range, with front-upper-left and back-lower-right corners, or even a list of such ranges in the case of cell concatenation.

ReferenceOp ::= IntersectionOp | ReferenceConcatenationOp | RangeOp

IntersectionOp ::= '!'

ReferenceConcatenationOp ::= '~'

RangeOp ::= ':'

Table 1 - Operators defines the associativity and precedence of operators, from highest to lowest precedence.

Table 1 - Operators

Associativity	Operator(s)	Comments
left	:	Range.
left	!	Reference intersection ([.A1:.C4]![.B1:.B5] is [.B1:.B4]). Displayed as the space character in some implementations.
left	~	Reference union. Note: Displayed as the function parameter separator in some implementations.
right	+,-	Prefix unary operators, e.g., -5 or -[.A1]. Note that these have a different precedence than add and subtract.
left	%	Postfix unary operator % (divide by 100). Note that this is legal with expressions (e.g., [.B1]%).
left	^	Power (2 ^ 3 is 8).
left	*,/	Multiply, divide.
left	+,-	Binary operations add, subtract. Note that unary (prefix) + and - have a different precedence.
left	&	Binary operation string concatenation. Note that unary (prefix) + and - have a different precedence. Note that "&" shall be escaped when included in an XML document
left	=, <>, <, <=, >, >=	Comparison operators equal to, not equal to, less than, less than or equal to, greater than, greater than or equal to

Note 1: Prefix “-” has a higher precedence than “^”, “^” is left-associative, and reference intersection has a higher precedence than reference union.

Note 2: Prefix “+” and “-“ are defined to be right-associative. However, note that typical applications which implement at most the operators defined in this specification (as specified) may implement them as left-associative, because the calculated results will be identical.

Note 3: Precedence can be overridden by using parentheses, so "=2+3*4" computes to 14 but "=(2+3)*4" computes 20. Implementations should retain "unnecessary" parentheses and white space, since these are added by people to improve readability.

5.6 Functions and Function Parameters

Functions are called by name, followed by parentheses surrounding a list of parameters. Parameters are separated using the semicolon ';' (SEMICOLON, U+003B) character:

FunctionName ::= LetterXML (LetterXML | DigitXML |

'_' | '.' | CombiningCharXML)*

Where LetterXML, DigitXML, and CombiningCharXML are Letter, Digit, and CombiningChar as they are defined in [XML1.0].

Function names are case-insensitive.

Function calls shall be given a parameter list, though it may be empty. An empty list of parameters is considered a call with 0 parameters, not a call with one parameter that happens to be empty. TRUE() is syntactically a function call with 0 parameters. It is syntactically legitimate to provide empty parameters, though functions need not accept empty parameters unless otherwise noted:

ParameterList ::= /* empty */ |

Parameter (Separator EmptyOrParameter )* |

Separator EmptyOrParameter /* First param empty */

(Separator EmptyOrParameter )*

EmptyOrParameter ::= /* empty */ Whitespace* | Parameter

Parameter ::= Expression

Separator ::= ';'

5.7 Nonstandard Function Names

When writing a document using function(s) not defined in this specification, an evaluator shall include a prefix in such function names to identify the original definer of the function's semantics. When the origin of a function cannot be determined, producers may omit a prefix. Producers may use the prefix to differentiate between different definition types. Evaluators that do not support a function should compute its result as some Error value other than #N/A.

Note: Examples of implementation-defined functions include extension functions included with an implementation, user-defined functions written by users, and 3rd party functions distributed in libraries.

Note: Examples of such names include COM.MICROSOFT.CUBEMEMBER, ORG.OPENOFFICE.STYLE, ORG.GNUMERIC.RANDRAYLEIGH, and COM.LOTUS.V98.FOO.

Evaluators should avoid defining evaluator-unique functions beginning with a top-level domain name followed by a period. Evaluators should avoid defining application functions beginning with “NET.”, “COM.”, “ORG.”, or a country code followed by a period.

Evaluators that do not support a function should compute its result as some Error value other than #N/A.

5.8 References

References refer to a specific cell or set of cells. The syntax for a constant reference:

Reference ::= '[' (Source? RangeAddress) | ReferenceError ']'

RangeAddress ::=
SheetLocatorOrEmpty '.' Column Row (':' '.' Column Row )? |
SheetLocatorOrEmpty '.' Column ':' '.' Column |
SheetLocatorOrEmpty '.' Row ':' '.' Row |
SheetLocator '.' Column Row ':' SheetLocator '.' Column Row |
SheetLocator '.' Column ':' SheetLocator '.' Column |
SheetLocator '.' Row ':' SheetLocator '.' Row

SheetLocatorOrEmpty ::= SheetLocator | /* empty */

SheetLocator ::= SheetName ('.' SubtableCell)*

SheetName ::= QuotedSheetName | '$'? [^\]\. #$']+

QuotedSheetName ::= '$'? SingleQuoted

SubtableCell ::= ( Column Row ) | QuotedSheetName

ReferenceError ::= "#REF!"

Column ::= '$'? [A-Z]+

Row ::= '$'? [1-9] [0-9]*

Source ::= "'" IRI "'" "#"

CellAddress ::= SheetLocatorOrEmpty '.' Column Row /* Not used directly */

References always begin with '[' (LEFT SQUARE BRACKET, U+005B); this disambiguates cell addresses from function names and named expressions. SheetNames include single-quote“'” (APOSTROPHE, U+0027) characters by doubling them and having the entire name surrounded by single-quotes. Column labels shall be in uppercase. The syntax supports whole-row and whole-column references. A reference is of type Reference.

