Math

Round a number to the nearest integer.

round 1.0 == 1

round 1.2 == 1

round 1.5 == 2

round 1.8 == 2

round -1.2 == -1

round -1.5 == -1

round -1.8 == -2

Floor function, rounding down.

floor 1.0 == 1

floor 1.2 == 1

floor 1.5 == 1

floor 1.8 == 1

floor -1.2 == -2

floor -1.5 == -2

floor -1.8 == -2

Ceiling function, rounding up.

ceiling 1.0 == 1

ceiling 1.2 == 2

ceiling 1.5 == 2

ceiling 1.8 == 2

ceiling -1.2 == -1

ceiling -1.5 == -1

ceiling -1.8 == -1

Truncate a number, rounding towards zero.

truncate 1.0 == 1

truncate 1.2 == 1

truncate 1.5 == 1

truncate 1.8 == 1

truncate -1.2 == -1

truncate -1.5 == -1

truncate -1.8 == -1

Perform modular arithmetic. A common trick is to use (n mod 2) to detect even and odd numbers:

modBy 2 0 == 0

modBy 2 1 == 1

modBy 2 2 == 0

modBy 2 3 == 1

Our modBy function works in the typical mathematical way when you run into negative numbers:

List.map (modBy 4) [ -5, -4, -3, -2, -1,  0,  1,  2,  3,  4,  5 ]
--                 [  3,  0,  1,  2,  3,  0,  1,  2,  3,  0,  1 ]

Use remainderBy for a different treatment of negative numbers, or read Daan Leijen’s Division and Modulus for Computer Scientists for more information.

Get the remainder after division. Here are bunch of examples of dividing by four:

List.map (remainderBy 4) [ -5, -4, -3, -2, -1,  0,  1,  2,  3,  4,  5 ]
--                       [ -1,  0, -3, -2, -1,  0,  1,  2,  3,  0,  1 ]

Use modBy for a different treatment of negative numbers, or read Daan Leijen’s Division and Modulus for Computer Scientists for more information.

Get the absolute value of a number.

abs 16 == 16

abs -4 == 4

abs -8.5 == 8.5

abs 3.14 == 3.14

Take the square root of a number.

sqrt 4 == 2

sqrt 9 == 3

sqrt 16 == 4

sqrt 25 == 5

Calculate the logarithm of a number with a given base.

logBase 10 100 == 2

logBase 2 256 == 8

An approximation of e.

Convert degrees to standard Gren angles (radians).

degrees 180 == 3.141592653589793

Convert radians to standard Gren angles (radians).

radians pi == 3.141592653589793

Convert turns to standard Gren angles (radians). One turn is equal to 360°.

turns (1 / 2) == 3.141592653589793

An approximation of pi.

Figure out the cosine given an angle in radians.

cos (degrees 60) == 0.5000000000000001

cos (turns (1 / 6)) == 0.5000000000000001

cos (radians (pi / 3)) == 0.5000000000000001

cos (pi / 3) == 0.5000000000000001

Figure out the sine given an angle in radians.

sin (degrees 30) == 0.49999999999999994

sin (turns (1 / 12)) == 0.49999999999999994

sin (radians (pi / 6)) == 0.49999999999999994

sin (pi / 6) == 0.49999999999999994

Figure out the tangent given an angle in radians.

tan (degrees 45) == 0.9999999999999999

tan (turns (1 / 8)) == 0.9999999999999999

tan (radians (pi / 4)) == 0.9999999999999999

tan (pi / 4) == 0.9999999999999999

Figure out the arccosine for adjacent / hypotenuse in radians:

acos (1 / 2) == 1.0471975511965979 -- 60° or pi/3 radians

Figure out the arcsine for opposite / hypotenuse in radians:

asin (1 / 2) == 0.5235987755982989 -- 30° or pi/6 radians

This helps you find the angle (in radians) to an (x,y) coordinate, but in a way that is rarely useful in programming. You probably want atan2 instead!

This version takes y/x as its argument, so there is no way to know whether the negative signs comes from the y or x value. So as we go counter-clockwise around the origin from point (1,1) to (1,-1) to (-1,-1) to (-1,1) we do not get angles that go in the full circle:

atan (1 / 1) == 0.7853981633974483 --  45° or   pi/4 radians

atan (1 / -1) == -0.7853981633974483 -- 315° or 7*pi/4 radians

atan (-1 / -1) == 0.7853981633974483 --  45° or   pi/4 radians

atan (-1 / 1) == -0.7853981633974483 -- 315° or 7*pi/4 radians

Notice that everything is between pi/2 and -pi/2. That is pretty useless for figuring out angles in any sort of visualization, so again, check out atan2 instead!

This helps you find the angle (in radians) to an (x,y) coordinate. So rather than saying atan (y/x) you say atan2 y x and you can get a full range of angles:

atan2 1 1 == 0.7853981633974483 --  45° or   pi/4 radians

atan2 1 -1 == 2.356194490192345 -- 135° or 3*pi/4 radians

atan2 -1 -1 == -2.356194490192345 -- 225° or 5*pi/4 radians

atan2 -1 1 == -0.7853981633974483 -- 315° or 7*pi/4 radians