1. Introduction
The value definition field of each CSS property can contain keywords, data types (which appear between < and >), and information on how they can be combined. Generic data types (<length> being the most widely used) that can be used by many properties are described in this specification, while more specific data types (e.g., <spacinglimit>) are described in the corresponding modules.
1.1. Module Interactions
This module replaces and extends the data type definitions in [CSS21] sections 1.4.2.1, 4.3, and A.2.
2. Value Definition Syntax
The value definition syntax described here is used to define the set of valid values for CSS properties (and the valid syntax of many other parts of CSS). A value so described can have one or more components.
2.1. Component Value Types
Component value types are designated in several ways:
 keyword values (such as auto, disc, etc.),
which appear literally, without quotes (e.g.
auto
)  basic data types, which appear between < and > (e.g., <length>, <percentage>, etc.). For numeric data types, this type notation can annotate any range restrictions using the bracketed range notation described below.
 types that have the same range of values as a property bearing the same name (e.g., <'borderwidth'>, <'backgroundattachment'>, etc.). In this case, the type name is the property name (complete with quotes) between the brackets. Such a type does not include CSSwide keywords such as inherit, and also does not include any toplevel commaseparatedlist multiplier (i.e. if a property named pairing is defined as [ <customident> <integer>? ]#, then <'pairing'> is equivalent to [ <customident> <integer>? ], not [ <customident> <integer>? ]#).
 nonterminals that do not share the same name as a property. In this case, the nonterminal name appears between < and >, as in <spacinglimit>. Notice the distinction between <borderwidth> and <'borderwidth'>: the latter is defined as the value of the borderwidth property, the former requires an explicit expansion elsewhere. The definition of a nonterminal is typically located near its first appearance in the specification.
Some property value definitions also include the slash (/), the comma (,), and/or parentheses as literals. These represent their corresponding tokens. Other nonkeyword literal characters that may appear in a component value, such as “+”, must be written enclosed in single quotes.
Commas specified in the grammar are implicitly omissible in some circumstances, when used to separate optional terms in the grammar. Within a toplevel list in a property or other CSS value, or a function’s argument list, a comma specified in the grammar must be omitted if:
 all items preceding the comma have been omitted
 all items following the comma have been omitted
 multiple commas would be adjacent (ignoring white space/comments), due to the items between the commas being omitted.
example( first? , second? , third? )
Given this grammar, writing example(first, second, third) is valid, as is example(first, second) or example(first, third) or example(second). However, example(first, , third) is invalid, as one of those commas are no longer separating two options; similarly, example(,second) and example(first,) are invalid. example(first second) is also invalid, as commas are still required to actually separate the options.
If commas were not implicitly omittable, the grammar would have to be much more complicated to properly express the ways that the arguments can be omitted, greatly obscuring the simplicity of the feature.
All CSS properties also accept the CSSwide keyword values as the sole component of their property value.
For readability these are not listed explicitly in the property value syntax definitions.
For example, the full value definition of bordercolor is <color>{1,4}  inherit  initial  unset
(even though it is listed as <color>{1,4}
).
Note: This implies that, in general, combining these keywords with other component values in the same declaration results in an invalid declaration. For example, background: url(corner.png) norepeat, inherit; is invalid.
2.2. Component Value Combinators
Component values can be arranged into property values as follows:
 Juxtaposing components means that all of them must occur, in the given order.
 A double ampersand (&&) separates two or more components, all of which must occur, in any order.
 A double bar () separates two or more options: one or more of them must occur, in any order.
 A bar () separates two or more alternatives: exactly one of them must occur.
 Brackets ([ ]) are for grouping.
Juxtaposition is stronger than the double ampersand, the double ampersand is stronger than the double bar, and the double bar is stronger than the bar. Thus, the following lines are equivalent:
a b  c  d && e f [ a b ]  [ c  [ d && [ e f ]]]
For reorderable combinators (, &&), ordering of the grammar does not matter: components in the same grouping may be interleaved in any order. Thus, the following lines are equivalent:
a  b  c b  a  c
2.3. Component Value Multipliers
Every type, keyword, or bracketed group may be followed by one of the following modifiers:
 An asterisk (*) indicates that the preceding type, word, or group occurs zero or more times.
 A plus (+) indicates that the preceding type, word, or group occurs one or more times.
 A question mark (?) indicates that the preceding type, word, or group is optional (occurs zero or one times).
 A single number in curly braces ({A}) indicates that the preceding type, word, or group occurs A times.
 A commaseparated pair of numbers in curly braces ({A,B}) indicates that the preceding type, word, or group occurs at least A and at most B times. The B may be omitted ({A,}) to indicate that there must be at least A repetitions, with no upper bound on the number of repetitions.
 A hash mark (#) indicates that the preceding type, word, or group occurs one or more times, separated by comma tokens (which may optionally be surrounded by white space and/or comments). It may optionally be followed by the curly brace forms, above, to indicate precisely how many times the repetition occurs, like <length>#{1,4}.
 An exclamation point (!) after a group indicates that the group is required and must produce at least one value; even if the grammar of the items within the group would otherwise allow the entire contents to be omitted, at least one component value must not be omitted.
For repeated component values (indicated by *, +, or #), UAs must support at least 20 repetitions of the component. If a property value contains more than the supported number of repetitions, the declaration must be ignored as if it were invalid.
2.4. Combinator and Multiplier Patterns
There are a small set of common ways to combine multiple independent component values in particular numbers and orders. In particular, it’s common to want to express that, from a set of component value, the author must select zero or more, one or more, or all of them, and in either the order specified in the grammar or in any order.
All of these can be easily expressed using simple patterns of combinators and multipliers:
in order  any order  

zero or more  A? B? C?
 A?  B?  C?

one or more  [ A? B? C? ]!
 A  B  C

all  A B C
 A && B && C

Note that all of the "any order" possibilities are expressed using combinators, while the "in order" possibilities are all variants on juxtaposition.
2.5. Component Values and White Space
Unless otherwise specified, white space and/or comments may appear before, after, and/or between components combined using the above combinators and multipliers.
Note: In many cases, spaces will in fact be required between components in order to distinguish them from each other. For example, the value 1em2em would be parsed as a single <dimensiontoken> with the number 1 and the identifier em2em, which is an invalid unit. In this case, a space would be required before the 2 to get this parsed as the two lengths 1em and 2em.
2.6. Property Value Examples
Below are some examples of properties with their corresponding value definition fields
Property  Value definition field  Example value 

orphans  <integer>  3 
textalign  left  right  center  justify  center 
paddingtop  <length>  <percentage>  5% 
outlinecolor  <color>  invert  #fefefe 
textdecoration  none  underline  overline  linethrough  blink  overline underline 
fontfamily  [ <familyname>  <genericfamily> ]#  "Gill Sans", Futura, sansserif 
borderwidth  [ <length>  thick  medium  thin ]{1,4}  2px medium 4px 
boxshadow  [ inset? && <length>{2,4} && <color>? ]#  none  3px 3px rgba(50%, 50%, 50%, 50%), lemonchiffon 0 0 4px inset 
3. Combining Values: Interpolation, Addition, and Accumulation
Some procedures, for example transitions and animations, combine two CSS property values.
The following combining operations—
 interpolation

Given two property values V_{a} and V_{B},
produces an intermediate value V_{result} at a distance of p along the interval between V_{a} and V_{B} such that p = 0 produces V_{a} and p = 1 produces V_{B}.
The range of p is (−∞, ∞) due to the effect of timing functions. As a result, this procedure must also define extrapolation behavior for p outside [0, 1].
 addition

Given two property values V_{a} and V_{B},
returns the sum of the two properties, V_{result}.
For addition that is not commutative
(for example, matrix multiplication) V_{a} represents
the first term of the operation and V_{B} represents
the second.
Note: While addition can often be expressed in terms of the same weighted sum function used to define interpolation, this is not always the case. For example, interpolation of transform matrices involves decomposing and interpolating the matrix components whilst addition relies on matrix multiplication.
 accumulation