A ReferenceError provides information that a formula evaluates to an Error because of a particular reference having been invalidated by actions that occurred after the formula was validly created.

Columns are named by a sequence of one or more uppercase letters A-Z (U+0041 through U+005A). Columns are named A, B, C, ... X, Y, Z, AA, AB, AC, ... AY, AZ, BA, BB, BC, ... ZX, ZY, ZZ, AAA, AAB, AAC, AAZ, ABA, ABB, and so on.

If a RangeAddress does not contain a Column element or does not contain a Row element, it specifies a cell rectangle (4.8 Reference). If it contains Row elements, the cell rectangle starts on the first column and ends on the last column the evaluator supports. If it contains Column elements, the cell rectangle starts on the first row and ends on the last row the evaluator supports.

If in a RangeAddress the first part (left of ':' colon) contains a SheetLocator and the second part (right of ':' colon) does not contain a SheetLocator, the second part inherits the SheetLocator from the first part.

If a RangeAddress contains two different SheetLocators, it specifies a cell cuboid (4.8 Reference).

If a RangeAddress contains no SheetLocator, the current sheet local to the position where the expression is evaluated is referred.

A reference with an explicit row or column value beyond the capabilities of an evaluator shall be computed as an Error, and not as a reference.

Note that references can include a single embedded “:” separator. Evaluators should use references with embedded “:” separators inside the [..] markers, instead of the general-purpose “:” operator, when saving files, and, where there is a choice of cells to join, evaluators should choose the leftmost pair.

The optional Source expresses that the reference is to sheets and/or cells in a different location (possibly in a same-document fragment) from that for the formula in which the reference occurs. The optional Source is also used for locating Named Expressions (section 5.11).

The IRI portion of Source shall be an IRI reference [RFC3987] conforming to the general syntax IRI-reference rule (section 2.2 of [RFC3987]) after each pair of consecutive single-quote characters (APOSTROPHE, U+0027) is replaced by one single single-quote character.

Note: The escaping of single-quotes as paired single-quotes is because the IRI is enclosed in single quote characters of the Source.

Resolution of the [RFC3987] IRI reference is host-defined behavior. 3.4

5.9 Reference List

A reference list is the result of the Infix Operator Reference Concatenation 6.4.13 '~', the syntax is:

ReferenceList ::= Reference (Whitespace* ReferenceConcatenationOp Whitespace* Reference)*

A reference list can be passed as an argument to functions expecting a reference parameter where passing one reference results in an identical computation as an arbitrary sequence of single references occupying the identical cell range. A reference list cannot be converted to an array.

5.10 Quoted Label

5.10.1 General

A quoted label is Text contained in a table as cell content, either literally or as a formula result.

QuotedLabel ::= SingleQuoted

A quoted label identifies a column or a row, depending on the label range in which its text appears.

5.10.2 Lookup of Defined Labels

For a QuotedLabel, first the cells defined in column label ranges (cell ranges of the table:label-cell-range-address attribute (OpenDocument Part 3, 19.660) in the elements <table:label-range> (OpenDocument Part 3, 9.4.9) with attribute table:orientation (OpenDocument Part 3, 19.690.4) set to column) are searched for the string content of QuotedLabel (without the quotes). If found, the corresponding column's range of the data cell range of the table:data-cell-range-address attribute (OpenDocument Part 3, 19.612) is taken as a reference. If not found, the cells defined in row label ranges (attribute table:orientation set to row) are searched and if found the corresponding row's range of the data cell range is taken. Label ranges of the current formula's sheet take precedence over label ranges of other sheets if a name occurs in both.

5.10.3 Automatic Lookup of Labels

For a QuotedLabel where no defined label is found, an automatic lookup is performed on the sheet where the formula cell resides, if that document setting is enabled (HOST-AUTOMATIC-FIND-LABELS value true ).

Matches to the upper left of the formula cell are preferred over other matches, followed by matches with a smaller distance. The following algorithm is used:

Cells on the same sheet as the formula cell are examined column-wise from left to right whether they contain the text of QuotedLabel (without the quotes). If more than one cell match, the distance and direction from the formula cell's position is taken into account. The distance is calculated by Distance= ColumnDifference*ColumnDifference+ RowDifference*Row Difference using an idealized layout of square cells. For the direction, during the run two independent match positions are remembered each time Distance is smaller than a previous Distance: Match2 for positions right of and/or below the formula position (FormulaColumn < MatchColumn || FormulaRow < MatchRow), Match1 for all others (not right of and not below). Match1 also holds the very first match, in case there is only one match or all matches are somewhere below or right of the formula cell. After having found the smallest distances the conditions are:

1.If Match1 has the smallest distance, that match is taken.
2.Else, Match2 (right and/or below) has the smallest or an equal distance:
1. 2.1 A match to the upper left (FormulaColumn >= Match1Column && FormulaRow >= Match1Row) takes precedence over matches to other directions.
2. 2.2 Else, if there is no match to the upper left:
  1. 2.2.1 If Match1 is somewhere right of the formula cell (FormulaColumn < Match1Column) it was the first match found in column-wise lookup.
    1. 2.2.1.1 If Match2 is above the formula cell (FormulaRow >= Match2Row) it is to the upper right of the formula cell and either nearer than Match1 or Match1 is below. Match2 is taken.
    2. 2.2.1.2 Else Match2 is below the formula cell and Match1 is taken.
  2. 2.2.2 Else (Match1 not right of the formula cell => two matches below or below and right) the match with the smallest distance is taken.

