package base

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package base
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Floating-point representation and utilities.

If using 32-bit OCaml, you cannot quite assume operations act as you'd expect for IEEE 64-bit floats. E.g., one can have let x = ~-. (2. ** 62.) in x = x -. 1. evaluate to false while let x = ~-. (2. ** 62.) in let y = x -. 1 in x = y evaluates to true. This is related to 80-bit registers being used for calculations; you can force representation as a 64-bit value by let-binding.

type t = float
include Floatable.S with type t := t
val of_float : float -> t
val to_float : t -> float

max and min will return nan if either argument is nan.

The validate_* functions always fail if class is Nan or Infinite.

include Identifiable.S with type t := t
val hash_fold_t : Hash.state -> t -> Hash.state
val hash : t -> Hash.hash_value
include Sexpable.S with type t := t
val t_of_sexp : Sexplib0.Sexp.t -> t
val sexp_of_t : t -> Sexplib0.Sexp.t
include Stringable.S with type t := t
val of_string : string -> t
include Comparable.S with type t := t
include Comparisons.S with type t := t
include Comparisons.Infix with type t := t
val (>=) : t -> t -> bool
val (<=) : t -> t -> bool
val (=) : t -> t -> bool
val (>) : t -> t -> bool
val (<) : t -> t -> bool
val (<>) : t -> t -> bool
val equal : t -> t -> bool
val compare : t -> t -> int

compare t1 t2 returns 0 if t1 is equal to t2, a negative integer if t1 is less than t2, and a positive integer if t1 is greater than t2.

val min : t -> t -> t
val max : t -> t -> t
val ascending : t -> t -> int

ascending is identical to compare. descending x y = ascending y x. These are intended to be mnemonic when used like List.sort ~compare:ascending and List.sort ~cmp:descending, since they cause the list to be sorted in ascending or descending order, respectively.

val descending : t -> t -> int
val between : t -> low:t -> high:t -> bool

between t ~low ~high means low <= t <= high

val clamp_exn : t -> min:t -> max:t -> t

clamp_exn t ~min ~max returns t', the closest value to t such that between t' ~low:min ~high:max is true.

Raises if not (min <= max).

val clamp : t -> min:t -> max:t -> t Or_error.t
include Comparator.S with type t := t
type comparator_witness
val validate_lbound : min:t Maybe_bound.t -> t Validate.check
val validate_ubound : max:t Maybe_bound.t -> t Validate.check
val validate_bound : min:t Maybe_bound.t -> max:t Maybe_bound.t -> t Validate.check
include Pretty_printer.S with type t := t
val pp : Formatter.t -> t -> unit
include Comparable.With_zero with type t := t
val validate_positive : t Validate.check
val validate_non_negative : t Validate.check
val validate_negative : t Validate.check
val validate_non_positive : t Validate.check
val is_positive : t -> bool
val is_non_negative : t -> bool
val is_negative : t -> bool
val is_non_positive : t -> bool
include Invariant.S with type t := t
val invariant : t -> unit
val validate_ordinary : t Validate.check

validate_ordinary fails if class is Nan or Infinite.

val nan : t
val infinity : t
val neg_infinity : t
val max_value : t

Equal to infinity.

val min_value : t

Equal to neg_infinity.

val zero : t
val one : t
val minus_one : t
val pi : t

The constant pi.

val sqrt_pi : t

The constant sqrt(pi).

val sqrt_2pi : t

The constant sqrt(2 * pi).

val euler : t

Euler-Mascheroni constant (γ).

val epsilon_float : t

The difference between 1.0 and the smallest exactly representable floating-point number greater than 1.0. That is:

epsilon_float = (one_ulp `Up 1.0) -. 1.0

This gives the relative accuracy of type t, in the sense that for numbers on the order of x, the roundoff error is on the order of x *. float_epsilon.

See also: Machine epsilon.

val max_finite_value : t
  • min_positive_subnormal_value = 2 ** -1074
  • min_positive_normal_value = 2 ** -1022
val min_positive_subnormal_value : t
val min_positive_normal_value : t
val to_int64_preserve_order : t -> int64 option

An order-preserving bijection between all floats except for nans, and all int64s with absolute value smaller than or equal to 2**63 - 2**52. Note both 0. and -0. map to 0L.

val to_int64_preserve_order_exn : t -> int64
val of_int64_preserve_order : int64 -> t

Returns nan if the absolute value of the argument is too large.

val one_ulp : [ `Up | `Down ] -> t -> t

The next or previous representable float. ULP stands for "unit of least precision", and is the spacing between floating point numbers. Both one_ulp `Up infinity and one_ulp `Down neg_infinity return a nan.

