4 Functions and operators on numerics

This section specifies arithmetic operators on the numeric datatypes defined in [XML Schema Part 2: Datatypes Second Edition].

4.1 Numeric types

The operators described in this section are defined on the following atomic types. Each type whose name is indented is derived from the type whose name appears nearest above with one less level of indentation.

xs:decimal  
xs:integer
xs:float  
xs:double  

They also apply to types derived by restriction from the above types.

The type xs:numeric is defined as a union type whose member types are (in order) xs:double, xs:float, and xs:decimal. This type is implicitly imported into the static context, so it can also be used in defining the signature of user-written functions. Apart from the fact that it is implicitly imported, it behaves exactly like a user-defined type with the same definition. This means, for example:

Note:

This specification uses [IEEE 754-2008] arithmetic for xs:float and xs:double values. One consequence of this is that some operations result in the value NaN (not-a number), which has the unusual property that it is not equal to itself. 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 754-2008]. A conformant implementation must respect these semantics. In consequence, the expression -0.0e0 (which is actually a unary minus operator applied to an xs:double value) will always return negative zero: see 4.2.8 op:numeric-unary-minus. As a concession to implementations that rely on implementations of XSD 1.0, however, when casting from string to double the lexical form -0 may be converted to positive zero, though negative zero is recommended.

XML Schema 1.1 introduces support for positive and negative zero as distinct values, and also uses the [IEEE 754-2008] semantics for comparisons involving NaN.

4.2 Arithmetic operators on numeric values

The following functions define the semantics of arithmetic operators defined in [XQuery 4.1: An XML Query Language] and [XML Path Language (XPath) 4.0] on these numeric types.

Operator Meaning
op:numeric-add Addition
op:numeric-subtract Subtraction
op:numeric-multiply Multiplication
op:numeric-divide Division
op:numeric-integer-divide Integer division
op:numeric-mod Modulus
op:numeric-unary-plus Unary plus
op:numeric-unary-minus Unary minus (negation)

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.1 Type Promotion XP31 and Section B.2 Operator Mapping XP31 describe the semantics of these operations in detail.

The result type of operations depends on their argument datatypes and is defined in the following table:

Operator Returns
op:operation(xs:integer, xs:integer) xs:integer (except for op:numeric-divide(integer, integer), which returns xs:decimal)
op:operation(xs:decimal, xs:decimal) xs:decimal
op:operation(xs:float, xs:float) xs:float
op:operation(xs:double, xs:double) xs:double
op:operation(xs:integer) xs:integer
op:operation(xs:decimal) xs:decimal
op:operation(xs:float) xs:float
op:operation(xs:double) xs:double

These rules define any operation on any pair of arithmetic types. Consider the following example:

op:operation(xs:int, xs:double) => op:operation(xs:double, xs:double)

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 100. They may then derive fenceHeight using an enumeration to restrict the permitted set of values to, say, 36, 48 and 60.

op:operation(fenceHeight, xs:integer) => op:operation(xs:integer, xs:integer)

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 754-2008]. 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:

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.

For xs:decimal values, let N be the number of digits of precision supported by the implementation, and let M (M <= N) be the minimum limit on the number of digits required for conformance (18 digits for XSD 1.0, 16 digits for XSD 1.1). 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.

Note:

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 1e100 to 1e-100 requires 200 digits of precision for an accurate representation of the result.

The [IEEE 754-2008] 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(-1)). 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 underlying IEEE exception may be notified to the application or to the user by some ·implementation-defined· warning condition, but the observable effect on an application using the functions and operators defined in this specification is simply to return the defined result (typically -INF, +INF, or NaN) with no error.

The [IEEE 754-2008] 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.

4.2.1 op:numeric-add

Summary

Returns the arithmetic sum of its operands: ($arg1 + $arg2).

Operator Mapping

Defines the semantics of the "+" operator when applied to two numeric values

Signature
op:numeric-add(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

Notes

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.

4.2.2 op:numeric-subtract

Summary

Returns the arithmetic difference of its operands: ($arg1 - $arg2).

Operator Mapping

Defines the semantics of the "-" operator when applied to two numeric values.

Signature
op:numeric-subtract(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

Notes

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.

4.2.3 op:numeric-multiply

Summary

Returns the arithmetic product of its operands: ($arg1 * $arg2).

Operator Mapping

Defines the semantics of the "*" operator when applied to two numeric values.

Signature
op:numeric-multiply(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

Notes

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.

4.2.4 op:numeric-divide

Summary

Returns the arithmetic quotient of its operands: ($arg1 div $arg2).

Operator Mapping

Defines the semantics of the "div" operator when applied to two numeric values.

Signature
op:numeric-divide(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

As a special case, if the types of both $arg1 and $arg2 are xs:integer, then the return type is xs:decimal.

Error Conditions

A dynamic error is raised [err:FOAR0001] for xs:decimal and xs:integer operands, if the divisor is (positive or negative) zero.

Notes

For xs:float and xs:double operands, floating point division is performed as specified in [IEEE 754-2008]. 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.

4.2.5 op:numeric-integer-divide

Summary

Performs an integer division.

Operator Mapping

Defines the semantics of the "idiv" operator when applied to two numeric values.

Signature
op:numeric-integer-divide(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:integer
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

If $arg2 is INF or -INF, and $arg1 is not INF or -INF, then the result is zero.

Otherwise, subject to limits of precision and overflow/underflow conditions, the result is the largest (furthest from zero) xs:integer value $N such that the following expression is true:

fn:abs($N * $arg2) le fn:abs($arg1) 
               and fn:compare($N * $arg2, 0) eq fn:compare($arg1, 0).

Note:

The second term in this condition ensures that the result has the correct sign.

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.

Error Conditions

A dynamic error is raised [err:FOAR0001] if the divisor is (positive or negative) zero.

A dynamic error is raised [err:FOAR0002] if either operand is NaN or if $arg1 is INF or -INF.

Notes

Except in situations involving errors, loss of precision, or overflow/underflow, the result of $a idiv $b is the same as ($a div $b) cast as xs:integer.

The semantics of this function are different from integer division as defined in programming languages such as Java and C++.

Examples

The expression op:numeric-integer-divide(10,3) returns 3.

The expression op:numeric-integer-divide(3,-2) returns -1.

The expression op:numeric-integer-divide(-3,2) returns -1.

The expression op:numeric-integer-divide(-3,-2) returns 1.

The expression op:numeric-integer-divide(9.0,3) returns 3.

The expression op:numeric-integer-divide(-3.5,3) returns -1.

The expression op:numeric-integer-divide(3.0,4) returns 0.

The expression op:numeric-integer-divide(3.1E1,6) returns 5.

The expression op:numeric-integer-divide(3.1E1,7) returns 4.

4.2.6 op:numeric-mod

Summary

Returns the remainder resulting from dividing $arg1, the dividend, by $arg2, the divisor.

Operator Mapping

Defines the semantics of the "mod" operator when applied to two numeric values.

Signature
op:numeric-mod(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

The operation a mod b for operands that are xs:integer or xs:decimal, or types derived from them, produces a result such that (a idiv b)*b+(a mod b) is equal to a and the magnitude of the result is always less than the magnitude of b. 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 either operand is NaN, the result is NaN.

  • 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.

  • In the remaining cases, where neither positive or negative infinity, nor positive or negative zero, nor NaN is involved, the result obeys (a idiv b)*b+(a mod b) = a. Division is truncating division, analogous to integer division, not [IEEE 754-2008] rounding division i.e. additional digits are truncated, not rounded to the required precision.

Error Conditions

A dynamic error is raised [err:FOAR0001] for xs:integer and xs:decimal operands, if $arg2 is zero.

Examples

The expression op:numeric-mod(10,3) returns 1.