If the resulting cell is below or above another cell containing Text a row label is assumed, else a column label is assumed.

Note: Use of automatically looked up column or row labels in expressions impairs interoperability.

5.10.4 Implicit Intersection

For the reference resulting from a single QuotedLabel an implicit intersection is generated if the operator or function with which it is used expects a scalar value. The intersection is generated with the current formula's cell position, thus for a column label an intersection is generated with the formula cell's row, for a row label with the formula cell's column.

5.10.5 Automatic Range

When passed as a non-scalar argument (e.g. Array or NumberSequence) to a function, an automatically-looked-up column or row label (not defined label range) is converted to an automatic range reference that is adjusted each time the formula is interpreted. The range is generated from the column below a column label, or the row to the right of a row label, constructed by encompassing contiguous non-empty cells. An empty cell interrupts contiguousness, one empty cell directly below a column label cell or to the right of a row label cell is ignored.

Example:

Table 2 - Automatic Range

Row

Data

Expression

Result

Comment

1	Label
2
3	1
4	2
5
6	8
7
8	32

=SUM('Label')

Empty cell in row 2 is skipped (two empty cells in row 2 and 3 would not be skipped and would stop the automatic range), empty cell in row 5 stops the automatic range.

If any cell content is entered in row 5 the range is regenerated as follows:

Table 3 - Automatic Range

Row

Data

Expression

Result

Comment

1	Label
2
3	1
4	2
5	4
6	8
7
8	32

=SUM('Label')

Empty cell in row 2 is skipped, empty cell in row 7 stops the automatic range.

5.10.6 Automatic Intersection

An automatic intersection may be used to identify the intersection of two quoted labels. Note that this is different from the IntersectionOp, which takes two references instead of two labels:

AutomaticIntersection ::= QuotedLabel Whitespace* '!!' Whitespace* QuotedLabel

In an automatic intersection, one of the labels identifies a row, the other a column; they may be in either order. Each QuotedLabel is looked up as defined above under "Lookup of Defined Labels" and "Automatic Lookup of Labels". If two data cell ranges are found, the intersection of the column's data cell range and the row's data cell range is generated. If the intersection result is not exactly one cell, an Error is generated.

5.11 Named Expressions

A NamedExpression references another expression, possibly in a completely different spreadsheet or any other type of document that can be imported into a spreadsheet.

NamedExpression ::= SimpleNamedExpression |
SheetLocalNamedExpression | ExternalNamedExpression

SimpleNamedExpression ::= Identifier |
'$$' (Identifier | SingleQuoted)

SheetLocalNamedExpression ::=
QuotedSheetName '.' SimpleNamedExpression

ExternalNamedExpression ::=
Source (SimpleNamedExpression | SheetLocalNamedExpression)

Evaluators supporting named expressions shall support Simple Named Expressions that are global to all the sheets in a (spreadsheet) document in the current document. This is a named expression without a Source, QuotedSheetName, or SubtableCell. The type of a named expression is the type of the value that the named expression returns.

Named expressions are case-consistent, meaning that matching is done case-insensitive and identifiers can not differ ONLY in their case. Evaluators should write identifiers with identical case in all locations.

Evaluators may support Sheet-local Named Expressions that are local (attached) to individual sheets. In that case, a non-empty QuotedSheetName can be used to reference a sheet-specific named expression. The most specific named expression for a given expression is used. If the QuotedSheetName is empty, the search for the named expression begins with the current sheet, then up through the container(s) of the sheet (the same is true if the QuotedSheetName rule fragment is not included at all). If there is a non-empty QuotedSheetName, search begins with that named sheet, then up through its container(s) for the given name.

Note: There is no syntax for referencing a named expression without first looking at the current sheet's named expressions; where this is a problem, a user can define a blank sheet and reference that sheet as the starting location for finding the named expression.

If a sheetname is not empty, it shall be quoted using the single-quote character “'” (APOSTROPHE, U+0027). While both Source and QuotedSheetName can begin with the single-quote character, they are distinguished: after the closing single-quote character, a non-empty source shall have the '#' (NUMBER SIGN, U+0023) character as the next non-whitespace character; a non-empty sheetname shall be followed by the '.' (FULL STOP, U+002E) character as the next non-whitespace character.

Expressions should limit the names of their identifiers to only ([UNICODE]) letters, underscores, and digits, not including patterns that look like cell references or the words True or False.

Identifier ::= ( LetterXML

(LetterXML | DigitXML | '_' | CombiningCharXML)* )
- ( [A-Za-z]+[0-9]+ )
- ([Tt][Rr][Uu][Ee]) - ([Ff][Aa][Ll][Ss][Ee])

5.12 Constant Errors

Evaluators shall support the Error named #N/A. Evaluators may support other Errors. Evaluators may allow entry of errors directly, parse them and recognize them as Errors. Functions shall propagate Errors unless stated otherwise.

Error names shall have the following syntax:

Error ::= '#' [A-Z0-9]+ ([!?] | ('/' ([A-Z] | ([0-9] [!?]))))

Specific Errors are indicated by their corresponding names.

Table 4 is a list of Errors that are used by several existing implementations.

