XBRL Formula Requirements
Candidate Recommendation dated 20050621
This file:
FormulaReqCR20050621.htm
is nonnormative. The normative version will be found in the file
Name 
Contact 
Affiliation 
Walter Hamscher 
Standard Advantage / Consultant to PricewaterhouseCoopers 
Geoff Shuetrim 
KPMG LLP 

David vun Kannon 
KPMG LLP 
This is a Candidate Recommendation whose circulation is unlimited. It was approved for publication by the International Steering Committee 20050621. Comments should be directed to the editor and/or contributors by email. Recipients of this draft are invited to submit, with their comments, notification of any relevant patent rights of which they are aware and to provide supporting documentation.
3 Linkbaserelated requirements
4 Processingrelated Requirements
5 Expressionrelated Requirements
8 Result Expression Requirements
11 Rejected requirements and use cases
12 Document history (nonnormative)
13 Intellectual Property Status (nonnormative)
Appendix: Approval process (nonnormative)
Example 1: Example of an example
Example 2: Example of a counterexample
Example 3: Relationship of an instance DTS and DTS of a set of formulas
Example 4: Incompatible formulas are distinguishable.
Example 5: Prohibiting a formula.
Example 6: Result expression contains a conditional.
Example 7: Formula applies only to the latest period in an instance.
Example 8: Expression evaluation requires all facts to be bound.
Example 9: Formula uses a locally defined symbolic constant.
Example 10: Formula uses a globally defined symbolic constant.
Example 11: Formula uses a globally defined function.
Example 12: Different formulas bind the same name to different functions.
Example 13: Catching an exception
Example 14: Formula binds items that are pequal and uequal in identical contexts
Example 15: Formula binds items that are pequal, uequal and cequal
Example 16: Formula binds items that are pequal, uequal and contexts whose periods differ
Example 17: Formula binds items in contexts that match period endpoints
Example 18: Formula binds facts in different segments
Example 19: Formula binds facts in different scenarios
Example 20. Formula binds facts across segments of a common entity
Example 21. Formula produces a default fact in the absence of a matching fact
Example 22: Formula assumes default values in the absence of matching facts
Example 23: Formulas ignore Nil facts by default but may bind them
Example 24: Formula has a precondition on item values
Example 25: Formula applies only to certain time periods
Example 26: Precondition tests date items in historical period
Example 27: Formula does not bind duplicate facts.
Example 28: Formula produces a diagnostic message.
Example 29: Expression yields a new unit of measure.
Example 30: Formula determines the result units of an expression from the units of the bound facts.
Example 31: Formula overrides the natural result units of an expression.
Example 32: Expression result has limited precision.
Example 33: Expression result has limited number of decimal places.
Example 34: Formula produces a tuple.
Example 35. Formula merges items into an existing tuple.
Figure 1: Items Appearing in Examples
Figure 2: Units Appearing in Examples
Figure 3: Segments and Scenarios Appearing in Examples
Figure 4: Contexts Appearing in Examples
Figure 5: Formula applies only to certain units
To effectively exchange business reporting information between a producing and a consuming application often requires the applications to perform validation on data types and values, apply consistency checks, test data quality, augment the data with calculated values and possibly corrections, and to provide feedback to the producing application that indicates the nature and severity of problems encountered. A producing application may also add calculated values and perform tests on its own outputs before sending results to consuming applications. Applications based on the exchange of spreadsheets that contain both data and formulas are common but present maintenance problems; here the intent is explicitly to separate the representation and location of formulas and validation criteria from a given instance.
These different functions are characteristic of many XBRLenabled applications. That is because it is a goal of XBRL to allow applications to consume XBRL formatted business information no matter what its original source. Data entry validation (e.g., prohibiting dates such as February 31, 2001 from entering the system) is a familiar example of using a formula to flag errors, but the breadth and scope of formulas is much broader than this. Good information management practice generally recognizes that validations should be tested, adhered to, and corrections made as far “upstream” in an information supply chain as possible – recognizing, of course, that the same tests may be applied repeatedly as data makes its way from its many possible origins, to its many ultimate destinations.
Any general programming language could be used to perform computations on business data. However, there is regularity in business reporting information: regularity that is encoded in XBRL in instance constructs such as “facts,” “periods,” “entities,” and in taxonomies as items, definitions, and other relationships. Furthermore, applications that consume XBRL formatted data should be able to “publish the formulas” that govern documents containing XBRL data, so that producing applications can test and apply those formulas and thereby reduce delay, rework and retransmissions, and smooth and accelerate the flow of business reporting information.
Rule languages—in which rules are expressed as patterns to be matched and actions to be taken when the data matches the pattern—have a rich tradition and are nearly as old as computer science itself ([Newell, 1962]). They enjoyed a heyday in the 1980’s for the programming of “expert systems”, and today rule languages live on in languages such as Prolog, and in commercial products such as the Blaze Rule Engine [HNC] and JRules [ILOG]. Rule languages tend to be fairly compact and concise and resemble a database of logical sentences (“all people are mortal” “if X is a mountain, then the elevation of X must be positive”). They declare the rules to be obeyed, and an interpreter (or compiler) is responsible for efficiently executing the rules (matching patterns and executing actions) correctly when presented with incoming data.
Even if the goal of synthesizing information is set aside so as to merely detect violation of constraints, XML Schema [XSDL] itself is still not sufficient to this task because it allows the validation of individual data elements (“Revenues are a 12digit nonnegative number”) and structural relationships (e.g., “an invoice must contain a header, a list of items, and an amount”) but does not express constraints between elements, also known as coconstraints (“if more than 10 dependents are claimed then Schedule K must be completed”). Besides, the nature of rules is that they generally compute a whole series of results, some of which may be considered fatal errors, others as warnings, others as merely informational; an XML Schema validator mainly detects fatal errors. Other general programming languages, including XML Schema Transformation [XSLT], could be pressed into service since they have the requisite patternaction structure, but these neither take any account nor take any advantage of the constrained nature of XBRL instances, taxonomies, and so on.
XBRLspecific formulas are well motivated. The language would be, like XBRL itself, suitable for publishing and transporting between applications; it would exploit the XBRL language itself, and provide a concise and well constrained way of expressing common rules such as “Net Receivables cannot be negative,” “Net revenues are the difference between Gross revenues and Gross expenses,” “Unless income for the previous quarter was zero, income growth for the quarter is the ratio of the change in income between the current and previous quarter, divided by the income of the previous quarter,” and so on. The earliest discussions of XBRL acknowledged the need to express not only specific numeric facts (“1988 Revenue for TLA Inc. was $40m”), but also relationships among those facts (“TLA Inc. revenue consists of TBD Inc. revenue plus BFD Ltd. revenue less eliminations”) and general relationships (“The write down allowance for an asset in any year after 1993 is limited to 25% of its original purchase price”). There has now been sufficient experimentation and implementation experience with XBRL for the exchange of business reporting information in live and planned applications to have illustrated both the need for an extension to the language to meet this need, as well as illustrating deficiencies in schemes used to date and the typical patterns of usage and the limitations on what that language actually needs to cover. This information is advantageous to have because it limits the scope and guides the design of the language.
Terminology used in XBRL frequently overlaps with terminology from other fields. The terminology used in this document is summarised in Table 1.
Table 1: Terminology
abstract element, bind, concept, concrete element, context, Discoverable Taxonomy Set (DTS), duplicate items, duplicate tuples, element, entity, equal, essence concept, fact, instance, item, least common ancestor, linkbase, period, taxonomy, tuple, unit, taxonomy schema, child, parent, sibling, grandparent, uncle, ancestor, XBRL instance, cequal, pequal, sequal, uequal, vequal, xequal, minimally conforming XBRL processor, fully conforming XBRL processor. 
As defined in [XBRL]. 

