package tablecloth-base

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Module Tablecloth.FloatSource

Functions for working with floating point numbers.

A module for working with floating-point numbers.

Valid syntax for floats includes:

  0.
  42.
  42.0
  3.14
  -0.1234
  123_456.123_456
  6.022e23   (* = (6.022 * 10^23) *)
  6.022e+23  (* = (6.022 * 10^23) *)
  1.602e-19  (* = (1.602 * 10^-19) *)
  1e3        (* = (1 * 10 ** 3) = 1000. *)

Without opening this module you can use the . suffixed operators e.g

 1. +. 2. /. 0.25 *. 2. = 17.

But by opening this module locally you can use the un-suffixed operators

Float.((10.0 - 1.5 / 0.5) ** 3.0) = 2401.0

Historical Note: The particular details of floats (e.g. NaN) are specified by IEEE 754 which is literally hard-coded into almost all CPUs in the world.

Sourcetype t = float

Constants

Sourceval zero : t

The literal 0.0 as a named value

Sourceval one : t

The literal 1.0 as a named value

Sourceval nan : t

NaN as a named value. NaN stands for not a number.

Note comparing values with Float.nan will always return false even if the value you are comparing against is also NaN.

For detecting Nan you should use Float.is_nan

Examples

  let is_not_a_number x = Float.(x = nan) in

  is_not_a_number nan = false
Sourceval infinity : t

Positive infinity.

Float.log ~base:10.0 0.0 = Float.infinity
Sourceval negative_infinity : t

Negative infinity, see Float.infinity

Sourceval e : t

An approximation of Euler's number.

Sourceval pi : t

An approximation of pi.

Sourceval epsilon : t

The smallest interval between two representable numbers.

Sourceval largest_value : t

The largest (furthest from zero) representable positive float.

Sourceval smallest_value : t

The smallest representable positive float. The closest to zero without actually being zero.

Sourceval maximum_safe_integer : t

For floats greater than maximum_safe_integer, it no longer holds that Float.(n + 1.) > n.

Sourceval minimum_safe_integer : t

For floats less than minimum_safe_integer, it no longer holds that Float.(n - 1.) < n.

Create

Sourceval from_int : int -> t

Convert an Int to a float.

Examples

  Float.from_int 5 = 5.0
  Float.from_int 0 = 0.0
  Float.from_int (-7) = (-7.0)
Sourceval from_string : string -> t option

Convert a String to a float.

Parses nan and infinity case-insensitive.

Examples

Float.from_string "4.667" = Some 4.667
Float.from_string "-4.667" = Some (-4.667)
Float.from_string "Hamster" = None
Float.from_string "NaN" = Some Float.nan
Float.from_string "nan" = Some Float.nan
Float.from_string "Infinity" = Some Float.infinity

Basic arithmetic and operators

Sourceval add : t -> t -> t

Addition for floating point numbers.

Although ints and floats support many of the same basic operations such as addition and subtraction you cannot add an int and a float directly which means you need to use functions like Int.to_float to convert both values to the same type.

So if you needed to add a List.length to a float for some reason, you could:

Float.add 3.14 (Int.to_float (List.length [1;2;3])) = 6.14

or

Float.(round 3.5 |> toInt) + List.length [1;2;3] = 6

Languages like Java and JavaScript automatically convert int values to float values when you mix and match. This can make it difficult to be sure exactly what type of number you are dealing with and cause unexpected behavior.

OCaml has opted for a design that makes all conversions explicit.

Examples

  Float.add 3.14 3.14 = 6.28
  Float.(3.14 + 3.14 = 6.28)
Sourceval (+) : t -> t -> t

See Float.add

Sourceval subtract : t -> t -> t

Subtract numbers.

Alternatively the - operator can be used

Examples

Float.subtract 4.0 3.0 = 1.0
Float.(4.0 - 3.0) = 1.0
Sourceval (-) : t -> t -> t

See Float.subtract

Sourceval multiply : t -> t -> t

Multiply numbers.

Alternatively the * operator can be used

Examples

Float.multiply 2.0 7.0 = 14.0
Float.(2.0 * 7.0) = 14.0
Sourceval (*) : t -> t -> t

See Float.multiply.

Sourceval divide : t -> by:t -> t

Floating-point division:

Alternatively the / operator can be used

Examples

Float.divide 3.14 ~by:2.0 = 1.57
Float.(3.14 / 2.0) = 1.57
Sourceval (/) : t -> t -> t

See Float.divide

Sourceval power : base:t -> exponent:t -> t

Exponentiation, takes the base first, then the exponent.

Alternatively the ** operator can be used

Examples

Float.power ~base:7.0 ~exponent:3.0 = 343.0
Float.(7.0 ** 3.0) = 343.0
Sourceval (**) : t -> t -> t

See Float.power

Sourceval negate : t -> t

Flips the 'sign' of a float so that positive floats become negative and negative integers become positive. Zero stays as it is.