Table 4 - Possible Other Constant Error Names

Name	Comments
#DIV/0!	Attempt to divide by zero, including division by an empty cell. ERROR.TYPE of 2 6.13.11
#NAME?	Unrecognized/deleted name. ERROR.TYPE of 5.
#N/A	Not available. ISNA() applied to this value will return TRUE. Lookup functions which failed, and NA(), return this value. ERROR.TYPE of 7.
#NULL!	Intersection of ranges produced zero cells. ERROR.TYPE of 1.
#NUM!	Failed to meet domain constraints (e.g., input was too large or too small). ERROR.TYPE of 6.
#REF!	Reference to invalid cell (e.g., beyond the application’s abilities). ERROR.TYPE of 4.
#VALUE!	Parameter is wrong type. ERROR.TYPE of 3.

Evaluators may implement other Errors.

An unknown Error name shall be mapped into an Error supported by the evaluator when read (e.g., the application's equivalent of #NAME?), though an evaluator may warn the user if this has or will take place. It is desirable to preserve the original specific Error name when writing an Error back out, where possible, but for Errors other than #N/A evaluators may write a different Error for a formula than they did when reading it. Whitespace shall not be included in an Error name.

Evaluators should use a human-comprehensible name, not a numeric id, for Error names they write.

5.13 Inline Arrays

Inline arrays are enclosed with curly braces. Inside, they contain one or more rows, with each row separated by a row separator:

Array ::= '{' MatrixRow ( RowSeparator MatrixRow )* '}'

MatrixRow ::= Expression ( ';' Expression )*

RowSeparator ::= '|'

Evaluators that support inline arrays shall accept a matrix with one or more rows, each with one or more columns, with the same number of columns in each row, with constant values for each expression. Evaluators that do not support inline arrays, or cannot support a particular use permitted by this syntax, should compute an Error value for such arrays. An inline array is of type Array.

Note: Expression authors should be aware that use of Expression other than constant Number or constant String may impair interoperability.

5.14 Whitespace

Whitespace ::= #x20 | #x09 | #x0a | #x0d

For calculation purposes, whitespace is ignored unless it is inside the contents of string constants or text surrounded by single quotes. Evaluators shall ignore any whitespace characters before and/or after any operators, constant numbers, constant strings, constant errors, inline arrays, parentheses used for controlling precedence, and the closing parenthesis of a function call. Whitespace shall be ignored following the initial equal sign(s). Whitespace shall be ignored just before a function name, but whitespace shall not separate a function name from its initial opening parenthesis. Whitespace shall not be used in the interior of a terminating grammar rule (a rule that references no other rule other than character sets, internally or externally-defined), unless specifically permitted by the terminating grammar rule, since these rules define the lexical properties of a component. Evaluators shall not write formulas with whitespace embedded in any unquoted identifier, constant Number, or constant Error. Evaluators shall treat SPACE (U+0020), CHARACTER TABULATION (U+0009), LINE FEED (U+000A), and CARRIAGE RETURN (U+000D) as whitespace characters.

An embedded line break shall be represented by a single LINE FEED character (U+000A), not by a carriage return-linefeed pair. When embedded in an XML attribute the linefeed character is represented as “
”.

Evaluators should retain whitespace entered by the original formula creator and use it when saving or presenting the formula, and should not add additional whitespace unless directed to do so during the process of editing a formula.

6 Standard Operators and Functions

6.1 General

OpenFormula defines commonly used operators and functions.

Function names ignore case. Evaluators should write function names in all uppercase letters when writing OpenFormula formulas.

Unless otherwise noted, if any value being provided is an Error, the result is an Error; if more than one Error is provided, one of them is returned (evaluators should return the leftmost Error result).

6.2 Common Template for Functions and Operators

For every function or operator, the following are defined in this specification:

•Name: The function/operator name.
•Summary: One sentence briefly describing the function or operator.
•Syntax:
- –Parameter names are shown in order, with each parameter prefixed by the type or pseudotype of that parameter. If the type has multiple names separated by “|”, then any of those types are permitted.
- –A { ... } indicates a list of zero or more parameters, separated by the function parameter separator character.
- –A { ... } followed by a superscripted + indicates a list of one or more parameters, separated by the function parameter separator character.
- –Components surrounded by [ ... ] are optional. Optional components may be omitted.
- –An optional parameter followed by the = symbol has the default value given after the equal sign.
- –Parameters are separated with a semicolon (";"), as per the OpenFormula expression syntax 5.6.
  When a function is given a value of a different type, the parameters are first converted using the implicit conversion rules before the function operates on its parameters.

Evaluators may extend functions by permitting fewer or additional parameters, which documents may use. Extended functions may result in a lack of interoperability.

•Returns: Return type (e.g., Number, Text, Logical, Reference).
•Constraints: A description of constraints, in addition to the constraints imposed by the parameter types. If there are no additional constraints beyond those imposed by the parameter types, this is "None". If a constraint is not met, the function/operator shall return an Error unless otherwise noted.
•Semantics: This text describes what the function/operator does.
If a parameter is a pseudotype, but the provided value fails to meet the requirements for that type, the behavior is implementation-defined.
Note: Functions and operators are defined by mathematical formulas or by an OpenFormula formula. Formulas define the correct result, and not the algorithm for calculation. Since computing systems have limited precision and range of numbers, some functions cannot or should not be naively implemented as their formulas suggest. This specification defines the mathematically correct answer, and allows implementors to choose the best algorithm that will meet that definition.
•Comment: Explanatory comment.
•See also A list of related operators and functions.

The implicit conversion operators omit many of these items, e.g., the syntax (since there is none).