Given two property values V_{a} and V_{B},
returns the result, V_{result},
of combining the two operands
such that V_{B} is treated as a delta from V_{a}.
For accumulation that is not commutative
(for example, accumulation of mismatched transform lists) V_{a} represents the first term of the operation
and V_{B} represents the second.
Note: For many types of animation such as numbers or lengths, accumulation is defined to be identical to addition.
A common case where the definitions differ is for listbased types where addition may be defined as appending to a list whilst accumulation may be defined as componentbased addition. For example, the filter list values blur(2) and blur(3), when added together would produce blur(2) blur(3), but when accumulated would produce blur(5).
These operations are only defined on computed values. (As a result, it is not necessary to define, for example, how to add a <length> value of 15pt with 5em since such values will be resolved to their canonical unit before being passed to any of the above procedures.)
If a value type does not define a specific procedure for addition or is defined as not additive, its addition operation is simply V_{result} = V_{a}.
If a value types does not define a specific procedure for accumulation, its accumulation operation is identical to addition.
4. Textual Data Types
The textual data types include various keywords and identifiers as well as strings (<string>) and URLs (<url>). Aside from the casing of predefined keywords or as explicitly defined for a given property, no normalization is performed, not even Unicode normalization: the specified and computed value of a property are exactly the provided Unicode values after parsing (which includes character set conversion and escaping). [UNICODE] [CSS3SYN]
CSS identifiers, generically denoted by <ident>, consist of a sequence of characters conforming to the <identtoken> grammar. [CSS3SYN] Identifiers cannot be quoted; otherwise they would be interpreted as strings. CSS properties accept two classes of identifiers: predefined keywords and authordefined identifiers.
Note: The <ident> production is not meant for property value definitions—
All textual data types interpolate as discrete and are not additive.
4.1. Predefined Keywords
In the value definition fields, keywords with a predefined meaning appear literally. Keywords are CSS identifiers and are interpreted ASCII caseinsensitively (i.e., [az] and [AZ] are equivalent).
Value: collapse  separate
And here is an example of its use:
table { bordercollapse: separate }
4.1.1. CSSwide keywords: initial, inherit and unset
As defined above, all properties accept the CSSwide keywords, which represent value computations common to all CSS properties.
The initial keyword represents the value specified as the property’s initial value. The inherit keyword represents the computed value of the property on the element’s parent. The unset keyword acts as either inherit or initial, depending on whether the property is inherited or not. All of these keywords are normatively defined in the Cascade module. [CSS3CASCADE]
Other CSS specifications can define additional CSSwide keywords.
4.2. Authordefined Identifiers: the <customident> type
Some properties accept arbitrary authordefined identifiers as a component value. This generic data type is denoted by <customident>, and represents any valid CSS identifier that would not be misinterpreted as a predefined keyword in that property’s value definition. Such identifiers are fully casesensitive (meaning they’re compared by codepoint), even in the ASCII range (e.g. example and EXAMPLE are two different, unrelated userdefined identifiers).
The CSSwide keywords are not valid <customident>s.
The default keyword is reserved
and is also not a valid <customident>.
Specifications using <customident> must specify clearly
what other keywords are excluded from <customident>, if any—
When parsing positionallyambiguous keywords in a property value, a <customident> production can only claim the keyword if no other unfulfilled production can claim it.
Note: When designing grammars with <customident>, the <customident> should always be "positionally unambiguous", so that it’s impossible to conflict with any keyword values in the property.
4.3. Quoted Strings: the <string> type
Strings are denoted by <string> and consist of a sequence of characters delimited by double quotes or single quotes. They correspond to the <stringtoken> production in the CSS Syntax Module [CSS3SYN].
"\""
or as "\22"
).
Analogously for single quotes ('\''
or '\27'
).
content: "this is a 'string'."; content: "this is a \"string\"."; content: 'this is a "string".'; content: 'this is a \'string\'.'
It is possible to break strings over several lines, for aesthetic or other reasons, but in such a case the newline itself has to be escaped with a backslash (\). The newline is subsequently removed from the string. For instance, the following two selectors are exactly the same:
Since a string cannot directly represent a newline, to include a newline in a string, use the escape "\A". (Hexadecimal A is the line feed character in Unicode (U+000A), but represents the generic notion of "newline" in CSS.)
4.4. Resource Locators: the <url> type
The url() functional notation, denoted by <url>, represents a URL, which is a pointer to a resource. The typical syntax of a <url> is:
<url> = url( <string> <urlmodifier>* )
body { background: url("http://www.example.com/pinkish.gif") }
A <url> may alternately be written without quotation marks around the URL itself, in which case it is speciallyparsed as a <urltoken> [CSS3SYN].
background: url("http://www.example.com/pinkish.gif"); background: url(http://www.example.com/pinkish.gif);
Note: This unquoted syntax cannot accept a <urlmodifier> argument and has extra escaping requirements: parentheses, whitespace characters, single quotes (') and double quotes (") appearing in a URL must be escaped with a backslash, e.g. url(open\(parens), url(close\)parens). (In quoted <string> url()s, only newlines and the character used to quote the string need to be escaped.) Depending on the type of URL, it might also be possible to write these characters as URLescapes (e.g. url(open%28parens) or url(close%29parens)) as described in [URL].
Some CSS contexts (such as @import) also allow a <url> to be represented by a bare <string>, without the url() wrapper. In such cases the string behaves identically to a url() function containing that string.
@import url("basetheme.css"); @import "basetheme.css";
4.4.1. Relative URLs
In order to create modular style sheets that are not dependent on the absolute location of a resource, authors should use relative URLs. Relative URLs (as defined in [URL]) are resolved to full URLs using a base URL. RFC 3986, section 3, defines the normative algorithm for this process. For CSS style sheets, the base URL is that of the style sheet itself, not that of the styled source document. Style sheets embedded within a document have the base URL associated with their container.
When a <url> appears in the computed value of a property, it is resolved to an absolute URL, as described in the preceding paragraph. The computed value of a URL that the UA cannot resolve to an absolute URL is the specified value.
body { background: url("tile.png") }
is located in a style sheet designated by the URL:
http://www.example.org/style/basic.css
The background of the source document’s <body>
will be tiled with whatever image is described by the resource designated by the URL:
http://www.example.org/style/tile.png
The same image will be used regardless of the URL of the source document containing the <body>
.
4.4.1.1. Fragment URLs
To work around some common eccentriticites in browser URL handling, CSS has special behavior for fragmentonly urls.
If a url()’s value starts with a U+0023 NUMBER SIGN (#
) character,
parse it as per normal for URLs,
but additionally set the local url flag of the url().
When matching a url() with the local url flag set, ignore everything but the URL’s fragment, and resolve that fragment against the current document that relative URLs are resolved against. This reference must always be treated as samedocument (rather than crossdocument).
When serializing a url() with the local url flag set, it must serialize as just the fragment.
What “browser eccentricities”?
Theoretically, browsers should reresolve any relative URLs,
including fragmentonly URLs,
whenever the document’s base URL changes
(such as through mutation of the base
element,
or calling pushState()
).
In many cases they don’t, however,
and so without special handling,
fragmentonly URLs will suddenly become crossdocument references
(pointing at the previous base URL)
and break in many of the places they’re used.
Since fragmentonly URLs express a clear semantic of wanting to refer to the current document regardless of what its current URL is, this hack preserves the expected behavior at least in these cases.
4.4.2. Empty URLs
If the value of the url() is the empty string (like url("") or url()), the url must resolve to an invalid resource (similar to what the url about:invalid does).
Note: This matches the behavior of empty urls for embedded resources elsewhere in the web platform, and avoids excess traffic rerequesting the stylesheet or host document due to editting mistakes leaving the url() value empty, which are almost certain to be invalid resources for whatever the url() shows up in. Linking on the web platform does allow empty urls, so if/when CSS gains some functionality to control hyperlinks, this restriction can be relaxed in those contexts.
4.4.3. URL Modifiers
The url() function supports specifying additional <urlmodifier>s, which change the meaning or the interpretation of the URL somehow. A <urlmodifier> is either an <ident> or a functional notation.
This specification does not define any <urlmodifier>s, but other specs may do so.
Note: A <url> that is either unquoted or not wrapped in url() notation cannot accept any <urlmodifier>s.
5. Numeric Data Types
Numeric data types are used to represent quantities, indexes, positions, and other such values. Although many syntactic variations can exist in expressing the quantity (numeric aspect) in a given numeric value, the specified and computed value do not distinguish these variations: they represent the value’s abstract quantity, not its syntactic representation.
The numeric data types include <integer>, <number>, <percentage>, and various dimensions including <length>, <angle>, <time>, <frequency>, and <resolution>.
Note: While generalpurpose dimensions are defined here, some other modules define additional data types (e.g. [cssgrid1] introduces fr units) whose usage is more localized.
5.1. Range Restrictions and Range Definition Notation
Properties can restrict numeric values to some range.
If the value is outside the allowed range,
then unless otherwise specified,
the declaration is invalid and must be ignored.
Range restrictions can be annotated in the numeric type notation
using CSS bracketed range notation—[min,max]
—
Note: CSS values generally do not allow open ranges; thus only squarebracket notation is used.
CSS theoretically supports infinite precision and infinite ranges for all value types; however in reality implementations have finite capacity. UAs should support reasonably useful ranges and precisions. Range extremes that are ideally unlimited are indicated using ∞ or −∞ as appropriate.
If no range is indicated,
either by using the bracketed range notation or in the property description,
then [−∞,∞]
is assumed.
Note: At the time of writing,
the bracketed range notation is new;
thus in most CSS specifications
any range limitations are described only in prose.
(For example, “Negative values are not allowed” or
“Negative values are invalid”
indicate a [0,∞]
range.)
This does not make them any less binding.
5.2. Integers: the <integer> type
Integer values are denoted by <integer>.
When written literally, an integer is one or more decimal digits 0 through 9 and corresponds to a subset of the <numbertoken> production in the CSS Syntax Module [CSS3SYN]. The first digit of an integer may be immediately preceded by  or + to indicate the integer’s sign.
5.2.1. Combination of <integer>
Interpolation of <integer> is defined as V_{result} = round((1  p) × V_{a} + p × V_{b}); that is, interpolation happens in the real number space as for <number>s, and the result is converted to an <integer> by rounding to the nearest integer, with values halfway between adjacent integers rounded towards positive infinity.
Addition of <number> is defined as V_{result} = V_{a} + V_{b}
5.3. Real Numbers: the <number> type
Number values are denoted by <number>, and represent real numbers, possibly with a fractional component.
When written literally, a number is either an integer, or zero or more decimal digits followed by a dot (.) followed by one or more decimal digits and optionally an exponent composed of "e" or "E" and an integer. It corresponds to the <numbertoken> production in the CSS Syntax Module [CSS3SYN]. As with integers, the first character of a number may be immediately preceded by  or + to indicate the number’s sign.
The value <zero> represents a literal number with the value 0. Expressions that merely evaluate to a <number> with the value 0 (for example, calc(0)) do not match <zero>; only literal <numbertoken>s do.
5.3.1. Combination of <number>
Interpolation of <number> is defined as V_{result} = (1  p) × V_{a} + p × V_{b}
Addition of <number> is defined as V_{result} = V_{a} + V_{b}
5.4. Numbers with Units: dimension values
The general term dimension refers to a number with a unit attached to it; and is denoted by <dimension>.
When written literally, a dimension is a number immediately followed by a unit identifier, which is an identifier. It corresponds to the <dimensiontoken> production in the CSS Syntax Module [CSS3SYN]. Like keywords, unit identifiers are ASCII caseinsensitive.
CSS uses <dimension>s to specify distances (<length>), durations (<time>), frequencies (<frequency>), resolutions (<resolution>), and other quantities.
5.4.1. Compatible Units
When serializing computed values [CSSOM], compatible units (those related by a static multiplicative factor, like the 96:1 factor between px and in, or the the computed fontsize factor between em and px) are converted into a single canonical unit. Each group of compatible units defines which among them is the canonical unit that will be used for serialization.
When serializing resolved values that are used values, all value types (percentages, numbers, keywords, etc.) that represent lengths are considered compatible with lengths. Likewise any future API that returns used values must consider any values represent distances/durations/frequencies/etc. as compatible with the relevant class of dimensions, and canonicalize accordingly.
5.4.2. Combination of Dimensions
Interpolation of compatible dimensions (for example, two <length> values) is defined as V_{result} = (1  p) × V_{a} + p × V_{b}
Addition of compatible dimensions is defined as V_{result} = V_{a} + V_{b}
5.5. Percentages: the <percentage> type
Percentage values are denoted by <percentage>, and indicates a value that is some fraction of another reference value.
When written literally, a percentage consists of a number immediately followed by a percent sign %. It corresponds to the <percentagetoken> production in the CSS Syntax Module [CSS3SYN].
Percentage values are always relative to another quantity, for example a length. Each property that allows percentages also defines the quantity to which the percentage refers. This quantity can be a value of another property for the same element, the value of a property for an ancestor element, a measurement of the formatting context (e.g., the width of a containing block), or something else.
5.5.1. Combination of <percentage>
Interpolation of <percentage> is defined as V_{result} = (1  p) × V_{a} + p × V_{b}
Addition of <percentage> is defined as V_{result} = V_{a} + V_{b}
5.6. Mixing Percentages and Dimensions
In cases where a <percentage> can represent the same quantity as a dimension in the same component value position, and can therefore be combined with them in a calc() expression, the following convenience notations may be used in the property grammar:
 <lengthpercentage>

