Another consequence is that some operations return the value negative zero. This differs from [XML Schema Part 2: Datatypes Second Edition] which defines NaN as being equal to itself and defines only a single zero in the value space. The text accompanying several functions defines behavior for both positive and negative zero inputs and outputs in the interest of alignment with [IEEE ]. A conformant implementation must respect these semantics.

In consequence, the expression As a concession to implementations that rely on implementations of XSD 1. XML Schema 1. The following functions define the semantics of arithmetic operators defined in [XQuery 3. The parameters and return types for the above operators are in most cases declared to be of type xs:numeric , which permits the basic numeric types: xs:integer , xs:decimal , xs:float and xs:double , and types derived from them.

In general the two-argument functions require that both arguments are of the same primitive type, and they return a value of this same type. The exceptions are op:numeric-divide , which returns an xs:decimal if called with two xs:integer operands, and op:numeric-integer-divide which always returns an xs:integer. If the two operands of an arithmetic expression are not of the same type, subtype substitution and numeric type promotion are used to obtain two operands of the same type.

Section B. The result type of operations depends on their argument datatypes and is defined in the following table:. These rules define any operation on any pair of arithmetic types.

Consider the following example:. For this operation, xs:int must be converted to xs:double. This can be done, since by the rules above: xs:int can be substituted for xs:integer , xs:integer can be substituted for xs:decimal , xs:decimal can be promoted to xs:double. As far as possible, the promotions should be done in a single step. Specifically, when an xs:decimal is promoted to an xs:double , it should not be converted to an xs:float and then to xs:double , as this risks loss of precision.

As another example, a user may define height as a derived type of xs:integer with a minimum value of 20 and a maximum value of They may then derive fenceHeight using an enumeration to restrict the permitted set of values to, say, 36, 48 and fenceHeight can be substituted for its base type height and height can be substituted for its base type xs:integer.

The basic rules for addition, subtraction, and multiplication of ordinary numbers are not set out in this specification; they are taken as given. In the case of xs:double and xs:float the rules are as defined in [IEEE ]. The rules for handling division and modulus operations, as well as the rules for handling special values such as infinity and NaN , and exception conditions such as overflow and underflow, are described more explicitly since they are not necessarily obvious.

On overflow and underflow situations during arithmetic operations conforming implementations must behave as follows:. For xs:float and xs:double operations, overflow behavior must be conformant with [IEEE ].

This specification allows the following options:. Raising a dynamic error [ err:FOAR ] via an overflow trap. For xs:float and xs:double operations, underflow behavior must be conformant with [IEEE ]. Raising a dynamic error [ err:FOAR ] via an underflow trap. Returning 0. For xs:decimal operations, overflow behavior must raise a dynamic error [ err:FOAR ]. On underflow, 0. For xs:integer operations, implementations that support limited-precision integer operations must select from the following options:.

They may choose to always raise a dynamic error [ err:FOAR ]. They may provide an · implementation-defined · mechanism that allows users to choose between raising an error and returning a result that is modulo the largest representable integer value.

See [ISO ]. The functions op:numeric-add , op:numeric-subtract , op:numeric-multiply , op:numeric-divide , op:numeric-integer-divide and op:numeric-mod are each defined for pairs of numeric operands, each of which has the same type: xs:integer , xs:decimal , xs:float , or xs:double.

The functions op:numeric-unary-plus and op:numeric-unary-minus are defined for a single operand whose type is one of those same numeric types. For xs:float and xs:double arguments, if either argument is NaN , the result is NaN. Then for addition, subtraction, and multiplication operations, the returned result should be accurate to N digits of precision, and for division and modulus operations, the returned result should be accurate to at least M digits of precision.

The actual precision is · implementation-defined ·. If the number of digits in the mathematical result exceeds the number of digits that the implementation retains for that operation, the result is truncated or rounded in an · implementation-defined · manner.

This Recommendation does not specify whether xs:decimal operations are fixed point or floating point. In an implementation using floating point it is possible for very simple operations to require more digits of precision than are available; for example adding 1e to 1e requires digits of precision for an accurate representation of the result.

The [IEEE ] specification also describes handling of two exception conditions called divideByZero and invalidOperation. The IEEE divideByZero exception is raised not only by a direct attempt to divide by zero, but also by operations such as log 0. The IEEE invalidOperation exception is raised by attempts to call a function with an argument that is outside the function's domain for example, sqrt -1 or log Although IEEE defines these as exceptions, it also defines "default non-stop exception handling" in which the operation returns a defined result, typically positive or negative infinity, or NaN.

With this function library, these IEEE exceptions do not cause a dynamic error at the application level; rather they result in the relevant function or operator returning the defined non-error result.

The [IEEE ] specification distinguishes two NaN values, a quiet NaN and a signaling NaN. These two values are not distinguishable in the XDM model: the value spaces of xs:float and xs:double each include only a single NaN value. This does not prevent the implementation distinguishing them internally, and triggering different · implementation-defined · warning conditions, but such distinctions do not affect the observable behavior of an application using the functions and operators defined in this specification.

For xs:float or xs:double values, if one of the operands is a zero or a finite number and the other is INF or -INF , INF or -INF is returned. If both operands are INF , INF is returned. If both operands are -INF , -INF is returned. If one of the operands is INF and the other is -INF , NaN is returned. Defines the semantics of the "-" operator when applied to two numeric values. For xs:float or xs:double values, if one of the operands is a zero or a finite number and the other is INF or -INF , an infinity of the appropriate sign is returned.