6.3 Implicit Conversion Operators

6.3.1 General

Any given function or operand takes 0 or more parameters, and each of those parameters has an expected type. The expected type can be one of the base types, identified above. It can also be of some conversion type that controls conversion, e.g., Any means that no conversion is done (it can be of any type); NumberSequence causes a conversion to an ordered sequence of zero or more numbers. If the passed-in type does not match the expected type, an attempt is made to automatically convert the value to the expected type. An Error is returned if the type cannot be converted (this can never happen if the expected type is Any). Unless otherwise noted, any conversion operation applied to a value of type Error returns the same value.

6.3.2 Conversion to Scalar

To convert to a scalar, if the value is of type:

•Number, Logical, or Text, return the value.
•reference to a single cell: obtain the value of the referenced cell, and return that value.
•reference to more than one cell: do an implied intersection, 6.3.3, to determine which single cell to use, then handle as a reference to a single cell.

6.3.3 Implied intersection

In some cases a reference to a single cell is needed, but a reference to multiple cells is provided. In this case an "implied intersection" is performed. To perform an implied intersection:

•Compute the union of cells contained in the current row and current column of the formula being computed.
•Intersect this with the provided reference to multiple cells
•If a single cell is referenced; return it; otherwise, return an Error.

6.3.4 Force to array context (ForceArray)

A ForceArray attribute forces calculation of the argument's expression into non-scalar array mode. This means that no implied intersection is performed, instead where a reference to a single cell is expected and multiple cells are provided, iteration over the multiple cells is performed and results are stored in an array that is passed on.

6.3.5 Conversion to Number

If the expected type is Number, then if the value is of type:

•Number, return it.
•Logical, return 0 if FALSE, 1 if TRUE.
•Text: The specific conversion is implementation-defined; an evaluator may return 0, an Error value, or the results of its attempt to convert the Text value to a Number (and fall back to 0 or Error if it fails to do so). Evaluators may apply VALUE 6.13.34 or some other function to do this conversion, should they choose to do so. Conversion depends on the actual locale the application runs in, especially if group or decimal separators are involved.
•Reference: If the reference covers more than one cell, do an implied intersection to determine which cell to use. Then obtain the value of the single cell and perform the rules as above. If the calculation setting “precision-as-shown” is true, then convert the number to the closest possible representation of the displayed number. If the cell is empty (blank), use 0 (zero) as the value. Evaluators may choose to convert references to Text in a different manner than they handle converting embedded Text to a Number.

6.3.6 Conversion to Integer

If the expected type is Integer for a function or operator, apply the “Conversion to Number” operation. 6.3.5 Then, if the result is a Number but not an integer, apply the specific conversion from Number to integer specified by that particular function/operator. If the function or operator does not specify any particular conversion operation, then the conversion from a non-integer Number into an integer is implementation-defined.

Many different conversions from a non-integer number into an integer are possible. The conversion direction may be towards negative infinity, towards positive infinity, towards zero, away from zero, towards the nearest even number, or towards the nearest odd number. A conversion can select the nearest integer, the nearest even or odd integer, or the “next” integer in the given direction if it is not already an integer. If a conversion selects the nearest integer, a direction is still needed (for when a value is halfway between two integers). In this specification, this conversion is referred to as “rounding” or “truncation”; these terms by themselves do not specify any specific operation.

If a function specifies its rounding operation using a series of capital letters, the function defined in this specification for that function is used to do the conversion to integer. Common such functions are:

•INT, which if given non-integer rounds down to the next integer towards negative infinity, regardless of whether or not it is the closest integer.
•ROUND, which if given non-integer rounds to the nearest integer. If the input number is halfway between integers, it rounds away from zero.
•TRUNC, which if given non-integer rounds towards zero, regardless of whether or not that integer is the closest integer.

6.3.7 Conversion to NumberSequence

If the expected type is NumberSequence, then if value is of type:

•Number, Text, or Logical, handle as Conversion to Number 6.3.5 (creating a sequence of length 1).
•reference, create a sequence of numbers from the values of the referenced cells that only includes the values of type Number or Error. Thus, Empty cells and Text that could be converted into a value are not included in a number sequence. If the Logical type is a distinguished type from the Number type, it should not be included in the sequence of numbers; if the Logical type is not a distinguished type, then such values will (of course) be included in the number sequence.

6.3.8 Conversion to NumberSequenceList

Identical to Conversion to NumberSequence 6.3.7, with the addition that instead of a Reference also a ReferenceList is accepted as argument. Each Reference in the list is converted to a NumberSequence in the order of occurrence.

6.3.9 Conversion to DateSequence

Identical to Conversion to NumberSequence 6.3.7 except that each element in the list represents a serial date value of subtype Date.

6.3.10 Conversion to Complex Number

An evaluator may accept complex numbers as Text, Number, or a different distinguishable type.

If the value is:

•Number that is not complex, use the Number with 0 as the imaginary part.
•Text, attempt to convert to complex number using VALUE 6.13.34. If it is a number that is not complex, use it. If the text matches one of these patterns, accept it:

([+-]?Number [+-])?Number[ij]

[+-]?Number[ij]

•Logical, convert to Number and then handle as Number.
•reference: Convert to Scalar 6.3.2, then use the rules above. If the reference is to an empty cell, consider it equal to 0.