Equivalent to
[ <length>  <percentage> ]
, where the <percentage> will resolve to a <length>.  <frequencypercentage>

Equivalent to
[ <frequency>  <percentage> ]
, where the <percentage> will resolve to a <frequency>.  <anglepercentage>

Equivalent to
[ <angle>  <percentage> ]
, where the <percentage> will resolve to an <angle>.  <timepercentage>

Equivalent to
[ <time>  <percentage> ]
, where the <percentage> will resolve to a <time>.
On the other hand, the second and third arguments of the hsl() function can only be expressed as <percentage>s. Although calc() productions are allowed in their place, they can only combine percentages with themselves, as in calc(10% + 20%).
Note: Specifications should never alternate <percentage> in place of a dimension in a grammar unless they are compatible.
Note: More <TYPEpercentage> productions can be added in the future as needed. A <numberpercentage> will never be added, as <number> and <percentage> can’t be combined in calc().
5.6.1. Combination of Percentage and Dimension Mixes
Interpolation of percengagedimension value combinations (e.g. <lengthpercentage>, <frequencypercentage>, <anglepercentage>, <timepercentage> or equivalent notations) is defined as
 equivalent to interpolation of <length> if both V_{a} and V_{b} are pure <length> values
 equivalent to interpolation of <percentage> if both V_{a} and V_{b} are pure <percentage> values
 equivalent to converting both values into a calc() expression representing the sum of the dimension type and a percentage (each possibly zero) and interpolating each component individually (as a <length>/<frequency>/<angle>/<time> and as a <percentage>, respectively)
Addition of <percentage> is defined the same as interpolation except by adding each component rather than interpolating it.
6. Distance Units: the <length> type
Lengths refer to distance measurements and are denoted by <length> in the property definitions. A length is a dimension.
For zero lengths the unit identifier is optional (i.e. can be syntactically represented as the <number> 0). However, if a 0 could be parsed as either a <number> or a <length> in a property (such as lineheight), it must parse as a <number>.
Properties may restrict the length value to some range. If the value is outside the allowed range, the declaration is invalid and must be ignored.
While some properties allow negative length values, this may complicate the formatting and there may be implementationspecific limits. If a negative length value is allowed but cannot be supported, it must be converted to the nearest value that can be supported.
In cases where the used length cannot be supported, user agents must approximate it in the actual value.
There are two types of length units: relative and absolute.
6.1. Relative Lengths
Relative length units specify a length relative to another length. Style sheets that use relative units can more easily scale from one output environment to another.
The relative units are:
unit  relative to 

em  font size of the element 
ex  xheight of the element’s font 
cap  cap height (the nominal height of capital letters) of the element’s font 
ch  average character advance of a narrow glyph in the element’s font, as represented by the “0” (ZERO, U+0030) glyph 
ic  average character advance of a fullwidth glyph in the element’s font, as represented by the “水” (CJK water ideograph, U+6C34) glyph 
rem  font size of the root element 
lh  line height of the element 
rlh  line height of the root element 
vw  1% of viewport’s width 
vh  1% of viewport’s height 
vi  1% of viewport’s size in the root element’s inline axis 
vb  1% of viewport’s size in the root element’s block axis 
vmin  1% of viewport’s smaller dimension 
vmax  1% of viewport’s larger dimension 
Child elements do not inherit the relative values as specified for their parent; they inherit the computed values.
6.1.1. Fontrelative Lengths: the em, ex, cap, ch, ic, rem, lh, rlh units
The fontrelative lengths refer to the font metrics of the element on which they are used—
 em unit
 Equal to the computed value of the fontsize property of the element on which it is used.
 ex unit
 Equal to the used xheight of the first available font [CSS3FONTS]. The xheight is so called because it is often equal to the height of the lowercase "x". However, an ex is defined even for fonts that do not contain an "x". The xheight of a font can be found in different ways. Some fonts contain reliable metrics for the xheight. If reliable font metrics are not available, UAs may determine the xheight from the height of a lowercase glyph. One possible heuristic is to look at how far the glyph for the lowercase "o" extends below the baseline, and subtract that value from the top of its bounding box. In the cases where it is impossible or impractical to determine the xheight, a value of 0.5em must be assumed.
 cap unit
 Equal to the used capheight of the first available font [CSS3FONTS]. The capheight is so called because it is approximately equal to the height of a capital Latin letter. However, a cap is defined even for fonts that do not contain Latin letters. The capheight of a font can be found in different ways. Some fonts contain reliable metrics for the capheight. If reliable font metrics are not available, UAs may determine the capheight from the height of an uppercase glyph. One possible heuristic is to look at how far the glyph for the uppercase “O” extends below the baseline, and subtract that value from the top of its bounding box. In the cases where it is impossible or impractical to determine the capheight, the font’s ascent must be used.
 ch unit

Equal to the used advance measure of the “0” (ZERO, U+0030) glyph
in the font used to render it.
(The advance measure of a glyph is its advance width or height,
whichever is in the inline axis of the element.)
This measurement is an approximation (and in monospace fonts, an exact measure) of a single narrow glyph’s advance measure, thus allowing measurements based on an expected glyph count.
Note: The advance measure of a glyph depends on writingmode and textorientation as well as font settings, texttransform, and any other properties that affect glyph selection or orientation.
In the cases where it is impossible or impractical to determine the measure of the “0” glyph, it must be assumed to be 0.5em wide by 1em tall. Thus, the ch unit falls back to 0.5em in the general case, and to 1em when it would be typeset upright (i.e. writingmode is verticalrl or verticallr and textorientation is upright).
 ic unit

Equal to the used advance measure of the “水” (CJK water ideograph, U+6C34) glyph
found in the font used to render it.
This measurement is a typically an exact measure (in the few fonts with proportional fullwidth glyphs, an approximation) of a single fullwidth glyph’s advance measure, thus allowing measurements based on an expected glyph count.
In the cases where it is impossible or impractical to determine the ideographic advance measure, it must be assumed to be 1em.
 rem unit
 Equal to the computed value of fontsize on the root element. When specified on the fontsize property of the root element, the rem units refer to the property’s initial value.
 lh unit
 Equal to the computed value of the lineheight property of the element on which it is used, converting normal to an absolute length by using only the metrics of the first available font.
 rlh unit

Equal to the computed value of lineheight property on the root element,
converting normal to an absolute length as above.
Note: Setting the height of an element using either the lh or the rlh units does not enable authors to control the actual number of lines in that element. These units only enable length calculations based on the theoretical size of an ideal empty line; the size of actual lines boxes may differ based on their content. In cases where an author wants to limit the number of actual lines in an element, the maxlines property can be used instead.
We can potentially add more typographic units, like cicero, didot, etc. They’re just absolute units, and so can be done with the existing units, but is there enough desire for them (potentially for printing usecases) that it would be worth adding them? Or should we just wait for Houdini Custom Units?
When used outside the context of an element
(such as in media queries),
these units refer to the metrics corresponding
to the initial values of the font and lineheight properties.
When used in the value of the fontsize property on the element they refer to,
they resolve against the computed metrics of the parent element—
6.1.2. Viewportpercentage Lengths: the vw, vh, vi, vb, vmin, vmax units
The viewportpercentage lengths are relative to the size of the initial containing block. When the height or width of the initial containing block is changed, they are scaled accordingly. However, when the value of overflow on the root element is auto, any scroll bars are assumed not to exist. Note that the initial containing block’s size is affected by the presence of scrollbars on the viewport.
For paged media, the exact definition of the viewportpercentage lengths is deferred to [CSS3PAGE].
 vw unit
 Equal to 1% of the width of the initial containing block.
 vh unit
 Equal to 1% of the height of the initial containing block.
 vi unit
 Equal to 1% of the size of the initial containing block in the direction of the root element’s inline axis.
 vb unit
 Equal to 1% of the size of the initial containing block in the direction of the root element’s block axis.
 vmin unit
 Equal to the smaller of vw or vh.
 vmax unit
 Equal to the larger of vw or vh.
In situations where there is no root element or it hasn’t yet been styled (such as when evaluating media queries), the vi and vb units use the initial value of the writingmode property to determine which axis they correspond to.
6.2. Absolute Lengths: the cm, mm, Q, in, pt, pc, px units
The absolute length units are fixed in relation to each other and anchored to some physical measurement. They are mainly useful when the output environment is known. The absolute units consist of the physical units (in, cm, mm, pt, pc, Q) and the visual angle/pixel unit (px):
unit  name  equivalence 