The expression op:numeric-mod(6,-2) returns 0.

The expression op:numeric-mod(4.5,1.2) returns 0.9.

The expression op:numeric-mod(1.23E2, 0.6E1) returns 3.0E0.

4.2.7 op:numeric-unary-plus

Summary

Returns its operand with the sign unchanged: (+ $arg).

Operator Mapping

Defines the semantics of the unary "+" operator applied to a numeric value.

Signature
op:numeric-unary-plus(
$arg as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

The returned value is equal to $arg, and is an instance of xs:integer, xs:decimal, xs:double, or xs:float depending on the type of $arg.

Notes

Because function conversion rules are applied in the normal way, the unary + operator can be used to force conversion of an untyped node to a number: the result of +@price is the same as xs:double(@price) if the type of @price is xs:untypedAtomic.

4.2.8 op:numeric-unary-minus

Summary

Returns its operand with the sign reversed: (- $arg).

Operator Mapping

Defines the semantics of the unary "-" operator when applied to a numeric value.

Signature
op:numeric-unary-minus(
$arg as xs:numeric
) as xs:numeric
Rules

General rules: see 4.2 Arithmetic operators on numeric values.

The returned value is an instance of xs:integer, xs:decimal, xs:double, or xs:float depending on the type of $arg.

For xs:integer and xs:decimal arguments, 0 and 0.0 return 0 and 0.0, respectively. For xs:float and xs:double arguments, NaN returns NaN, 0.0E0 returns -0.0E0 and vice versa. INF returns -INF. -INF returns INF.

4.3 Comparison operators on numeric values

The six value comparison operators eq, ne, lt, le, gt, and ge are defined in terms of two underlying functions: op:numeric-equal and op:numeric-less-than. These functions are defined to operate on values of the same type.

If the arguments are of different types, then one of them is converted to the type of the other using the following rules, in order:

  1. If one operand is xs:float then it is converted to xs:double.

  2. If one operand is xs:double and the other is xs:decimal, then:

    1. If the xs:double operand is -INF, +INF, or NaN, then the xs:decimal value is cast to xs:double and the values are compared as doubles.

    2. Otherwise, the xs:double value is converted to a decimal number with no rounding or loss of precision and the values are compared acccording to their mathematical values.

      Note:

      Every instance of xs:float, xs:double, and xs:decimal, other than the values -INF, +INF, and NaN, can be represented exactly as a decimal number provided enough digits are available both before and after the decimal point. The effect of this rule is that the comparison operators are transitive.

      In edge cases this rule may give a different result from earlier releases. For example in XPath 3.1 (depending on the implementation-defined precision), the values xs:decimal('1.0000000000100000000001') and xs:double('1.00000000001') might compare as equal; in this version of the specification, they are not equal.

      This change removes the problems caused for fn:distinct-values and xsl:for-each-group as a result of non-transitivity, and it aligns the semantics of the eq operator (used also in fn:distinct-values, fn:index-of, and fn:deep-equal) with the semantics of the op:same-key comparison used for maps.

This specification defines the following comparison operators on numeric values. Each comparison operator returns a boolean value. If either, or both, operands are NaN, false is returned.

Function Meaning
op:numeric-equal Returns true if and only if the value of $arg1 is equal to the value of $arg2.
op:numeric-less-than Returns true if and only if $arg1 is numerically less than $arg2.

4.3.1 op:numeric-equal

Summary

Returns true if and only if the value of $arg1 is equal to the value of $arg2.

Operator Mapping

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".

Signature
op:numeric-equal(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:boolean
Rules

General rules: see 4.3 Comparison operators on numeric values.

The two arguments will always have been converted so they have the same primitive type (both xs:float, both xs:double, or both xs:decimal. The values are compared using the equality rules defined in [XML Schema Part 2: Datatypes Second Edition].

Notes

Positive and negative zero compare equal. If $arg1 or $arg2 is NaN, the function returns false.

4.3.2 op:numeric-less-than

Summary

Returns true if and only if $arg1 is numerically less than $arg2.

Operator Mapping

Defines the semantics of the lt operator when applied to two numeric values, and is also used in defining the semantics of le, gt, and ge.

Signature
op:numeric-less-than(
$arg1 as xs:numeric,
$arg2 as xs:numeric
) as xs:boolean
Rules

General rules: see 4.3 Comparison operators on numeric values.

The two arguments will always have been converted so they have the same primitive type (both xs:float, both xs:double, or both xs:decimal. The values are compared using the ordering rules defined in [XML Schema Part 2: Datatypes Second Edition]. If the ordering relation of the two values is imcomparable (which happens when one or both of them is NaN, then the result is false.

Notes

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. Positive and negative zero compare equal. If $arg1 or $arg2 is NaN, the function returns false.

4.4 Functions on numeric values

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

Function Meaning
fn:abs Returns the absolute value of $value.
fn:ceiling Rounds $value upwards to a whole number.
fn:floor Rounds $value downwards to a whole number.
fn:round Rounds a value to a specified number of decimal places, rounding upwards if two such values are equally near.
fn:round-half-to-even Rounds a value to a specified number of decimal places, rounding to make the last digit even if two such values are equally near.
fn:is-NaN Returns true if the argument is the xs:float or xs:double value NaN.

Note:

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.

Other ways of rounding midway values can be achieved as follows:

  • Towards negative infinity: -fn:round(-$x)

  • Away from zero: fn:round(fn:abs($x))*fn:compare($x,0)

  • Towards zero: fn:abs(fn:round(-$x))*-fn:compare($x,0)

4.4.1 fn:abs

Summary

Returns the absolute value of $value.

Signature
fn:abs(
$value as xs:numeric?
) as xs:numeric?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

General rules: see 4.4 Functions on numeric values.

If $value is negative the function returns -$value, otherwise it returns $value.

For the four types xs:float, xs:double, xs:decimal and xs:integer, it is guaranteed that if the type of $value is an instance of type T then the result will also be an instance of T. The result may also be an instance of a type derived from one of these four by restriction. For example, if $value is an instance of xs:positiveInteger then the value of $value may be returned unchanged.

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.

Examples

The expression fn:abs(10.5) returns 10.5.

The expression fn:abs(-10.5) returns 10.5.

4.4.2 fn:ceiling

Summary

Rounds $value upwards to a whole number.

Signature
fn:ceiling(
$value as xs:numeric?
) as xs:numeric?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

General rules: see 4.4 Functions on numeric values.

The function returns the smallest (closest to negative infinity) number with no fractional part that is not less than $value.

For the four types xs:float, xs:double, xs:decimal and xs:integer, it is guaranteed that if the type of $value is an instance of type T then the result will also be an instance of T. The result may also be an instance of a type derived from one of these four by restriction. For example, if $value is an instance of xs:decimal then the result may be an instance of xs:integer.

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.

Examples

The expression fn:ceiling(10.5) returns 11.

The expression fn:ceiling(-10.5) returns -10.

4.4.3 fn:floor

Summary

Rounds $value downwards to a whole number.

Signature
fn:floor(
$value as xs:numeric?
) as xs:numeric?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

General rules: see 4.4 Functions on numeric values.

The function returns the largest (closest to positive infinity) number with no fractional part that is not greater than $value.

For the four types xs:float, xs:double, xs:decimal and xs:integer, it is guaranteed that if the type of $value is an instance of type T then the result will also be an instance of T. The result may also be an instance of a type derived from one of these four by restriction. For example, if $value is an instance of xs:decimal then the result may be an instance of xs:integer.

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.

Examples

The expression fn:floor(10.5) returns 10.

The expression fn:floor(-10.5) returns -11.

4.4.4 fn:round

Summary

Rounds a value to a specified number of decimal places, rounding upwards if two such values are equally near.

