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Rationals.

This modules builds arbitrary precision rationals on top of arbitrary integers from module Z.

This file is part of the Zarith library http://forge.ocamlcore.org/projects/zarith . It is distributed under LGPL 2 licensing, with static linking exception. See the LICENSE file included in the distribution.

Copyright (c) 2010-2011 Antoine Miné, Abstraction project. Abstraction is part of the LIENS (Laboratoire d'Informatique de l'ENS), a joint laboratory by: CNRS (Centre national de la recherche scientifique, France), ENS (École normale supérieure, Paris, France), INRIA Rocquencourt (Institut national de recherche en informatique, France).

A rational is represented as a pair numerator/denominator, reduced to have a non-negative denominator and no common factor. This form is canonical (enabling polymorphic equality and hashing). The representation allows three special numbers: `inf`

(1/0), `-inf`

(-1/0) and `undef`

(0/0).

`make num den`

constructs a new rational equal to `num`

/`den`

. It takes care of putting the rational in canonical form.

`val zero : t`

`val one : t`

`val minus_one : t`

0, 1, -1.

`val inf : t`

1/0.

`val minus_inf : t`

-1/0.

`val undef : t`

0/0.

`val of_int : int -> t`

`val of_int32 : int32 -> t`

`val of_int64 : int64 -> t`

`val of_nativeint : nativeint -> t`

Conversions from various integer types.

`val of_ints : int -> int -> t`

Conversion from an `int`

numerator and an `int`

denominator.

`val of_float : float -> t`

Conversion from a `float`

. The conversion is exact, and maps NaN to `undef`

.

`val of_string : string -> t`

Converts a string to a rational. Plain integers, `/`

separated integer ratios (with optional sign), decimal point and scientific notations are understood. Additionally, the special `inf`

, `-inf`

, and `undef`

are recognized (they can also be typeset respectively as `1/0`

, `-1/0`

, `0/0`

).

Rationals can be categorized into different kinds, depending mainly on whether the numerator and/or denominator is null.

`val is_real : t -> bool`

Whether the argument is non-infinity and non-undefined.

`val sign : t -> int`

Returns 1 if the argument is positive (including inf), -1 if it is negative (including -inf), and 0 if it is null or undefined.

`compare x y`

compares `x`

to `y`

and returns 1 if `x`

is strictly greater that `y`

, -1 if it is strictly smaller, and 0 if they are equal. This is a total ordering. Infinities are ordered in the natural way, while undefined is considered the smallest of all: undef = undef < -inf <= -inf < x < inf <= inf. This is consistent with OCaml's handling of floating-point infinities and NaN.

OCaml's polymorphic comparison will NOT return a result consistent with the ordering of rationals.

Equality testing. Unlike `compare`

, this follows IEEE semantics: `undef`

<> `undef`

.

`val to_int : t -> int`

`val to_int32 : t -> int32`

`val to_int64 : t -> int64`

`val to_nativeint : t -> nativeint`

Convert to integer by truncation. Raises a `Divide_by_zero`

if the argument is an infinity or undefined. Raises a `Z.Overflow`

if the result does not fit in the destination type.

`val to_string : t -> string`

Converts to human-readable, base-10, `/`

-separated rational.

`val to_float : t -> float`

Converts to a floating-point number, using the current floating-point rounding mode. With the default rounding mode, the result is the floating-point number closest to the given rational; ties break to even mantissa.

In all operations, the result is `undef`

if one argument is `undef`

. Other operations can return `undef`

: such as `inf`

-`inf`

, `inf`

*0, 0/0.

`val print : t -> unit`

Prints the argument on the standard output.

`val output : out_channel -> t -> unit`

Prints the argument on the specified channel. Also intended to be used as `%a`

format printer in `Printf.printf`

.

`val sprint : unit -> t -> string`

To be used as `%a`

format printer in `Printf.sprintf`

.

`val pp_print : Format.formatter -> t -> unit`

Prints the argument on the specified formatter. Also intended to be used as `%a`

format printer in `Format.printf`

.

Classic prefix and infix `int`

operators are redefined on `t`

.

`val (~$) : int -> t`

Conversion from `int`

.

`val (//) : int -> int -> t`

Creates a rational from two `int`

s.

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