Alternatively an operator is available

Examples

Float.(~- 4.0) = (-4.0)
  Float.negate 8. = (-8.)
  Float.negate (-7.) = 7.
  Float.negate 0. = 0.
Sourceval (~-) : t -> t

See Float.negate

Sourceval absolute : t -> t

Get the absolute value of a number.

Examples

  Float.absolute 8. = 8.
  Float.absolute (-7.) = 7.
  Float.absolute 0. = 0.
Sourceval maximum : t -> t -> t

Returns the larger of two floats, if both arguments are equal, returns the first argument.

If either (or both) of the arguments are NaN, returns NaN

Examples

Float.maximum 7. 9. = 9.
Float.maximum (-4.) (-1.) = (-1.)
Float.(is_nan (maximum 7. nan)) = true
Sourceval minimum : t -> t -> t

Returns the smaller of two floats, if both arguments are equal, returns the first argument.

If either (or both) of the arguments are NaN, returns NaN.

Examples

Float.minimum 7.0 9.0 = 7.0
Float.minimum (-4.0) (-1.0) = (-4.0)
Float.(is_nan (minimum 7. nan)) = true
Sourceval clamp : t -> lower:t -> upper:t -> t

Clamps n within the inclusive lower and upper bounds.

Exceptions

Throws an Invalid_argument exception if lower > upper.

Examples

Float.clamp ~lower:0. ~upper:8. 5. = 5.
Float.clamp ~lower:0. ~upper:8. 9. = 8.
Float.clamp ~lower:(-10.) ~upper:(-5.) 5. = (-5.)

Fancier math

Sourceval square_root : t -> t

Take the square root of a number.

square_root returns NaN when its argument is negative. See Float.nan for more.

Examples

Float.square_root 4.0 = 2.0
Float.square_root 9.0 = 3.0
Sourceval log : t -> base:t -> t

Calculate the logarithm of a number with a given base.

Examples

Float.log ~base:10. 100. = 2.
Float.log ~base:2. 256. = 8.

Query

Sourceval is_nan : t -> bool

Determine whether a float is an undefined or unrepresentable number.

Note: this function is more useful than it might seem since NaN does not equal NaN:

Float.(nan = nan) = false

Examples

Float.is_nan (0.0 /. 0.0) = true
Float.(is_nan (square_root (-1.0))) = true
Float.is_nan (1.0 /. 0.0) = false  (* Float.infinity {b is} a number *)
Float.is_nan 1. = false
Sourceval is_finite : t -> bool

Determine whether a float is finite number. True for any float except Infinity, -Infinity or NaN

Notice that NaN is not finite!

Examples

Float.is_finite (0. /. 0.) = false
Float.(is_finite (square_root (-1.))) = false
Float.is_finite (1. /. 0.) = false
Float.is_finite 1. = true
Float.(is_finite nan) = false
Sourceval is_infinite : t -> bool

Determine whether a float is positive or negative infinity.

Examples

Float.is_infinite (0. /. 0.) = false
Float.(is_infinite (square_root (-1.))) = false
Float.is_infinite (1. /. 0.) = true
Float.is_infinite 1. = false
Float.(is_infinite nan) = false
Sourceval is_integer : t -> bool

Determine whether the passed value is an integer.

Examples

Float.is_integer 4.0 = true
Float.is_integer Float.pi = false
Sourceval is_safe_integer : t -> bool

Determine whether the passed value is a safe integer (number between -(2**53 - 1) and 2**53 - 1).

Examples

Float.is_safe_integer 4.0 = true
Float.is_safe_integer Float.pi = false
Float.(is_safe_integer (maximum_safe_integer + 1.)) = false
Sourceval in_range : t -> lower:t -> upper:t -> bool

Checks if a float is between lower and up to, but not including, upper.

If lower is not specified, it's set to to 0.0.

Exceptions

Throws an Invalid_argument exception if lower > upper

Examples

Float.in_range ~lower:2. ~upper:4. 3. = true
Float.in_range ~lower:1. ~upper:2. 2. = false
Float.in_range ~lower:5.2 ~upper:7.9 9.6 = false

Angles

Sourcetype radians = float

This type is just an alias for float.

Its purpose is to make understanding the signatures of the following functions a little easier.

Sourceval hypotenuse : t -> t -> t

hypotenuse x y returns the length of the hypotenuse of a right-angled triangle with sides of length x and y, or, equivalently, the distance of the point (x, y) to (0, 0).

Examples

Float.hypotenuse 3. 4. = 5.
Sourceval degrees : t -> radians

Converts an angle in degrees to Float.radians.

