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More Math Functions: Draft 2.2

(Issue)

This proposal adds the following members to the built-in sass:math module.

Table of Contents

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?

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.

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.

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:

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.

  • 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:

    (...)

  • Otherwise, if declaration is within one or more blocks associated with @if, @each, @for, and/or @while rules and no other blocks:

    (...)

  • 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.

  • 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.

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.

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.

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.

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.

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.

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.

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.
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