If both operands are INF or -INF , NaN is returned. If one of the operands is INF and the other is -INF , an infinity of the appropriate sign is returned. For xs:float or xs:double values, if one of the operands is a zero and the other is an infinity, NaN is returned.

If one of the operands is a non-zero number and the other is an infinity, an infinity with the appropriate sign is returned. Defines the semantics of the "div" operator when applied to two numeric values. A dynamic error is raised [ err:FOAR ] for xs:decimal and xs:integer operands, if the divisor is positive or negative zero. For xs:float and xs:double operands, floating point division is performed as specified in [IEEE ]. A positive number divided by positive zero returns INF.

A negative number divided by positive zero returns -INF. Division by negative zero returns -INF and INF , respectively. Positive or negative zero divided by positive or negative zero returns NaN. Also, INF or -INF divided by INF or -INF returns NaN. Defines the semantics of the "idiv" operator when applied to two numeric values.

The implementation may adopt a different algorithm provided that it is equivalent to this formulation in all cases where · implementation-dependent · or · implementation-defined · behavior does not affect the outcome, for example, the implementation-defined precision of the result of xs:decimal division.

A dynamic error is raised [ err:FOAR ] if the divisor is positive or negative zero. Defines the semantics of the "mod" operator when applied to two numeric values. This identity holds even in the special case that the dividend is the negative integer of largest possible magnitude for its type and the divisor is -1 the remainder is 0. It follows from this rule that the sign of the result is the sign of the dividend. For xs:float and xs:double operands the following rules apply:.

If the dividend is positive or negative infinity, or the divisor is positive or negative zero 0 , or both, the result is NaN. If the dividend is finite and the divisor is an infinity, the result equals the dividend.

If the dividend is positive or negative zero and the divisor is finite, the result is the same as the dividend. Division is truncating division, analogous to integer division, not [IEEE ] rounding division i. additional digits are truncated, not rounded to the required precision. Defines the semantics of the unary "-" operator when applied to a numeric value.

For xs:integer and xs:decimal arguments, 0 and 0. For xs:float and xs:double arguments, NaN returns NaN , 0. INF returns -INF. This specification defines the following comparison operators on numeric values. Comparisons take two arguments of the same type.

If the arguments are of different types, one argument is promoted to the type of the other as described above in 4. Each comparison operator returns a boolean value. If either, or both, operands are NaN , false is returned. Defines the semantics of the "eq" operator when applied to two numeric values, and is also used in defining the semantics of "ne", "le" and "ge". General rules: see 4. For xs:float and xs:double values, positive zero and negative zero compare equal. INF equals INF , and -INF equals -INF.

NaN does not equal itself. Defines the semantics of the "lt" operator when applied to two numeric values, and is also used in defining the semantics of "le". For xs:float and xs:double values, positive infinity is greater than all other non- NaN values; negative infinity is less than all other non- NaN values. Defines the semantics of the "gt" operator when applied to two numeric values, and is also used in defining the semantics of "ge".

The following functions are defined on numeric types. Each function returns a value of the same type as the type of its argument.

For xs:float and xs:double arguments, if the argument is "NaN", "NaN" is returned. Except for fn:abs , for xs:float and xs:double arguments, if the argument is positive or negative infinity, positive or negative infinity is returned. fn:round and fn:round-half-to-even produce the same result in all cases except when the argument is exactly midway between two values with the required precision.

The result may also be an instance of a type derived from one of these four by restriction. For xs:float and xs:double arguments, if the argument is positive zero or negative zero, then positive zero is returned. If the argument is positive or negative infinity, positive infinity is returned.

For xs:float and xs:double arguments, if the argument is positive zero, then positive zero is returned. If the argument is negative zero, then negative zero is returned. If the argument is less than zero and greater than -1, negative zero is returned. Rounds a value to a specified number of decimal places, rounding upwards if two such values are equally near.

For other values, the argument is cast to xs:decimal using an implementation of xs:decimal that imposes no limits on the number of digits that can be represented.

The function is applied to this xs:decimal value, and the resulting xs:decimal is cast back to xs:float or xs:double as appropriate to form the function result. For arguments of type xs:float and xs:double the results may be counter-intuitive. For example, consider round The result is not This is because the xs:double written as The expression fn:round Not the possible alternative, Rounds a value to a specified number of decimal places, rounding to make the last digit even if two such values are equally near.

If two such values are equally near e. If the argument is NaN , positive or negative zero, or positive or negative infinity, then the result is the same as the argument.

In all other cases, the argument is cast to xs:decimal using an implementation of xs:decimal that imposes no limits on the number of digits that can be represented. If the resulting xs:decimal value is zero, then positive or negative zero is returned according to the sign of the original argument. This function is typically used in financial applications where the argument is of type xs:decimal. For example, consider round-half-to-even xs:float This is because the conversion of the xs:float value represented by the literal The expression fn:round-half-to-even 3.

It is possible to convert strings to values of type xs:integer , xs:float , xs:decimal , or xs:double using the constructor functions described in 18 Constructor functions or using cast expressions as described in 19 Casting. In addition the fn:number function is available to convert strings to values of type xs:double. It differs from the xs:double constructor function in that any value outside the lexical space of the xs:double datatype is converted to the xs:double value NaN.

Calling the zero-argument version of the function is defined to give the same result as calling the single-argument version with the context item. That is, fn:number is equivalent to fn:number. If the conversion to xs:double fails, the xs:double value NaN is returned. As a consequence of the rules given above, a type error occurs if the context item cannot be atomized, or if the result of atomizing the context item is a sequence containing more than one atomic value.