Examples

Float.degrees 180. = Float.pi
Float.degrees 360. = Float.pi *. 2.
Float.degrees 90. = Float.pi /. 2.
Sourceval radians : t -> radians

Convert a Float.t to radians.

Note This function doesn't actually do anything to its argument, but can be useful to indicate intent when inter-mixing angles of different units within the same function.

Examples

Float.(radians pi) = 3.141592653589793
Sourceval turns : t -> radians

Convert an angle in turns into Float.radians.

One turn is equal to 360 degrees.

Examples

Float.(turns (1. / 2.)) = Float.pi
Float.(turns 1. = degrees 360.)

Polar coordinates

Sourceval from_polar : (float * radians) -> float * float

Convert polar coordinates (radius, radians) to Cartesian coordinates (x,y).

Examples

Float.(from_polar (square_root 2., degrees 45.)) = (1., 1.)
Sourceval to_polar : (float * float) -> float * radians

Convert Cartesian coordinates (x, y) to polar coordinates (radius, radians).

Examples

Float.to_polar (-1.0, 0.0) = (1.0, Float.pi)
Float.to_polar (3.0, 4.0) = (5.0, 0.9272952180016122)
Float.to_polar (5.0, 12.0) = (13.0, 1.1760052070951352)
Sourceval cos : radians -> t

Figure out the cosine given an angle in radians.

Examples

Float.(cos (degrees 60.)) = 0.5000000000000001
Float.(cos (radians (pi / 3.))) = 0.5000000000000001
Sourceval acos : radians -> t

Figure out the arccosine for adjacent / hypotenuse in radians:

Examples

Float.(acos (radians 1.0 / 2.0)) = Float.radians 1.0471975511965979 (* 60 degrees or pi/3 radians *)
Sourceval sin : radians -> t

Figure out the sine given an angle in radians.

Examples

Float.(sin (degrees 30.)) = 0.49999999999999994
Float.(sin (radians (pi / 6.))) = 0.49999999999999994
Sourceval asin : radians -> t

Figure out the arcsine for opposite / hypotenuse in radians:

Examples

Float.(asin (1.0 / 2.0)) = 0.5235987755982989 (* 30 degrees or pi / 6 radians *)
Sourceval tan : radians -> t

Figure out the tangent given an angle in radians.

Examples

Float.(tan (degrees 45.)) = 0.9999999999999999
Float.(tan (radians (pi / 4.))) = 0.9999999999999999
Float.(tan (pi / 4.)) = 0.9999999999999999
Sourceval atan : t -> 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:

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 Float.atan2 instead!

Examples

Float.atan (1. /. 1.) = 0.7853981633974483  (* 45 degrees or pi/4 radians *)
Float.atan (1. /. -1.) = (-0.7853981633974483)  (* 315 degrees or 7 * pi / 4 radians *)
Float.atan (-1. /. -1.) = 0.7853981633974483 (* 45 degrees or pi/4 radians *)
Float.atan (-1. /.  1.) = (-0.7853981633974483) (* 315 degrees or 7 * pi/4 radians *)
Sourceval atan2 : y:t -> x:t -> radians

This helps you find the angle (in radians) to an (x, y) coordinate.

So rather than Float.(atan (y / x)) you can Float.atan2 ~y ~x and you can get a full range of angles:

Examples

Float.atan2 ~y:1. ~x:1. = 0.7853981633974483  (* 45 degrees or pi/4 radians *)
Float.atan2 ~y:1. ~x:(-1.) = 2.3561944901923449  (* 135 degrees or 3 * pi/4 radians *)
Float.atan2 ~y:(-1.) ~x:(-1.) = (-2.3561944901923449) (* 225 degrees or 5 * pi/4 radians *)
Float.atan2 ~y:(-1.) ~x:1. = (-0.7853981633974483) (* 315 degrees or 7 * pi/4 radians *)

Rounding

Sourcetype direction = [
  1. | `Zero
  2. | `AwayFromZero
  3. | `Up
  4. | `Down
  5. | `Closest of [ `Zero | `AwayFromZero | `Up | `Down | `ToEven ]
]

The possible directions availible when doing Float.round.

See Float.round for what each variant represents.

Sourceval round : ?direction:direction -> t -> t

Round a number, by default to the to the closest int with halves rounded `Up (towards positive infinity).

Other rounding strategies are available by using the optional ~direction labelelled.