must, must not, required, shall, shall not, should, should not, may, optional 
See [RFC2119] for definitions of these and other terms. These include, in particular:


expression 
An expression using constants, variables, arithmetic, logical, and functional operators. 

formula 
An expression along with criteria that indicate the domain of each variable in that expression and how they are to be bound. 

rule 
The term “rule” is not used in this version of the requirement, although it appeared previously as a synonym for “formula”. 

variable 
A “variable” appears in expressions as a symbol that the application of a formula binds to a value, so that the expressions of that formula can be evaluated. 

argument 
An “argument” of a formula is a formal parameter bound to some portion of an input instance. Some of the arguments will be identified as variables to be used in expressions. 
The following highlighting is used for positive examples:
Example 1: Example of an example

Counterexamples indicate incorrect or unsupported examples:
Example 2: Example of a counterexample

Nonnormative editorial comments are denoted as follows and removed from final recommendations:
WcH: This highlighting indicates editorial comments about the current draft, prefixed by the editor’s initials.
Italics are used for rhetorical emphasis only and do not convey any special normative meaning.
Distinctively bracketed and coloured numbers {1.2.3} refer to that particular numbered section of the XBRL 2.1 Specification Recommendation [XBRL].
This document is normative in the sense that any formula specification recommendations by XBRL International MUST satisfy the requirements as they are stated here.
This document version depends upon XBRL 2.1 Specification Recommendation [XBRL].
XBRL Specification 2.1 does not depend in any way upon this document.
The official language of XBRL International’s own work products is English and the preferred spelling convention is UK English. Unless otherwise stated, XBRL specifications must not require XBRL users to use English in documentation, item, tuple, entity, scenario or any other elements.
The general use case for an XBRL formula specification is the externalisation and publication of a set of formulas that will be applied by a consuming application:
· A financial regulator collecting quarterly balance sheets and income statements;
· A statistical agency collecting market valuations of various asset categories;
· A tax authority accepting electronic tax filings;
· A stock exchange accepting registration requests;
· A bank evaluating loan applications or checking loan covenants;
More specifically, there are general use cases that these consuming applications may have:
1. Test XBRL instances for consistency and signalling the failure of a formula, in the same sense that an inconsistent set of fact bindings for the summationitem arcs of a calculation linkbase signals an inconsistency.
2. Test XBRL instances for consistency and provide detailed messages for each formula that failed. In this case, general XML output for further processing and rendering as appropriate by an application is likely to be sufficient.
3. Creating a tuplefree XBRL instance based on existing XBRL instances. In this case, it is sensible for a formula processor to produce a valid XBRL instance. If producing applications verify that their own output will be accepted by those consuming applications, significant efficiencies are possible, particularly in a distributed environment such as the Internet.
4. Transforming, adjusting, or composing XBRL instances. By contrast with use case 3 in which fact order is not relevant, element order can be significant in tuples.
Use case 4 encompasses use case 3, which encompasses use case 2, which encompasses use case 1.
The examples are expressed as “consistent” and “inconsistent” fact sets for use cases 1 and 2, and as “before” and “after” fact sets and instances for use cases 3 and 4, along with a structured description of the relevant generalisation of the behaviour illustrated by that case.
The examples shown in the requirements all depend on a single taxonomy shown in Figure 1 consisting mainly of financial position and performance items.
Figure 1: Items Appearing in Examples
@name 
@type 
@periodType 
Label (en, standard) 
Assets 
monetaryItemType 
instant 
Assets 
CurrentAssets 
monetaryItemType 
instant 
Current Assets 
FixedAssets 
monetaryItemType 
instant 
Fixed Assets 
Equity 
monetaryItemType 
instant 
Equity 
Shares 
sharesItemType 
instant 
Shares 
Price 
monetaryItemType 
instant 
Price 
Earnings 
monetaryItemType 
duration 
Earnings 
AvgShares 
sharesItemType 
duration 
Average Shares 
PE 
pureItemType 
instant 
PriceEarnings Ratio 
ROE 
pureItemType 
instant 
Return on Equity 
EPS 
decimalItemType 
duration 
Earnings per Share 
AssetsEquity 
pureItemType 
instant 
AssetstoEquity Ratio 
Rating 
integerItemType 
instant 
Rating 
AssetsOkay 
booleanItemType 
instant 
Assets Okay 
EquityOkay 
booleanItemType 
instant 
Equity Okay 
AssetsMessage 
stringItemType 
instant 
Assets Message 
AssetsLB 
booleanItemType 
instant 
Assets in Lower Bound 
AssetsUB 
booleanItemType 
instant 
Assets in Upper Bound 
AssetsLarge 
booleanItemType 
instant 
Assets not too Large 
AssetsSmall 
booleanItemType 
instant 
Assets not too Small 
IntangiblesPatents 
monetaryItemType 
instant 
Intangibles (Patents) 
IntangiblesClass 
stringItemType 
instant 
Intangibles Class 
IntangibleGross 
monetaryItemType 
instant 
Intangible Gross 
IntangibleReserve 
monetaryItemType 
instant 
Intangible Reserve 
IntangibleNet 
monetaryItemType 
instant 
Intangible Net 
IntangibleAsset 
tupleType 

Intangible Asset 
AutoWriteDown 
tupleType 

Automobile WriteDown 
AutoName 
tokenItemType 
duration 
Automobile Name 
AcquisitionDate 
dateItemType 
duration 
Acquisition Date 
WriteDownAllowance 
monetaryItemType 
duration 
WriteDown Allowance 
WDVBroughtForward 
monetaryItemType 
duration 
WriteDown Value Brought Forward 
UsefulLife 
durationItemType 
duration 
Useful Life 
All facts in the examples have a unitRef and contextRef drawn from the units and contexts shown in Figure 2 and Figure 4, respectively. The namespace prefix si refers to international standard scientific units.
Figure 2: Units Appearing in Examples
@id 
measure 
usd 
iso4217:USD 
gbp 
iso4217:GBP 
pure 
xbrli:pure 
shs 
xbrli:shares 
usdsh 
iso4217:USD/xbrli:shares 
gbpsh 
iso4217:GBP/xbrli:shares 
year 
si:year 
The namespace prefix co with elements aaa and bbb are used to distinguish segments of entity 444. The elements actual, budgeted and variance are used to distinguish scenarios.
Figure 3: Segments and Scenarios Appearing in Examples
prefix 
element 
co: 
<aaa/> 
co: 
<bbb/> 
co: 
<actual/> 
co: 
<budgeted/> 
co: 
<variance/> 
Figure 4: Contexts Appearing in Examples
@id 
entity/ @identifier 
entity/ segment 
period/ instant 
period/ startDate 
period/ endDate 
scenario 
c01 
333 