XSD 1. It is · implementation-defined · whether XSD 1. Generally fn:number returns NaN rather than raising a dynamic error if the argument cannot be converted to xs:double.

However, a type error is raised in the usual way if the supplied argument cannot be atomized or if the result of atomization does not match the required argument type. Assume that the context item is the xs:string value " 15 ". Then fn:number returns 1. Formats an integer according to a given picture string, using the conventions of a given natural language if specified.

The two-argument form of this function is · deterministic · , · context-dependent · , and · focus-independent ·. It depends on default language.

The three-argument form of this function is · deterministic · , · context-independent · , and · focus-independent ·. The rules that follow describe how non-negative numbers are output. The primary format token is always present and must not be zero-length. If the string contains one or more semicolons then everything that precedes the last semicolon is taken as the primary format token and everything that follows is taken as the format modifier; if the string contains no semicolon then the entire picture is taken as the primary format token, and the format modifier is taken to be absent which is equivalent to supplying a zero-length string.

A decimal-digit-pattern made up of optional-digit-signs , mandatory-digit-signs , and grouping-separator-signs. A mandatory-digit-sign is a · character · in Unicode category Nd. All mandatory-digit-signs within the format token must be from the same digit family, where a digit family is a sequence of ten consecutive characters in Unicode category Nd , having digit values 0 through 9.

Within the format token, these digits are interchangeable: a three-digit number may thus be indicated equivalently by , , or a grouping-separator-sign is a non-alphanumeric character, that is a · character · whose Unicode category is other than Nd , Nl , No , Lu , Ll , Lt , Lm or Lo. If it contains a digit but does not match this pattern, a dynamic error is raised [ err:FODF ].

If a semicolon is to be used as a grouping separator, then the primary format token as a whole must be followed by another semicolon, to ensure that the grouping separator is not mistaken as a separator between the primary format token and the format modifier. There must be at least one mandatory-digit-sign. There may be zero or more optional-digit-signs , and if present these must precede all mandatory-digit-signs. There may be zero or more grouping-separator-signs.

A grouping-separator-sign must not appear at the start or end of the decimal-digit-pattern , nor adjacent to another grouping-separator-sign. The corresponding output format is a decimal number, using this digit family, with at least as many digits as there are mandatory-digit-signs in the format token.

Thus, a format token 1 generates the sequence 0 1 The position of grouping separators within the format token, counting backwards from the last digit, indicates the position of grouping separators to appear within the formatted number, and the character used as the grouping-separator-sign within the format token indicates the character to be used as the corresponding grouping separator in the formatted number. More specifically, the position of a grouping separator is the number of optional-digit-signs and mandatory-digit-signs appearing between the grouping separator and the right-hand end of the primary format token.

Every positive integer multiple of G that is less than the number of optional-digit-signs and mandatory-digit-signs in the primary format token is the position of a grouping separator. The grouping separator template is a possibly infinite set of position, character pairs. If grouping separators are regular, then the grouping separator template contains one pair of the form n×G, C for every positive integer n where G is the grouping size and C is the grouping character.

Otherwise when grouping separators are not regular , the grouping separator template contains one pair of the form P, C for every grouping separator found in the primary formatting token, where C is the grouping separator character and P is its position. If there are no grouping separators, then the grouping separator template is an empty set. Let S 1 be the result of formatting the supplied number in decimal notation as if by casting it to xs:string.

Let S 2 be the result of padding S 1 on the left with as many leading zeroes as are needed to ensure that it contains at least as many digits as the number of mandatory-digit-signs in the primary format token. Let S 3 be the result of replacing all decimal digits in S 2 with the corresponding digits from the selected digit family. Let S 4 be the result of inserting grouping separators into S 3 : for every position P , character C pair in the grouping separator template where P is less than the number of digits in S 3 , insert character C into S 3 at position P , counting from the right-hand end.

Let S 5 be the result of converting S 4 into ordinal form, if an ordinal modifier is present, as described below. The format token A , which generates the sequence A B C Z AA AB AC The format token a , which generates the sequence a b c z aa ab ac The format token i , which generates the sequence i ii iii iv v vi vii viii ix x The format token I , which generates the sequence I II III IV V VI VII VIII IX X The format token w , which generates numbers written as lower-case words, for example in English, one two three four The format token W , which generates numbers written as upper-case words, for example in English, ONE TWO THREE FOUR The format token Ww , which generates numbers written as title-case words, for example in English, One Two Three Four Any other format token, which indicates a numbering sequence in which that token represents the number 1 one but see the note below.

It is · implementation-defined · which numbering sequences, additional to those listed above, are supported. If an implementation does not support a numbering sequence represented by the given token, it must use a format token of 1. In some traditional numbering sequences additional signs are added to denote that the letters should be interpreted as numbers; these are not included in the format token. An example see also the example below is classical Greek where a dexia keraia x, ʹ and sometimes an aristeri keraia x, ͵ is added.

For all format tokens other than a decimal-digit-pattern , there may be · implementation-defined · lower and upper bounds on the range of numbers that can be formatted using this format token; indeed, for some numbering sequences there may be intrinsic limits.

For the numbering sequences described above any upper bound imposed by the implementation must not be less than one thousand and any lower bound must not be greater than 1. Numbers that fall outside this range must be formatted using the format token 1.

The above expansions of numbering sequences for format tokens such as a and i are indicative but not prescriptive. There are various conventions in use for how alphabetic sequences continue when the alphabet is exhausted, and differing conventions for how roman numerals are written for example, IV versus IIII as the representation of the number 4.