Examples

  Float.round 1.2 = 1.0
  Float.round 1.5 = 2.0
  Float.round 1.8 = 2.0
  Float.round (-1.2) = (-1.0)
  Float.round (-1.5) = (-1.0)
  Float.round (-1.8) = (-2.0)

Towards zero

  Float.round ~direction:`Zero 1.2 = 1.0
  Float.round ~direction:`Zero 1.5 = 1.0
  Float.round ~direction:`Zero 1.8 = 1.0
  Float.round ~direction:`Zero (-1.2) = (-1.0)
  Float.round ~direction:`Zero (-1.5) = (-1.0)
  Float.round ~direction:`Zero (-1.8) = (-1.0)

Away from zero

  Float.round ~direction:`AwayFromZero 1.2 = 2.0
  Float.round ~direction:`AwayFromZero 1.5 = 2.0
  Float.round ~direction:`AwayFromZero 1.8 = 2.0
  Float.round ~direction:`AwayFromZero (-1.2) = (-2.0)
  Float.round ~direction:`AwayFromZero (-1.5) = (-2.0)
  Float.round ~direction:`AwayFromZero (-1.8) = (-2.0)

Towards infinity

This is also known as Float.ceiling.

  Float.round ~direction:`Up 1.2 = 2.0
  Float.round ~direction:`Up 1.5 = 2.0
  Float.round ~direction:`Up 1.8 = 2.0
  Float.round ~direction:`Up (-1.2) = (-1.0)
  Float.round ~direction:`Up (-1.5) = (-1.0)
  Float.round ~direction:`Up (-1.8) = (-1.0)

Towards negative infinity

This is also known as Float.floor.

List.map  ~f:(Float.round ~direction:`Down) [-1.8; -1.5; -1.2; 1.2; 1.5; 1.8] = [-2.0; -2.0; -2.0; 1.0; 1.0; 1.0]

To the closest integer

Rounding a number x to the closest integer requires some tie-breaking for when the fraction part of x is exactly 0.5.

Halves rounded towards zero

List.map  ~f:(Float.round ~direction:(`Closest `Zero)) [-1.8; -1.5; -1.2; 1.2; 1.5; 1.8] = [-2.0; -1.0; -1.0; 1.0; 1.0; 2.0]

Halves rounded away from zero

This method is often known as commercial rounding.

List.map  ~f:(Float.round ~direction:(`Closest `AwayFromZero)) [-1.8; -1.5; -1.2; 1.2; 1.5; 1.8] = [-2.0; -2.0; -1.0; 1.0; 2.0; 2.0]

Halves rounded down

List.map  ~f:(Float.round ~direction:(`Closest `Down)) [-1.8; -1.5; -1.2; 1.2; 1.5; 1.8] = [-2.0; -2.0; -1.0; 1.0; 1.0; 2.0]

Halves rounded up

This is the default.

Float.round 1.5 is the same as Float.round ~direction:(`Closest `Up) 1.5

Halves rounded towards the closest even number

  Float.round ~direction:(`Closest `ToEven) (-1.5) = (-2.0)
  Float.round ~direction:(`Closest `ToEven) (-2.5) = (-2.0)
Sourceval floor : t -> t

Floor function, equivalent to Float.round ~direction:`Down.

Examples

  Float.floor 1.2 = 1.0
  Float.floor 1.5 = 1.0
  Float.floor 1.8 = 1.0
  Float.floor (-1.2) = (-2.0)
  Float.floor (-1.5) = (-2.0)
  Float.floor (-1.8) = (-2.0)
Sourceval ceiling : t -> t

Ceiling function, equivalent to Float.round ~direction:`Up.

Examples

  Float.ceiling 1.2 = 2.0
  Float.ceiling 1.5 = 2.0
  Float.ceiling 1.8 = 2.0
  Float.ceiling (-1.2) = (-1.0)
  Float.ceiling (-1.5) = (-1.0)
  Float.ceiling (-1.8) = (-1.0)
Sourceval truncate : t -> t

Ceiling function, equivalent to Float.round ~direction:`Zero.

Examples

  Float.truncate 1.0 = 1.
  Float.truncate 1.2 = 1.
  Float.truncate 1.5 = 1.
  Float.truncate 1.8 = 1.
  Float.truncate (-1.2) = (-1.)
  Float.truncate (-1.5) = (-1.)
  Float.truncate (-1.8) = (-1.)

Convert

Sourceval to_int : t -> int option

Converts a float to an Int by ignoring the decimal portion. See Float.truncate for examples.

Returns None when trying to round a float which can't be represented as an int such as Float.nan or Float.infinity or numbers which are too large or small.

You probably want to use some form of Float.round prior to using this function.

Examples

Float.(to_int 1.6) = (Some 1)
Float.(to_int 2.0) = (Some 2)
Float.(to_int 5.683) = (Some 5)
Float.(to_int nan) = None
Float.(to_int infinity) = None
Float.(round 1.6 |> to_int) = Some 2
Sourceval to_string : t -> string

Convert a float to a String.

The behaviour of this function is platform specific.

Compare

Sourceval equal : t -> t -> bool

Test two floats for equality.

Sourceval compare : t -> t -> int

Compare two floats.