gg 1.0.0 · OCaml Package

Floating point number utilities.

This module defines a few useful constants, functions, predicates and comparisons on floating point numbers. The formatters output a lossless textual representation of floats.

Quick recall on OCaml's floating point representation.

Warning. This module existed before Stdlib.Float was introduced in OCaml 4.07.0. Since Gg 1.0.0, the module now includes Stdlib.Float and some values initially provided by Gg are now provided by Stdlib.Float, see the release notes of the package for a precise account of the changes.

`Stdlib.Float`

include module type of Float

val zero : float

The floating point 0.

since 4.08.0

val one : float

The floating-point 1.

since 4.08.0

val minus_one : float

The floating-point -1.

since 4.08.0

val neg : float -> float

Unary negation.

val add : float -> float -> float

Floating-point addition.

val sub : float -> float -> float

Floating-point subtraction.

val mul : float -> float -> float

Floating-point multiplication.

val div : float -> float -> float

Floating-point division.

val fma : float -> float -> float -> float

fma x y z returns x * y + z, with a best effort for computing this expression with a single rounding, using either hardware instructions (providing full IEEE compliance) or a software emulation.

On 64-bit Cygwin, 64-bit mingw-w64 and MSVC 2017 and earlier, this function may be emulated owing to known bugs on limitations on these platforms. Note: since software emulation of the fma is costly, make sure that you are using hardware fma support if performance matters.

since 4.08.0

val rem : float -> float -> float

rem a b returns the remainder of a with respect to b. The returned value is a -. n *. b, where n is the quotient a /. b rounded towards zero to an integer.

val succ : float -> float

succ x returns the floating point number right after x i.e., the smallest floating-point number greater than x. See also next_after.

since 4.08.0

val pred : float -> float

pred x returns the floating-point number right before x i.e., the greatest floating-point number smaller than x. See also next_after.

since 4.08.0

val abs : float -> float

abs f returns the absolute value of f.

val infinity : float

Positive infinity.

val neg_infinity : float

Negative infinity.

val nan : float

A special floating-point value denoting the result of an undefined operation such as 0.0 /. 0.0. Stands for 'not a number'. Any floating-point operation with nan as argument returns nan as result. As for floating-point comparisons, =, <, <=, > and >= return false and <> returns true if one or both of their arguments is nan.

val pi : float

The constant pi.

val max_float : float

The largest positive finite value of type float.

val min_float : float

The smallest positive, non-zero, non-denormalized value of type float.

val epsilon : float

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

val is_finite : float -> bool

is_finite x is true if and only if x is finite i.e., not infinite and not nan.

since 4.08.0

val is_infinite : float -> bool

is_infinite x is true if and only if x is infinity or neg_infinity.

since 4.08.0

val is_nan : float -> bool

is_nan x is true if and only if x is not a number (see nan).

since 4.08.0

val is_integer : float -> bool

is_integer x is true if and only if x is an integer.

since 4.08.0

val of_int : int -> float

Convert an integer to floating-point.

val to_int : float -> int

Truncate the given floating-point number to an integer. The result is unspecified if the argument is nan or falls outside the range of representable integers.

val of_string : string -> float

Convert the given string to a float. The string is read in decimal (by default) or in hexadecimal (marked by 0x or 0X). The format of decimal floating-point numbers is [-] dd.ddd (e|E) [+|-] dd , where d stands for a decimal digit. The format of hexadecimal floating-point numbers is [-] 0(x|X) hh.hhh (p|P) [+|-] dd , where h stands for an hexadecimal digit and d for a decimal digit. In both cases, at least one of the integer and fractional parts must be given; the exponent part is optional. The _ (underscore) character can appear anywhere in the string and is ignored. Depending on the execution platforms, other representations of floating-point numbers can be accepted, but should not be relied upon.

raises Failure
if the given string is not a valid representation of a float.

val of_string_opt : string -> float option

Same as of_string, but returns None instead of raising.

val to_string : float -> string

Return a string representation of a floating-point number.

This conversion can involve a loss of precision. For greater control over the manner in which the number is printed, see Printf.