val of_int : int -> t

Note that this doesn't round trip in either direction. For example, Float.to_int (Float.of_int max_int) <> max_int.

val to_int : t -> int
val of_int63 : Int63.t -> t
val of_int64 : int64 -> t
val to_int64 : t -> int64
val round : ?dir:[ `Zero | `Nearest | `Up | `Down ] -> t -> t

round rounds a float to an integer float. iround{,_exn} rounds a float to an int. Both round according to a direction dir, with default dir being `Nearest.

      | `Down    | rounds toward Float.neg_infinity                             |
      | `Up      | rounds toward Float.infinity                                 |
      | `Nearest | rounds to the nearest int ("round half-integers up")         |
      | `Zero    | rounds toward zero                                           |

iround_exn raises when trying to handle nan or trying to handle a float outside the range [float min_int, float max_int).

Here are some examples for round for each direction:

      | `Down    | [-2.,-1.)   to -2. | [-1.,0.)   to -1. | [0.,1.) to 0., [1.,2.) to 1. |
      | `Up      | (-2.,-1.]   to -1. | (-1.,0.]   to -0. | (0.,1.] to 1., (1.,2.] to 2. |
      | `Zero    | (-2.,-1.]   to -1. | (-1.,1.)   to 0.  | [1.,2.) to 1.                |
      | `Nearest | [-1.5,-0.5) to -1. | [-0.5,0.5) to 0.  | [0.5,1.5) to 1.              |

For convenience, versions of these functions with the dir argument hard-coded are provided. If you are writing performance-critical code you should use the versions with the hard-coded arguments (e.g. iround_down_exn). The _exn ones are the fastest.

The following properties hold:

  • of_int (iround_*_exn i) = i for any float i that is an integer with min_int <= i <= max_int.
  • round_* i = i for any float i that is an integer.
  • iround_*_exn (of_int i) = i for any int i with -2**52 <= i <= 2**52.
val iround : ?dir:[ `Zero | `Nearest | `Up | `Down ] -> t -> int option
val iround_exn : ?dir:[ `Zero | `Nearest | `Up | `Down ] -> t -> int
val round_towards_zero : t -> t
val round_down : t -> t
val round_up : t -> t
val round_nearest : t -> t

Rounds half integers up.

val round_nearest_half_to_even : t -> t

Rounds half integers to the even integer.

val iround_towards_zero : t -> int option
val iround_down : t -> int option
val iround_up : t -> int option
val iround_nearest : t -> int option
val iround_towards_zero_exn : t -> int
val iround_down_exn : t -> int
val iround_up_exn : t -> int
val iround_nearest_exn : t -> int
val int63_round_down_exn : t -> Int63.t
val int63_round_up_exn : t -> Int63.t
val int63_round_nearest_exn : t -> Int63.t
val iround_lbound : t

If f <= iround_lbound || f >= iround_ubound, then iround* functions will refuse to round f, returning None or raising as appropriate.

val iround_ubound : t
val round_significant : float -> significant_digits:int -> float

round_significant x ~significant_digits:n rounds to the nearest number with n significant digits. More precisely: it returns the representable float closest to x rounded to n significant digits. It is meant to be equivalent to sprintf "%.*g" n x |> Float.of_string but faster (10x-15x). Exact ties are resolved as round-to-even.

However, it might in rare cases break the contract above.

It might in some cases appear as if it violates the round-to-even rule:

let x = 4.36083208835;;
let z = 4.3608320883;;
assert (z = fast_approx_round_significant x ~sf:11)

But in this case so does sprintf, since x as a float is slightly under-represented:

sprintf "%.11g" x = "4.3608320883";;
sprintf "%.30g" x = "4.36083208834999958014577714493"

More importantly, round_significant might sometimes give a different result than sprintf ... |> Float.of_string because it round-trips through an integer. For example, the decimal fraction 0.009375 is slightly under-represented as a float:

sprintf "%.17g" 0.009375 = "0.0093749999999999997"

But:

0.009375 *. 1e5 = 937.5

Therefore:

round_significant 0.009375 ~significant_digits:3 = 0.00938

whereas:

sprintf "%.3g" 0.009375 = "0.00937"

In general we believe (and have tested on numerous examples) that the following holds for all x:

let s = sprintf "%.*g" significant_digits x |> Float.of_string in
s = round_significant ~significant_digits x
|| s = round_significant ~significant_digits (one_ulp `Up x)
|| s = round_significant ~significant_digits (one_ulp `Down x)

Also, for float representations of decimal fractions (like 0.009375), round_significant is more likely to give the "desired" result than sprintf ... |> of_string (that is, the result of rounding the decimal fraction, rather than its float representation). But it's not guaranteed either--see the 4.36083208835 example above.

val round_decimal : float -> decimal_digits:int -> float

round_decimal x ~decimal_digits:n rounds x to the nearest 10**(-n). For positive n it is meant to be equivalent to sprintf "%.*f" n x |> Float.of_string, but faster.