Signatures
fn:round(
$value as xs:numeric?
) as xs:numeric?
fn:round(
$value as xs:numeric?,
$precision as xs:integer
) as xs:numeric?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

General rules: see 4.4 Functions on numeric values.

The function returns the nearest (that is, numerically closest) value to $value that is a multiple of ten to the power of minus $precision. If two such values are equally near (for example, if the fractional part in $value is exactly .5), the function returns the one that is closest to positive infinity.

For the four types xs:float, xs:double, xs:decimal and xs:integer, it is guaranteed that if the type of $value is an instance of type T then the result will also be an instance of T. The result may also be an instance of a type derived from one of these four by restriction. For example, if $value is an instance of xs:decimal and $precision is less than one, then the result may be an instance of xs:integer.

The single-argument version of this function produces the same result as the two-argument version with $precision=0 (that is, it rounds to a whole number).

When $value is of type xs:float and xs:double:

  1. If $value is NaN, positive or negative zero, or positive or negative infinity, then the result is the same as the argument.

  2. 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. If the resulting xs:decimal value is zero, then positive or negative zero is returned according to the sign of $value.

Notes

This function is typically used with a non-zero $precision in financial applications where the argument is of type xs:decimal. For arguments of type xs:float and xs:double the results may be counter-intuitive. For example, consider round(35.425e0, 2). The result is not 35.43, as might be expected, but 35.42. This is because the xs:double written as 35.425e0 has an exact value equal to 35.42499999999..., which is closer to 35.42 than to 35.43.

Examples

The expression fn:round(2.5) returns 3.0.

The expression fn:round(2.4999) returns 2.0.

The expression fn:round(-2.5) returns -2.0. (Not the possible alternative, -3).

The expression fn:round(1.125, 2) returns 1.13.

The expression fn:round(8452, -2) returns 8500.

The expression fn:round(3.1415e0, 2) returns 3.14e0.

4.4.5 fn:round-half-to-even

Summary

Rounds a value to a specified number of decimal places, rounding to make the last digit even if two such values are equally near.

Signatures
fn:round-half-to-even(
$value as xs:numeric?
) as xs:numeric?
fn:round-half-to-even(
$value as xs:numeric?,
$precision as xs:integer
) as xs:numeric?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

General rules: see 4.4 Functions on numeric values.

The function returns the nearest (that is, numerically closest) value to $value that is a multiple of ten to the power of minus $precision. If two such values are equally near (e.g. if the fractional part in $value is exactly .500...), the function returns the one whose least significant digit is even.

For the four types xs:float, xs:double, xs:decimal and xs:integer, it is guaranteed that if the type of $value is an instance of type T then the result will also be an instance of T. The result may also be an instance of a type derived from one of these four by restriction. For example, if $value is an instance of xs:decimal and $precision is less than one, then the result may be an instance of xs:integer.

The first signature of this function produces the same result as the second signature with $precision=0.

For arguments of type xs:float and xs:double:

  1. If the argument is NaN, positive or negative zero, or positive or negative infinity, then the result is the same as the argument.

  2. 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. 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. If the resulting xs:decimal value is zero, then positive or negative zero is returned according to the sign of the original argument.

Notes

This function is typically used in financial applications where the argument is of type xs:decimal. For arguments of type xs:float and xs:double the results may be counter-intuitive. For example, consider round-half-to-even(xs:float(150.015), 2). The result is not 150.02 as might be expected, but 150.01. This is because the conversion of the xs:float value represented by the literal 150.015 to an xs:decimal produces the xs:decimal value 150.014999389..., which is closer to 150.01 than to 150.02.

Examples

The expression fn:round-half-to-even(0.5) returns 0.0.

The expression fn:round-half-to-even(1.5) returns 2.0.

The expression fn:round-half-to-even(2.5) returns 2.0.

The expression fn:round-half-to-even(3.567812e+3, 2) returns 3567.81e0.

The expression fn:round-half-to-even(4.7564e-3, 2) returns 0.0e0.

The expression fn:round-half-to-even(35612.25, -2) returns 35600.

4.4.6 fn:is-NaN

Summary

Returns true if the argument is the xs:float or xs:double value NaN.

Signature
fn:is-NaN(
$value as xs:anyAtomicType
) as xs:boolean
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

The function returns true if the argument is the xs:float or xs:double value NaN; otherwise it returns false.

Examples

The expression fn:is-NaN(23) returns false().

The expression fn:is-NaN("NaN") returns false().

The expression fn:is-NaN(fn:number("twenty-three")) returns true().

The expression fn:is-NaN(math:sqrt(-1)) returns true().

History

New in 4.0. Accepted 2022-09-20.

4.5 Parsing numbers

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.

Function Meaning
fn:number Returns the value indicated by $value or, if $value is not specified, the context item after atomization, converted to an xs:double.

4.5.1 fn:number

Summary

Returns the value indicated by $value or, if $value is not specified, the context item after atomization, converted to an xs:double.

Signatures
fn:number() as xs:double
fn:number(
$value as xs:anyAtomicType? := .
) as xs:double
Properties

The zero-argument form of this function is ·deterministic·, ·context-dependent·, and ·focus-dependent·.

The one-argument form of this function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

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(.), as defined by the rules that follow.

If $value is the empty sequence or if $value cannot be converted to an xs:double, the xs:double value NaN is returned.

Otherwise, $value is converted to an xs:double following the rules of 19.1.2.2 Casting to xs:double. If the conversion to xs:double fails, the xs:double value NaN is returned.

Error Conditions

A dynamic error is raised [err:XPDY0002]XP if $value is omitted and the context item is absentDM40.

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.

Notes

XSD 1.1 allows the string +INF as a representation of positive infinity; XSD 1.0 does not. It is ·implementation-defined· whether XSD 1.1 is supported.

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.

Examples

The expression fn:number($item1/quantity) returns 5.0e0.

The expression fn:number($item2/description) returns xs:double('NaN').

Assume that the context item is the xs:string value "15". Then fn:number() returns 1.5e1.

4.6 Formatting integers

Function Meaning
fn:format-integer Formats an integer according to a given picture string, using the conventions of a given natural language if specified.

4.6.1 fn:format-integer

Summary

Formats an integer according to a given picture string, using the conventions of a given natural language if specified.

Signatures
fn:format-integer(
$value as xs:integer?,
$picture as xs:string
) as xs:string
fn:format-integer(
$value as xs:integer?,
$picture as xs:string,
$lang as xs:string?
) as xs:string
Properties

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·.

Rules

If $value is an empty sequence, the function returns a zero-length string.

In all other cases, the $picture argument describes the format in which $value is output.

The rules that follow describe how non-negative numbers are output. If the value of $value is negative, the rules below are applied to the absolute value of $value, and a minus sign is prepended to the result.

The value of $picture consists of a primary format token, optionally followed by a format modifier. 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).

The primary format token is classified as one of the following:

  1. A decimal-digit-pattern made up of optional-digit-signs, mandatory-digit-signs, and grouping-separator-signs.

    • The optional-digit-sign is the character "#".

    • 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 000, 001, or 999.

    • 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 the primary format token contains at least one Unicode digit then it is taken as a decimal digit pattern, and in this case it must match the regular expression ^((\p{Nd}|#|[^\p{N}\p{L}])+?)$. If it contains a digit but does not match this pattern, a dynamic error is raised [err:FODF1310].

    Note:

    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 2 ... 10 11 12 ..., and a format token 01 (or equivalently, 00 or 99) generates the sequence 00 01 02 ... 09 10 11 12 ... 99 100 101. A format token of &#x661; (Arabic-Indic digit one) generates the sequence ١ then ٢ then ٣ ...