This function is an alias for Stdlib.string_of_float.

type fpclass = fpclass =

| FP_normal
(*
Normal number, none of the below
*)
| FP_subnormal
(*
Number very close to 0.0, has reduced precision
*)
| FP_zero
(*
Number is 0.0 or -0.0
*)
| FP_infinite
(*
Number is positive or negative infinity
*)
| FP_nan
(*
Not a number: result of an undefined operation
*)

The five classes of floating-point numbers, as determined by the classify_float function.

val classify_float : float -> fpclass

Return the class of the given floating-point number: normal, subnormal, zero, infinite, or not a number.

val pow : float -> float -> float

Exponentiation.

val sqrt : float -> float

Square root.

val cbrt : float -> float

Cube root.

since 4.13.0

val exp : float -> float

Exponential.

val exp2 : float -> float

Base 2 exponential function.

since 4.13.0

val log : float -> float

Natural logarithm.

val log10 : float -> float

Base 10 logarithm.

val log2 : float -> float

Base 2 logarithm.

since 4.13.0

val expm1 : float -> float

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

val log1p : float -> float

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

val cos : float -> float

Cosine. Argument is in radians.

val sin : float -> float

Sine. Argument is in radians.

val tan : float -> float

Tangent. Argument is in radians.

val acos : float -> float

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 : float -> float

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 : float -> float

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

val atan2 : float -> float -> float

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 : float -> float -> float

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. If one of x or y is infinite, returns infinity even if the other is nan.

val cosh : float -> float

Hyperbolic cosine. Argument is in radians.

val sinh : float -> float

Hyperbolic sine. Argument is in radians.

val tanh : float -> float

Hyperbolic tangent. Argument is in radians.

val acosh : float -> float

Hyperbolic arc cosine. The argument must fall within the range [1.0, inf]. Result is in radians and is between 0.0 and inf.

since 4.13.0

val asinh : float -> float

Hyperbolic arc sine. The argument and result range over the entire real line. Result is in radians.

since 4.13.0

val atanh : float -> float

Hyperbolic arc tangent. The argument must fall within the range [-1.0, 1.0]. Result is in radians and ranges over the entire real line.

since 4.13.0

val erf : float -> float

Error function. The argument ranges over the entire real line. The result is always within [-1.0, 1.0].

since 4.13.0

val erfc : float -> float

Complementary error function (erfc x = 1 - erf x). The argument ranges over the entire real line. The result is always within [-1.0, 1.0].

since 4.13.0

val trunc : float -> float

trunc x rounds x to the nearest integer whose absolute value is less than or equal to x.

since 4.08.0

val round : float -> float

round x rounds x to the nearest integer with ties (fractional values of 0.5) rounded away from zero, regardless of the current rounding direction. If x is an integer, +0., -0., nan, or infinite, x itself is returned.

On 64-bit mingw-w64, this function may be emulated owing to a bug in the C runtime library (CRT) on this platform.

since 4.08.0

val ceil : float -> float

Round above to an integer value. ceil f returns the least integer value greater than or equal to f. The result is returned as a float.

val floor : float -> float

Round below to an integer value. floor f returns the greatest integer value less than or equal to f. The result is returned as a float.

val next_after : float -> float -> float

next_after x y returns the next representable floating-point value following x in the direction of y. More precisely, if y is greater (resp. less) than x, it returns the smallest (resp. largest) representable number greater (resp. less) than x. If x equals y, the function returns y. If x or y is nan, a nan is returned. Note that next_after max_float infinity = infinity and that next_after 0. infinity is the smallest denormalized positive number. If x is the smallest denormalized positive number, next_after x 0. = 0.

since 4.08.0

val copy_sign : float -> float -> float

copy_sign 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 sign_bit : float -> bool

sign_bit x is true if and only if the sign bit of x is set. For example sign_bit 1. and signbit 0. are false while sign_bit (-1.) and sign_bit (-0.) are true.

since 4.08.0

val frexp : float -> float * 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 ldexp : float -> int -> float

ldexp x n returns x *. 2 ** n.

val modf : float -> float * float

modf f returns the pair of the fractional and integral part of f.

type t = float

An alias for the type of floating-point numbers.

val compare : t -> t -> int

compare x y returns 0 if x is equal to y, a negative integer if x is less than y, and a positive integer if x is greater than y. compare treats nan as equal to itself and less than any other float value. This treatment of nan ensures that compare defines a total ordering relation.

val equal : t -> t -> bool

The equal function for floating-point numbers, compared using compare.

val min : t -> t -> t

min x y returns the minimum of x and y. It returns nan when x or y is nan. Moreover min (-0.) (+0.) = -0.