20030101 
20031231 

c02 
333 

20021231 



c03 
333 

20031231 



c04 
444 


20030101 
20031231 

c05 
444 

20031231 



c06 
333 

20031231 



c07 
444 
<geo>ON</geo> 

20030101 
20031231 

c08 
444 
<geo>ON</geo> 
20031231 



c09 
444 
<geo>MI</geo> 

20030101 
20031231 

c10 
444 
<geo>MI</geo> 
20031231 



c11 
444 

20031231 


<co:actual/> 
c12 
444 

20031231 


<co:budgeted/> 
c13 
444 

20031231 


<co:variance/> 
c14 
444 


19930101 
19931231 

c15 
444 


19950101 
19951231 

c16 
333 

20011231 



c17 
444 
<lob>paper</lob> 

20030101 
20031231 

c18 
444 
<lob>paper</lob> 
20031231 



c19 
444 
<lob>plastic</lob> 

20030101 
20031231 

c20 
444 
<lob>plastic</lob> 
20031231 



Note that contexts c06 and c03 are sequal.
The data type of the content in any UsefulLife fact is assumed to be yearMonthDuration; the XBRL durationItemType does not specify whether fact content is yearMonthDuration or dayTimeDuration.
Each fact in this document also has a unique identifier such as f42; the identifiers are not relevant in any way to processing the facts or formulas, but the identifiers allow them to be referred to in the text.
Expressions are written in XML Path Language 2.0 [XPATH2] and are meant to be transparent for the example at hand, but are not prescriptive as to the expression language to be required. Some cases use the syntax of XML Query Language 1.0 [XQuery] to declare functions; as stated in [XPATH2], “XPath is designed to be embedded in a host language such as XSLT 2.0 or XQuery… XPath per se does not provide a way to declare functions, but a host language may provide such a mechanism.” The host language chosen in the examples is XQuery because it allows users to declare functions of their own.
The requirements here specify the linkbase features that formulas must support in order to integrate fully with the rest of XBRL.
All these requirements are common to all four use cases.
It must be possible to define a set of formulas in such a way that it can be part of one or more discoverable taxonomy sets {1.4}.
Formulas refer to items and tuples defined in taxonomies {3}. Therefore it must be possible for a set of formulas to rely upon the presences of specific taxonomy schemas and linkbases in its processing environment.
It is not sufficient to rely on the schemaRef and linkbaseRef elements of an instance because the formulas may be computing results that are elements from an entirely different taxonomy schema. Example 3 shows a formula linkbase that has inputs from the taxonomy schema MDRM.xsd but computes results that are items in Form031results.xsd. An instance with a schemaRef only to MDRM.xsd is not sufficient for execution of the formulas. In effect the execution of formulas requires a DTS that is the union of its own DTS and the DTS of the instance in question.
Example 3: Relationship of an instance DTS and DTS of a set of formulas

Formulas maintained in the same taxonomy may provide alternative, possibly incompatible definitions for the same item. A formula is a relationship among two or more facts and so a set of formulas must use the same xlink:role attribute as used in XBRL to indicate which relationships participate in the same networks of relationships based on the value of the role {5.2}.
Example 4: Incompatible formulas are distinguishable.
EPS for some purposes is computed using the average number of shares during the earnings period, and for other purposes using the number of shares at the end of the period.
Formulas:
item 
test 
expression 
type 
matching 
EPS 
“method 1” 
($Earnings div $AvgShares) 
decimals=7 
pequal, cequal. 
EPS 
“method 2” 
($Earnings div $Shares) 
decimals=7 
pequal; Shares context period/instant = Earnings context period/endDate. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f29 
EPS 
c01 
usdsh 
INF 
0.2 
Inconsistent Facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f82 
EPS 
c01 
usdsh 
9 
.183333333 
Use cases 3 and 4:
Initial facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f17 
Shares 
c03 
shs 
INF 
60000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f29 
EPS 
c01 
usdsh 
INF 
0.2 
f67 
EPS 
c01 
usdsh 
9 
0.183333333 
In this example, only one of the two results should be computed, depending on whether the (set of) formulas labelled “method 1” or “method 2” is selected for processing. The xlink:role attribute may be an appropriate way to model this, since the evaluation of the condition does not depend in any way on the facts in the instance.
An extension taxonomy MAY prohibit a formula in a base taxonomy {3.5.3.9.7.5}. The requirement does imply that formulas require a base/extension scheme like that used in taxonomies.
In order for one formula to prohibit another they must be identical, not just sequal. Assuming that a formula set is implemented as a linkbase, a prohibiting arc MUST connect the same XML fragments as the original arc, and both the original and the prohibiting arc MAY connect multiple resources (formulas, variables, constants) by making the ID attribute required.
Example 5: Prohibiting a formula.
An extension {3.2} {3.5.3.9.7.5} of a set of formulas may need to change the method by which an item is computed in a base set of formulas. In this example, the simpler calculation of EPS is in the base set of formulas:
Formulas:
item 
test 
expression 
type 
matching 
EPS 

($Earnings div $Shares) 
decimals=7 
pequal; Shares context period/instant = Earnings context period/endDate. 
An extension set of formulas then prohibits that formula, and asserts a different one:
Formulas:
item 
test 
expression 
type 
matching 





EPS 

($Earnings div $AvgShares) 
decimals=7 
pequal, cequal. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f17 
Shares 
c03 
shs 
INF 
60000 
f29 
EPS 
c01 
usdsh 
INF 
0.2 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f17 
Shares 
c03 
shs 
INF 
60000 
f83 
EPS 
c01 
usdsh 
9 
.1833333333 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
f17 
Shares 
c03 
shs 
INF 
60000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f29 
EPS 
c01 
usdsh 
INF 
0.2 
In this example, the computation based on average number of shares is used and only one value of EPS is calculated.
Humanreadable documentation that explains meaning of a formula in multiple languages may be included in each formula or set of formulas.
This is a general principle applicable to most if not all representations. By analogy with the unit element in XBRL and the unitRef attribute, where a set of formulas is likely to share common information, the specification will use syntax that allows it to be specified just once and referred to in many formulas.
Formulas in an instance are believed to irrevocably render XBRL instances useless from an archival standpoint because of differences in the results that different processors would derive results. The treatment of formulas for archiving must be the same as for calculation arcs.
The requirements here describe the processing semantics of formulas relative to input XBRL instances and outputs.
Authors may write formulas in such a way as to presume an XBRLvalid instance with respect to a known taxonomy schema.
This requirement is common to use cases 1, 2, 3, and 4.
One and only one processing iteration shall be performed in satisfying this requirement.
This requirement is common to use cases 1, 2, 3, and 4.
Nonlocal properties of XBRL validity that would apply to the output instance—such as the testing of the requires‑element constraints {5.2.6.2.4}—can be difficult to guarantee on a local, incremental basis, so that in practice a formula processor would almost certainly require an XBRL validity checker.
One and only one processing iteration shall be performed in satisfying this requirement.
This requirement is common to use cases 3 and 4.
Authors may write formulas in such a way as to provide multiple ways to derive a given fact. In practice authors should avoid writing formulas that bind the same set of facts to produce the same result facts, since the result facts may be duplicates or even contradictory.
This requirement is common to use cases 3 and 4.
This requirement is common to use cases 1, 2 and 3, and does not apply to use case 4.
The firing order of formulas is implementation dependent. This cannot affect the semantics of the outcome, given requirement 6.7 below, “Formulas MUST only bind facts that are explicitly present in the input”
Formulas are to be processed one instance at a time. Processing the instances is implementation dependent and cannot affect the semantics of the outcome. There is no dependency between any formula and any other formula.
If formulas had priorities then they could be used to order formula application in cases where more than one formula applies, but since all must be evaluated, and the order of output elements is not relevant in XBRL, the XBRL semantics would not be impacted.
The requirements here refer to the common features needed in all three of:
· expressions that may be used to bind arguments to facts;
· expressions that may form a Boolean precondition of the formula after arguments are bound; and
· expressions that yield the result of processing a formula (use cases 2, 3 and 4).
Where it is not clear from context elsewhere in the document, these are called binding expressions, precondition expressions, and result expressions, respectively.
From an adoption standpoint, it is not desirable to allow multiple expression languages since that would unnecessarily increase the implementation burden on a compliant formula processor.
Other standards (e.g., XSL and XPointer) have frameworks in which different expressions or scripting languages may be used; new recommendations incrementally extend the set of expressions that must be supported.
Therefore, the requirement that there must be only one expression language MAY be relaxed in a future version of the formula specification.
A “programmer of average skill” means a person familiar with infix and functional notations and able to write spreadsheet formulas, SQL queries, or expressions in a 3GL or 4GL programming language.
This requirement will be satisfied if the expression language is a subset of a widely used language such as ECMAScript or XPATH 1.0. Expression syntaxes ruled out by this requirement include those used by APL, FORTH, LISP, PROLOG, etc.
Factbinding expressions, conditional expressions and result expressions must be able to operate (for example) on the units of measure, the contexts, and other parts of the input instance. This is believed to be difficult for the formula specification and therefore for an implementation, to prevent this accessibility than it is for an implementation to require it.
Expressions must be able to express any constant from the value space of XML Schema primitive data types [SCHEMA‑2].
Expressions must be able to express the following operations:
· Addition and subtraction of values;
· Division and modulus of values, both integer and real;
· Multiplication of values;
· Determine maxima and minima of a sequence of values;
· The range of string matching and modification operations made possible by regular expressions;
· All of the following relational operations, =, <, >, <=, >=, and != on numeric items and =, != on nonnumeric items.
Formulas must be able to express the following operations:
· If, Then, Else
· Elseif
· Switch / Case
Conditional expressions and nested conditionals may test several conditions in sequence.
Example 6: Result expression contains a conditional.
In this example a continuous P/E ratio is mapped to one of three discrete values of Rating.
Formulas:
item 
test 
result expression 
type 
matching 
Rating 