Sometimes alphabetic sequences are used that omit letters such as i and o. This specification does not prescribe the detail of any sequence other than those sequences consisting entirely of decimal digits. Many numbering sequences are language-sensitive. This applies especially to the sequence selected by the tokens w , W and Ww.

It also applies to other sequences, for example different languages using the Cyrillic alphabet use different sequences of characters, each starting with the letter x Cyrillic capital letter A. If the argument is specified, the value should be either an empty sequence or a value that would be valid for the xml:lang attribute see [Extensible Markup Language XML 1.

Note that this permits the identification of sublanguages based on country codes from ISO as well as identification of dialects and regions within a country. The set of languages for which numbering is supported is · implementation-defined ·. That is, if it is present it must consist of one or more of the following, in order:.

either c or o , optionally followed by a sequence of characters enclosed between parentheses, to indicate cardinal or ordinal numbering respectively, the default being cardinal numbering. either a or t , to indicate alphabetic or traditional numbering respectively, the default being · implementation-defined ·. If the o modifier is present, this indicates a request to output ordinal numbers rather than cardinal numbers.

For example, in English, when used with the format token 1 , this outputs the sequence 1st 2nd 3rd 4th The string of characters between the parentheses, if present, is used to select between other possible variations of cardinal or ordinal numbering sequences. The interpretation of this string is · implementation-defined ·.

No error occurs if the implementation does not define any interpretation for the defined string. If ordinal numbering is not supported for the combination of the format token, the language, and the string appearing in parentheses, the request is ignored and cardinal numbers are generated instead.

The use of the a or t modifier disambiguates between numbering sequences that use letters. In many languages there are two commonly used numbering sequences that use letters. One numbering sequence assigns numeric values to letters in alphabetic sequence, and the other assigns numeric values to each letter in some other manner traditional in that language.

In English, these would correspond to the numbering sequences specified by the format tokens a and i. In some languages, the first member of each sequence is the same, and so the format token alone would be ambiguous.

In the absence of the a or t modifier, the default is · implementation-defined ·. A dynamic error is raised [ err:FODF ] if the format token is invalid, that is, if it violates any mandatory rules indicated by an emphasized must or required keyword in the above rules.

For example, the error is raised if the primary format token contains a digit but does not match the required regular expression. Note the careful distinction between conditions that are errors and conditions where fallback occurs. The principle is that an error in the syntax of the format picture will be reported by all processors, while a construct that is recognized by some implementations but not others will never result in an error, but will instead cause a fallback representation of the integer to be used.

If grouping-separator-signs appear at regular intervals within the format token, then the sequence is extrapolated to the left, so grouping separators will be used in the formatted number at every multiple of N. For example, if the format token is 0' then the number one million will be formatted as 1'' , while the number fifteen will be formatted as 0' The only purpose of optional-digit-signs is to mark the position of grouping-separator-signs.

For example, if the format token is ' 0 then the number one million will be formatted as 1'' , while the number fifteen will be formatted as A grouping separator is included in the formatted number only if there is a digit to its left, which will only be the case if either a the number is large enough to require that digit, or b the number of mandatory-digit-signs in the format token requires insignificant leading zeros to be present.

Grouping separators are not designed for effects such as formatting a US telephone number as In general they are not suitable for such purposes because a only single characters are allowed, and b they cannot appear at the beginning or end of the number. Numbers will never be truncated. Given the decimal-digit-pattern 01 , the number three hundred will be output as , despite the absence of any optional-digit-sign.

In some languages, the form of numbers especially ordinal numbers varies depending on the grammatical context: they may have different genders and may decline with the noun that they qualify.

In such cases the string appearing in parentheses after the letter c or o may be used to indicate the variation of the cardinal or ordinal number required. The way in which the variation is indicated will depend on the conventions of the language. For inflected languages that vary the ending of the word, the approach recommended in the previous version of this specification was to indicate the required ending, preceded by a hyphen: for example in German, appropriate values might be o -e , o -er , o -es , o -en.

format-integer , 'w' might return "one hundred and twenty-three". format-integer 14, 'Ww;o -e ', 'de' might return "Vierzehnte". This function can be used to format any numeric quantity, including an integer. For integers, however, the fn:format-integer function offers additional possibilities.

Decimal formats are defined in the static context, and the way they are defined is therefore outside the scope of this specification. XSLT and XQuery both provide custom syntax for creating a decimal format. The static context provides a set of decimal formats. One of the decimal formats is unnamed, the others if any are identified by a QName. There is always an unnamed decimal format available, but its contents are · implementation-defined ·. Each decimal format provides a set of named properties, described in the following table:.

A phrase such as "The minus-sign XP31 character" is to be read as "the character assigned to the minus-sign XP31 property in the relevant decimal format within the static context". For any named or unnamed decimal format, the properties representing characters used in a · picture string · must have distinct values. These properties are decimal-separator XP31 , grouping-separator XP31 , exponent-separator XP31 , percent XP31 , per-mille XP31 , digit XP31 , and pattern-separator XP Furthermore, none of these properties may be equal to any · character · in the · decimal digit family ·.

Returns a string containing a number formatted according to a given picture string, taking account of decimal formats specified in the static context. The two-argument form of this function is · deterministic · , · context-independent · , and · focus-independent ·. The three-argument form of this function is · deterministic · , · context-dependent · , and · focus-independent ·. It depends on decimal formats, and namespaces.