since 4.08.0

val max : float -> float -> float

max x y returns the maximum of x and y. It returns nan when x or y is nan. Moreover max (-0.) (+0.) = +0.

since 4.08.0

val min_max : float -> float -> float * float

min_max x y is (min x y, max x y), just more efficient.

since 4.08.0

val min_num : t -> t -> t

min_num x y returns the minimum of x and y treating nan as missing values. If both x and y are nan, nan is returned. Moreover min_num (-0.) (+0.) = -0.

since 4.08.0

val max_num : t -> t -> t

max_num x y returns the maximum of x and y treating nan as missing values. If both x and y are nan nan is returned. Moreover max_num (-0.) (+0.) = +0.

since 4.08.0

val min_max_num : float -> float -> float * float

min_max_num x y is (min_num x y, max_num x y), just more efficient. Note that in particular min_max_num x nan = (x, x) and min_max_num nan y = (y, y).

since 4.08.0

val hash : t -> int

The hash function for floating-point numbers.

module Array : sig ... end

Float arrays with packed representation.

module ArrayLabels : sig ... end

Float arrays with packed representation (labeled functions).

Constants

val e : float

The constant e.

val two_pi : float

2 *. pi, two times pi.

val pi_div_2 : float

pi /. 2.

val pi_div_4 : float

pi /. 4.

val inv_pi : float

1 /. pi.

val max_sub_float : float

The greatest positive subnormal floating point number.

val min_sub_float : float

The smallest positive subnormal floating point number.

val max_frac_float : float

The greatest positive floating point number with a fractional part (the float before 2⁵²). Any number outside [-max_frac_float;max_frac_float] is an integer.

val max_int_arith : float

The greatest positive floating point number (2⁵³) such that any integer in the range [-max_int_arith;max_int_arith] is represented exactly. Integer arithmetic can be performed exactly in this interval.

Functions

Note. If applicable, a function taking NaNs returns a NaN unless otherwise specified.

val deg_of_rad : float -> float

deg_of_rad r is r radians in degrees.

val rad_of_deg : float -> float

rad_of_deg d is d degrees in radians.

val wrap_angle : float -> float

wrap_angle r is the angle r in the interval [-pi;pi[.

val random : ?min:float -> len:float -> unit -> float

random min len () is a random float in the interval [min;min+len] (min defaults to 0.). Uses the standard library's default Random state for the generation.

Warning. The float generated by a given state may change in future versions of the library.

val srandom : Random.State.t -> ?min:float -> len:float -> unit -> float

srandom state min len () is like random but uses state for the generation.

Warning. The float generated by a given state may change in future versions of the library.

val mix : float -> float -> float -> float

mix x y t is the linear interpolation x +. t *. (y -. x).

val step : float -> float -> float

step edge x is 0. if x < edge and 1. otherwise. The result is undefined on NaNs.

val smooth_step : float -> float -> float -> float

smooth_step e0 e1 x is 0. if x <= e0, 1. if x >= e1 and cubic hermite interpolation between 0. and 1. otherwise. The result is undefined on NaNs.

val clamp : min:float -> max:float -> float -> float

clamp min max x is min if x < min, max if x > max and x otherwise. The result is undefined on NaNs and if min > max.

val remap : x0:float -> x1:float -> y0:float -> y1:float -> float -> float

remap x0 x1 y0 y1 v applies to v the affine transform that maps x0 to y0 and x1 to y1. If the transform is undefined (x0 = x1 and y0 <> y1) the function returns y0 for any v.

val int_of_round : float -> int

int_of_round x is truncate (round v). The result is undefined on NaNs and infinities.

val round_dfrac : int -> float -> float

round_dfrac d x rounds x to the dth decimal fractional digit. Ties are rounded towards positive infinity. If x is an infinity, returns x. The result is only defined for 0 <= d <= 16.

val round_dsig : int -> float -> float

round_dsig d x rounds the normalized decimal significand of x to the dth decimal fractional digit. Ties are rounded towards positive infinity. The result is NaN on infinities. The result only defined for 0 <= d <= 16.