6.3.11 Conversion to ComplexSequence

If the expected type is ComplexSequence, then if value is of type:

•Number, Text, or Logical, handle as Conversion to Complex Number (creating a sequence of length 1).
•Reference, create a sequence of complex numbers from the values of the referenced cells that only includes the values of type Number, Text, and Error. Empty cells are not included in a complex number sequence. If the Logical type is a distinguished type from the Number type, it should not be included in the sequence of numbers; if the Logical type is not a distinguished type, then such values will (of course) be included in the number sequence.

6.3.12 Conversion to Logical

If the expected type is Logical, then if value is of type:

•Number, return TRUE for nonzero and FALSE for 0.
•Text: The specific conversion is implementation-defined; an evaluator may return FALSE, an Error value, or the results of its attempt to convert the Text value (ignoring case) to a Logical value (and fall back to FALSE or Error if it fails to do so). Conversion depends on the actual locale the evaluator runs in.
•Logical, return it.
•Reference, convert to scalar and then perform as above. If the reference is to an empty cell, consider it FALSE.

6.3.13 Conversion to LogicalSequence

If the expected type is LogicalSequence, then if value is of type:

•Number or Logical, handle as Conversion to Logical (creating a sequence of length 1).
•Reference, create a sequence of logical values from the values of the referenced cells that only includes the values of type Logical and Error. If the Logical type is not a distinguished type, then include values of type Number, converting each to a Logical value as described in Conversion to Logical. Empty cells are not included in a LogicalSequence.

6.3.14 Conversion to Text

If the expected type is Text, then if value is of type:

•Number, transform into Text (with no whitespace).
•Text, return it.
•Logical, return "TRUE" if it is true and "FALSE" if it is false.
•Reference: perform conversion to scalar. If the referenced cell is empty, treat as an empty string (a text value with length 0). Then perform as above.

6.3.15 Conversion to DateParam

If the expected type is the pseudotype DateParam, then if value is of type:

•Number, return it.
•Text, pass to DATEVALUE 6.10.4, and if non-Error, return it. If DATEVALUE would return an Error, an evaluator may attempt to convert to a Number in other ways (such as by calling VALUE 6.13.34); this is implementation-defined. If the evaluator cannot convert to Number, it returns an Error.
•Logical, the result is implementation-defined, either a Number or Error
•Reference: perform conversion to scalar, then perform as above. If the cell is empty, return 0.

6.3.16 Conversion to TimeParam

If the expected type is the pseudotype TimeParam, then if value is of type:

•Number, return it.
•Text, pass to TIMEVALUE 6.10.18, and if non-Error, return it. If TIMEVALUE would return an Error, an evaluator may attempt to convert to a Number in other ways (such as by calling VALUE 6.13.34); this is implementation-defined. If the evaluator cannot convert to Number, it returns an Error.
•Logical, the result is implementation-defined, either a Number or Error
•Reference: perform conversion to scalar, then perform as above. If the cell is empty, return 0.

6.4 Standard Operators

6.4.1 General

The functions defined under standard operators differ from other functions only by their frequency of use. That frequency of use has lead to the colloquial terminology, standard operators.

Format	Returns the number of
y	Years
m	Months. If there is not a complete month between the dates, 0 will be returned.
d	Days
md	Days, ignoring months and years
ym	Months, ignoring years
yd	Days, ignoring years

Weekday Type	1	2	3	11	12	13	14	15	16	17
Sunday	1	7	6	7	6	5	4	3	2	1
Monday	2	1	0	1	7	6	5	4	3	2
Tuesday	3	2	1	2	1	7	6	5	4	3
Wednesday	4	3	2	3	2	1	7	6	5	4
Thursday	5	4	3	4	3	2	1	7	6	5
Friday	6	5	4	5	4	3	2	1	7	6
Saturday	7	6	5	6	5	4	3	2	1	7

Mode	Effect
0 or missing	Data converted to number using VALUE in the number style's locale of the default table cell style
1	Data converted to number using VALUE in the English-US (en_US) locale
2	Data retrieved as text (not converted to number)

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly
12	Monthly

Info_Type	Comment
COL	Returns the column number of the cell.
ROW	Returns the row number of the cell.
SHEET	Returns the sheet number of the cell.
ADDRESS	Returns the absolute address of the cell. The sheet name is included if given in the reference and does not reference the same sheet as the sheet the expression is evaluated upon. For an external reference a Source as specified in the syntax rules for References 5.8 is included.
FILENAME	Returns the file name of the file that contains the cell as an IRI. If the file is newly created and has not yet been saved, the file name is empty text “”.
CONTENTS	Returns the contents of the cell, without formatting properties.
COLOR	Returns 1 if color formatting is set for negative value in this cell; otherwise returns 0
FORMAT	Returns a text string which shows the number format of the cell. ,(comma) = number with thousands separator F = number without thousands separator C = currency format S = exponential representation P = percentage To indicate the number of decimal places after the decimal separator, a number is given right after the above characters. D1 = MMM-D-YY, MM-D-YY and similar formats D2 = DD-MM D3 = MM-YY D4 = DD-MM-YYYY HH:MM:SS D5 = MM-DD D6 = HH:MM:SS AM/PM D7 = HH:MM AM/PM D8 = HH:MM:SS D9 = HH:MM G = All other formats - (Minus) at the end = negative numbers in the cell have color setting () (brackets) at the end = this cell has the format settings with parentheses for positive or all values
TYPE	Returns the text value corresponding to the type of content in the cell: “b” : blank or empty cell content “l” : label or text cell content “v” : number value cell content
WIDTH	Returns the column width of the cell. The unit is the width of one zero (0) character in default font size.
PROTECT	Returns the protection status of the cell: 1 = cell is protected 0 = cell is unprotected
PARENTHESES	Returns 1 if the cell has the format settings with parentheses for positive or all values, otherwise returns 0
PREFIX	Returns single character text strings corresponding to the alignment of the cell. “'” (APOSTROPHE, U+0027) = left alignment '"' (QUOTATION MARK, U+0022) = right alignment “^” (CIRCUMFLEX ACCENT, U+005E) = centered alignment “\” (REVERSE SOLIDUS, U+005C) = filled alignment otherwise, returns empty string "".