cm  centimeters  1cm = 96px/2.54 
mm  millimeters  1mm = 1/10th of 1cm 
Q  quartermillimeters  1Q = 1/40th of 1cm 
in  inches  1in = 2.54cm = 96px 
pc  picas  1pc = 1/6th of 1in 
pt  points  1pt = 1/72th of 1in 
px  pixels  1px = 1/96th of 1in 
h1 { margin: 0.5in } /* inches */ h2 { lineheight: 3cm } /* centimeters */ h3 { wordspacing: 4mm } /* millimeters */ h3 { letterspacing: 1Q } /* quartermillimeters */ h4 { fontsize: 12pt } /* points */ h4 { fontsize: 1pc } /* picas */ p { fontsize: 12px } /* px */
All of the absolute length units are compatible, and px is their canonical unit.
For a CSS device, these dimensions are anchored either
 by relating the physical units to their physical measurements, or
 by relating the pixel unit to the reference pixel.
For print media at typical viewing distances, the anchor unit should be one of the physical units (inches, centimeters, etc). For screen media (including highresolution devices), lowresolution devices, and devices with unusual viewing distances), it is recommended instead that the anchor unit be the pixel unit. For such devices it is recommended that the pixel unit refer to the whole number of device pixels that best approximates the reference pixel.
Note: If the anchor unit is the pixel unit, the physical units might not match their physical measurements. Alternatively if the anchor unit is a physical unit, the pixel unit might not map to a whole number of device pixels.
Note: This definition of the pixel unit and the physical units differs from previous versions of CSS. In particular, in previous versions of CSS the pixel unit and the physical units were not related by a fixed ratio: the physical units were always tied to their physical measurements while the pixel unit would vary to most closely match the reference pixel. (This change was made because too much existing content relies on the assumption of 96dpi, and breaking that assumption broke the content.)
Note: Units are ASCII caseinsensitive and serialize as lower case, for example 1Q serializes as 1q.
The reference pixel is the visual angle of one pixel on a device with a pixel density of 96dpi and a distance from the reader of an arm’s length. For a nominal arm’s length of 28 inches, the visual angle is therefore about 0.0213 degrees. For reading at arm’s length, 1px thus corresponds to about 0.26 mm (1/96 inch).
The image below illustrates the effect of viewing distance on the size of a reference pixel: a reading distance of 71 cm (28 inches) results in a reference pixel of 0.26 mm, while a reading distance of 3.5 m (12 feet) results in a reference pixel of 1.3 mm.
This second image illustrates the effect of a device’s resolution on the pixel unit: an area of 1px by 1px is covered by a single dot in a lowresolution device (e.g. a typical computer display), while the same area is covered by 16 dots in a higher resolution device (such as a printer).
7. Other Quantities
7.1. Angle Units: the <angle> type and deg, grad, rad, turn units
Angle values are <dimension>s denoted by <angle>. The angle unit identifiers are:
 deg
 Degrees. There are 360 degrees in a full circle.
 grad
 Gradians, also known as "gons" or "grades". There are 400 gradians in a full circle.
 rad
 Radians. There are 2π radians in a full circle.
 turn
 Turns. There is 1 turn in a full circle.
For example, a right angle is 90deg or 100grad or 0.25turn or approximately 1.57rad.
All <angle> units are compatible, and deg is their canonical unit.
For example, in the lineargradient() function, the <angle> that determines the direction of the gradient is interpreted as a bearing angle.
Note: For legacy reasons, some uses of <angle> allow a bare 0 to mean 0deg. This is not true in general, however, and will not occur in future uses of the <angle> type.
7.2. Duration Units: the <time> type and s, ms units
Time values are dimensions denoted by <time>. The time unit identifiers are:
 s
 Seconds.
 ms
 Milliseconds. There are 1000 milliseconds in a second.
All <time> units are compatible, and s is their canonical unit.
Properties may restrict the time value to some range. If the value is outside the allowed range, the declaration is invalid and must be ignored.
7.3. Frequency Units: the <frequency> type and Hz, kHz units
Frequency values are dimensions denoted by <frequency>. The frequency unit identifiers are:
 Hz
 Hertz. It represents the number of occurrences per second.
 kHz
 KiloHertz. A kiloHertz is 1000 Hertz.
For example, when representing sound pitches, 200Hz (or 200hz) is a bass sound, and 6kHz (or 6khz) is a treble sound.
All <frequency> units are compatible, and hz is their canonical unit.
Note: Units are ASCII caseinsensitive and serialize as lower case, for example 1Hz serializes as 1hz.
7.4. Resolution Units: the <resolution> type and dpi, dpcm, dppx units
Resolution units are dimensions denoted by <resolution>. The resolution unit identifiers are:
 dpi
 Dots per inch.
 dpcm
 Dots per centimeter.
 dppx
 x
 Dots per px unit.
The <resolution> unit represents the size of a single "dot" in a graphical representation by indicating how many of these dots fit in a CSS in, cm, or px. For uses, see e.g. the resolution media query in [MEDIAQ] or the imageresolution property defined in [CSS3IMAGES].
All <resolution> units are compatible, and dppx is their canonical unit.
Note that due to the 1:96 fixed ratio of CSS in to CSS px, 1dppx is equivalent to 96dpi. This corresponds to the default resolution of images displayed in CSS: see imageresolution.
@media (minresolution: 2dppx) { ... }
8. Data Types Defined Elsewhere
Some data types are defined in their own modules. This example talks about some of the most common ones used across several specifications.
8.1. Colors: the <color> type
The <color> data type is defined in [CSS3COLOR]. UAs that support CSS Color Level 3 or its successor must interpret <color> as defined therein.
8.1.1. Combination of <color>
Interpolation of <color> is defined as the independent interpolation of each component (red, green, blue, alpha) as a <number>. Interpolation is done between premultiplied colors (that is, colors for which the red, green, and blue components specified have been multiplied by the alpha).
Addition of <number> is likewise defined as the independent addition of each component as a <number> in premultiplied space.
Computed value needs to be able to represent
combinations of currentColor and an actual color.
Consider the value of textemphasiscolor in div { textemphasis: circle; transition: all 2s; }
See Issue 445.
div:hover { textemphasiscolor: lime; }
em { color: red; }
8.2. Images: the <image> type
The <image> data type is defined in [CSS3IMAGES]. UAs that support CSS Images Level 3 or its successor must interpret <image> as defined therein. UAs that do not yet support CSS Images Level 3 must interpret <image> as <url>.
8.2.1. Combination of <image>
Note: Interpolation of <image> is defined in CSS Images 3 §6 Interpolation.
Images are not additive.
8.3. 2D Positioning: the <position> type
The <position> value specifies the position of a object area (e.g. background image) inside a positioning area (e.g. background positioning area). It is interpreted as specified for backgroundposition. [CSS3BACKGROUND]
<position> = [ [ left  center  right ]  [ top  center  bottom ]  [ left  center  right  <lengthpercentage> ] [ top  center  bottom  <lengthpercentage> ]?  [ [ left  right ] <lengthpercentage> ] && [ [ top  bottom ] <lengthpercentage> ] ]
Note: The backgroundposition property also accepts a threevalue syntax. This has been disallowed generically because it creates parsing ambiguities when combined with other length or percentage components in a property value.
The canonical order when serializing is the horizontal component followed by the vertical component.
When specified in a grammar alongside other keywords, <length>s, or <percentage>s, <position> is greedily parsed; it consumes as many components as possible.
8.3.1. Combination of <position>
Interpolation of <position> is defined as the independent interpolation of each component (x, y) normalized as an offset from the top left corner as a <lengthpercentage>.
Addition of <position> is likewise defined as the independent addition each component (x, y) normalized as an offset from the top left corner as a <lengthpercentage>.
9. Functional Notations
A functional notation is a type of component value that can represent more complex types or invoke special processing. The syntax starts with the name of the function immediately followed by a left parenthesis (i.e. a <functiontoken>) followed by the argument(s) to the notation followed by a right parenthesis. White space is allowed, but optional, immediately inside the parentheses. Functions can take multiple arguments, which are formatted similarly to a CSS property value.
Some legacy functional notations, such as rgba(), use commas unnecessarily, but generally commas are only used to separate items in a list, or pieces of a grammar that would be ambiguous otherwise. If a comma is used to separate arguments, white space is optional before and after the comma.
background: url(http://www.example.org/image); color: rgb(100, 200, 50 ); content: counter(listitem) ". "; width: calc(50%  2em);
The math functions are defined in § 11 Mathematical Expressions.
9.1. Toggling Between Values: toggle()
The toggle() expression allows descendant elements to cycle over a list of values instead of inheriting the same value.
<em>
elements italic in general,
but makes them normal if they’re inside something that’s italic:
em { fontstyle: toggle(italic, normal); }
ul { liststyletype: toggle(disc, circle, square, box); }
The syntax of the toggle() expression is:
toggle( <togglevalue># )
where <togglevalue> is any CSS value that is valid where the expression is placed, and that doesn’t contain any toplevel commas. If any of the values inside are not valid, then the entire toggle() expression is invalid. The toggle() expression may be used as the value of any property, but must be the only component in that property’s value.
The toggle() notation is not allowed to be nested; nor may it contain attr() or calc() notations. Declarations containing such constructs are invalid.
backgroundposition: 10px toggle(50px, 100px);
/* toggle() must be the sole value of the property */
liststyletype: toggle(disc, 50px);
/* 50px isn’t a valid value of 'liststyletype' */
To determine the computed value of toggle(), first evaluate each argument as if it were the sole value of the property in which toggle() is placed to determine the computed value that each represents, called C_{n} for the nth argument to toggle(). Then, compare the property’s inherited value with each C_{n}. For the earliest C_{n} that matches the inherited value, the computed value of toggle() is C_{n+1}. If the match was the last argument in the list, or there was no match, the computed value of toggle() is the computed value that the first argument represents.
Note: This means that repeating values in a toggle() shortcircuits the list. For example toggle(1em, 2em, 1em, 4em) will be equivalent to toggle(1em, 2em).
Note: That toggle() explicitly looks at the computed value of the parent, so it works even on noninherited properties. This is similar to the inherit keyword, which works even on noninherited properties.
Note: That the computed value of a property is an abstract set of values, not a particular serialization [CSS21], so comparison between computed values should always be unambiguous and have the expected result. For example, a Level 2 backgroundposition computed value is just two offsets, each represented as an absolute length or a percentage, so the declarations backgroundposition: top center and backgroundposition: 50% 0% produce identical computed values. If the "Computed Value" line of a property definition seems to define something ambiguous or overly strict, please provide feedback so we can fix it.
If toggle() is used on a shorthand property, it sets each of its longhands to a toggle() value with arguments corresponding to what the longhand would have received had each of the original toggle() arguments been the sole value of the shorthand.
margin: toggle(1px 2px, 4px, 1px 5px 4px);
is equivalent to the following longhand declarations:
margintop: toggle(1px, 4px, 1px); marginright: toggle(2px, 4px, 5px); marginbottom: toggle(1px, 4px, 4px); marginleft: toggle(2px, 4px, 5px);
Note that, since 1px appears twice in the top margin and 4px appears twice in bottom margin, they will cycle between only two values while the left and right margins cycle through three. In other words, the declarations above will yield the same computed values as the longhand declarations below:
margintop: toggle(1px, 4px); marginright: toggle(2px, 4px, 5px); marginbottom: toggle(1px, 4px); marginleft: toggle(2px, 4px, 5px);
which may not be what was intended.
10. Attribute References: the attr() function
The attr() function substitutes the value of an attribute on an element into a property, similar to how the var() function substitutes a custom property value into a function.
attr() = attr( <wqname> <attrtype>? , <declarationvalue>?) <attrtype> = string  url  ident  color  number  percentage  length  angle  time  frequency  flex  <dimensionunit>
The <dimensionunit> production matches a literal "%" character (that is, a <delimtoken> with a value of "%") or an ident whose value is any of the CSS units for <length>, <angle>, <time>, <frequency>, or <flex> values (such as px or ms).
The arguments of attr() are:
 <wqname>