if ($PE lt 5) then 1 else if ($PE gt 10) then 2 else 3 
INF 

Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f22 
PE 
c02 
pure 
INF 
6.5 
f39 
Rating 
c03 
pure 
INF 
3 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f21 
PE 
c03 
pure 
INF 
56 
f84 
Rating 
c03 
pure 
INF 
2 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f21 
PE 
c03 
pure 
INF 
56 
f22 
PE 
c02 
pure 
INF 
6.5 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f39 
Rating 
c03 
pure 
INF 
3 
f40 
Rating 
c02 
pure 
INF 
1 
In principle, any conditional expression could be recast as a series of separate formulas with disjoint conditions (this example would require three formulas, with one having the condition “5 ≤ PE and PE < 10”).
Expressions may determine the latest and earliest startDate, endDate and instant appearing in the instance. This could also include the use of date expressions that include a function such as now() which would return the current instant as an ISO 8601 string.
Example 7: Formula applies only to the latest period in an instance.
Facts may match the formula only if they fall within a time period that is described as the “latest” period in the instance.
Formulas:
item 
test 
expression 
type 
matching 
AssetEquity 
($Equity gt 0) 
$Assets div $Equity 
precision=18 
pequal, uequal, cequal; Context must have a period that is the latest occurring in the input instance. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f01 
Assets 
c02 
usd 
INF 
100000 
f03 
Equity 
c02 
usd 
INF 
17000 
f86 
AssetEquity 
c02 
usd 
9 
58.82352941 
f13 
Assets 
c03 
usd 
INF 
43000 
f15 
Equity 
c03 
usd 
INF 
26000 
f31 
AssetEquity 
c03 
pure 
18 
1.653846153846153846 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f13 
Assets 
c03 
usd 
INF 
43000 
f15 
Equity 
c03 
usd 
INF 
26000 
f87 
AssetEquity 
c03 
pure 
9 
58.83252941 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f01 
Assets 
c02 
usd 
INF 
100000 
f03 
Equity 
c02 
usd 
INF 
17000 
f13 
Assets 
c03 
usd 
INF 
43000 
f15 
Equity 
c03 
usd 
INF 
26000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f31 
AssetEquity 
c03 
pure 
18 
1.653846153846153846 
In this example, one pair of facts (f01 and f03) evaluates the condition to false and the other (f13 and f15) evaluates it to true. The latest period among the inputs is the instance 2003‑12‑31.
This applies to use cases 1, 2, 3, and 4.
The presence or absence of a fact may be tested.
See Example 6 for an example in a conditional expression, Example 24 for an example in binding expressions; Example 23 also indicates that Nil items may be tested for.
This applies to use cases 1, 2, 3, and 4.
Although in principle, expressions such as or(x,y,z) could be evaluated lefttoright and with x="true", neither y nor z would need to be bound, formula processing does not support this.
Example 8: Expression evaluation requires all facts to be bound.
Formulas:
item 
test 
expression 
type 
matching 
AssetsOkay 

$AssetsUB and $AssetsLB 
Boolean 
pequal, cequal. 
Facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f49 
AssetsUB 
c03 


false 
Result:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
In this example, even though AssetsLB is not strictly needed to evaluate the expression, the output still should not contain the result that AssetsOkay is false
A formula will bind arguments to facts, and substitution of the arguments as variables in the expression allows the expression to be evaluated. The formula specification must indicate, in particular, what result if any is produced when the expression throws an exception when evaluated.
A try/catch mechanism would allow the formula author to control the behaviour on a casebycase basis. This presumably though not always will include the generation of a humanreadable error message.
This applies to use cases 1, 2, 3, and 4.
Formulas often refer to constants that may appear locally.
Example 9: Formula uses a locally defined symbolic constant.
In this formula, lowerTolerance and upperTolerance are constants bound to “500” and “500” respectively.
Formulas:
item 
test 
expression 
type 
matching 
AssetsOkay 

($lowerTolerance lt ($Assets  ($CurrentAssets + $FixedAssets)) and (($Assets – ($CurrentAssets + $FixedAssets) lt $upperTolerance) 

pequal, uequal and cequal; 
Use cases 1 and 2 (AssetsOkay is not an explicit concept):
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f48 
Assets 
c03 
usd 
INF 
43400 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Use cases 3 and 4 (AssetsOkay is an explicit concept):
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Result facts:
@id 
item 
@contextRef 


[content] 
f48 
AssetsOkay 
c03 


false 
Sets of formulas often refer to constants that appear in several related formulas; there must be a way to define a constant whose scope is an entire set of formulas. There must be a means to limit the scope of such a constant, such as by using the xlink:role attribute.
Example 10: Formula uses a globally defined symbolic constant.
Here, tolerance is bound to “500” globally and so has the identical value in two different formulas, one for testing the upper bound (UB) and one for the lower bound (LB).
Formulas:
item 
test 
expression 
type 
matching 
AssetsUB 

($Assets  ($CurrentAssets + $FixedAssets)) lt $tolerance 
Boolean 
pequal, uequal and cequal; 
AssetsLB 

$tolerance lt ($Assets  ($CurrentAssets + $FixedAssets)) 
Boolean 
pequal, uequal and cequal; 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
f49 
AssetsUB 
c03 


false 
f50 
AssetsLB 
c03 


true 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
f89 
AssetsUB 
c03 


true 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Result facts:
@id 
item 
@contextRef 


[content] 
f49 
AssetsUB 
c03 


false 
f50 
AssetsLB 
c03 


true 
Any function that is not “built in” to the expression language – for example, trigonometric functions – would either have to be defined repeatedly in every formula that needed it, or an intermediate item defined in the taxonomy whose only role would be to hold the result of the formula and then append that result to an instance so that its value could be bound in all formulas that may need it. Examination of UK Inland Revenue Tax Computation use cases, similar items on the FFIEC 031 and 041 forms, and consideration of many financial analysis routines, reveals that the same formulas are used again and again.
Example 11: Formula uses a globally defined function.
In this formula, there is a reference to an externally defined function:
declare function my:withinTolerance( $x as xdt:anyAtomicType, $y as xdt:anyAtomicType ) as xdt:boolean* { fn:abs( $x  $y) lt 500 }
Formulas:
item 
test 
expression 
type 
matching 
AssetsOkay 

my:withinTolerance ($Assets,($CurrentAssets + $FixedAssets)) 
Boolean 
pequal, uequal and cequal; 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Result facts:
@id 
item 
@contextRef 


[content] 
f48 
AssetsOkay 
c03 


false 
There must be a way to limit the scope of such a function definition to be usable only within a certain set of formulas, such as by using the xlink:role attribute.
Example 12: Different formulas bind the same name to different functions.
In two different formulas, there are references to externally defined functions that apply in different rules within the same set. The rules are not distinguished by an arithmetic test but rather by an indicator such as xlink:role.
scope 
function 
Taxation Enforcement 
declare function tax:withinTolerance( $x as xdt:anyAtomicType, $y as xdt:anyAtomicType ) as xdt:boolean* { fn:abs( $x  $y) lt 500 }

