# More Math Functions: Draft 2.2 *[(Issue)](https://github.com/sass/sass/issues/851)* This proposal adds the following members to the built-in `sass:math` module. ## Table of Contents * [Background](#background) * [Summary](#summary) * [Semantics](#semantics) * [Built-in Module Variables](#built-in-module-variables) * [Variables](#variables) * [`$e`](#e) * [`$pi`](#pi) * [Functions](#functions) * [`clamp()`](#clamp) * [`hypot()`](#hypot) * [Exponentiation](#exponentiation) * [`log()`](#log) * [`pow()`](#pow) * [`sqrt()`](#sqrt) * [Trigonometry](#trigonometry) * [`cos()`](#cos) * [`sin()`](#sin) * [`tan()`](#tan) * [`acos()`](#acos) * [`asin()`](#asin) * [`atan()`](#atan) * [`atan2()`](#atan2) * [Edge cases](#edge-cases) ## Background > This section is non-normative. Sass recently implemented a module system with a new built-in `sass:math` module. The demand for built-in math functions can now be fulfilled safely by implementing them inside this module. None of these new functions will be made available on the global namespace. ## Summary > This section is non-normative. This proposal defines Sassified versions of all the mathematical functions in the [CSS Values and Units 4 Draft], as well as logarithms and the constants `e` and `pi`. Each function is basically equivalent to its mathematical form, with stricter unit handling. Proper unit handling prevents these functions from creating meaningless units. For instance, consider `(1px)^(1/3)`—what does the unit `px^(1/3)` mean? [CSS Values and Units 4 Draft]: https://drafts.csswg.org/css-values-4/#math To avoid issues like this, the exponential functions—`log()`, `pow()`, `sqrt()`— accept only a unitless number as input, and output a unitless number. The trig functions—`cos()`, `sin()`, `tan()`—accept a SassScript number with a unit, as long as that unit is an [angle] type. If the input is a unitless number, it is treated as though it were in `rad`. These functions output a unitless number. [angle]: https://drafts.csswg.org/css-values-4/#angles The inverse trig functions—`acos()`, `asin()`, `atan()`—accept a unitless number and output a SassScript number in `deg`. `atan2()` is similar, but it accepts two unitless numbers. `clamp()` accepts three SassScript numbers with [compatible] units: the minimum value, preferred value, and maximum value. This function "clamps" the preferred value in between the minimum and maximum values, while preserving their units appropriately. For example, `clamp(1in, 15cm, 12in)` outputs `15cm`, whereas `clamp(1in, 1cm, 12in)` outputs `1in`. [compatible]: ../spec/built-in-modules/math.md#compatible `hypot()` accepts `n` SassScript numbers with compatible units, and outputs the length of the `n`-dimensional vector that has components equal to each of the inputs. Since the inputs' units may all be different, the output takes the unit of the first input. ## Semantics ### Built-in Module Variables Variables defined in built-in modules are not modifiable. As such, this proposal modifies the semantics of [Executing a Variable Declaration] within the [Variables spec] to read as follows: [Executing a Variable Declaration]: ../spec/variables.md#executing-a-variable-declaration [Variables spec]: ../spec/variables.md To execute a `VariableDeclaration` `declaration`: * Let `value` be the result of evaluating `declaration`'s `Expression`. * Let `name` be `declaration`'s `Variable`. * **Let `resolved` be the result of [resolving a variable] named `name`.** [resolving a variable]: ../spec/modules.md#resolving-a-member * If `name` is a `NamespacedVariable` and `declaration` has a `!global` flag, throw an error. * **Otherwise, if `resolved` is a variable from a built-in module, throw an error.** * Otherwise, if `declaration` is outside of any block of statements, *or* `declaration` has a `!global` flag, *or* `name` is a `NamespacedVariable`: * ~~Let `resolved` be the result of [resolving a variable] named `name` using `file`, `uses`, and `import`.