The effect of the two-argument form of the function is equivalent to calling the three-argument form with an empty sequence as the value of the third argument. The syntax of the picture string is described in 4. Note that if an xs:decimal is supplied, it is not automatically promoted to an xs:double , as such promotion can involve a loss of precision. A lexical QName, which is expanded using the statically known namespaces.

The default namespace is not used no prefix means no namespace. A URIQualifiedName using the syntax Q{uri}local , where the URI can be zero-length to indicate a name in no namespace. The evaluation of the fn:format-number function takes place in two phases, an analysis phase described in 4. The analysis phase takes as its inputs the · picture string · and the variables derived from the relevant decimal format in the static context, and produces as its output a number of variables with defined values.

The formatting phase takes as its inputs the number to be formatted and the variables produced by the analysis phase, and produces as its output a string containing a formatted representation of the number.

The result of the function is the formatted string representation of the supplied number. If the processor is able to detect the error statically for example, when the argument is supplied as a string literal , then the processor may optionally signal this as a static error.

A string is an ordered sequence of characters, and this specification uses terms such as "left" and "right", "preceding" and "following" in relation to this ordering, irrespective of the position of the characters when visually rendered on some output medium. Both in the picture string and in the result string, digits with higher significance that is, representing higher powers of ten always precede digits with lower significance, even when the rendered text flow is from right to left.

The following examples assume a default decimal format in which the chosen digits are the ASCII digits , the decimal separator is ". The expression format-number The following example assumes the existence of a decimal format named 'ch' in which the grouping separator is ʹ and the decimal separator is · :.

The following examples assume that the exponent separator is in decimal format 'fortran' is 'E':. The expression format-number 0. This differs from the format-number function previously defined in XSLT 2. The digits will all be from the same decimal digit family, specifically, the sequence of ten consecutive digits starting with the digit assigned to the zero-digit property.

This change is to align format-number which previously used '' with format-dateTime which used ''. The picture string is a sequence of · characters · , in which the characters assigned to the properties decimal-separator XP31 , exponent-separator XP31 , grouping-separator XP31 , and digit XP31 , and pattern-separator XP31 and the members of the · decimal digit family · , are classified as active characters, and all other characters including the values of the properties percent XP31 and per-mille XP31 are classified as passive characters.

A dynamic error is raised [ err:FODF ] if the · picture string · does not conform to the following rules. Note that in these rules the words "preceded" and "followed" refer to characters anywhere in the string, they are not to be read as "immediately preceded" and "immediately followed". A picture-string consists either of a sub-picture, or of two sub-pictures separated by the pattern-separator XP31 character.

A picture-string must not contain more than one instance of the pattern-separator XP31 character. If the picture-string contains two sub-pictures, the first is used for positive and unsigned zero values and the second for negative values. A sub-picture must not contain more than one instance of the decimal-separator XP31 character.

A sub-picture must not contain more than one instance of the percent XP31 or per-mille XP31 characters, and it must not contain one of each.

The mantissa part of a sub-picture defined below must contain at least one character that is either an · optional digit character · or a member of the · decimal digit family ·. A sub-picture must not contain a passive character that is preceded by an active character and that is followed by another active character. A sub-picture must not contain a grouping-separator XP31 character that appears adjacent to a decimal-separator XP31 character, or in the absence of a decimal-separator XP31 character, at the end of the integer part.

A sub-picture must not contain two adjacent instances of the grouping-separator XP31 character. The integer part of a sub-picture defined below must not contain a member of the · decimal digit family · that is followed by an instance of the · optional digit character ·. The fractional part of a sub-picture defined below must not contain an instance of the · optional digit character · that is followed by a member of the · decimal digit family ·.

A character that matches the exponent-separator XP31 property is treated as an exponent-separator-sign if it is both preceded and followed within the sub-picture by an active character. Otherwise, it is treated as a passive character. A sub-picture must not contain more than one character that is treated as an exponent-separator-sign. A sub-picture that contains a percent XP31 or per-mille XP31 character must not contain a character treated as an exponent-separator-sign.

If a sub-picture contains a character treated as an exponent-separator-sign then this must be followed by one or more characters that are members of the · decimal digit family · , and it must not be followed by any active character that is not a member of the · decimal digit family ·. The mantissa part of the sub-picture is defined as the part that appears to the left of the exponent-separator-sign if there is one, or the entire sub-picture otherwise.

The exponent part of the subpicture is defined as the part that appears to the right of the exponent-separator-sign ; if there is no exponent-separator-sign then the exponent part is absent. The integer part of the sub-picture is defined as the part that appears to the left of the decimal-separator XP31 character if there is one, or the entire mantissa part otherwise.

The fractional part of the sub-picture is defined as that part of the mantissa part that appears to the right of the decimal-separator XP31 character if there is one, or the part that appears to the right of the rightmost active character otherwise. The fractional part may be zero-length.

This phase of the algorithm analyzes the · picture string · and the properties from the selected decimal format in the static context, and it has the effect of setting the values of various variables, which are used in the subsequent formatting phase. These variables are listed below. Each is shown with its initial setting and its datatype.

Several variables are associated with each sub-picture. If there are two sub-pictures, then these rules are applied to one sub-picture to obtain the values that apply to positive and unsigned zero numbers, and to the other to obtain the values that apply to negative numbers. If there is only one sub-picture, then the values for both cases are derived from this sub-picture. The integer-part-grouping-positions is a sequence of integers representing the positions of grouping separators within the integer part of the sub-picture.

For each grouping-separator XP31 character that appears within the integer part of the sub-picture, this sequence contains an integer that is equal to the total number of · optional digit character · and · decimal digit family · characters that appear within the integer part of the sub-picture and to the right of the grouping-separator XP31 character.