Gives the name of the attribute being referenced.
If no namespace is specified (just an identifier is given, like attr(foo)), the null namespace is implied. (This is usually what’s desired, as namespaced attributes are rare. In particular, HTML and SVG do not contain namespaced attributes.) As with attribute selectors, the casesensitivity of <wqname> depends on the document language.
If attr() is used in a property applied to an element, it references the attribute of the given name on that element; if applied to a pseudoelement, the attribute is looked up on the pseudoelement’s originating element.
 <attrtype>

Specifies what kind of CSS value the attribute’s value will be interpreted into (the attr()’s substitution value) and what, if any, special parsing will be done to the value.
The possible values and their behavior are defined in § 10.1 attr() Types.
Defaults to string if omitted.
 <declarationvalue>

Specifies a fallback value for the attr(), which will be substituted instead of the attribute’s value if the attribute is missing or fails to parse as the specified type.
If the <attrtype> argument is string, defaults to the empty string if omitted; otherwise, defaults to the guaranteedinvalid value if omitted.
If a property contains one or more attr() functions, and those functions are syntactically valid, the entire property’s grammar must be assumed to be valid at parse time. It is only syntaxchecked at computedvalue time, after attr() functions have been substituted.
10.1. attr() Types
The behavior of the attr() function depends partially on the value of the <attrtype> argument:
 string

The substitution value is a CSS string, whose value is the literal value of the attribute. (No CSS parsing or "cleanup" of the value is performed.)
No value triggers fallback.
 url

The substitution value is a CSS <url> value, whose url is the literal value of the attribute. (No CSS parsing or "cleanup" of the value is performed.)
Note: If url() was syntactically capable of containing functions, attr(foo url) would be identical to url(attr(foo string)).
No value triggers fallback.
 ident

The substitution value is a CSS <customident>, whose value is the literal value of the attribute, with leading and trailing ASCII whitespace stripped. (No CSS parsing of the value is performed.)
If the attribute value, after trimming, is the empty string, there is instead no substitution value.
If the <customident>’s value is a CSSwide keyword or default, there is instead no substitution value.
 color

Parse a component value from the attribute’s value. If the result is a <hexcolor> or a named color ident, the substitution value is that result as a <color>.
Otherwise there is no substitution value.
 number

Parse a component value from the attribute’s value. If the result is a <numbertoken>, the result is the substitution value.
Otherwise, there is no substitution value.
 percentage

Parse a component value from the attribute’s value. If the result is a <percentagetoken>, the result is the substitution value.
Otherwise, there is no substitution value.
 length
 angle
 time
 frequency
 flex

Parse a component value from the attribute’s value. If the result is a <dimensiontoken> whose unit matches the given type, the result is the substitution value.
Otherwise, there is no substitution value.
 <dimensionunit>

Parse a component value from the attribute’s value. If the result is a <numbertoken>, the substitution value is a dimension with the result’s value, and the given unit.
Otherwise, there is no substitution value.
Do we want to allow math functions as attr values
for all the numeric types?
And color functions for "color"?
I think we do,
but I’d have to check the contents to make sure they don’t contain further reference functions; foo="rgb(var(red), 0, 0)"
needs to be illegal for attr(foo color).
<stock> <wood length="12"/> <wood length="5"/> <metal length="19"/> <wood length="4"/> </stock> stock::before { display: block; content: "To scale, the lengths of materials in stock are:"; } stock > * { display: block; width: attr(length em, 0px); height: 1em; border: solid thin; margin: 0.5em; } wood { background: orange url(wood.png); } metal { background: silver url(metal.png); }
10.2. attr() Substitution
attr() and var() substitute at the same time, so I should probably rewrite substitute a var() to be more generally about "substitute a reference" and just use that for both of these functions.
attr() functions are substituted at computedvalue time. If a declaration, once all attr() functions are substituted in, does not match its declared grammar, the declaration is invalid at computedvalue time.
To substitute an attr():

If the attr() function has a substitution value, replace the attr() function by the substitution value.

Otherwise, if the atr() function has a fallback value as its last argument, replace the attr() function by the fallback value. If there are any var() or attr() references in the fallback, substitute them as well.

Otherwise, the property containing the attr() function is invalid at computedvalue time.
11. Mathematical Expressions
The math functions (calc(), clamp(), sin(), and others defined in this chapter) allow numeric CSS values to be written as mathematical expressions.
A math function represents a numeric value, one of:
...or the <lengthpercentage>/etc mixed types, and can be used wherever such a value would be valid.
11.1. Basic Arithmetic: calc()
The calc() function is a math function that allows basic arithmetic to be performed on numerical values, using addition (+), subtraction (), multiplication (*), division (/), and parentheses.
A calc() function contains a single calculation, which is a sequence of values interspersed with operators, and possibly grouped by parentheses (matching the <calcsum> grammar), which represents the result of evaluating the expression using standard operator precedence rules. (* and / bind tighter than + and , and operators are otherwise evaluated lefttoright.) The calc() function represents the result of its contained calculation.
Components of a calculation can be literal values (such as 5px), other math functions, or other expressions, such as attr(), that evaluate to a valid argument type (like <length>).
Math functions can be used to combine value that use different units. In this example the author wants the margin box of each section to take up 1/3 of the space, so they start with 100%/3, then subtract the element’s borders and margins. (boxsizing can automatically achieve this effect for borders and padding, but a math function is needed if you want to include margins.)
section { float : left; margin : 1 em ; border : solid1 px ; width : calc ( 100 % / 3 2 * 1em  2 * 1px); }
Similarly, in this example the gradient will show a color transition only in the first and last 20px of the element:
.fade { backgroundimage : lineargradient ( silver0 % , white20 px , whitecalc ( 100 % 20 px ), silver100 % ); }
Math functions can also be useful just to express values in a more natural, readable fashion, rather than as an obscure decimal. For example, the following sets the fontsize so that exactly 35em fits within the viewport, ensuring that roughly the same amount of text always fills the screen no matter the screen size.
:root { fontsize : calc ( 100 vw / 35 ); }
Functionalitywise, this is identical to just writing fontsize: 2.857vw, but then the intent (that 35em fills the viewport) is much less clear to someone reading the code; the later reader will have to reverse the math themselves to figure out that 2.857 is meant to approximate 100/35.
Standard mathematical precedence rules for the operators apply: calc(2 + 3 * 4) is equal to 14, not 20.
Parentheses can be used to manipulate precedence: calc((2 + 3) * 4) is instead equal to 20.
Parentheses and nesting additional calc() functions are equivalent; the preceding expression could equivalently have been written as calc(calc(2 + 3) * 4). This can be useful when building up values piecemeal via var(), such as in the following example:
.aspectratiobox { ar : calc ( 16 / 9 ); w : calc ( 100 % / 3 ); h : calc ( var ( w) / var ( ar)); width : var ( w); height : var ( h); }
Altho ar could have been written as simply ar: (16 / 9);, w is used both on its own (in width) and as a calc() component (in h), so it has to be written as a full calc() function itself.
11.2. Comparison Functions: min(), max(), and clamp()
The comparison functions of min(), max(), and clamp() compare multiple calculations and represent the value of one of them.
The min() or max() functions contain one or more commaseparated calculations, and represent the smallest (most negative) or largest (most positive) of them, respectively.
The clamp() function takes three calculations—
min(), max(), and clamp() can be used to make sure a value doesn’t exceed a "safe" limit: For example, "responsive type" that sets fontsize with viewport units might still want a minimum size to ensure readability:
.type { /* Set fontsize to 10x the average of vw and vh, but don’t let it go below 12px. */ fontsize : max ( 10 * (1vw + 1vh) / 2, 12px); }
Note: Full math expressions are allowed in each of the arguments; there’s no need to nest a calc() inside! You can also provide more than two arguments, if you have multiple constraints to apply.
.type { /* Force the fontsize to stay between 12px and 100px */ fontsize : clamp ( 12 px , 10 * (1vw + 1vh) / 2, 100px); }
If alternate resolution mechanics are desired they can be achieved by combining clamp() with min() or max():
 To have MAX win over MIN:

