Floating-point arithmetic¶

OCaml's floating-point numbers follow the IEEE 754 standard, using double precision (64 bits) numbers. Floating-point operations never raise an exception on overflow, underflow, division by zero, etc. Instead, special IEEE numbers are returned as appropriate, such as infinity for 1.0 /. 0.0 , neg_infinity for -1.0 /. 0.0 , and nan ('not a number') for 0.0 /. 0.0 . These special numbers then propagate through floating-point computations as expected: for instance, 1.0 /. infinity is 0.0 , and any arithmetic operation with nan as argument returns nan as result.

Since 4.07.0

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. 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 Float.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 Float.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 iff x is finite i.e., not infinite and not Float.nan .

Since 4.08.0

val is_infinite : float -> bool

is_infinite x is true iff x is Float.infinity or Float.neg_infinity .

Since 4.08.0

val is_nan : float -> bool

is_nan x is true iff x is not a number (see Float.nan ).

Since 4.08.0

val is_integer : float -> bool

is_integer x is true iff 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 the string representation of a floating-point number.

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 Float.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 exp : float -> float

Exponential.

val log : float -> float

Natural logarithm.

val log10 : float -> float

Base 10 logarithm.

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 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.

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 iff 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 Float.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

module ArrayLabels : sig end