Category	Meaning	Type
"directory"	Current directory. This shall be formatted so file names can be appended to the result (e.g., on POSIX and Windows systems it shall end with the separator “/” or “\” respectively).	Text
"memavail"	Amount of memory “available”, in bytes. On many modern (virtual memory) systems this value is not really available, but a system should return 0 if it is known that there is no more memory available, and greater than 0 otherwise	Number
"memused"	Amount of memory used, in bytes, by the data	Number
"numfile"	Number of active worksheets in files	Number
"osversion"	Operating system version	Text
"origin"	The top leftmost visible cell's absolute reference prefixed with “$A:”. In locales where cells are ordered right-to-left, the top rightmost visible cell is used instead.	Text
"recalc"	Current recalculation mode. If the locale is English, this is either "Automatic" or "Manual" (the exact text depends on the locale)	Text
"release"	The version of the implementation.	Text
"system"	The type of the operating system.	Text
"totmem"	Total memory available in bytes, including the memory already used.	Number

Format	Example	Comment
MM/DD/YYYY	5/21/2006	LOCALE-DEPENDENT; Long year format with slashes.
MM/DD/YY	5/21/06	LOCALE-DEPENDENT; Short year format with slashes
MM-DD-YYYY	5-21-2006	LOCALE-DEPENDENT; Long year format with dashes (short year may be supported, but it may also be used for years less than 100 .
mmm DD, YYYY	Oct 29, 2006	LOCALE-DEPENDENT; Short alphabetic month day, year. Note: mmm depends on the locale's language.
DD mmm YYYY	29 Oct 2006	LOCALE-DEPENDENT; Short alphabetic day month year
mmmmm DD, YYYY	October 29, 2006	LOCALE-DEPENDENT; Long alphabetic month day, year
DD mmmmm YYYY	29 October 2006	LOCALE-DEPENDENT; Long alphabetic day month year

Abs	Meaning	A1Style = TRUE	A1Style = FALSE
1	fully absolute	$A$1	R1C1
2	row absolute, column relative	A$1	R1C[1]
3	row relative, column absolute	$A1	R[1]C1
4	fully relative	A1	R[1]C[1]

	col_B	col_C	col_D	col_E	col_F
row_2	1		1	2	3
row_3	1	1	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;F$2)
row_4	=B2+B3	2	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;F$2)
row_5	=B2*B3+B4	3	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;F$2)
		4	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;F$2)

Unit Prefix	Description	Prefix Value
"Y"	yotta	1E+24
"Z"	zetta	1E+21
"E"	exa	1E+18
"P"	peta	1E+15
"T"	tera	1E+12
"G"	giga	1E+09
"M"	mega	1E+06
"k"	kilo	1E+03
"h"	hecto	1E+02
"da" or "e"	deka ( Note: “e” is not a standard SI prefix	1E+01
"d"	deci	1E-01
"c"	centi	1E-02
"m"	milli	1E-03
"u"	micro Note: this is “u”, not the standard SI µ	1E-06
"n"	nano	1E-09
"p"	pico	1E-12
"f"	femto	1E-15
"a"	atto	1E-18
"z"	zepto	1E-21
"y"	yocto	1E-24

Binary Unit Prefix	Description	Prefix Value	Derived from
"Yi"	yobi	2^80 = 1 208 925 819 614 629 174 706 176	yotta
"Zi"	zebi	2^70 = 1 180 591 620 717 411 303 424	zetta
"Ei"	exbi	2^60 = 1 152 921 504 606 846 976	exa
"Pi"	pebi	2^50 = 1 125 899 906 842 624	peta
"Ti"	tebi	2^40 = 1 099 511 627 776	tera
"Gi"	gibi	2^30 = 1 073 741 824	giga
"Mi"	mebi	2^20 = 1 048 576	mega
"Ki"	kibi	2^10 = 1024	kilo

From	To	Rate	Currency	Decimals
"EUR"	"ATS"	13.7603	Austrian Schilling	2
"EUR"	"BEF"	40.3399	Belgian Franc	0
"EUR"	"DEM"	1.95583	German Mark	2
"EUR"	"ESP"	166.386	Spanish Peseta	0
"EUR"	"FIM"	5.94573	Finnish Markka	2
"EUR"	"FRF"	6.55957	French Franc	2
"EUR"	"IEP"	0.787564	Irish Pound	2
"EUR"	"ITL"	1936.27	Italian Lira	0
"EUR"	"LUF"	40.3399	Luxembourg Franc	0
"EUR"	"NLG"	2.20371	Dutch Guilder	2
"EUR"	"PTE"	200.482	Portuguese Escudo	2
"EUR"	"GRD"	340.750	Greek Drachma	2
"EUR"	"SIT"	239.640	Slovenian Tolar	2
“EUR”	“MTL”	0.429300	Maltese Lira	2
“EUR”	“CYP”	0.585274	Cypriot Pound	2
"EUR"	"SKK"	30.1260	Slovak Koruna	2