clamp(min(MIN, MAX), VAL, MAX). If you want to avoid repeating the MAX calculation, you can just reverse the nesting of functions that clamp() is defined against—
min(MAX, max(MIN, VAL)).  To have MAX and MIN "swap" when they’re in the wrong order:

clamp(min(MIN, MAX), VAL, max(MIN, MAX)). Unfortunately, there’s no easy way to do this without repeating the MIN and MAX terms.
11.3. Trigonometric Functions: sin(), cos(), tan(), asin(), acos(), atan(), and atan2()
The trigonometric functions—
The sin(A), cos(A), and tan(A) functions all contain a single calculation which must resolve to either a <number> or an <angle>, and compute their corresponding function by interpreting the result of their argument as radians. (That is, sin(45deg), sin(.125turn), and sin(3.14159 / 4) all represent the same value, approximately .707.) They all represent a <number>; sin() and cos() will always return a number between −1 and 1, while tan() can return any number between −∞ and +∞. (See § 11.6 Type Checking for details on how math functions handle ∞.)
The asin(A), acos(A), and atan(A) functions are the "arc" or "inverse" trigonometric functions, representing the inverse function to their corresponding "normal" trig functions. All of them contain a single calculation which must resolve to a <number>, and compute their corresponding function, interpreting their result as a number of radians, representing an <angle>. The angle returned by asin() must be normalized to the range [90deg, 90deg]; the angle returned by acos() to the range [0deg, 180deg]; and the angle returned by atan() to the range [90deg, 90deg].
The atan2(A, B) function contains two commaseparated calculations, A and B. A and B can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid. The function returns the <angle> between the positive Xaxis and the point (B,A). The returned angle must be normalized to the interval (180deg, 180deg] (that is, greater than 180deg, and less than or equal to 180deg).
Note: atan2(Y, X) is generally equivalent to atan(Y / X), but it gives a better answer when the point in question may include negative components. atan2(1, 1), corresponding to the point (1, 1), returns 135deg, distinct from atan2(1, 1), corresponding to the point (1, 1), which returns 45deg. In contrast, atan(1 / 1) and atan(1 / 1) both return45deg, because the internal calculation resolves to 1 for both.
11.3.1. Argument Ranges
In sin(A), cos(A), or tan(A), if A is infinite, the result is NaN. (See § 11.6 Type Checking for details on how math functions handle NaN.)
In sin(A) or tan(A), if A is 0⁻, the result is 0⁻.
In tan(A), if A is one of the asymptote values (such as 90deg, 270deg, etc), the result must be +∞ for 90deg and all values a multiple of 360deg from that (such as 270deg or 450deg), and −∞ for 90deg and all values a multiple of 360deg from that (such as 450deg or 270deg).
Note: This is only relevant for units that can exactly represent the asymptotic values, such as deg or grad. rad cannot, and so whether the result is a very large negative or positive value can depend on rounding and precise details of how numbers are internally stored. It’s recommended you don’t depend on this behavior if using such units.
In asin(A) or acos(A), if A is less than 1 or greater than 1, the result is NaN.
In acos(A), if A is exactly 1, the result is 0.
In asin(A) or atan(A), if A is 0⁻, the result is 0⁻.
In atan(A), if A is +∞, the result is 90deg; if A is −∞, the result is 90deg.
In atan2(Y, X), the following table gives the results for all unusual argument combinations:
X  

−∞  finite  0⁻  0⁺  +finite  +∞  
Y  −∞  135deg  90deg  90deg  90deg  90deg  45deg 
finite  180deg  (normal)  90deg  90deg  (normal)  0⁻deg  
0⁻  180deg  180deg  180deg  0⁻deg  0⁻deg  0⁻deg  
0⁺  180deg  180deg  180deg  0⁺deg  0⁺deg  0⁺deg  
+finite  180deg  (normal)  90deg  90deg  (normal)  0⁺deg  
+∞  135deg  90deg  90deg  90deg  90deg  45deg 
Note: All of these behaviors are intended to match the "standard" definitions of these functions as implemented by most programming languages, in particular as implemented in JS.
Note: The behavior of tan(90deg), while not constrained by JS behavior (because the JS function’s input is in radians, and one cannot perfectly express a value of π/2 in JS numbers), is defined so that roundtripping of values works; tan(atan(1 / 0)) yields +∞, tan(atan(1 / 0)) yields −∞, atan(tan(90deg)) yields 90deg, and atan(tan(90deg)) yields 90deg.
11.4. Exponential Functions: pow(), sqrt(), hypot()
The exponential functions—
The pow(A, B) function contains two commaseparated calculations A and B, both of which must resolve to <number>s, and returns the result of raising A to the power of B, returning the value as a <number>.
The sqrt(A) function contains a single calculation which must resolve to a <number>, and returns the square root of the value as a <number>. (sqrt(X) and pow(X, .5) are basically equivalent, differing only in some errorhandling; sqrt() is a common enough function that it is provided as a convenience.)
The hypot(A, …) function contains one or more commaseparated calculations, and returns the length of an Ndimensional vector with components equal to each of the calculations. (That is, the square root of the sum of the squares of its arguments.) The argument calculations can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid; the result will have the same type as the arguments.
Why does hypot() allow dimensions (values with units), but pow() and sqrt() only work on numbers?
You are allowed to write expressions like hypot(30px, 40px), which resolves to 50px, but you aren’t allowed to write the expression sqrt(pow(30px, 2) + pow(40px, 2)), despite the two being equivalent in most mathematical systems.
There are two reasons for this: numeric precision in the exponents, and clashing expectations from authors.
First, numerical precision. For a type to match a CSS production like <length>, it needs to have a single unit with its exponent set to exactly 1. Theoretically, expressions like pow(pow(30px, 3), 1/3) should result in exactly that: the inner pow(30px, 3) would resolve to a value of 27000 with a type of «[ "length" → 3 ]» (aka <length>³), and then the pow(X, 1/3) would cuberoot the value back down to 30 and multiply the exponent by 1/3, giving «[ "length" → 1 ]», which matches <length>.
In the realm of pure mathematics, that’s guaranteed to work out;
in the realworld of computers using binary floatingpoint arithmetic,
in some cases the powers might not exactly cancel out,
leaving you with an invalid math function for confusing, hardtotrackdown reasons.
(For a JS example,
evaluate Math.pow(Math.pow(30, 10/3), .1+.1+.1)
;
the result is not exactly 30,
because .1+.1+.1
is not exactly 3/10.
Instead, (10/3) * (.1 + .1 + .1)
is slightly greater than 1.)
Requiring authors to cast their value down into a number, do all the math on the raw number, then finally send it back to the desired unit, while inconvenient, ensures that numerical precision won’t bite anyone: calc(pow(pow(30px / 1px, 3), 1/3) * 1px) is guaranteed to resolve to a <length>, with a value that, if not exactly 30, is at least very close to 30, even if numerical precision actually prevents the powers from exactly canceling.
Second, clashing expectations. It’s not uncommon for authors to expect pow(30px, 2) to result in 900px (such as in this Sass issue); that is, just squaring the numerical value and leaving the unit alone. This, however, means the result is dependent on what unit you’re expressing the argument in; if 1em is 16px, then pow(1em, 2) would give 1em, while pow(16px, 2) would give 256px, or 16em, which are very different values for what should otherwise be identical input arguments! This sort of input dependency is troublesome for CSS, which generally allows values to be canonicalized freely; it also makes more complex expressions like pow(2em + 10px, 2) difficult to interpret.
Again, requiring authors to cast their value down into a number and then back up again into the desired unit sidesteps these issues; pow(30, 2) is indeed 900, and the author can interpret that however they wish.
On the other hand, hypot() doesn’t suffer from these problems. Numerical precision in units isn’t a concern, as the inputs and output all have the same type. The result isn’t unitdependent, either, due to the nature of the operation; hypot(3em, 4em) and hypot(48px, 64px) both result in the same length when 1em equals 16px: 5em or 80px. Thus it’s fine to let author use dimensions directly in hypot().
These sizes can be easily written into custom properties like:
:root { h6 : calc ( 1 rem * pow(1.5, 1)); h5: calc(1rem * pow(1.5, 0)); h4: calc(1rem * pow(1.5, 1)); h3: calc(1rem * pow(1.5, 2)); h2: calc(1rem * pow(1.5, 3)); h1: calc(1rem * pow(1.5, 4)); }
...rather than writing out the values in precalculated numbers like 5.0625rem (what calc(1rem * pow(1.5, 4)) resolves to) which have less clear provenance when encountered in a stylesheet.
With more arguments, it gives the size of the main diagonal of a box whose side lengths are given by the arguments. This can be useful for transformrelated things, giving the distance that an element will actually travel when it’s translated by a particular X, Y, and Z amount.
For example, hypot(30px, 40px) resolves to 50px, which is indeed the distance between an element’s starting and ending positions when it’s translated by a translate(30px, 40px) transform. If an author wanted elements to get smaller as they moved further away from their starting point (drawing some sort of word cloud, for example), they could then use this distance in their scaling factor calculations.
11.4.1. Argument Ranges
In pow(A, B), if A is negative and finite, and B is finite, B must be an integer, or else the result is NaN.
If A or B are infinite or 0, the following tables give the results:
A is −∞  A is 0⁻  A is 0⁺  A is +∞  

B is −finite  0⁻ if B is an odd integer, 0⁺ otherwise  −∞ if B is an odd integer, +∞ otherwise  +∞  0⁺ 
B is 0  always 1  
B is +finite  −∞ if B is an odd integer, +∞ otherwise  0⁻ if B is an odd integer, 0⁺ otherwise  0⁺  +∞ 
A is < 1  A is 1  1 < A < 1  A is 1  A is > 1  