There is a positive integer G the grouping size such that the position of every grouping-separator in the integer part of the sub-picture is a positive integer multiple of G. Every position in the integer part of the sub-picture that is a positive integer multiple of G is occupied by a grouping-separator.

If the grouping is regular, then the integer-part-grouping-positions sequence contains all integer multiples of G as far as necessary to accommodate the largest possible number. The minimum-integer-part-size is an integer indicating the minimum number of digits that will appear to the left of the decimal-separator character.

It is initially set to the number of · decimal digit family · characters found in the integer part of the sub-picture, but may be adjusted as described below. There is no maximum integer part size. All significant digits in the integer part of the number will be displayed, even if this exceeds the number of · optional digit character · and · decimal digit family · characters in the subpicture. The scaling factor is a non-negative integer used to determine the scaling of the mantissa in exponential notation.

It is set to the number of · decimal digit family · characters found in the integer part of the sub-picture. The prefix is set to contain all passive characters in the sub-picture to the left of the leftmost active character. If the picture string contains only one sub-picture, the prefix for the negative sub-picture is set by concatenating the minus-sign XP31 character and the prefix for the positive sub-picture if any , in that order. The fractional-part-grouping-positions is a sequence of integers representing the positions of grouping separators within the fractional part of the sub-picture.

For each grouping-separator XP31 character that appears within the fractional part of the sub-picture, this sequence contains an integer that is equal to the total number of · optional digit character · and · decimal digit family · characters that appear within the fractional part of the sub-picture and to the left of the grouping-separator XP31 character. There is no need to extrapolate grouping positions on the fractional side, because the number of digits in the output will never exceed the number of · optional digit character · and · decimal digit family · characters in the fractional part of the sub-picture.

The minimum-fractional-part-size is set to the number of · decimal digit family · characters found in the fractional part of the sub-picture. The maximum-fractional-part-size is set to the total number of · optional digit character · and · decimal digit family · characters found in the fractional part of the sub-picture. If the effect of the above rules is that minimum-integer-part-size and maximum-fractional-part-size are both zero, then an adjustment is applied as follows:. This has the effect that with the picture.

e9 , the value 0. This has the effect that with the picture , the value 0. There is at least one · optional digit character · in the integer part of the sub-picture. If after making the above adjustments the minimum-integer-part-size and the minimum-fractional-part-size are both zero, then the minimum-fractional-part-size is set to 1 one.

The minimum-exponent-size is set to the number of · decimal digit family · characters found in the exponent part of the sub-picture if present, or zero otherwise. The rules for the syntax of the picture string ensure that if an exponent separator is present, then the minimum-exponent-size will always be greater than zero.

The suffix is set to contain all passive characters to the right of the rightmost active character in the sub-picture. If there is only one sub-picture, then all variables for positive numbers and negative numbers will be the same, except for prefix : the prefix for negative numbers will be preceded by the minus-sign XP31 character. This section describes the second phase of processing of the fn:format-number function.

This phase takes as input a number to be formatted referred to as the input number , and the variables set up by analyzing the decimal format in the static context and the · picture string · , as described above.

The result of this phase is a string, which forms the return value of the fn:format-number function. If the input number is NaN not a number , the result is the value of the pattern separator XP31 property with no prefix or suffix.

In the rules below, the positive sub-picture and its associated variables are used if the input number is positive, and the negative sub-picture and its associated variables are used if it is negative. For xs:double and xs:float , negative zero is taken as negative, positive zero as positive. For xs:decimal and xs:integer , the positive sub-picture is used for zero.

If the sub-picture contains a percent XP31 character, the adjusted number is the input number multiplied by If the sub-picture contains a per-mille XP31 character, the adjusted number is the input number multiplied by If the multiplication causes numeric overflow, no error occurs, and the adjusted number is positive or negative infinity as appropriate. If the adjusted number is positive or negative infinity, the result is the concatenation of the appropriate prefix , the value of the infinity XP31 property, and the appropriate suffix.

If the minimum exponent size is non-zero, then the adjusted number is scaled to establish a mantissa and an integer exponent. The mantissa and exponent are chosen such that all the following conditions are true:. The primitive type of the mantissa is the same as the primitive type of the adjusted number integer, decimal, float, or double.

The mantissa multiplied by ten to the power of the exp onent is equal to the adjusted number. The mantissa is less than 10 N , and at least 10 N-1 , where N is the scaling factor. If the minimum exponent size is zero, then the mantissa is the adjusted number and there is no exponent. The mantissa is converted if necessary to an xs:decimal value, using an implementation of xs:decimal that imposes no limits on the totalDigits or fractionDigits facets.

If there are several such values that are numerically equal to the mantissa bearing in mind that if the mantissa is an xs:double or xs:float , the comparison will be done by converting the decimal value back to an xs:double or xs:float , the one that is chosen should be one with the smallest possible number of digits not counting leading or trailing zeroes whether significant or insignificant. For example, 1. This value is then rounded so that it uses no more than maximum-fractional-part-size digits in its fractional part.

The rounded number is defined to be the result of converting the mantissa to an xs:decimal value, as described above, and then calling the function fn:round-half-to-even with this converted number as the first argument and the maximum-fractional-part-size as the second argument, again with no limits on the totalDigits or fractionDigits in the result.

The absolute value of the rounded number is converted to a string in decimal notation, using the digits in the · decimal digit family · to represent the ten decimal digits, and the decimal-separator XP31 character to separate the integer part and the fractional part.