Securities Enforcement 
declare function sec:withinTolerance( $x as xdt:anyAtomicType, $y as xdt:anyAtomicType ) as xdt:boolean* { fn:abs( $x  $y) lt 5000000 } 
Formulas:
item 
test 
expression 
type 
matching 
AssetsOkay 
$Enforcement eq “Securities Enforcement” 
tax:withinTolerance ($Assets,($CurrentAssets + $FixedAssets)) 
Boolean 
pequal, uequal and cequal; 
EarningsOkay 
$Enforcement eq “Taxation Enforcement” 
sec:withinTolerance ($Equity,$EquityPrev + $Earnings) 
Boolean 
pequal, uequal; “EquityPrev” refers to a context in the year previous to that of “Equity”.
The durationtype period of “Earnings” must begin at the instant represented by “EquityPrev”. 
Both computations have 1000dollar errors but only one of them results in a false value.
Use cases 1 and 2 (AssetsOkay and EarningsOkay are not explicit concepts):
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f03 
Equity 
c02 
usd 
INF 
17000 
f09 
Earnings 
c01 
usd 
INF 
9000 
f11 
Equity 
c03 
usd 
INF 
27000 
Use cases 3 and 4 (AssetsOkay and EarningsOkay are explicit concepts):
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
f03 
Equity 
c02 
usd 
INF 
17000 
f09 
Earnings 
c01 
usd 
INF 
9000 
f11 
Equity 
c03 
usd 
INF 
27000 
Result facts:
@id 
item 
@contextRef 


[content] 
f48 
AssetsOkay 
c03 


True 
f74 
EarningsOkay 
c03 


False 
The expression language must support a try/catch or other exception handling mechanism and allow the expression to produce a result. The following is only an example result and is not meant to be normative for all formulas.
Example 13: Catching an exception
Formulas:
item 
test 
expression 
matching 
EPS 

Try ($Earnings div $AvgShares) Catch Exception Return <EPS>0</EPS> 
pequal, cequal. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f72 
AvgShares 
c01 
shs 
INF 
0 
f90 
EPS 
c01 
usdsh 
INF 
0 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f72 
AvgShares 
c01 
shs 
INF 
0 
f37 
EPS 
c01 
usdsh 
c01 
0.16363636 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f72 
AvgShares 
c01 
shs 
INF 
0 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f90 
EPS 
c01 
usdsh 
INF 
0 
Note that the units, context and precision of the result remain the same as if the calculation had not thrown an exception; only its content is different.
The requirements here refer to the features needed to bind facts to the arguments for the formula as they appear in the condition and result expressions.
The facts bound in a formula may have different contexts.
Contexts are related to one another using the following orderings:
· The relationship “after” that partially orders periods;
· The subset relationship “during” between periods;
· The subset relationship implied between an entity and its segments {4.7.3};
· The subset relationship implied between an empty (universal) scenario and a scenario with additional discriminators {4.7.4}.
Example 14: Formula binds items that are pequal and uequal in identical contexts
An important type of formula involves a mathematical operator applied to a pair of items when they are pequal, cequal and uequal.
Formulas:
item 
test 
expression 
type[1] 
matching 
Assets 

$CurrentAssets + $FixedAssets 
INF 
pequal, cequal and uequal 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f34 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
43000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f34 
FixedAssets 
c03 
usd 
INF 
35000 
f48 
Assets 
c03 
usd 
INF 
43400 
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f34 
FixedAssets 
c03 
usd 
INF 
35000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f13 
Assets 
c03 
usd 
INF 
43000 
Example 15: Formula binds items that are pequal, uequal and cequal
A variation of Example 14 is when items are cequal without the contexts being identical.
Formulas:
item 
test 
expression 
type 
matching 
Assets 

$CurrentAssets + $FixedAssets 
INF 
pequal, cequal and uequal 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c06 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
43000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c06 
usd 
INF 
35000 
f48 
Assets 
c03 
usd 
INF 
43400 
Use cases 3 and 4:
Facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c06 
usd 
INF 
35000 
Result:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f13 
Assets 
c03 
usd 
INF 
43000 
The output context is sequal to c03 and c06. Formula processors should behave consistently and the specification must define which context to use on output.
Example 16: Formula binds items that are pequal, uequal and contexts whose periods differ
Formulas must be able to express relations that cross periods. In this example the value of any two values for “Shares” that differ by one year are averaged during the intervening duration.
Formulas:
item 
test 
expression 
type 
matching 
AvgShares 

($SharesNext + $Shares) div 2 
INF 
pequal, uequal; “SharesNext” is in a context offset by one year from “Shares”. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f05 
Shares 
c02 
shs 
INF 
50000 
f17 
Shares 
c03 
shs 
INF 
60000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f05 
Shares 
c02 
shs 
INF 
50000 
f17 
Shares 
c03 
shs 
INF 
60000 
f12 
AvgShares 
c01 
shs 
INF 
5000 
Use cases 3 and 4:
Facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f05 
Shares 
c02 
shs 
INF 
50000 
f17 
Shares 
c03 
shs 
INF 
60000 
The output context, because it is not present among the input facts, must be synthesised by the formula. In this example it happens to be sequal to c01. Note that the two inputs are in “instant” contexts while the output is a “duration” context whose start and end dates are equal to those of the input instants (taking into account differences in the way startDate and endDate default the time of day {4.7.2}).
Result:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f11 
AvgShares 
sequal to c01 
shs 
INF 
55000 
The matching of input periods may involve matching of endpoints. More generally, endpoints may match in any relationship expressible by using comparison operators (equal, less than, greater than), arithmetic offsets (plus, minus), and constants (dates, datetimes, durations).
Example 17: Formula binds items in contexts that match period endpoints
This example is a simple movement analysis on the value of Equity.
Formulas:
item 
test 
expression 
type 
matching 
Equity 

$EquityPrev + $Earnings 
INF 
pequal, uequal; “EquityPrev” refers to a context in the year previous to that of “Equity”.
The durationtype period of “Earnings” must begin at the instant represented by “EquityPrev”. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f03 
Equity 
c02 
usd 
INF 
17000 
f09 
Earnings 
c01 
usd 
INF 
9000 
f11 
Equity 
c03 
usd 
INF 
26000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f03 
Equity 
c02 
usd 
INF 
17000 
f09 
Earnings 
c01 
usd 
INF 
9000 
f94 
Equity 
c03 
usd 
INF 
0 
Use cases 3 and 4:
Facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f03 
Equity 
c02 
usd 
INF 
17000 
f09 
Earnings 
c01 
usd 
INF 
9000 
Result:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f11 
Equity 
sequal to c03 
usd 
INF 
26000 
This requirement also implies that the expression language SHOULD provide a native library of date manipulation functions such as those specified for XQuery and XPath [XQPFO].
Example 18: Formula binds facts in different segments
A fixed asset breakdown between an entity (444) and its segments (aaa and bbb) has the formula shown below, and the two figures 50,000 and 80,000 summing to 130,000.
The presumption of this use case is that if there is no segment element, this corresponds to the “universal” segment, i.e., the entire entity.
Formulas:
item 
test 
expression 
type 
matching 
FixedAssets 