B is +∞  result is +∞  result is NaN  result is 0⁺  result is NaN  result is +∞ 
B is −∞  result is 0⁺  result is NaN  result is +∞  result is NaN  result is 0⁺ 
In sqrt(A), if A is +∞, the result is +∞. If A is 0⁻, the result is 0⁻. If A is less than 0, the result is NaN.
In hypot(A, …), if any of the inputs are infinite, the result is +∞.
(See § 11.6 Type Checking for details on how math functions handle NaN and infinities.)
The only divergences from the behavior of the equivalent JS functions are that NaN is "infectious" in every function, forcing the function to return NaN if any argument calculation is NaN.
Details of the JS Behavior
There are two cases in JS where a NaN is not "infectious" to the math function it finds itself in:

Math
will return. hypot( Infinity , NaN )
.Infinity 
Math
will return. pow( NaN , 0 )
.1
The logic appears to be that,
if you replace the NaN with any Number,
the return value will be the same.
However, this logic is not applied consistently to the Math
functions: Math
returns
, not
;
the same is true of Math
.
Because this is an error corner case, JS isn’t consistent on the matter, and NaN recognition/handling of calculations is likely done at a higher CSS level rather than in the internal math functions anyway, consistency in CSS was chosen to be more important, so all functions were defined to have "infection" NaN.
11.5. Syntax
The syntax of a math function is:
<calc()> = calc( <calcsum> ) <min()> = min( <calcsum># ) <max()> = max( <calcsum># ) <clamp()> = clamp( <calcsum>#{3} ) <sin()> = sin( <calcsum> ) <cos()> = cos( <calcsum> ) <tan()> = tan( <calcsum> ) <asin()> = asin( <calcsum> ) <acos()> = acos( <calcsum> ) <atan()> = atan( <calcsum> ) <atan2()> = atan2( <calcsum>, <calcsum> ) <pow()> = pow( <calcsum>, <calcsum> ) <sqrt()> = sqrt( <calcsum> ) <hypot()> = hypot( <calcsum># ) <calcsum> = <calcproduct> [ [ '+'  '' ] <calcproduct> ]* <calcproduct> = <calcvalue> [ [ '*'  '/' ] <calcvalue> ]* <calcvalue> = <number>  <dimension>  <percentage>  ( <calcsum> )
In addition, whitespace is required on both sides of the + and  operators. (The * and / operaters can be used without white space around them.)
Several of the math functions above have additional constraints on what their <calcsum> arguments can contain. Check the definitions of the individual functions for details.
UAs must support calculations of at least 20 <calcvalue> terms. If a calculation contains more than the supported number of terms, it must be treated as if it were invalid.
11.6. Type Checking
A math function can be many possible types, such as <length>, <number>, etc., depending on the calculations it contains, as defined below. It can be used anywhere a value of that type is allowed.
Additionally, math functions that resolve to <number> can be used in any place that only accepts <integer>. (It gets rounded to the nearest integer, as specified in § 11.9 Range Checking.)
Operators form subexpressions, which gain types based on their arguments.
Note: In previous versions of this specification, multiplication and division were limited in what arguments they could take, to avoid producing more complex intermediate results (such as 1px * 1em, which is <length>²) and to make divisionbyzero detectable at parse time. This version now relaxes those restrictions.

At a + or  subexpression, attempt to add the types of the left and right arguments. If this returns failure, the entire calculation’s type is failure. Otherwise, the subexpression’s type is the returned type.

At a * subexpression, multiply the types of the left and right arguments. The subexpression’s type is the returned result.

At a / subexpression, let left type be the result of finding the types of its left argument, and right type be the result of finding the types of its right argument and then inverting it.
The subexpression’s type is the result of multiplying the left type and right type.

Anything else is a terminal value, whose type is determined based on its CSS type:
 <number>
 <integer>

the type is «[ ]» (empty map)
 <length>

the type is «[ "length" → 1 ]»
 <angle>

the type is «[ "angle" → 1 ]»
 <time>

the type is «[ "time" → 1 ]»
 <frequency>

the type is «[ "frequency" → 1 ]»
 <resolution>

the type is «[ "resolution" → 1 ]»
 <flex>

the type is «[ "flex" → 1 ]»
 <percentage>

If, in the context in which the math function containing this calculation is placed, <percentage>s are resolved relative to another type of value (such as in width, where <percentage> is resolved against a <length>), and that other type is not <number>, the type is determined as the other type.
Otherwise, the type is «[ "percent" → 1 ]».
 anything else

The calculation’s type is failure.
In all cases, the associated percent hint is null.
Math functions themselves have types, according to their contained calculations:

The type of a calc() expression is the type of its contained calculation.

The type of a min(), max(), or clamp() expression is the result of adding the types of its commaseparated calculations.

The type of a sin(), cos(), or tan() expression is «[ "number" → 1 ]».

The type of an asin(), acos(), atan(), or atan2() expression is «[ "angle" → 1 ]».

The type of a pow() or sqrt() expression is «[ "number" → 1 ]».

The type of a hypot() expression is the result of adding the types of its commaseparated calculations.
For each of the above, if the type is failure, the math function is invalid.
A math function resolves to <number>, <length>, <angle>, <time>, <frequency>, <resolution>, <flex>, or <percentage> according to which of those productions its type matches. (These categories are mutually exclusive.) If it can’t match any of these, the math function is invalid.
Division by zero is possible, which introduces certain complications. Math functions follow IEEE754 semantics for these operations:

Dividing a positive value by zero produces +∞.

Dividing a negative value by zero produces −∞.

Adding or subtracting ±∞ to anything produces the appropriate infinity, unless a following rule would define it as producing NaN.

Multiplying any value by ±∞ produces the appropriate infinity, unless a following rule would define it as producing NaN.

Dividing any value by ±∞ produces zero, unless a following rule would define it as producing NaN.

Dividing zero by zero, dividing ±∞ by ±∞, multiplying 0 by ±∞, adding +∞ to −∞ (or the equivalent subtractions) produces NaN.

Any operation with at least one NaN argument produces NaN.
Additionally, IEEE754 introduces the concept of "negative zero", which must be tracked within a calculation and between nested calculations:

Negative zero (0⁻) can be produced literally by negating a zero (0), or by a multiplication or division that produces zero with exactly one negative argument (such as 5 * 0 or 1 / (1 / 0)).
Note: Note that, outside of math functions, 0 just produces a "standard" zero, identical to 0—
CSS as a whole doesn’t recognize the concept of signed zeros. Negative zeros also don’t escape a math function; as detailed below, they’re "censored" away into an "unsigned" zero. 
0 + 0 or 0  0 produces 0⁻. All other additions or subtractions that would produce a zero produce 0⁺.

Multiplying or dividing 0⁻ with a positive number (including 0⁺) produces a negative result (either 0⁻ or −∞), while multiplying or dividing 0⁻ with a negative number produces a positive result.
(In other words, multiplying or dividing with 0⁻ follows standard sign rules.)

When comparing 0⁺ and 0⁻, 0⁻ is less than 0⁺. For example, min(0, 0) must produce 0⁻, max(0, 0) must produce 0⁺, and clamp(0, 0, 1) must produce 0⁺.
If a toplevel calculation (a math function not nested inside of another math function) would produce a NaN, it instead produces +∞. If a toplevel calculation would produce 0⁻, it instead produces the standard "unsigned" zero.
On the other hand, calc(1 / calc(5 * 0)) produces −∞,
same as calc(1 / (5 * 0))—
Note: Algebraic simplifications do not affect the validity of a math function or its resolved type. For example, calc(5px  5px + 10s) and calc(0 * 5px + 10s) are both invalid due to the attempt to add a length and a time.
Note: Note that <percentage>s relative to <number>s,
such as in opacity,
are not combinable with those numbers—
Note: Because <numbertoken>s are always interpreted as <number>s or <integer>s, "unitless 0" <length>s aren’t supported in math functions. That is, width: calc(0 + 5px); is invalid, because it’s trying to add a <number> to a <length>, even though both width: 0; and width: 5px; are valid.
Note: Altho there are a few properties in which a bare <number> becomes a <length> at usedvalue time (specifically, lineheight and tabsize), <number>s never become "lengthlike" in calc(). They always stay as <number>s.
11.7. Internal Representation
The internal representation of a math function is a calculation tree: a tree where the branch nodes are operator nodes corresponding either to math functions (such as Min, Cos, Sqrt, etc) or to operators in a calculation (Sum, Product, Negate, and Invert, the calcoperator nodes), and the leaf nodes are either numeric values (such as numbers, dimensions, and percentages) or nonmath functions that resolve to a numeric type.
Math functions are turned into calculation trees depending on the function:
 calc()

The internal representation of a calc() function is the result of parsing a calculation from its argument.
 any other math function

The internal representation is an operator node with the same name as the function, whose children are the result of parsing a calculation from each of the function’s arguments, in the order they appear.

Discard any <whitespacetoken>s from values.

An item in values is an “operator” if it’s a <delimtoken> with the value "+", "", "*", or "/". Otherwise, it’s a “value”.

Collect children into Product and Invert nodes.
For every consecutive run of value items in values separated by "*" or "/" operators:

For each "/" operator in the run, replace its righthand value item rhs with an Invert node containing rhs as its child.

Replace the entire run with a Product node containing the value items of the run as its children.


Collect children into Sum and Negate nodes.

For each "" operator item in values, replace its righthand value item rhs with a Negate node containing rhs as its child.

If values has only one item, and it is a Product node or a parenthesized simple block, replace values with that item.
Otherwise, replace values with a Sum node containing the value items of values as its children.


At this point values is a tree of Sum, Product, Negate, and Invert nodes, with other types of values at the leaf nodes. Process the leaf nodes.
For every leaf node leaf in values:

If leaf is a parenthesized simple block, replace leaf with the result of parsing a calculation from leaf’s contents.

If leaf is a math function, replace leaf with the internal representation of that math function.