This string must always contain a decimal-separator XP31 , and it must contain no leading zeroes and no trailing zeroes. The value zero will at this stage be represented by a decimal-separator XP31 on its own. If the number of digits to the left of the decimal-separator XP31 character is less than minimum-integer-part-size , leading zero digit XP31 characters are added to pad out to that size. If the number of digits to the right of the decimal-separator XP31 character is less than minimum-fractional-part-size , trailing zero digit XP31 characters are added to pad out to that size.

For each integer N in the integer-part-grouping-positions list, a grouping-separator XP31 character is inserted into the string immediately after that digit that appears in the integer part of the number and has N digits between it and the decimal-separator XP31 character, if there is such a digit.

For each integer N in the fractional-part-grouping-positions list, a grouping-separator XP31 character is inserted into the string immediately before that digit that appears in the fractional part of the number and has N digits between it and the decimal-separator XP31 character, if there is such a digit.

If there is no decimal-separator XP31 character in the sub-picture, or if there are no digits to the right of the decimal-separator character in the string, then the decimal-separator character is removed from the string it will be the rightmost character in the string. If an exponent exists, then the string produced from the mantissa as described above is extended with the following, in order: a the exponent-separator XP31 character; b if the exponent is negative, the minus-sign XP31 character; c the value of the exponent represented as a decimal integer, extended if necessary with leading zeroes to make it up to the minimum exponent size , using digits taken from the · decimal digit family ·.

The result of the function is the concatenation of the appropriate prefix , the string conversion of the number as obtained above, and the appropriate suffix. The functions in this section perform trigonometric and other mathematical calculations on xs:double values. They are provided primarily for use in applications performing geometrical computation, for example when generating SVG graphics. Functions are provided to support the six most commonly used trigonometric calculations: sine, cosine and tangent, and their inverses arc sine, arc cosine, and arc tangent.

Other functions such as secant, cosecant, and cotangent are not provided because they are easily computed in terms of these six. The functions in this section with the exception of math:pi are specified by reference to [IEEE ] , where they appear as Recommended operations in section 9. IEEE defines these functions for a variety of floating point formats; this specification defines them only for xs:double values.

The IEEE specification applies with the following caveats:. IEEE states that the preferred quantum is language-defined. In this specification, it is · implementation-defined ·.

IEEE states that certain functions should raise the inexact exception if the result is inexact. In this specification, this exception if it occurs does not result in an error.

Any diagnostic information is outside the scope of this specification. IEEE defines various rounding algorithms for inexact results, and states that the choice of rounding direction, and the mechanisms for influencing this choice, are language-defined.

In this specification, the rounding direction and any mechanisms for influencing it are · implementation-defined ·. Certain operations such as taking the square root of a negative number are defined in IEEE to signal the invalid operation exception and return a quiet NaN. In this specification, such operations return NaN and do not raise an error. The same policy applies to operations such as taking the logarithm of zero that raise a divide-by-zero exception. Operations whose mathematical result is greater than the largest finite xs:double value are defined in IEEE to signal the overflow exception; operations whose mathematical result is closer to zero than the smallest non-zero xs:double value are similarly defined in IEEE to signal the underflow exception.

The treatment of these exceptions in this specification is defined in 4. Returns an approximation to the mathematical constant π.

This function returns the xs:double value whose lexical representation is 3. The treatment of overflow and underflow is defined in 4. The expression math:exp 1 returns 2. The treatment of divideByZero and invalidOperation exceptions is defined in 4. The effect is that if the argument is zero, the result is -INF , and if it is negative, the result is NaN.

The treatment of the divideByZero and invalidOperation exceptions is defined in 4. Some of the consequences are illustrated in the examples below. The expression math:pow -0e0, Odd-valued whole numbers are treated specially.

The expression math:pow -0e0, 3. The expression math:pow The treatment of the invalidOperation exception is defined in 4. The effect is that if the argument is less than zero, the result is NaN. Negative zero is the only case where the result can have negative sign. The treatment of the invalidOperation and underflow exceptions is defined in 4. The expression math:sin math:pi div 2 returns 1. The expression math:sin -math:pi div 2 returns The expression math:cos math:pi div 2 returns 0.

The expression math:cos -math:pi div 2 returns 0. The expression math:tan math:pi div 4 returns 1. The expression math:tan -math:pi div 4 returns The expression 1 div math:tan math:pi div 2 returns 0. But because math:pi div 2 returns an approximation, the result of math:tan math:pi div 2 will be a large but finite number. The expression 1 div math:tan -math:pi div 2 returns But because -math:pi div 2 returns an approximation, the result of math:tan -math:pi div 2 will be a large but finite negative number.

The expression math:asin 1. The expression math:asin The expression math:acos 0 returns 1. The expression math:acos The treatment of the underflow exception is defined in 4. The expression math:atan 1.

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GPS Visualizer is an online utility that creates maps and profiles from geographic data. It is free and easy to use, yet powerful and extremely customizable. Input can be in the form of GPS data tracks and waypoints , driving routes, street addresses, or simple coordinates. Use it to see where you've been, plan where you're going, or quickly visualize geographic data scientific observations, events, business locations, customers, real estate, geotagged photos, etc.

To set more options, use the detailed input pages:. Google Earth KML. Plot data points. Profiles elevation, etc. Convert to GPX. Convert to plain text. Sandbox drawing.