Function	Exclude hidden by filter	Exclude hidden by filter or collapsed
AVERAGE	1	101
COUNT	2	102
COUNTA	3	103
MAX	4	104
MIN	5	105
PRODUCT	6	106
STDEV	7	107
STDEVP	8	108
SUM	9	109
VAR	10	110
VARP	11	111

bn	bn-1	…	b1	a
$s_{b_{n}}$	$s_{b_{n - 1}}$	…	$s_{b_{1}}$	$s_{a}$
$R^{2}$	$s_{e}$
F	df
SSreg	SSresid

$e^{b_{n}}$	$e^{b_{n - 1}}$	…	$e^{b_{1}}$	$e^{a}$
$s_{b_{n}}$	$s_{b_{n - 1}}$	…	$s_{b_{1}}$	$s_{a}$
$R^{2}$	$s_{e}$
F	df
SSreg	SSresid

Roman Numeral	Value	Unicode Code Point
I	1	U+0049
V	5	U+0056
X	10	U+0058
L	50	U+004C
C	100	U+0043
D	500	U+0044
M	1000	U+004D

Format	Meaning
0 or omitted (or TRUE)	Only subtract powers of 10, not L or V, and only if the next number is not more than 10 times greater. A number following the larger one shall be smaller than the subtracted number. Also known as classic.
1	Powers of 10, and L and V may be subtracted, only if the next number is not more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
2	Powers of 10 and L, but not V, may be subtracted, also if the next number is more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
3	Powers of 10, and L and V may be subtracted, also if the next number is more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
4 (or FALSE)	Produce the fewest Roman digits possible. Also known as simplified.

From Unicode Character (c)	To Unicode Character	Comment
U+30a1 ≤ c ≤ U+30aa if c%2==0	(c - 0x30a2) / 2 + 0xff71	katakana a-o
U+30a1 ≤ c ≤ U+30aa if c%2==1	(c - 0x30a1) / 2 + 0xff67	katakana small a-o
U+30ab ≤ c ≤ U+30c2 if c%2==1	(c - 0x30ab) / 2 + 0xff76	katakana ka-chi
U+30ab ≤ c ≤ U+30c2 if c%2==0	(c - 0x30ac) / 2 + 0xff76 followed by 0xff9e	katakana ga-dhi
U+30c3	0xff6f	katakana small tsu
U+30c4 ≤ c ≤ U+30c9 if c%2==0	(c - 0x30c4) / 2 + 0xff82	katakana tsu-to
U+30c4 ≤ c ≤ U+30c9 if c%2==1	(c - 0x30c5) / 2 + 0xff82 followed by 0xff9e	katakana du-do
U+30ca ≤ c ≤ U+30ce	c - 0x30ca + 0xff85	katakana na-no
U+30cf ≤ c ≤ U+30dd if c%3==0	(c - 0x30cf) / 3 + 0xff8a	katakana ha-ho
U+30cf ≤ c ≤ U+30dd if c%3==1	(c - 0x30d0) / 3 + 0xff8a followed by 0xff9e	katakana ba-bo
U+30cf ≤ c ≤ U+30dd if c%3==2	(c - 0x30d1) / 3 + 0xff8a followed by 0xff9f	katakana pa-po
U+30de ≤ c ≤ U+30e2	c - 0x30de + 0xff8f	katakana ma-mo
U+30e3 ≤ c ≤ U+30e8 if c%2==0	(c - 0x30e4) / 2 + 0xff94)	katakana ya-yo
U+30e3 ≤ c ≤ U+30e8 if c%2==1	(c - 0x30e3) / 2 + 0xff6c	katakana small ya-yo
U+30e9 ≤ c ≤ U+30ed	c - 0x30e9 + 0xff97	katakana ra-ro
U+30ef	U+ff9c	katakana wa
U+30f2	U+ff66	katakana wo
U+30f3	U+ff9d	katakana nn
U+ff01 <= c <= U+ff5e	c - 0xff01 + 0x0021	ASCII characters
U+2015	U+ff70	HORIZONTAL BAR => HALFWIDTH KATAKANA-HIRAGANA PROLONGED SOUND MARK
U+2018	U+0060	LEFT SINGLE QUOTATION MARK => GRAVE ACCENT
U+2019	U+0027	RIGHT SINGLE QUOTATION MARK => APOSTROPHE
U+201d	U+0022	RIGHT DOUBLE QUOTATION MARK => QUOTATION MARK
U+3001	U+ff64	IDEOGRAPHIC COMMA
U+3002	U+ff61	IDEOGRAPHIC FULL STOP
U+300c	U+ff62	LEFT CORNER BRACKET
U+300d	U+ff63	RIGHT CORNER BRACKET
U+309b	U+ff9e	KATAKANA-HIRAGANA VOICED SOUND MARK
U+309c	U+ff9f	KATAKANA-HIRAGANA SEMI-VOICED SOUND MARK
U+30fb	U+ff65	KATAKANA MIDDLE DOT
U+30fc	U+ff70	KATAKANA-HIRAGANA PROLONGED SOUND MARK
U+ffe5	U+005c	FULLWIDTH YEN SIGN => REVERSE SOLIDUS "\"
Note 1: The "\" (REVERSE SOLIDUS, U+005C) is a specialty that gets displayed as a Yen sign with some Japanese fonts, which is a legacy of code-page 932.