Warning. The current implementation overflows on large x and d.

val round_zero : eps:float -> float -> float

round_zero eps x is 0. if abs_float x < eps and x otherwise. The result is undefined if eps is NaN.

val chop : eps:float -> float -> float

chop eps x is round x if abs_float (x -. round x) < eps and x otherwise. The result is undefined if eps is NaN.

val sign : float -> float

sign x is 1. if x > 0., 0. if x = 0., -1. if x < 0.

val nan_with_payload : int -> float

nan_with_payload payload is a NaN whose 51 lower significand bits are defined by the 51 lower (or less, as int allows) bits of payload.

val nan_payload : float -> int

nan_payload x is the 51 lower significand bits (or less, as int allows) of the NaN x.

Raises Invalid_argument if x is not a NaN.

Predicates and comparisons

val is_zero : eps:float -> float -> bool

is_zero eps x is true if abs_float x < eps and false otherwise. The result is undefined if eps is NaN.

val equal_tol : eps:float -> float -> float -> bool

equal_tol eps x y is true iff |x - y| <= eps * max (1,|x|,|y|). On special values the function behaves like compare x y = 0. The condition turns into an absolute tolerance test for small magnitudes and a relative tolerance test for large magnitudes.

val compare_tol : eps:float -> float -> float -> int

compare_tol ~eps x y is 0 iff equal_tol ~eps x y is true and Stdlib.compare x y otherwise.

Formatters

val pp : Format.formatter -> float -> unit

pp ppf x formats a lossless textual representation of x on ppf using "%h". Since 1.0.0, before this was the slower legacy_pp whose output differs on the representation of nan, infinities, or zeros.

Deprecated

val fmax : float -> float -> float

Deprecated use max_num.

deprecated Use Float.max_num instead.

val fmin : float -> float -> float

Deprecated use min_num.

deprecated Use Float.min_num instead.

val is_inf : float -> bool

Deprecated use is_infinite.

deprecated Use Float.is_infinite instead.

val is_int : float -> bool

Deprecated use is_integer.

deprecated Use Float.is_integer instead.

val legacy_pp : Format.formatter -> float -> unit

Deprecated use pp.

pp_legacy ppf x prints a lossless textual representation of x on ppf.

Normals are represented by "[-]0x1.<f>p<e>" where <f> is the significand bits in hexadecimal and <e> the unbiased exponent in decimal.
Subnormals are represented by "[-]0x0.<f>p-1022" where <f> is the significand bits in hexadecimal.
NaNs are represented by "[-]nan(0x<p>)" where <p> is the payload in hexadecimal.
Infinities and zeroes are represented by "[-]inf" and "[-]0.".

This format should be compatible with recent implementations of strtod and hence with float_of_string (but negative NaNs seem to be problematic to get back).

deprecated Use Float.pp instead (some values may render differently).

Quick recall on OCaml's `float`s

An OCaml float is an IEEE-754 64 bit double precision binary floating point number. The 64 bits are laid out as follows :

+----------------+-----------------------+-------------------------+
| sign s (1 bit) | exponent e (11 bits)  | significand t (52 bits) |
+----------------+-----------------------+-------------------------+
               63|62                   52|51                      0|

The value represented depends on s, e and t :

sign   exponent       significand   value represented           meaning
-------------------------------------------------------------------------
s      0              0             -1^s * 0                    zero
s      0              t <> 0        -1^s * 0.t * 2^-1022        subnormal
s      0 < e < 2047   f             -1^s * 1.t * 2^(e - 1023)   normal
s      2047           0             -1^s * infinity             infinity
s      2047           t <> 0        NaN                         not a number

There are two zeros, a positive and a negative one but both are deemed equal by = and Stdlib.compare. A NaN is never equal (=) to itself or to another NaN however Stdlib.compare asserts any NaN to be equal to itself and to any other NaN.

The bit layout of a float can be converted to an int64 and back using Int64.bits_of_float and Int64.float_of_bits.

The bit 51 of a NaN is used to distinguish between quiet (bit set) and signaling NaNs (bit cleared); the remaining 51 lower bits of the significand are the NaN's payload which can be used to store diagnostic information. These features don't seem to used in OCaml.

The significand of a floating point number is made of 53 binary digits (don't forget the implicit digit), this corresponds to log₁₀(2⁵³) ~ 16 decimal digits.

Only float values in the interval ]-2⁵²;2⁵²[ may have a fractional part. Float.max_frac_float is the greatest positive float with a fractional part.

Any integer value in the interval [-2⁵³;2⁵³] can be represented exactly by a float value. Integer arithmetic performed in this interval is exact. Float.max_int_arith is 2⁵³.

package gg

Stdlib.Float