All the considerations mentioned in round_significant apply (both functions use the same code path).

val is_nan : t -> bool
val is_inf : t -> bool

Includes positive and negative Float.infinity.

min_inan and max_inan return, respectively, the min and max of the two given values, except when one of the values is a nan, in which case the other is returned. (Returns nan if both arguments are nan.)

val min_inan : t -> t -> t
val max_inan : t -> t -> t
val (+) : t -> t -> t
val (-) : t -> t -> t
val (/) : t -> t -> t
val (*) : t -> t -> t
val (**) : t -> t -> t
val (~-) : t -> t
module Parts : sig ... end

Returns the fractional part and the whole (i.e., integer) part. For example, modf (-3.14) returns { fractional = -0.14; integral = -3.; }!

val modf : t -> Parts.t
val mod_float : t -> t -> t

mod_float x y returns a result with the same sign as x. It returns nan if y is 0. It is basically

let mod_float x y = x -. float(truncate(x/.y)) *. y

not

let mod_float x y = x -. floor(x/.y) *. y

and therefore resembles mod on integers more than %.

val add : t -> t -> t

Ordinary functions for arithmetic operations

These are for modules that inherit from t, since the infix operators are more convenient.

val sub : t -> t -> t
val neg : t -> t
val scale : t -> t -> t
val abs : t -> t
module O : sig ... end

A sub-module designed to be opened to make working with floats more convenient.

module O_dot : sig ... end

Similar to O, except that operators are suffixed with a dot, allowing one to have both int and float operators in scope simultaneously.

val to_string : t -> string

to_string x builds a string s representing the float x that guarantees the round trip, that is such that Float.equal x (Float.of_string s).

It usually yields as few significant digits as possible. That is, it won't print 3.14 as 3.1400000000000001243. The only exception is that occasionally it will output 17 significant digits when the number can be represented with just 16 (but not 15 or less) of them.

val to_string_hum : ?delimiter:char -> ?decimals:int -> ?strip_zero:bool -> t -> string

Pretty print float, for example to_string_hum ~decimals:3 1234.1999 = "1_234.200" to_string_hum ~decimals:3 ~strip_zero:true 1234.1999 = "1_234.2" . No delimiters are inserted to the right of the decimal.

val to_padded_compact_string : t -> string

Produce a lossy compact string representation of the float. The float is scaled by an appropriate power of 1000 and rendered with one digit after the decimal point, except that the decimal point is written as '.', 'k', 'm', 'g', 't', or 'p' to indicate the scale factor. (However, if the digit after the "decimal" point is 0, it is suppressed.)

The smallest scale factor that allows the number to be rendered with at most 3 digits to the left of the decimal is used. If the number is too large for this format (i.e., the absolute value is at least 999.95e15), scientific notation is used instead. E.g.:

  • to_padded_compact_string (-0.01) = "-0 "
  • to_padded_compact_string 1.89 = "1.9"
  • to_padded_compact_string 999_949.99 = "999k9"
  • to_padded_compact_string 999_950. = "1m "

In the case where the digit after the "decimal", or the "decimal" itself is omitted, the numbers are padded on the right with spaces to ensure the last two columns of the string always correspond to the decimal and the digit afterward (except in the case of scientific notation, where the exponent is the right-most element in the string and could take up to four characters).

  • to_padded_compact_string 1. = "1 "
  • to_padded_compact_string 1.e6 = "1m "
  • to_padded_compact_string 1.e16 = "1.e+16"
  • to_padded_compact_string max_finite_value = "1.8e+308"

Numbers in the range -.05 < x < .05 are rendered as "0 " or "-0 ".

Other cases:

  • to_padded_compact_string nan = "nan "
  • to_padded_compact_string infinity = "inf "
  • to_padded_compact_string neg_infinity = "-inf "

Exact ties are resolved to even in the decimal:

  • to_padded_compact_string 3.25 = "3.2"
  • to_padded_compact_string 3.75 = "3.8"
  • to_padded_compact_string 33_250. = "33k2"
  • to_padded_compact_string 33_350. = "33k4"

to_padded_compact_string is defined in terms of to_padded_compact_string_custom below as

let to_padded_compact_string t =
  to_padded_compact_string_custom t ?prefix:None
    ~kilo:"k" ~mega:"m" ~giga:"g" ~tera:"t" ~peta:"p"
    ()
val to_padded_compact_string_custom : t -> ?prefix:string -> kilo:string -> mega:string -> giga:string -> tera:string -> ?peta:string -> unit -> string

Similar to to_padded_compact_string but allows the user to provide different abbreviations. This can be useful to display currency values, e.g. $1mm3, where prefix="$", mega="mm".

val int_pow : t -> int -> t

int_pow x n computes x ** float n via repeated squaring. It is generally much faster than **.