$FixedAssetsaaa + $FixedAssetsbbb 
INF 
pequal, uequal; FixedAssets(aaa) and FixedAssets(bbb) contexts are s‑equal except for the segment identifier (aaa and bbb, respectively). 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
f43 
FixedAssets 
c05 
usd 
INF 
130000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
f92 
FixedAssets 
c05 
usd 
INF 
260000 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f43 
FixedAssets 
c05 
usd 
INF 
130000 
Note that the XML elements aaa and bbb, which are to be used in the input instance, must be present in a namespace accessible to the formulas, just like the item names and other parts of a taxonomy.
Example 19: Formula binds facts in different scenarios
The Earnings of an entity are reported as actual and budgeted in different scenarios, and in yet a third scenario the variance figure is to be computed from them.
There is no presumption in this use case is that if there is no scenario element that it means some kind of “universal” scenario; rather, scenario is simply unspecified.
Note that the XML elements actual, budgeted and variance, which are to be used in the input instance, must be present in a namespace accessible to the formulas, just like the item names and other parts of a taxonomy.
Formulas:
item 
test 
expression 
type 
matching 
Earnings(variance) 

$EarningsActual – $EarningsBudgeted 
INF 
pequal, uequal; Earnings(actual), Earnings(variance) and Earnings(budgeted) contexts are s‑equal except for the scenario identifier. 
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f44 
Earnings 
c11 
usd 
INF 
12000 
f45 
Earnings 
c12 
usd 
INF 
14000 
f46 
Earnings 
c13 
usd 
INF 
2000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f44 
Earnings 
c11 
usd 
INF 
12000 
f45 
Earnings 
c12 
usd 
INF 
14000 
f93 
Earnings 
c13 
usd 
INF 
30000 
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f44 
Earnings 
c11 
usd 
INF 
12000 
f45 
Earnings 
c12 
usd 
INF 
14000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f46 
Earnings 
c13 
usd 
INF 
2000 
Example 20. Formula binds facts across segments of a common entity
Two fixed asset breakdowns between:
· an entity (444);
· its geographic segments (<geo>ON</geo> and <geo>MI</geo>); and
· line of business segments (<lob>paper</lob> and <lob>plastic</lob>)
have the formula shown below, with the two sets of figures 50,000 and 80,000 and 40,000 and 90,000 separately summing to 130,000.
Formulas:
item 
condition 
expression 
type 
matching 
FixedAssets 

sum of FixedAssets* 
INF 
pequal, uequal; FixedAssets* contexts have a common entity, common segment element and distinct segment element contents, FixedAssets* contexts have s‑equal scenarios; Result context entity has no segment element. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
f75 
FixedAssets 
c18 
usd 
INF 
40000 
f76 
FixedAssets 
c19 
usd 
INF 
90000 
f43 
FixedAssets 
c05 
usd 
INF 
130000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
f75 
FixedAssets 
c18 
usd 
INF 
40000 
f76 
FixedAssets 
c19 
usd 
INF 
90000 
f92 
FixedAssets 
c05 
usd 
INF 
260000 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f41 
FixedAssets 
c08 
usd 
INF 
50000 
f42 
FixedAssets 
c10 
usd 
INF 
80000 
f75 
FixedAssets 
c18 
usd 
INF 
40000 
f76 
FixedAssets 
c19 
usd 
INF 
90000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f43 
FixedAssets 
c05 
usd 
INF 
130000 
f77 
FixedAssets 
c05 
usd 
INF 
130000 
By contrast with the previous example, in this example the formula is not limited to known segment names. Rather, the formula match criteria MUST be written in conjunction with a design of segment child elements that ensures distinct child elements of segment are treated as orthogonal and distinct element contents are comprehensive and non‑overlapping
See Example 31.
The condition on the facts may allow precision, decimals, or both, and comparative operators on their values.
Formulas MUST be able to use predicates that use paths with the parentchild and other XPATH axes in order to filter input facts. These paths MAY refer to facts already bound.
For example, an input may be filtered according to a criterion such as “the same context as that other argument, except one quarter earlier.”
There is no requirement that formulas automatically match to facts derived from previous formulas matched and evaluated.
This requirement does not apply in use cases 1 and 2.
Example 21. Formula produces a default fact in the absence of a matching fact
When Current Assets are known in a given context but Fixed Assets are not, a formula author may want to assert a default value (zero) for the item FixedAssets.
The conditions expressed in this formula would be evaluated sequentially in the sense that the formula only applies in a context where CurrentAssets is already bound. The function fn:not is used as the predicate "unbound”, in effect assuming that “unbound” variables are initially bound to the empty sequence until bound to something else by the formula processor".
Note also that fact f43 will appear in the output, not the input, so that any formulas that required input facts for CurrentAssets and FixedAssets in cequal contexts would require additional formula processor iterations.
Formulas:
item 
test 
expression 
type 
matching 
FixedAssets 
$CurrentAssets and fn:not($FixedAssets) 
0 
INF 
pequal, uequal and cequal; 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f43 
FixedAssets 
c03 
usd 
INF 
0 
Example 22: Formula assumes default values in the absence of matching facts
When values are not known in a given context, a formula author may assume a default value (zero) for items, in this example, default values of zero for CurrentAssets and FixedAssets.
Formulas:
item 
test 
expression 
type 
matching 
Assets 

$CurrentAssets + $FixedAssets 
INF 
pequal, cequal and uequal; If CurrentAssets unbound then CurrentAssets = 0; If FixedAssets unbound then FixedAssets = 0; If both are unbound the formula produces no result. 
Use cases 1 and 2:
This set of facts covers each of the three possible binding cases in the formula.
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f77 
CurrentAssets 
c02 
gbp 
INF 
2000 
f65 
FixedAssets 
c16 
gbp 
INF 
5000 
f13 
Assets 
c03 
usd 
INF 
43000 
f64 
Assets 
c02 
gbp 
INF 
2000 
f66 
Assets 
c16 
gbp 
INF 
5000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f77 
CurrentAssets 
c02 
gbp 
INF 
2000 
f65 
FixedAssets 
c16 
gbp 
INF 
5000 
f48 
Assets 
c03 
usd 
INF 
43400 
f02 
Assets 
c02 
gbp 
INF 
43000 
f91 
Assets 
c16 
gbp 
INF 
10000 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f77 
CurrentAssets 
c02 
gbp 
INF 
2000 
f65 
FixedAssets 
c16 
gbp 
INF 
5000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f13 
Assets 
c03 
usd 
INF 
43000 
f64 
Assets 
c02 
gbp 
INF 
2000 
f66 
Assets 
c16 
gbp 
INF 
5000 
Example 23: Formulas ignore Nil facts by default but may bind them
Formula authors must have the choice whether or not to bind expressions with nil facts. At least for the purpose of describing formulas in these examples the assumption is that the default is that nil facts are not bound, and so nothing appears in the “test” column.
The Assets rule should produce no results. Furthermore, if the formula had instead assigned a default value of zero to CurrentAssets then it should behave the same as Example 22 above. This suggests that this example represents a somewhat more robust treatment than use case Error! Reference source not found., which would involve creating a new fact that would then be a duplicate {4.10} of the nil item.
The Rating formula should produce one item as a result.
Formulas:
item 
test 
expression 
type 
matching 
Assets 

$CurrentAssets + $FixedAssets 
INF 
pequal, cequal and uequal. 
Rating 
element(my:PE, xbrli:decimalItemType nillable) 
if ( $PE[@xsi:nil = “true”]) then 0 else if ($PE lt 5) then 1 else if ($PE lt 10) then 2 else 3 
INF 

Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f68 
CurrentAssets 
c03 
usd 
INF 
(xsi:nil) 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
43000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f70 
PE 
c02 
pure 
INF 
(xsi:nil) 
f40 
Rating 
c02 
pure 
INF 
1 
Use cases 3 and 4:
Facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f68 
CurrentAssets 
c03 
usd 
INF 
(xsi:nil) 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f70 
PE 
c02 
pure 
INF 
(xsi:nil) 
Result:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f71 
Rating 
c02 
pure 
INF 
0 
For example, an expression may apply only when the value of a certain item is nonzero in the bound context. The requirement is that there be a way to “abort” the application of a formula after all inputs are bound to facts, but before its expression is evaluated.
This does not require that the matching and binding criteria of any argument be able to reference the value of any other bound argument. For example, a formula that binds argument “S” to the “SignatureDate” of a financial statement does not need to be able to use the date “S” to find the Revenue value as of date “S”. In general, computed offsets (e.g., the binding of one argument determines which of many different time periods the other arguments are bound in) do not have to be supported.
Setting a precondition on an item value may be needed to ensure the expression is meaningful and will not cause an evaluation error.
Example 24: Formula has a precondition on item values
This is an example in which the condition of one pair of facts (f19 and f29) evaluates to true, the other pair (f20 and f30) to false.
Formulas:
item 
test 
expression 
type 
matching 
PE 
($EPS gt 0) 
($Price div $EPS) 
INF 
pequal, uequal; Context of “Price” has a period that is the instant which ends that of “EPS”. 
Use cases 1 and 2:
(PE would normally have a data type of nonnegative decimal, making these particular examples somewhat artificial):
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f20 
Price 
c03 
gbpsh 
INF 
5.2 
f30 
EPS 
c01 
gbpsh 
INF 
0.8 
f22 
PE 
c03 
pure 
INF 
56 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f19 
Price 
c03 
usdsh 
INF 
11.2 
f29 
EPS 
c01 
usdsh 
INF 
.2 
f23 
PE 
c03 
pure 
INF 
56 
Use cases 3 and 4:
Facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f19 
Price 
c03 
usdsh 
INF 
11.2 
f29 
EPS 
c01 
usdsh 
INF 
.2 
f20 
Price 
c03 
gbpsh 
INF 
5.2 
f30 
EPS 
c01 
gbpsh 
INF 
0.8 
Result:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f21 
PE 
c03 
pure 
INF 
56 
Example 25: Formula applies only to certain time periods
Facts may match the formula only if they fall within a named time period. Note that because every item has a fixed periodType, it is not necessary to be able to specify the kind of period of which the fact is asserted, because that is implicit in the item itself.
In this example, one pair of facts (f01 and f03) evaluates the condition to true and the other (f13 and f15) evaluates it to false.
Formulas:
item 
test 
expression 
type 
matching 
AssetEquity 
($Equity gt 0) 
$Assets div $Equity 
precision=10 
pequal, uequal, cequal; Context must have a period whose endpoint is strictly before 2003‑01‑01. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f13 
Assets 
c03 
usd 
INF 
43000 
f15 
Equity 
c03 
usd 
INF 
26000 
f87 
AssetEquity 
c03 
pure 
10 
58.82352941 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f01 
Assets 
c02 
usd 
INF 
100000 
f03 
Equity 
c02 
usd 
INF 
17000 
f86 
AssetEquity 
c02 
pure 
10 
58.82352941 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f01 
Assets 
c02 
usd 
INF 
100000 
f03 
Equity 
c02 
usd 
INF 
17000 
f13 
Assets 
c03 
usd 
INF 
43000 
f15 
Equity 
c03 
usd 
INF 
26000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f31 
AssetEquity 
c02 
pure 
10 
5.882352941 
Example 26: Precondition tests date items in historical period
Setting a precondition on an item value may be part of defining its scope of applicability.
In this example, historical data (Acquisition Cost, Useful Life, and Acquisition Date) of an asset determines its write down allowance for all future periods. Acquisitions made after 11 March 1992 are subject to a GBP 2000 maximum writedown per year; hence the acquisition date of 10 October 1993 means that after two years only GBP 4000 has been written down, not 2*(20000/6) had the acquisition been made (say) 1991.
In this example the context of the WriteDown Allowance is determined from the context of a nonzero WriteDown Value Brought Forward. The item UsefulLife is a durationItemType but the result of the expression is cast to the same type and unit as AcquisitionCost.
Formulas:
item 
test 
expression 
type 
matching 
WriteDownAllowance 
($AcquisitionDate lt 19920311) and ($WDVBroughtForward gt 0) 
$AcquisitionCost div fn:getyearsfromyearMonthDuration($UsefulLife) 
INF 
pequal, cequal; Result context is that of a nonzero WDVBroughtForward whose period is greater than or equal to the input facts’ end date. 
WriteDownAllowance 
($AcquisitionDate ge 19920311) and ($WDVBroughtForward gt 0) 
fn:min(2000, $AcquisitionCost div fn:getyearsfromyearMonthDuration($UsefulLife)) 
INF 
pequal, cequal; Result context is that of a nonzero WDVBroughtForward whose period is greater than or equal to the input facts’ end date. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f57 
AutoWriteDown 



(tuple) 
f58 
AssetName 
c14 


571XVH 
f59 
AcquisitionDate 
c14 


19931010 
f60 
AcquisitionCost 
c14 
gbp 
INF 
20000 
f61 
UsefulLife 
c14 


6 
f62 
WDVBroughtForward 
c15 
gbp 
INF 
16000 
f63 
WriteDownAllowance 
c15 
gbp 
INF 
2000 
Inconsistent facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f57 
AutoWriteDown 



(tuple) 
f58 
AssetName 
c14 


571XVH 
f59 
AcquisitionDate 
c14 


19931010 
f60 
AcquisitionCost 
c14 
gbp 
INF 
20000 
f61 
UsefulLife 
c14 


6 
f62 
WDVBroughtForward 
c15 
gbp 
INF 
16000 
f80 
WriteDownAllowance 
c15 
gbp 
INF 
2400 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f57 
AutoWriteDown 



(tuple) 
f58 
AssetName 
c14 


571XVH 
f59 
AcquisitionDate 
c14 


19931010 
f60 
AcquisitionCost 
c14 
gbp 
INF 
20000 
f61 
UsefulLife 
c14 


6Y 
f62 
WDVBroughtForward 
c15 
gbp 
INF 
16000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f63 
WriteDownAllowance 
c15 
gbp 
INF 
2000 
Figure 5: Formula applies only to certain units
Facts may bind in the formula only if the units match.
Formulas:
item 
test 
expression 
type 
matching 
AssetEquity 
($Equity gt 0) 
$Assets div $Equity 
precision=10 
pequal, uequal, cequal; Assets context must have unit/measure = ISO4217:USD 
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f01 
Assets 
c02 
usd 
INF 
100000 
f03 
Equity 
c02 
usd 
INF 
17000 
f02 
Assets 
c02 
gbp 
INF 
43000 
f04 
Equity 
c02 
gbp 
INF 
26000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f31 
AssetEquity 
c02 
pure 
10 
5.882352941 
In this example, one pair of facts (f01 and f03) evaluates the condition to true and the other (f02 and f04) evaluates it to false.
The motivation for this is that some formulas will require multiple XBRL instances, if only because a given instance may require more than one formula processing iteration. Because all identifiers—particularly the id attributes of unit and context elements—cannot be duplicated in an XML element, simple concatenation within a parent element may not be significantly easier than actually merging the two instances properly to create a single XBRLvalid instance. Nevertheless the requirement remains.
A formula may match the same facts to different arguments and apply multiple times.
When duplicate facts appear in an instance, a single formula could:
(a) repeatedly bind different duplicate facts and reevaluate its expression repeatedly,
(b) follow the approach used in the calculation linkbase and derive no facts, or
(c) take some other approach.
The specification MUST indicate what facts if any are bound in this situation. The specification MUST use an approach consistent with interpretation of XBRL 2.1 {5.2.5.2}.
Example 27: Formula does not bind duplicate facts.
In XBRL 2.1 {5.2.5.2} duplicate facts do not bind in summationitem calculations (“A calculation binds for a summation item if it has no duplicates in the XBRL instance...”) or if the facts have nil values (“Items with nil values do not participate in calculation bindings”). The same approach applies consistently in formulas.
This formula produces no results because f33 and f69 satisfy the XBRL 2.1 “duplicates” predicate. Furthermore, this case is distinct from the absence of CurrentAssets and therefore a formula specifying a default value should not appl
Formulas:
item 
test 
expression 
type 
matching 
Assets 