    The grouping-separator-signs are handled as follows:

    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.

    2. 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.

    3. Grouping separators are defined to be regular if the following conditions apply:

      1. There is at least one grouping separator.

      2. Every grouping separator is the same character (call it C).

      3. There is a positive integer G (the grouping size) such that:

        1. The position of every grouping separator is an integer multiple of G, and

        2. 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.

    4. The grouping separator template is a (possibly infinite) set of (position, character) pairs.

    5. 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.

    6. 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.

    7. Note:

      If there are no grouping separators, then the grouping separator template is an empty set.

    The number is formatted as follows:

    1. Let S1 be the result of formatting the supplied number in decimal notation as if by casting it to xs:string.

    2. Let S2 be the result of padding S1 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.

    3. Let S3 be the result of replacing all decimal digits (0-9) in S2 with the corresponding digits from the selected digit family.

    4. Let S4 be the result of inserting grouping separators into S3: for every (position P, character C) pair in the grouping separator template where P is less than the number of digits in S3, insert character C into S3 at position P, counting from the right-hand end.

    5. Let S5 be the result of converting S4 into ordinal form, if an ordinal modifier is present, as described below.

    6. The result of the function is then S5.

  2. The format token A, which generates the sequence A B C ... Z AA AB AC....

  3. The format token a, which generates the sequence a b c ... z aa ab ac....

  4. The format token i, which generates the sequence i ii iii iv v vi vii viii ix x ....

  5. The format token I, which generates the sequence I II III IV V VI VII VIII IX X ....

  6. The format token w, which generates numbers written as lower-case words, for example in English, one two three four ...

  7. The format token W, which generates numbers written as upper-case words, for example in English, ONE TWO THREE FOUR ...

  8. The format token Ww, which generates numbers written as title-case words, for example in English, One Two Three Four ...

  9. 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.

    Note:

    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 (x0374, ʹ) and sometimes an aristeri keraia (x0375, ͵) 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 example, the format token &#x2460; (circled digit one, ①) has a range imposed by the Unicode character repertoire — zero to 20 in Unicode versions prior to 3.2, or zero to 50 in subsequent versions. For the numbering sequences described above any upper bound imposed by the implementation must not be less than 1000 (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 #x410 (Cyrillic capital letter A). In such cases, the $lang argument specifies which language's conventions are to be used. 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) 1.0 (Fifth Edition)]). Note that this permits the identification of sublanguages based on country codes (from ISO 3166-1) as well as identification of dialects and regions within a country.

The set of languages for which numbering is supported is ·implementation-defined·. If the $lang argument is absent, or is set to an empty sequence, or is invalid, or is not a language supported by the implementation, then the number is formatted using the default language from the dynamic context.

The format modifier must be a string that matches the regular expression ^([co](\(.+\))?)?[at]?$. 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 ..., and when used with the format token w outputs the sequence first second third fourth ....

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.

It is ·implementation-defined· what combinations of values of the format token, the language, and the cardinal/ordinal modifier are supported. 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·.

Error Conditions

A dynamic error is raised [err:FODF1310] 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.

Notes
  1. 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.

  2. The following notes apply when a decimal-digit-pattern is used:

    1. 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 0'000 then the number one million will be formatted as 1'000'000, while the number fifteen will be formatted as 0'015.

    2. 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 1'000'000, while the number fifteen will be formatted as 15. 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.

    3. Grouping separators are not designed for effects such as formatting a US telephone number as (365)123-9876. 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.

    4. Numbers will never be truncated. Given the decimal-digit-pattern 01, the number three hundred will be output as 300, despite the absence of any optional-digit-sign.

  3. The following notes apply when ordinal numbering is selected using the o modifier.

    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).

    Another approach, which might usefully be adopted by an implementation based on the open-source ICU localization library [ICU], or any other library making use of the Unicode Common Locale Data Repository [Unicode CLDR], is to allow the value in parentheses to be the name of a registered numbering rule set for the language in question, conventionally prefixed with a percent sign: for example, o(%spellout-ordinal-masculine), or c(%spellout-cardinal-year).

Examples

The expression format-integer(123, '0000') returns "0123".

format-integer(123, 'w') might return "one hundred and twenty-three"

Ordinal numbering in Italian: The specification "1;o(-º)" with $lang equal to it, if supported, should produce the sequence:

1º 2º 3º 4º ...

The specification "Ww;o" with $lang equal to it, if supported, should produce the sequence:

Primo Secondo Terzo Quarto Quinto ...

The expression format-integer(21, '1;o', 'en') returns "21st".

format-integer(14, 'Ww;o(-e)', 'de') might return "Vierzehnte"

The expression format-integer(7, 'a') returns "g".

The expression format-integer(57, 'I') returns "LVII".

The expression format-integer(1234, '#;##0;') returns "1;234".

4.7 Formatting numbers

This section defines a function for formatting decimal and floating point numbers.

Function Meaning
fn:format-number Returns a string containing a number formatted according to a given picture string, taking account of decimal formats specified in the static context.

Note:

This function can be used to format any numeric quantity, including an integer. For integers, however, the fn:format-integer function offers additional possibilities. Note also that the picture strings used by the two functions are not 100% compatible, though they share some options in common.

4.7.1 Defining a decimal format

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:

Name Type Usage (non-normative)
decimal-separator A single ·character· Defines the character used to represent the decimal point (typically ".") both in the picture string and in the formatted number.
grouping-separator A single ·character· Defines the character used to separate groups of digits (typically ",") both in the picture string and in the formatted number.
exponent-separator A single ·character· Defines the character used to separate the mantissa from the exponent in scientific notation (typically "e") both in the picture string and in the formatted number.
infinity A ·string· Defines the string used to represent the value positive or negative infinity in the formatted number (typically "Infinity")
minus-sign A single ·character· Defines the character used as a minus sign in the formatted number if there is no subpicture for formatting negative numbers (typically "-", x2D)
NaN A ·string· Defines the string used to represent the value NaN in the formatted number
percent A single ·character· Defines the character used as a percent sign (typically "%") both in the picture string and in the formatted number
per-mille A single ·character· Defines the character used as a per-mille sign (typically "‰", x2030) both in the picture string and in the formatted number
zero-digit A single ·character·, which must be a character in Unicode category Nd with decimal digit value 0 (zero) Defines the characters used in the picture string to represent a mandatory digit: for example, if the zero-digit is "0" then any of the digits "0" to "9" may be used (interchangeably) in the picture string to represent a mandatory digit, and in the formatted number the characters "0" to "9" will be used to represent the digits zero to nine.
digit A single ·character· Defines the character used in the picture string to represent an optional digit (typically "#")
pattern-separator A single ·character· Defines the character used in the picture string to separate the positive and negative subpictures (typically ";")

Note:

A phrase such as "The minus-signXP31 character" is to be read as "the character assigned to the minus-signXP31 property in the relevant decimal format within the static context".

[Definition] The decimal digit family of a decimal format is the sequence of ten digits with consecutive Unicode ·codepoints· starting with the character that is the value of the zero-digitXP31 property.

[Definition] The optional digit character is the character that is the value of the digitXP31 property.

For any named or unnamed decimal format, the properties representing characters used in a ·picture string· must have distinct values. These properties are decimal-separatorXP31 , grouping-separatorXP31, exponent-separatorXP31, percentXP31, per-milleXP31, digitXP31, and pattern-separatorXP31. Furthermore, none of these properties may be equal to any ·character· in the ·decimal digit family·.

4.7.2 fn:format-number

Summary

Returns a string containing a number formatted according to a given picture string, taking account of decimal formats specified in the static context.