~~ (...) * Otherwise, if `declaration` is within one or more blocks associated with `@if`, `@each`, `@for`, and/or `@while` rules *and no other blocks*: * ~~Let `resolved` be the result of [resolving a variable] named `name`.~~ (...) * ~~Otherwise, if no block containing `declaration` has a [scope] with a variable named `name`, set the innermost block's scope's variable `name` to `value`.~~ [scope]: ../spec/spec.md#scope * **Otherwise, if `resolved` is null, get the innermost block containing `declaration` and set its scope's variable `name` to `value`.** * ~~Otherwise, let `scope` be the scope of the innermost block such that `scope` already has a variable named `name`.~~ * **Otherwise, set `resolved`'s value to `value`.** ## Variables ### `$e` Equal to the value of the mathematical constant `e` with a precision of 10 digits after the decimal point: `2.7182818285`. ### `$pi` Equal to the value of the mathematical constant `pi` with a precision of 10 digits after the decimal point: `3.1415926536`. ## Functions ### `clamp()` ``` clamp($min, $number, $max) ``` * If the units of `$min`, `$number`, and `$max` are not compatible with each other, throw an error. * If some arguments have units and some do not, throw an error. * If `$min >= $max`, return `$min`. * If `$number <= $min`, return `$min`. * If `$number >= $max`, return `$max`. * Return `$number`. ### `hypot()` ``` hypot($numbers...) ``` * If all numbers are not compatible with each other, throw an error. * If some numbers have units and some do not, throw an error. * If all numbers are unitless, the return value is unitless. * Otherwise, the return value takes the unit of the leftmost number. * If any number equals `Infinity` or `-Infinity`, return `Infinity`. * Return the square root of the sum of the squares of each number. ### Exponentiation #### `log()` ``` log($number, $base: null) ``` * If `$number` has units, throw an error. * If `$base` is null: * If `$number < 0`, return `NaN` as a unitless number. * If `$number == 0`, return `-Infinity` as a unitless number. * If `$number == Infinity`, return `Infinity` as a unitless number. * Return the [natural log] of `$number`, as a unitless number. * Otherwise, return the natural log of `$number` divided by the natural log of `$base`, as a unitless number. [natural log]: https://en.wikipedia.org/wiki/Natural_logarithm #### `pow()` ``` pow($base, $exponent) ``` * If `$base` or `$exponent` has units, throw an error. * If `$exponent == 0`, return `1` as a unitless number. * Otherwise, if `$exponent == Infinity` or `$exponent == -Infinity`: * If `$base == 1` or `$base == -1`, return `NaN` as a unitless number. * If `$base < -1` or `$base > 1` and if `$exponent > 0`, *or* if `$base > -1` and `$base < 1` and `$exponent < 0`, return `Infinity` as a unitless number. * Return `0` as a unitless number. * Otherwise: * If `$base < 0` and `$exponent` is not an integer, return `NaN` as a unitless number. * If `$base == 0` and `$exponent < 0`, or if `$base == Infinity` and `$exponent > 0`, return `Infinity` as a unitless number. * If `$base == -0` and `$exponent < 0`, or if `$base == -Infinity` and `$exponent > 0`: * If `$exponent` is an odd integer, return `-Infinity` as a unitless number. * Return `Infinity` as a unitless number. * If `$base == 0` and `$exponent > 0`, or if `$base == Infinity` and `$exponent < 0`, return `0` as a unitless number. * If `$base == -0` and `$exponent > 0`, or if `$base == -Infinity` and `$exponent < 0`: * If `$exponent` is an odd integer, return `-0` as a unitless number. * Return `0` as a unitless number. * Return `$base` raised to the power of `$exponent`, as a unitless number. #### `sqrt()` ``` sqrt($number) ``` * If `$number` has units, throw an error. * If `$number < 0`, return `NaN` as a unitless number. * If `$number == -0`, return `-0` as a unitless number. * If `$number == 0`, return `0` as a