KML overlays. GPS Visualizer is a free service and hopefully always will be; however, if you find it interesting, time-saving, or just plain fun, you can say "thanks" -- and encourage further development -- by clicking the button above and making a contribution via credit card or PayPal. Or, you could send an Amazon wish list item.

wpt , Furuno , Garmin Fitness. fit , Garmin Forerunner. gdb , Geocaching. loc , Google Sheets , iGO. trk , Lowrance. usr , Microsoft Excel spreadsheets. xlsx , NMEA data , OziExplorer. sdf , Timex Trainer , TomTom. pgl , U-blox. ubx , XML feeds , and of course tab-delimited or comma-separated text. No ads? No problem! You can support GPS Visualizer by making a donation with PayPal instead.

click here for example maps. Get started now! Upload a GPS file:. Choose an output format: Leaflet Map Google Maps Google Earth PNG map JPEG map SVG drawing elevation profile GPX file plain text table.

Convert to GPX Convert to plain text Sandbox drawing Geocoding KML overlays. Help keep GPS Visualizer free GPS Visualizer is a free service and hopefully always will be; however, if you find it interesting, time-saving, or just plain fun, you can say "thanks" -- and encourage further development -- by clicking the button above and making a contribution via credit card or PayPal.

com's Connected Car Assistant device and smartphone app can now be converted or mapped with GPS Visualizer. Full documentation of the new feature is available here. are from the OfflineMaps Android app are now supported by GPS Visualizer. The map input forms now give you a choice between Google Maps or Leaflet for browser-viewable HTML maps.

es , are now available in GPS Visualizer's Google Maps. GPS Visualizer will now try to repair those problematic files. tsk extension from XCSoar open-source gliding computer software. Supply a "rotation" field in your input file to make them rotate.

The overzoomed levels show up as gray bars in the zoom control. FSH" has been improved greatly, thanks to Bernhard Fischer's parsefsh utility.

gdb files directly — rather than relying on GPSBabel as an intermediary. This should ensure compatibility with newer. gdb variations. osm XML files downloaded from OpenStreetMap. org can also be used. Read more about it in the new Elevation Gain Tutorial. kmz file that can be used as a Garmin custom map. But if you need more control, you can still supply CSS directives if you want.

Make sure to include a unit, or it'll work the way it always has and just split your profile into that many sections. ods files from OpenOffice. CMT files from Coxmate GPS devices are now supported by GPS Visualizer. Topo maps from IGN. fr have also been added to the Sandbox , but IGN maps are only available in your own maps if you get an API Key from IGN.

This feature has been implemented because OpenStreetMap's data import function requires time stamps for some reason. Not all files will be able to be fixed, but prematurely truncated files the most common error should be able to be handled. com ; look for the Wikiloc "share" link after you create a GPS Visualizer map or profile. ubx binary files from GPS devices that use U-Blox chips.

To access the polygon tool, click the "trk" label under the track tool, and it will change its function. Note that you'll have to export to KML to see the difference; GPX does not support polygons. Seamaster Pro. stk files from Pinpoint Systems' Seamaster Pro software. Note that these files do not include any time information.

BIN waypoint files from Furuno GPS units are now supported by GPS Visualizer. log files from Sygic GPS navigation software are now supported by GPS Visualizer. log files from the "mini " dashboard camera and possibly others in the mini 08xx series. This could be helpful if you want to simply place a geo-located label on a map without an associated graphic.

More global coverage may be coming soon. org as a background option for European data. The contour interval is the same as 4UMaps and OpenCycleMap, but for large areas, the colored relief shading in OpenTopoMaps looks nice.

TXT files from the LawTraK LawMate GPS logger are now supported. TRK and. MRK files from Maptech Terrain Navigator. Here's an example. New server! Because the new machine can handle a little more stress, the upload limit in the input forms has been raised. WPT from Fugawi software can now be read by GPS Visualizer. Swiss Map Mobile. XOL files created by SwissTopo's "Swiss Map Mobile" software. sbn and. sbp tracklogs from Locosys Genie GPS devices, and.

btk tracklogs and. gtk waypoint files from the Bushnell Backtrack D-Tour. cpo files from the EnerGymPro GPS sport watch, and. xml files from CycleOps' VirtualTraining software. SRW files are supported by GPS Visualizer. log files from TrueCam dashboard cameras can now be read by GPS Visualizer. It's based on the batch geocoding utility , so it's entirely browser-based and a bit slow, but it might be helpful if you need a few high-latitude or alpine elevations that aren't covered by GPS Visualizer's built-in DEM database.

A detailed explanation is available in the Tutorials section. Yahoo's free geocoding service no longer exists. rt3" can now be read by GPS Visualizer. If you have older GPS Visualizer maps that need to be upgraded, read the v2 to v3 migration guide. kmz file with GPS Visualizer, and your waypoints use standard Garmin icons , those icons will now be displayed in Google Earth.

olg files "OakLog" ;. trk files from a Czech-language Android app called SmartMaps Navigator, and ancient "WayPtDB. dat" files from some Northstar marine GPS units. VRTP track logs from the ViewRanger app for Android devices can now be read by GPS Visualizer.

Compare Brokers Bonuses Low Deposit Brokers Demo Accounts. Two function signatures are defined to be the same if the declared result types are identical and the declared argument types are pairwise identical. In the first signature, the parameter is omitted and the argument defaults to the context item, referred to as. mmo files ; Nokia Sports Tracker XML export files; and a weird old Garmin text format whose header begins with "Garmin:Trk:L". Note that if an xs:decimal is supplied, it is not automatically promoted to an xs:double , as such promotion can involve a loss of precision. You have a lot of useful information that you need to analyse. The Mac App Store features rich editorial content and great apps for Mac.

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