$CurrentAssets + $FixedAssets 
INF 
pequal, cequal and uequal. 
Use cases 1 and 2:
Consistent facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f69 
CurrentAssets 
c03 
usd 
INF 
10000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f48 
Assets 
c03 
usd 
INF 
43400 
Use cases 3 and 4:
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f69 
CurrentAssets 
c03 
usd 
INF 
10000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
This requirement is a logical consequence of other requirements such as 6.5 above, “Formulas MAY filter input facts depending on their surrounding tuple structure”
This requirement is a logical consequence of 6.12 above. Requirement 5.4 above, “Expressions may include constants.”, presumes that constants consist only of numbers, strings and other primitive data types, however, if formula arguments are bound to facts then constants must also be able to represent facts having a precision, context, and units.
Arguments bound in a formula may be bound to a sequence of facts for execution. Formulas such as “the sum of all revenues for years before 2001” and “the sum of values of the children of the item” are allowed. This will require operations on vectors and matrices spanning any number of contexts, which in turn requires a definition of the “best” or “maximal” match in order to avoid a combinatorial explosion. For example, if argument X is defined in six periods, then the “moving average of X” must be defined with respect to a fixed number of periods anyway (2 periods? 3 periods?), and this can adequately be expressed using formulas with a fixed number of input arguments. By contrast, if X is “the sum of X for all child entities,” all could mean “all child entities for which X is known in each known period” or “all child entities for which X is known in all periods.”
Instead of producing only a Boolean result as in use cases 3 or 4, or simply signalling an overall failure as in use case 1, in use case 3 a formula can produce a more detailed warning or explanation of a problem.
Example 28: Formula produces a diagnostic message.
Formulas:
item 
test 
expression 
type 
matching 
AssetsMessage 
fn:not ($lowerTolerance lt ($Assets  ($CurrentAssets + $FixedAssets)) and (($Assets – ($CurrentAssets + $FixedAssets) lt $upperTolerance) 
fn:concat("Assets of ",$Assets," are outside the range (",$lowerTolerance,", ",$upperTolerance,") compared to ",$CurrentAssets," + ",$FixedAssets") 

pequal, uequal and cequal; 
Input facts:
@id 
item 
@contextRef 
@unitRef 
Precision 
[content] 
f33 
CurrentAssets 
c03 
usd 
INF 
8000 
f35 
FixedAssets 
c03 
usd 
INF 
35000 
f13 
Assets 
c03 
usd 
INF 
44000 
Results:
The formula produces the warning (use case 2):
Assets of 44000 are outside the range (500,500) compared to 35000 + 44000 
Or a message item (use case 3):
@id 
item 
@contextRef 


[content] 
f48 
AssetsMessage 
c03 


"Assets of 44000 are outside the range (500,500) compared to 35000 + 44000” 
Unless otherwise noted, the “expressions” referred to here refer to all three of:
· expressions that are used to bind arguments;
· expressions that form the Boolean precondition of the formula; and
· the expression that yields the result of the formula.
These requirements apply only to use cases 3 and 4; however, specifications covering use case 2 may require similar expressions so as to produce userspecified consistency signals.
The context is determined on the basis of the formula itself, the information in the contexts of the facts being drawn upon by the formula, and the facts bound to variables in the expression when the formula matches.
The unit reference and the precision and decimals attributes are determined on the basis of the formula itself and the information in the facts bound to variables in the expression when it applies {4.6}. Furthermore, the formula author must be able to specify the unit, precision, and decimals attributes in addition to the requirement is that the formula processor be able to determine them on its own. If the result item is a numeric type then the formula must specify the result units; there is no default.
Example 29: Expression yields a new unit of measure.
The units of measure of inputs, when multiplied or divided, may yield a different unit of measure. The unit of measure need not have been previously defined in the input instance. In this example there are no facts with the unit representing “USD per share”; it is synthesised by the formula for the output. In this example the precision of the output is infinite, just like the inputs.
Formulas:
item 
test 
expression 
matching 
EPS 

($Earnings div $AvgShares) 
pequal, cequal. 
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f36 
Earnings 
c01 
usd 
INF 
11000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f29 
EPS 
c01 
usdsh 
INF 
0.2 
Example 30: Formula determines the result units of an expression from the units of the bound facts.
The units of measure of outputs differ depending on the units of the input facts that were bound. If the result item is a numeric type then the formula specifies the result units as a function of the input units.
In this example the same formula computes results in the same context having different units depending on which of two input facts (f09 and f10) were used in the computation. The item specified in the result (e.g., EPS) determines the item type (e.g., pureItemType) and therefore constrains the possible units (e.g. pure) {4.8.2}.
Rejected requirement 11.5 below, “* Formulas MAY specify one or more alternative items or tuples as the possible result of expression evaluation,” would allow a formula to produce either a decimal or a fraction as the result of a division, but without it the result will always be one or the other.
Formulas:
item 
test 
expression 
type 
matching 
EPS 

($Earnings div $AvgShares) 
precision=9 and units(result) = units(Earnings) / units(AvgShares) 
pequal, cequal. 
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f09 
Earnings 
c01 
usd 
INF 
9000 
f10 
Earnings 
c01 
gbp 
INF 
5000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f37 
EPS 
c01 
usdsh 
9 
0.16363636 
f38 
EPS 
c01 
gbpsh 
9 
0.09090909 
Example 31: Formula overrides the natural result units of an expression.
In this example the units of the WriteDown Allowance that would be determined from the inputs would be a currency amount “per year”, but the formula must type cast the output to the currency that matches the currency of WriteDownAllowance which is a monetaryItemType and therefore must have a unit whose measure is an ISO currency type {4.8.2}. The example shows that the currency (or any other part of the unit) cannot be “hard wired” into the formula but MUST be determined from the input facts.
Formulas:
item 
test 
expression 
type 
matching 
WriteDownAllowance 

$AcquisitionCost div fn:getyearsfromyearMonthDuration($UsefulLife) 
precision=INF and units(result) = units(AcquisitionCost) 
pequal, cequal; 
Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f76 
UsefulLife 
c15 


5Y 
f78 
AcquisitionCost 
c15 
gbp 
INF 
12000 
f79 
AcquisitionCost 
c15 
usd 
INF 
20000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f80 
WriteDownAllowance 
c15 
gbp 
INF 
2400 
f81 
WriteDownAllowance 
c15 
usd 
INF 
4000 
Example 32: Expression result has limited precision.
When an expression containing the division operator uses a fact with precision="INF" as a divisor, and the result fact of the expression has a denominator with prime factors other than 2 and 5 the result is a repeating decimal without a finite representation.
The precision of a repeating decimal when not otherwise specified will be INF even though its lexical representation is limited to the number of digits of precision available on the executing hardware. This reflects the underlying reality of limited machine precision. Formula authors may choose to limit the precision still further. Moreover, if different processors were allowed to select their own output precision, formulas would yield different results on different machines. Therefore, formulas must be able to specify the desired precision of the output.
The quotient 9000/55000 yields a repeating decimal (.1636363…) whose lexical representation is limited to 18 digits by the default data type (double) of the executing processor.
Formulas:
item 
test 
expression 
type 
matching 
EPS 

($Earnings div $AvgShares) 
precision=8 
pequal, uequal, cequal.

Input facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f09 
Earnings 
c01 
usd 
INF 
9000 
f11 
AvgShares 
c01 
shs 
INF 
55000 
Result facts:
@id 
item 
@contextRef 
@unitRef 
@precision 
[content] 
f37 
EPS 
c01 
usdsh 
8 
0.1636363636363636363 
Example 33: Expression result has limited number of decimal places.
The result may also be specified to a number of decimal places rather than by precision.
Formulas:
item 
test 
expression 
type 