unitless number. * If `$number == Infinity`, return `Infinity` as a unitless number. * Return the square root of `$number`, as a unitless number. ### Trigonometry #### `cos()` ``` cos($number) ``` * If `$number` has units but is not an angle, throw an error. * If `$number` is unitless, treat it as though its unit were `rad`. * If `$number == Infinity` or `$number == -Infinity`, return `NaN` as a unitless number. * Return the [cosine] of `$number`, as a unitless number. [cosine]: https://en.wikipedia.org/wiki/Trigonometric_functions#Right-angled_triangle_definitions #### `sin()` ``` sin($number) ``` * If `$number` has units but is not an angle, throw an error. * If `$number` is unitless, treat it as though its unit were `rad`. * If `$number == Infinity` or `$number == -Infinity`, return `NaN` as a unitless number. * If `$number == -0`, return `-0` as a unitless number. * If `$number == 0`, return `0` as a unitless number. * Return the [sine] of `$number`, as a unitless number. [sine]: https://en.wikipedia.org/wiki/Trigonometric_functions#Right-angled_triangle_definitions #### `tan()` ``` tan($number) ``` * If `$number` has units but is not an angle, throw an error. * If `$number` is unitless, treat it as though its unit were `rad`. * If `$number == Infinity` or `$number == -Infinity`, return `NaN` as a unitless number. * If `$number == -0`, return `-0` as a unitless number. * If `$number == 0`, return `0` as a unitless number. * If `$number` is equivalent to `90deg +/- 360deg * n`, where `n` is any integer, return `Infinity` as a unitless number. * If `$number` is equivalent to `-90deg +/- 360deg * n`, where `n` is any integer, return `-Infinity` as a unitless number. * Return the [tangent] of `$number`, as a unitless number. [tangent]: https://en.wikipedia.org/wiki/Trigonometric_functions#Right-angled_triangle_definitions #### `acos()` ``` acos($number) ``` * If `$number` has units, throw an error. * If `$number < -1` or `$number > 1`, return `NaN` as a number in `deg`. * If `$number == 1`, return `0deg`. * Return the [arccosine] of `$number`, as a number in `deg`. [arccosine]: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Basic_properties #### `asin()` ``` asin($number) ``` * If `$number` has units, throw an error. * If `$number < -1` or `$number > 1`, return `NaN` as a number in `deg`. * If `$number == -0`, return `-0deg`. * If `$number == 0`, return `0deg`. * Return the [arcsine] of `$number`, as a number in `deg`. [arcsine]: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Basic_properties #### `atan()` ``` atan($number) ``` * If `$number` has units, throw an error. * If `$number == -0`, return `-0deg`. * If `$number == 0`, return `0deg`. * If `$number == -Infinity`, return `-90deg`. * If `$number == Infinity`, return `90deg`. * Return the [arctangent] of `$number`, as a number in `deg`. [arctangent]: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Basic_properties #### `atan2()` > `atan2($y, $x)` is distinct from `atan($y / $x)` because it preserves the > quadrant of the point in question. For example, `atan2(1, -1)` corresponds to > the point `(-1, 1)` and returns `135deg`. In contrast, `atan(1 / -1)` and > `atan(-1 / 1)` resolve first to `atan(-1)`, so both return `-45deg`. ``` atan2($y, $x) ``` * If `$y` and `$x` are not compatible, throw an error. * If `$y` has units and `$x` does not, or vice-versa, throw an error. * If the inputs match one of the following edge cases, return the provided number. Otherwise, return the [2-argument arctangent] of `$y` and `$x`, as a number in `deg`. [2-argument arctangent]: https://en.wikipedia.org/wiki/Atan2 ##### Edge cases
X
−Infinity -finite -0 0 finite Infinity
Y −Infinity -135deg -90deg -90deg -90deg -90deg -45deg
-finite -180deg -90deg -90deg -0deg
-0 -180deg -180deg -180deg -0deg -0deg -0deg
0 180deg 180deg 180deg 0deg 0deg 0deg
finite 180deg 90deg 90deg 0deg
Infinity 135deg 90deg 90deg 90deg 90deg 45deg