Note that int_pow x 0 always returns 1., even if x = nan. This coincides with x ** 0. and is intentional.

For n >= 0 the result is identical to an n-fold product of x with itself under *., with a certain placement of parentheses. For n < 0 the result is identical to int_pow (1. /. x) (-n).

The error will be on the order of |n| ulps, essentially the same as if you perturbed x by up to a ulp and then exponentiated exactly.

Benchmarks show a factor of 5-10 speedup (relative to **) for exponents up to about 1000 (approximately 10ns vs. 70ns). For larger exponents the advantage is smaller but persists into the trillions. For a recent or more detailed comparison, run the benchmarks.

Depending on context, calling this function might or might not allocate 2 minor words. Even if called in a way that causes allocation, it still appears to be faster than **.

val square : t -> t

square x returns x *. x.

val ldexp : t -> int -> t

ldexp x n returns x *. 2 ** n

val frexp : t -> t * int

frexp f returns the pair of the significant and the exponent of f. When f is zero, the significant x and the exponent n of f are equal to zero. When f is non-zero, they are defined by f = x *. 2 ** n and 0.5 <= x < 1.0.

val log10 : t -> t

Base 10 logarithm.

val expm1 : t -> t

expm1 x computes exp x -. 1.0, giving numerically-accurate results even if x is close to 0.0.

val log1p : t -> t

log1p x computes log(1.0 +. x) (natural logarithm), giving numerically-accurate results even if x is close to 0.0.

val copysign : t -> t -> t

copysign x y returns a float whose absolute value is that of x and whose sign is that of y. If x is nan, returns nan. If y is nan, returns either x or -. x, but it is not specified which.

val cos : t -> t

Cosine. Argument is in radians.

val sin : t -> t

Sine. Argument is in radians.

val tan : t -> t

Tangent. Argument is in radians.

val acos : t -> t

Arc cosine. The argument must fall within the range [-1.0, 1.0]. Result is in radians and is between 0.0 and pi.

val asin : t -> t

Arc sine. The argument must fall within the range [-1.0, 1.0]. Result is in radians and is between -pi/2 and pi/2.

val atan : t -> t

Arc tangent. Result is in radians and is between -pi/2 and pi/2.

val atan2 : t -> t -> t

atan2 y x returns the arc tangent of y /. x. The signs of x and y are used to determine the quadrant of the result. Result is in radians and is between -pi and pi.

val hypot : t -> t -> t

hypot x y returns sqrt(x *. x + y *. y), that is, 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 origin.

val cosh : t -> t

Hyperbolic cosine. Argument is in radians.

val sinh : t -> t

Hyperbolic sine. Argument is in radians.

val tanh : t -> t

Hyperbolic tangent. Argument is in radians.

val sqrt : t -> t

Square root.

val exp : t -> t

Exponential.

val log : t -> t

Natural logarithm.

module Class : sig ... end

Excluding nan the floating-point "number line" looks like:

val classify : t -> Class.t
val is_finite : t -> bool

is_finite t returns true iff classify t is in Normal; Subnormal; Zero;.

val sign : t -> Sign.t
  • deprecated [since 2016-01] Replace [sign] with [robust_sign] or [sign_exn]
val sign_exn : t -> Sign.t

The sign of a float. Both -0. and 0. map to Zero. Raises on nan. All other values map to Neg or Pos.

val sign_or_nan : t -> Sign_or_nan.t

The sign of a float, with support for NaN. Both -0. and 0. map to Zero. All NaN values map to Nan. All other values map to Neg or Pos.

val create_ieee : negative:bool -> exponent:int -> mantissa:Int63.t -> t Or_error.t

These functions construct and destruct 64-bit floating point numbers based on their IEEE representation with a sign bit, an 11-bit non-negative (biased) exponent, and a 52-bit non-negative mantissa (or significand). See Wikipedia for details of the encoding.

In particular, if 1 <= exponent <= 2046, then:

create_ieee_exn ~negative:false ~exponent ~mantissa
= 2 ** (exponent - 1023) * (1 + (2 ** -52) * mantissa)
val create_ieee_exn : negative:bool -> exponent:int -> mantissa:Int63.t -> t
val ieee_negative : t -> bool
val ieee_exponent : t -> int
val ieee_mantissa : t -> Int63.t
module Terse : sig ... end

S-expressions contain at most 8 significant digits.

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