Signatures
fn:format-number(
$value as xs:numeric?,
$picture as xs:string
) as xs:string
fn:format-number(
$value as xs:numeric?,
$picture as xs:string,
$decimal-format-name as union(xs:string, xs:QName)?
) as xs:string
Properties

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.

Rules

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 function formats $value as a string using the ·picture string· specified by the $picture argument and the decimal-format named by the $decimal-format-name argument, or the unnamed decimal-format, if there is no $decimal-format-name argument. The syntax of the picture string is described in 4.7.3 Syntax of the picture string.

The $value argument may be of any numeric data type (xs:double, xs:float, xs:decimal, or their subtypes including xs:integer). 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.

If the supplied value of the $value argument is an empty sequence, the function behaves as if the supplied value were the xs:double value NaN.

The value of $decimal-format-name, if present and non-empty, must be either an xs:QName, or a string which after removal of leading and trailing whitespace is in the form of an EQName as defined in the XPath 4.0 grammar, that is one of the following:

  • 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 decimal format that is used is the decimal format in the static context whose name matches $decimal-format-name if supplied, or the unnamed decimal format in the static context otherwise.

The evaluation of the fn:format-number function takes place in two phases, an analysis phase described in 4.7.4 Analyzing the picture string and a formatting phase described in 4.7.5 Formatting the number.

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.

Error Conditions

A dynamic error is raised [err:FODF1280] if the $decimal-format-name argument is supplied as an xs:string that is neither a valid lexical QName nor a valid URIQualifiedName, or if it uses a prefix that is not found in the statically known namespaces, or if the static context does not contain a declaration of a decimal-format with a matching expanded QName. 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.

Notes

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.

Examples

The following examples assume a default decimal format in which the chosen digits are the ASCII digits 0-9, the decimal separator is ".", the grouping separator is ",", the minus-sign is "-", and the percent-sign is "%".

The expression format-number(12345.6, '#,###.00') returns "12,345.60".

The expression format-number(12345678.9, '9,999.99') returns "12,345,678.90".

The expression format-number(123.9, '9999') returns "0124".

The expression format-number(0.14, '01%') returns "14%".

The expression format-number(-6, '000') returns "-006".

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 expression format-number(1234.5678, '#ʹ##0·00', 'ch') returns "1ʹ234·57".

The following examples assume that the exponent separator is in decimal format 'fortran' is 'E':

The expression format-number(1234.5678, '00.000E0', 'fortran') returns "12.346E2".

The expression format-number(0.234, '0.0E0', 'fortran') returns "2.3E-1".

The expression format-number(0.234, '#.00E0', 'fortran') returns "0.23E0".

The expression format-number(0.234, '.00E0', 'fortran') returns ".23E0".

4.7.3 Syntax of the picture string

Note:

This differs from the format-number function previously defined in XSLT 2.0 in that any digit can be used in the picture string to represent a mandatory digit: for example the picture strings '000', '001', and '999' are equivalent. 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 '000') with format-dateTime (which used '001').

[Definition] The formatting of a number is controlled by a picture string. The picture string is a sequence of ·characters·, in which the characters assigned to the properties decimal-separatorXP31 , exponent-separatorXP31, grouping-separatorXP31, and digitXP31, and pattern-separatorXP31 and the members of the ·decimal digit family·, are classified as active characters, and all other characters (including the values of the properties percentXP31 and per-milleXP31) are classified as passive characters.

A dynamic error is raised [err:FODF1310] 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-separatorXP31 character. A picture-string must not contain more than one instance of the pattern-separatorXP31 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-separatorXP31 character.

  • A sub-picture must not contain more than one instance of the percentXP31 or per-milleXP31 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-separatorXP31 character that appears adjacent to a decimal-separatorXP31 character, or in the absence of a decimal-separatorXP31 character, at the end of the integer part.

  • A sub-picture must not contain two adjacent instances of the grouping-separatorXP31 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-separatorXP31 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 percentXP31 or per-milleXP31 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-separatorXP31 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-separatorXP31 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.

4.7.4 Analyzing the picture string

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 variables are as follows:

  • 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-separatorXP31 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-separatorXP31 character.

    The grouping is defined to be regular if the following conditions apply:

    1. There is an least one grouping-separator in the integer part of the sub-picture.

    2. 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.

    3. 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.

    Note:

    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-signXP31 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-separatorXP31 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-separatorXP31 character.

    Note:

    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:

    • If an exponent separator is present then:

      • minimum-fractional-part-size is changed to 1 (one).

      • maximum-fractional-part-size is changed to 1 (one).

      Note:

      This has the effect that with the picture #.e9, the value 0.123 is formatted as 0.1e0

    • Otherwise:

      • minimum-integer-part-size is changed to 1 (one).

      Note:

      This has the effect that with the picture #, the value 0.23 is formatted as 0

  • If all the following conditions are true:

    • An exponent separator is present

    • The minimum-integer-part-size is zero

    • There is at least one ·optional digit character· in the integer part of the sub-picture

    then the minimum-integer-part-size is changed to 1 (one).

    Note:

    This has the effect that with the picture .9e9, the value 0.1 is formatted as .1e0, while with the picture #.9e9, it is formatted as 0.1e0

  • 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.

    Note:

    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.

Note:

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-signXP31 character.

4.7.5 Formatting the number

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.

The algorithm for this second stage of processing is as follows:

  1. If the input number is NaN (not a number), the result is the value of the pattern separatorXP31 property (with no prefix or suffix).

  2. 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.

  3. The adjusted number is determined as follows:

    • If the sub-picture contains a percentXP31 character, the adjusted number is the input number multiplied by 100.

    • If the sub-picture contains a per-milleXP31 character, the adjusted number is the input number multiplied by 1000.

    • Otherwise, the adjusted number is the input number.

    If the multiplication causes numeric overflow, no error occurs, and the adjusted number is positive or negative infinity as appropriate.

  4. If the adjusted number is positive or negative infinity, the result is the concatenation of the appropriate prefix, the value of the infinityXP31 property, and the appropriate suffix.

  5. If the minimum exponent size is non-zero, and the adjusted number 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 exponent is equal to the adjusted number.

    • The mantissa (unless it is zero) is less than 10N, and at least 10N-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.

    If the minimum exponent size is non-zero and the adjusted number is zero, then the mantissa is the adjusted number and the exponent is zero.

  6. 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, 1.0 is preferred to 0.9999999999, and 100000000 is preferred to 100000001. 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.

  7. 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-separatorXP31 character to separate the integer part and the fractional part. This string must always contain a decimal-separatorXP31, and it must contain no leading zeroes and no trailing zeroes. The value zero will at this stage be represented by a decimal-separatorXP31 on its own.

  8. If the number of digits to the left of the decimal-separatorXP31 character is less than minimum-integer-part-size, leading zero digitXP31 characters are added to pad out to that size.

  9. If the number of digits to the right of the decimal-separatorXP31 character is less than minimum-fractional-part-size, trailing zero digitXP31 characters are added to pad out to that size.

  10. For each integer N in the integer-part-grouping-positions list, a grouping-separatorXP31 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-separatorXP31 character, if there is such a digit.

  11. For each integer N in the fractional-part-grouping-positions list, a grouping-separatorXP31 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-separatorXP31 character, if there is such a digit.

  12. If there is no decimal-separatorXP31 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).

  13. If an exponent exists, then the string produced from the mantissa as described above is extended with the following, in order: (a) the exponent-separatorXP31 character; (b) if the exponent is negative, the minus-signXP31 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·.

  14. 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.

4.8 Trigonometric and exponential functions

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 754-2008], 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:

  1. IEEE states that the preferred quantum is language-defined. In this specification, it is ·implementation-defined·.

  2. 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.

  3. 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·.

  4. 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. Any diagnostic information is outside the scope of this specification.