Return the result of simplifying a calculation tree from values.
11.7.1. Simplification
Internal representations of math functions are eagerly simplified to the extent possible, using standard algebraic simplifications (distributing multiplication over sums, combining similar units, etc.).

If root is a numeric value:

If root is a percentage that will be resolved against another value, and there is enough information available to resolve it, do so, and express the resulting numeric value in the appropriate canonical unit. Return the value.

If root is a dimension that is not expressed in its canonical unit, and there is enough information available to convert it to the canonical unit, do so, and return the value.

Otherwise, return root.


If root is any other leaf node (not an operator node):

If there is enough information available to determine its numeric value, return its value, expressed in the value’s canonical unit.

Otherwise, return root.


At this point, root is an operator node. Simplify all the children of root.

If root is an operator node that’s not one of the calcoperator nodes, and all of its children are numeric values with enough information to computed the operation root represents, return the result of running root’s operation using its children, expressed in the result’s canonical unit.

If root is a Negate node:

If root’s child is a numeric value, return an equivalent numeric value, but with the value negated (0  value).

If root’s child is a Negate node, return the child’s child.

Return root.


If root is an Invert node:

If root’s child is a number (not a percentage or dimension) return the reciprocal of the child’s value.

If root’s child is an Invert node, return the child’s child.

Return root.


If root is a Sum node:

For each of root’s children that are Sum nodes, replace them with their children.

For each set of root’s children that are numeric values with identical units, remove those children and replace them with a single numeric value containing the sum of the removed nodes, and with the same unit.
(E.g. combine numbers, combine percentages, combine px values, etc.)

If root has only a single child at this point, return the child. Otherwise, return root.


If root is a Product node:

For each of root’s children that are Product nodes, replace them with their children.

If root has multiple children that are numbers (not percentages or dimensions), remove them and replace them with a single number containing the product of the removed nodes.

If root contains only two children, one of which is a number (not a percentage or dimension) and the other of which is a Sum whose children are all numeric values, multiply all of the Sum’s children by the number, then return the Sum.

If root contains only numeric values and/or Invert nodes containing numeric values, and multiplying the types of all the children (noting that the type of an Invert node is the inverse of its child’s type) results in a type that matches any of the types that a math function can resolve to, return the result of multiplying all the values of the children (noting that the value of an Invert node is the reciprocal of its child’s value), expressed in the result’s canonical unit.

Return root.

11.8. Computed Value
The computed value of a math function is its calculation tree simplified, using all the information available at computed value time. (Such as the em to px ratio, how to resolve percentages in some properties, etc.)
Where percentages are not resolved at computedvalue time, they are not resolved in math functions, e.g. calc(100%  100% + 1px) resolves to calc(0% + 1px), not to 1px. If there are special rules for computing percentages in a value (e.g. the height property), they apply whenever a math function contains percentages.
The calculation tree is again simplified at used value time; with used value time information, a math function always simplifies down to a single numeric value.
Due to this, backgroundposition computation preserves the percentage in a calc() whereas fontsize will compute such expressions directly into a length.
Given the complexities of width and height calculations on table cells and table elements, math expressions mixing both percentages and lengths for widths and heights on table columns, table column groups, table rows, table row groups, and table cells in both auto and fixed layout tables MUST be treated as if auto had been specified.
11.9. Range Checking
Parsetime rangechecking of values is not performed within math functions, and therefore outofrange values do not cause the declaration to become invalid. However, the value resulting from an expression must be clamped to the range allowed in the target context. Clamping is performed on computed values to the extent possible, and also on used values if computation was unable to sufficiently simplify the expression to allow rangechecking. (Clamping is not performed on specified values.)
Note: This requires all contexts accepting calc() to define their allowable values as a closed (not open) interval.
Note: By definition, ±∞ are outside the allowed range for any property, and will clamp to the minimum/maximum value allowed. Even for properties that explicitly allow infinity as a keyword value, such as animationiterationcount, will end up clamping ±∞, as math functions can’t resolve to keyword values; the numeric part of the property’s syntax still has a minimum/maximum value.
Additionally, if a math function that resolves to <number> is used somewhere that only accepts <integer>, the computed value and used value are rounded to the nearest integer, in the same manner as clamping, above. The rounding method must be the same as is used for animations of integer values.
width: calc(5px  10px); width: calc(5px); width: 0px;
Note however that width: 5px is not equivalent to width: calc(5px)! Outofrange values outside calc() are syntactically invalid, and cause the entire declaration to be dropped.
11.10. Serialization
This section is still under discussion.

If the root of the calculation tree fn represents is a numeric value (number, percentage, or dimension), and the serialization being produced is of a computed value or later, then clamp the value to the range allowed for its context (if necessary), then serialize the value as normal and return the result.

If fn represents an infinite or NaN value:

Let s be the string "calc(".

Create a numeric value in the canonical unit for fn’s type (such as px for <length>), with a value of 1 if fn represents +∞, a value of 1 if fn represents −∞, or a value of 0 if fn represents NaN. Serialize this numeric value and append it to s.

Append " / 0)" to s, and return s.


If the calculation tree’s root node is a numeric value, or a calcoperator node, let s be a string initially containing "calc(".
Otherwise, let s be a string initially containing the name of the root node, lowercased (such as "sin" or "max"), followed by a "(" (open parenthesis).

For each child of the root node, serialize the calculation tree. If a result of this serialization starts with a "(" (open parenthesis) and ends with a ")" (close parenthesis), remove those characters from the result. Concatenate all of the results using ", " (comma followed by space), then append the result to s.

Append ")" (close parenthesis) to s.

Return s.

Let root be the root node of the calculation tree.

If root is a numeric value, or a nonmath function, serialize root per the normal rules for it and return the result.

If root is anything but a Sum, Negate, Product, or Invert node, serialize a math function for the function corresponding to the node type, treating the node’s children as the function’s commaseparated calculation arguments, and return the result.

If root is a Negate node, let s be a string initially containing "(1 * ".
Serialize root’s child, and append it to s.
Append ")" to s, then return it.

If root is an Invert node, let s be a string initially containing "(1 / ".
Serialize root’s child, and append it to s.
Append ")" to s, then return it.

If root is a Sum node, let s be a string initially containing "(".
Serialize root’s first child, and append it to s.
For each child of root beyond the first:

If child is a Negate node, append "  " to s, then serialize the Negate’s child and append the result to s.

If child is a negative numeric value, append "  " to s, then serialize the negation of child as normal and append the result to s.

Otherwise, append " + " to s, then serialize child and append the result to s.
Finally, append ")" to s and return it.


If root is a Product node, let s be a string initially containing "(".
Serialize root’s first child, and append it to s.
For each child of root beyond the first:

If child is an Invert node, append " / " to s, then serialize the Invert’s child and append the result to s.

Otherwise, append " * " to s, then serialize child and append the result to s.
Finally, append ")" to s and return it.


Let ret be an empty list.

If nodes contains a number, remove it from nodes and append it to ret.

If nodes contains a percentage, remove it from nodes and append it to ret.

If nodes contains any dimensions, remove them from nodes, sort them by their units, ordered ASCII caseinsensitively, and append them to ret.

If nodes still contains any items, append them to ret in the same order.

Return ret.
A value like calc(20px + 0%) would serialize as calc(0% + 20px), maintaining both terms in the serialized value. (It’s important to maintain zerovalued terms, so the calc() doesn’t suddenly "change shape" in the middle of a transition when one of the values happens to have a zero value temporarily. This also removes the need to "pick a unit" when all the terms are zero.)
A value like calc(20px + 2em) would serialize as calc(2em + 20px) as a specified value (maintaining both units as they’re incompatible at specifiedvalue time, but sorting them alphabetically), or as something like 52px as a computed value (em values are converted to absolute lengths at computedvalue time, so assuming 1em = 16px, they combine into 52px, which then drops the calc() wrapper.)
See [CSSOM] for further information on serialization.
11.11. Combination of Math Functions
Interpolation of math functions, with each other or with numeric values and other numericvalued functions, is defined as V_{result} = calc((1  p) * V_{a} + p * V_{b}). (Simplification of the value might then reduce the expression to a smaller, simpler form.)
Addition of math functions, with each other or with numeric values and other numericvalued functions, is defined as V_{result} = calc(V_{a} + V_{b}). (Simplification of the value might then reduce the expression to a smaller, simpler form.)
12. Appendix A: IANA Considerations
12.1. Registration for the about:invalid
URL scheme
This sections defines and registers the about:invalid
URL,
in accordance with the registration procedure defined in [RFC6694].
The official record of this registration can be found at http://www.iana.org/assignments/abouturitokens/abouturitokens.xhtml.
Registered Token  invalid


Intended Usage  The about:invalid URL references a nonexistent document with a generic error condition.
It can be used when a URL is necessary, but the default value shouldn’t be resolveable as any type of document.

Contact/Change controller  CSS WG <wwwstyle@w3.org> (on behalf of W3C) 
Specification  CSS Values and Units Module Level 3 
Acknowledgments
Firstly, the editors would like to thank all of the contributors to the previous level of this module.
Secondly, we would like to acknowledge Koji Ishii and Xidorn Quan for their comments and suggestions, which have improved Level 4.
Changes
Changes since the 10 October 2018 Working Draft consist of synchronizing with the recent changes in CSS Level 3.
Changes since the 14 August 2018 Working Draft:
 Added rules for interpolation per value type, and clarified computed values.
Changes since Level 3:
 Added the vi, vb, ic, cap, lh and rlh units.
 Added min(), max(), and clamp() functional notations.
 Added unit arithmetic to calc().
 Added toggle() (punted from level 3 originally).
 Added unit algebra, cribbing from [csstypedom1].
 A noninteger in a calc() automatically rounds to the nearest integer when used where an <integer> is required.
 Defined serialization of math functions.
Security and Privacy Considerations
This specification mostly just defines units that are common to CSS specifications, and which present no security concerns.
Note: Does URL handling have a security concern? Probably.
This specification defines units that expose the user’s screen size and default font size, but both are trivially observable from JS, so they do not constitutate a new privacy risk.