  5. 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 4.2 Arithmetic operators on numeric values.

Function Meaning
math:pi Returns an approximation to the mathematical constant π.
math:exp Returns the value of ex where x is the argument value.
math:exp10 Returns the value of 10x, where x is the supplied argument value.
math:log Returns the natural logarithm of the argument.
math:log10 Returns the base-ten logarithm of the argument.
math:pow Returns the result of raising the first argument to the power of the second.
math:sqrt Returns the non-negative square root of the argument.
math:sin Returns the sine of the argument. The argument is an angle in radians.
math:cos Returns the cosine of the argument. The argument is an angle in radians.
math:tan Returns the tangent of the argument. The argument is an angle in radians.
math:asin Returns the arc sine of the argument.
math:acos Returns the arc cosine of the argument.
math:atan Returns the arc tangent of the argument.
math:atan2 Returns the angle in radians subtended at the origin by the point on a plane with coordinates (x, y) and the positive x-axis.

4.8.1 math:pi

Summary

Returns an approximation to the mathematical constant π.

Signature
math:pi() as xs:double
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

This function returns the xs:double value whose lexical representation is 3.141592653589793e0

Examples

The expression 2*math:pi() returns 6.283185307179586e0.

The expression 60 * (math:pi() div 180) converts an angle of 60 degrees to radians.

4.8.2 math:exp

Summary

Returns the value of ex where x is the argument value.

Signature
math:exp(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the mathematical constant e raised to the power of $value, as defined in the [IEEE 754-2008] specification of the exp function applied to 64-bit binary floating point values.

Notes

The treatment of overflow and underflow is defined in 4.2 Arithmetic operators on numeric values.

Examples

The expression math:exp(()) returns ().

The expression math:exp(0) returns 1.0e0.

The expression math:exp(1) returns 2.7182818284590455e0 (approximately).

The expression math:exp(2) returns 7.38905609893065e0.

The expression math:exp(-1) returns 0.36787944117144233e0.

The expression math:exp(math:pi()) returns 23.140692632779267e0.

The expression math:exp(xs:double('NaN')) returns xs:double('NaN').

The expression math:exp(xs:double('INF')) returns xs:double('INF').

The expression math:exp(xs:double('-INF')) returns 0.0e0.

4.8.3 math:exp10

Summary

Returns the value of 10x, where x is the supplied argument value.

Signature
math:exp10(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is ten raised to the power of $value, as defined in the [IEEE 754-2008] specification of the exp10 function applied to 64-bit binary floating point values.

Notes

The treatment of overflow and underflow is defined in 4.2 Arithmetic operators on numeric values.

Examples

The expression math:exp10(()) returns ().

The expression math:exp10(0) returns 1.0e0.

The expression math:exp10(1) returns 1.0e1.

The expression math:exp10(0.5) returns 3.1622776601683795e0.

The expression math:exp10(-1) returns 1.0e-1.

The expression math:exp10(xs:double('NaN')) returns xs:double('NaN').

The expression math:exp10(xs:double('INF')) returns xs:double('INF').

The expression math:exp10(xs:double('-INF')) returns 0.0e0.

4.8.4 math:log

Summary

Returns the natural logarithm of the argument.

Signature
math:log(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the natural logarithm of $value, as defined in the [IEEE 754-2008] specification of the log function applied to 64-bit binary floating point values.

Notes

The treatment of divideByZero and invalidOperation exceptions is defined in 4.2 Arithmetic operators on numeric values. The effect is that if the argument is zero, the result is -INF, and if it is negative, the result is NaN.

Examples

The expression math:log(()) returns ().

The expression math:log(0) returns xs:double('-INF').

The expression math:log(math:exp(1)) returns 1.0e0.

The expression math:log(1.0e-3) returns -6.907755278982137e0.

The expression math:log(2) returns 0.6931471805599453e0.

The expression math:log(-1) returns xs:double('NaN').

The expression math:log(xs:double('NaN')) returns xs:double('NaN').

The expression math:log(xs:double('INF')) returns xs:double('INF').

The expression math:log(xs:double('-INF')) returns xs:double('NaN').

4.8.5 math:log10

Summary

Returns the base-ten logarithm of the argument.

Signature
math:log10(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the base-10 logarithm of $value, as defined in the [IEEE 754-2008] specification of the log10 function applied to 64-bit binary floating point values.

Notes

The treatment of divideByZero and invalidOperation exceptions is defined in 4.2 Arithmetic operators on numeric values. The effect is that if the argument is zero, the result is -INF, and if it is negative, the result is NaN.

Examples

The expression math:log10(()) returns ().

The expression math:log10(0) returns xs:double('-INF').

The expression math:log10(1.0e3) returns 3.0e0.

The expression math:log10(1.0e-3) returns -3.0e0.

The expression math:log10(2) returns 0.3010299956639812e0.

The expression math:log10(-1) returns xs:double('NaN').

The expression math:log10(xs:double('NaN')) returns xs:double('NaN').

The expression math:log10(xs:double('INF')) returns xs:double('INF').

The expression math:log10(xs:double('-INF')) returns xs:double('NaN').

4.8.6 math:pow

Summary

Returns the result of raising the first argument to the power of the second.

Signature
math:pow(
$x as xs:double?,
$y as xs:numeric
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $x is the empty sequence, the function returns the empty sequence.

If $y is an instance of xs:integer, the result is $x raised to the power of $y as defined in the [IEEE 754-2008] specification of the pown function applied to a 64-bit binary floating point value and an integer.

Otherwise $y is converted to an xs:double by numeric promotion, and the result is $x raised to the power of $y as defined in the [IEEE 754-2008] specification of the pow function applied to two 64-bit binary floating point values.

Notes

The treatment of the divideByZero and invalidOperation exceptions is defined in 4.2 Arithmetic operators on numeric values. Some of the consequences are illustrated in the examples below.

Examples

The expression math:pow((), 93.7) returns ().

The expression math:pow(2, 3) returns 8.0e0.

The expression math:pow(-2, 3) returns -8.0e0.

The expression math:pow(2, -3) returns 0.125e0.

The expression math:pow(-2, -3) returns -0.125e0.

The expression math:pow(2, 0) returns 1.0e0.

The expression math:pow(0, 0) returns 1.0e0.

The expression math:pow(xs:double('INF'), 0) returns 1.0e0.

The expression math:pow(xs:double('NaN'), 0) returns 1.0e0.

The expression math:pow(-math:pi(), 0) returns 1.0e0.

The expression math:pow(0e0, 3) returns 0.0e0.

The expression math:pow(0e0, 4) returns 0.0e0.

The expression math:pow(-0e0, 3) returns -0.0e0.

The expression math:pow(0, 4) returns 0.0e0.

The expression math:pow(0e0, -3) returns xs:double('INF').

The expression math:pow(0e0, -4) returns xs:double('INF').

The expression math:pow(-0e0, -3) returns xs:double('-INF').

The expression math:pow(0, -4) returns xs:double('INF').

The expression math:pow(16, 0.5e0) returns 4.0e0.

The expression math:pow(16, 0.25e0) returns 2.0e0.

The expression math:pow(0e0, -3.0e0) returns xs:double('INF').

The expression math:pow(-0e0, -3.0e0) returns xs:double('-INF'). (Odd-valued whole numbers are treated specially).

The expression math:pow(0e0, -3.1e0) returns xs:double('INF').

The expression math:pow(-0e0, -3.1e0) returns xs:double('INF').

The expression math:pow(0e0, 3.0e0) returns 0.0e0.

The expression math:pow(-0e0, 3.0e0) returns -0.0e0. (Odd-valued whole numbers are treated specially).

The expression math:pow(0e0, 3.1e0) returns 0.0e0.

The expression math:pow(-0e0, 3.1e0) returns 0.0e0.

The expression math:pow(-1, xs:double('INF')) returns 1.0e0.

The expression math:pow(-1, xs:double('-INF')) returns 1.0e0.

The expression math:pow(1, xs:double('INF')) returns 1.0e0.

The expression math:pow(1, xs:double('-INF')) returns 1.0e0.

The expression math:pow(1, xs:double('NaN')) returns 1.0e0.

The expression math:pow(-2.5e0, 2.0e0) returns 6.25e0.

The expression math:pow(-2.5e0, 2.00000001e0) returns xs:double('NaN').

4.8.7 math:sqrt

Summary

Returns the non-negative square root of the argument.

Signature
math:sqrt(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the mathematical non-negative square root of $value as defined in the [IEEE 754-2008] specification of the squareRoot function applied to 64-bit binary floating point values.

Notes

The treatment of the invalidOperation exception is defined in 4.2 Arithmetic operators on numeric values. The effect is that if the argument is less than zero, the result is NaN.

If $value is positive or negative zero, positive infinity, or NaN, then the result is $value. (Negative zero is the only case where the result can have negative sign)

Examples

The expression math:sqrt(()) returns ().

The expression math:sqrt(0.0e0) returns 0.0e0.

The expression math:sqrt(-0.0e0) returns -0.0e0.

The expression math:sqrt(1.0e6) returns 1.0e3.

The expression math:sqrt(2.0e0) returns 1.4142135623730951e0.

The expression math:sqrt(-2.0e0) returns xs:double('NaN').

The expression math:sqrt(xs:double('NaN')) returns xs:double('NaN').

The expression math:sqrt(xs:double('INF')) returns xs:double('INF').

The expression math:sqrt(xs:double('-INF')) returns xs:double('NaN').

4.8.8 math:sin

Summary

Returns the sine of the argument. The argument is an angle in radians.

Signature
math:sin(
$radians as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $radians is the empty sequence, the function returns the empty sequence.

Otherwise the result is the sine of $radians (which is treated as an angle in radians) as defined in the [IEEE 754-2008] specification of the sin function applied to 64-bit binary floating point values.

Notes

The treatment of the invalidOperation and underflow exceptions is defined in 4.2 Arithmetic operators on numeric values.

If $radians is positive or negative zero, the result is $radians.

If $radians is positive or negative infinity, or NaN, then the result is NaN.

Otherwise the result is always in the range -1.0e0 to +1.0e0

Examples

The expression math:sin(()) returns ().

The expression math:sin(0) returns 0.0e0.

The expression math:sin(-0.0e0) returns -0.0e0.

The expression math:sin(math:pi() div 2) returns 1.0e0 (approximately).

The expression math:sin(-math:pi() div 2) returns -1.0e0 (approximately).

The expression math:sin(math:pi()) returns 0.0e0 (approximately).

The expression math:sin(xs:double('NaN')) returns xs:double('NaN').

The expression math:sin(xs:double('INF')) returns xs:double('NaN').

The expression math:sin(xs:double('-INF')) returns xs:double('NaN').

4.8.9 math:cos

Summary

Returns the cosine of the argument. The argument is an angle in radians.

Signature
math:cos(
$radians as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $radians is the empty sequence, the function returns the empty sequence.

If $radians is positive or negative infinity, or NaN, then the result is NaN.

Otherwise the result is the cosine of $θ (which is treated as an angle in radians) as defined in the [IEEE 754-2008] specification of the cos function applied to 64-bit binary floating point values.

Notes

The treatment of the invalidOperation exception is defined in 4.2 Arithmetic operators on numeric values.

If $radians is positive or negative zero, the result is $radians.

If $radiansis positive or negative infinity, or NaN, then the result is NaN.

Otherwise the result is always in the range -1.0e0 to +1.0e0

Examples

The expression math:cos(()) returns ().

The expression math:cos(0) returns 1.0e0.

The expression math:cos(-0.0e0) returns 1.0e0.

The expression math:cos(math:pi() div 2) returns 0.0e0 (approximately).

The expression math:cos(-math:pi() div 2) returns 0.0e0 (approximately).

The expression math:cos(math:pi()) returns -1.0e0 (approximately).

The expression math:cos(xs:double('NaN')) returns xs:double('NaN').

The expression math:cos(xs:double('INF')) returns xs:double('NaN').

The expression math:cos(xs:double('-INF')) returns xs:double('NaN').

4.8.10 math:tan

Summary

Returns the tangent of the argument. The argument is an angle in radians.

Signature
math:tan(
$radians as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $radians is the empty sequence, the function returns the empty sequence.

Otherwise the result is the tangent of $radians (which is treated as an angle in radians) as defined in the [IEEE 754-2008] specification of the tan function applied to 64-bit binary floating point values.

Notes

The treatment of the invalidOperation and underflow exceptions is defined in 4.2 Arithmetic operators on numeric values.

If $radians is positive or negative infinity, or NaN, then the result is NaN.

Examples

The expression math:tan(()) returns ().

The expression math:tan(0) returns 0.0e0.

The expression math:tan(-0.0e0) returns -0.0e0.

The expression math:tan(math:pi() div 4) returns 1.0e0 (approximately).

The expression math:tan(-math:pi() div 4) returns -1.0e0 (approximately).

The expression 1 div math:tan(math:pi() div 2) returns 0.0e0 (approximately). (Mathematically, tan(π/2) is positive infinity. 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 -0.0e0 (approximately). (Mathematically, tan(-π/2) is negative infinity. 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:tan(math:pi()) returns 0.0e0 (approximately).

The expression math:tan(xs:double('NaN')) returns xs:double('NaN').

The expression math:tan(xs:double('INF')) returns xs:double('NaN').

The expression math:tan(xs:double('-INF')) returns xs:double('NaN').

4.8.11 math:asin

Summary

Returns the arc sine of the argument.

Signature
math:asin(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the arc sine of $value as defined in the [IEEE 754-2008] specification of the asin function applied to 64-bit binary floating point values. The result is in the range -π/2 to +π/2 radians.

Notes

The treatment of the invalidOperation and underflow exceptions is defined in 4.2 Arithmetic operators on numeric values.

If $value is positive or negative zero, the result is $value.

If $value is NaN, or if its absolute value is greater than one, then the result is NaN.

In other cases the result is an xs:double value representing an angle θ in radians in the range -π/2 <= θ <= +π/2.

Examples

The expression math:asin(()) returns ().

The expression math:asin(0) returns 0.0e0.

The expression math:asin(-0.0e0) returns -0.0e0.

The expression math:asin(1.0e0) returns 1.5707963267948966e0 (approximately).

The expression math:asin(-1.0e0) returns -1.5707963267948966e0 (approximately).

The expression math:asin(2.0e0) returns xs:double('NaN').

The expression math:asin(xs:double('NaN')) returns xs:double('NaN').

The expression math:asin(xs:double('INF')) returns xs:double('NaN').

The expression math:asin(xs:double('-INF')) returns xs:double('NaN').

4.8.12 math:acos

Summary

Returns the arc cosine of the argument.

Signature
math:acos(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the arc cosine of $value, as defined in the [IEEE 754-2008] specification of the acos function applied to 64-bit binary floating point values. The result is in the range zero to +π radians.

Notes

The treatment of the invalidOperation exception is defined in 4.2 Arithmetic operators on numeric values.

If $value is NaN, or if its absolute value is greater than one, then the result is NaN.

In other cases the result is an xs:double value representing an angle θ in radians in the range 0 <= θ <= +π.

Examples

The expression math:acos(()) returns ().

The expression math:acos(0) returns 1.5707963267948966e0 (approximately).

The expression math:acos(-0.0e0) returns 1.5707963267948966e0 (approximately).

The expression math:acos(1.0e0) returns 0.0e0.

The expression math:acos(-1.0e0) returns 3.141592653589793e0 (approximately).

The expression math:acos(2.0e0) returns xs:double('NaN').

The expression math:acos(xs:double('NaN')) returns xs:double('NaN').

The expression math:acos(xs:double('INF')) returns xs:double('NaN').

The expression math:acos(xs:double('-INF')) returns xs:double('NaN').

4.8.13 math:atan

Summary

Returns the arc tangent of the argument.

Signature
math:atan(
$value as xs:double?
) as xs:double?
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

If $value is the empty sequence, the function returns the empty sequence.

Otherwise the result is the arc tangent of $value, as defined in the [IEEE 754-2008] specification of the atan function applied to 64-bit binary floating point values. The result is in the range -π/2 to +π/2 radians.

Notes

The treatment of the underflow exception is defined in 4.2 Arithmetic operators on numeric values.

If $value is positive or negative zero, the result is $value.

If $value is NaN then the result is NaN.

In other cases the result is an xs:double value representing an angle θ in radians in the range -π/2 <= θ <= +π/2.

Examples

The expression math:atan(()) returns ().

The expression math:atan(0) returns 0.0e0.

The expression math:atan(-0.0e0) returns -0.0e0.

The expression math:atan(1.0e0) returns 0.7853981633974483e0 (approximately).

The expression math:atan(-1.0e0) returns -0.7853981633974483e0 (approximately).

The expression math:atan(xs:double('NaN')) returns xs:double('NaN').

The expression math:atan(xs:double('INF')) returns 1.5707963267948966e0 (approximately).

The expression math:atan(xs:double('-INF')) returns -1.5707963267948966e0 (approximately).

4.8.14 math:atan2

Summary

Returns the angle in radians subtended at the origin by the point on a plane with coordinates (x, y) and the positive x-axis.

Signature
math:atan2(
$y as xs:double,
$x as xs:double
) as xs:double
Properties

This function is ·deterministic·, ·context-independent·, and ·focus-independent·.

Rules

The result is the value of atan2(y, x) as defined in the [IEEE 754-2008] specification of the atan2 function applied to 64-bit binary floating point values. The result is in the range -π to +π radians.

Notes

The treatment of the underflow exception is defined in 4.2 Arithmetic operators on numeric values. The following rules apply when the values are finite and non-zero, (subject to rules for overflow, underflow and approximation).

If either argument is NaN then the result is NaN.

If $x is positive, then the value of atan2($y, $x) is atan($y div $x).

If $x is negative, then:

  • If $y is positive, then the value of atan2($y, $x) is atan($y div $x) + π.

  • If $y is negative, then the value of atan2($y, $x) is atan($y div $x) - π.

Some results for special values of the arguments are shown in the examples below.

Examples

The expression math:atan2(+0.0e0, 0.0e0) returns 0.0e0.

The expression math:atan2(-0.0e0, 0.0e0) returns -0.0e0.

The expression math:atan2(+0.0e0, -0.0e0) returns 3.141592653589793e0.

The expression math:atan2(-0.0e0, -0.0e0) returns -3.141592653589793e0.

The expression math:atan2(-1, 0.0e0) returns -1.5707963267948966e0.

The expression math:atan2(+1, 0.0e0) returns 1.5707963267948966e0.

The expression math:atan2(-0.0e0, -1) returns -3.141592653589793e0.

The expression math:atan2(+0.0e0, -1) returns 3.141592653589793e0.

The expression math:atan2(-0.0e0, +1) returns -0.0e0.

The expression math:atan2(+0.0e0, +1) returns +0.0e0.

4.9 Random Numbers

Function Meaning
fn:random-number-generator Returns a random number generator, which can be used to generate sequences of random numbers.

4.9.1 fn:random-number-generator

Summary

Returns a random number generator, which can be used to generate sequences of random numbers.

Signatures
fn:random-number-generator() as item-type(rng)
fn:random-number-generator(
$seed as xs:anyAtomicType?
) as item-type(rng)
Properties

This function is ·deterministic·, ·context-independent·, ·focus-independent·, and ·higher-order·.

Rules

The function returns a random number generator. A random number generator is represented as a value of type rng, defined as follows:

record(
       number   as xs:double,
       next     as (function() as record(number, next, permute, *)),
       permute  as (function(item()*) as item()*),
       *
   )

That is, the result of the function is a map containing three entries. The keys of each entry are strings:

  1. The entry with key "number" holds a random number; it is an xs:double greater than or equal to zero (0.0e0), and less than one (1.0e0).

  2. The entry with key "next" is a zero-arity function that can be called to return another random number generator.

    The properties of this function are as follows:

    • name: absent

    • parameter names: ()

    • signature: () => map(xs:string, item())

    • non-local variable bindings: none

    • implementation: implementation-dependent

  3. The entry with key "permute" is a function with arity 1 (one), which takes an arbitrary sequence as its argument, and returns a random permutation of that sequence.

    The properties of this function are as follows:

    • name: absent

    • parameter names: ("arg")

    • signature: (item()*) => item()*

    • non-local variable bindings: none

    • implementation: implementation-dependent

Calling the fn:random-number-generator function with no arguments is equivalent to calling the single-argument form of the function with an implementation-dependent seed.

Calling the fn:random-number-generator function with an empty sequence as $seed is equivalent to calling the single-argument form of the function with an implementation-dependent seed.

If a $seed is supplied, it may be an atomic value of any type.

Both forms of the function are ·deterministic·: calling the function twice with the same arguments, within a single ·execution scope·, produces the same results.

The value of the number entry should be such that the distribution of numbers is uniform: for example, the probability of the number being in the range 0.1e0 to 0.2e0 is the same as the probability of its being in the range 0.8e0 to 0.9e0.

The function returned in the permute entry should be such that all permutations of the supplied sequence are equally likely to be chosen.

The map returned by the fn:random-number-generator function may contain additional entries beyond those specified here, but it must match the type item-type(rng) defined above. The meaning of any additional entries is ·implementation-defined·. To avoid conflict with any future version of this specification, the keys of any such entries should start with an underscore character.

Notes

It is not meaningful to ask whether the functions returned in the next and permute functions resulting from two separate calls with the same seed are "the same function", but the functions must be equivalent in the sense that calling them produces the same sequence of random numbers.

The repeatability of the results of function calls in different execution scopes is outside the scope of this specification. It is recommended that when the same seed is provided explicitly, the same random number sequence should be delivered even in different execution scopes; while if no seed is provided, the processor should choose a seed that is likely to be different from one execution scope to another. (The same effect can be achieved explicitly by using fn:current-dateTime() as a seed.)

The specification does not place strong conformance requirements on the actual randomness of the result; this is left to the implementation. It is desirable, for example, when generating a sequence of random numbers that the sequence should not get into a repeating loop; but the specification does not attempt to dictate this.

Examples

The following example returns a random permutation of the integers in the range 1 to 100: fn:random-number-generator()?permute(1 to 100)

The following example returns a 10% sample of the items in an input sequence $seq, chosen at random: fn:random-number-generator()?permute($seq)[position() = 1 to (count($seq) idiv 10)]

The following code defines a function that can be called to produce a random sequence of xs:double values in the range zero to one, of specified length:

declare %public function r:random-sequence($length as xs:integer) as xs:double* {
  r:random-sequence($length, fn:random-number-generator())
};

declare %private function r:random-sequence($length as xs:integer, 
                                            $G as record(number as xs:double, next as function(*), *)) {
  if ($length eq 0)
  then ()
  else ($G?number, r:random-sequence($length - 1, $G?next()))
};

r:random-sequence(200);