# Module Batteries.Big_int

    module Big_int: BatBig_int

    type big_int = Big_int.big_int

The type of big integers.

    val zero : big_int

    val zero_big_int : big_int

The big integer `0`.

    val one : big_int

    val unit_big_int : big_int

The big integer `1`.

###### Arithmetic operations

    val neg : big_int -> big_int

    val succ : big_int -> big_int

    val pred : big_int -> big_int

    val abs : big_int -> big_int

    val add : big_int -> big_int -> big_int

    val sub : big_int -> big_int -> big_int

    val mul : big_int -> big_int -> big_int

    val div : big_int -> big_int -> big_int

    val modulo : big_int -> big_int -> big_int

    val pow : big_int -> big_int -> big_int

    type t = big_int

    val (+) : t -> t -> t

    val (-) : t -> t -> t

    val ( * ) : t -> t -> t

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

    val ( ** ) : t -> t -> t

    val minus_big_int : big_int -> big_int

Unary negation.

    val abs_big_int : big_int -> big_int

Absolute value.

    val add_big_int : big_int -> big_int -> big_int

Addition.

    val succ_big_int : big_int -> big_int

Successor (add 1).

    val add_int_big_int : int -> big_int -> big_int

Addition of a small integer to a big integer.

    val sub_big_int : big_int -> big_int -> big_int

Subtraction.

    val pred_big_int : big_int -> big_int

Predecessor (subtract 1).

    val mult_big_int : big_int -> big_int -> big_int

Multiplication of two big integers.

    val mult_int_big_int : int -> big_int -> big_int

Multiplication of a big integer by a small integer

    val square_big_int : big_int -> big_int

Return the square of the given big integer

    val sqrt_big_int : big_int -> big_int

`sqrt_big_int a` returns the integer square root of `a`, that is, the largest big integer `r` such that `r * r <= a`.
Raises `Invalid_argument` if `a` is negative.

    val quomod_big_int : big_int ->
           big_int -> big_int * big_int

Euclidean division of two big integers. The first part of the result is the quotient, the second part is the remainder. Writing `(q,r) = quomod_big_int a b`, we have `a = q * b + r` and `0 <= r < |b|`.
Raises `Division_by_zero` if the divisor is zero.

    val div_big_int : big_int -> big_int -> big_int

Euclidean quotient of two big integers. This is the first result `q` of `quomod_big_int` (see above).

    val mod_big_int : big_int -> big_int -> big_int

Euclidean modulus of two big integers. This is the second result `r` of `quomod_big_int` (see above).

    val gcd_big_int : big_int -> big_int -> big_int

Greatest common divisor of two big integers.

    val power_int_positive_int : int -> int -> big_int

    val power_big_int_positive_int : big_int -> int -> big_int

    val power_int_positive_big_int : int -> big_int -> big_int

    val power_big_int_positive_big_int : big_int -> big_int -> big_int

Exponentiation functions. Return the big integer representing the first argument `a` raised to the power `b` (the second argument). Depending on the function, `a` and `b` can be either small integers or big integers.
Raises `Invalid_argument` if `b` is negative.

    val operations : t BatNumber.numeric

###### Generators

    val (--) : big_int -> big_int -> big_int BatEnum.t

    val (---) : big_int -> big_int -> big_int BatEnum.t

###### Comparisons and tests

    val compare : big_int -> big_int -> int

    val ord : big_int -> big_int -> BatOrd.order

    val equal : big_int -> big_int -> bool

    val sign_big_int : big_int -> int

Return `0` if the given big integer is zero, `1` if it is positive, and `-1` if it is negative.

    val compare_big_int : big_int -> big_int -> int

`compare_big_int a b` returns `0` if `a` and `b` are equal, `1` if `a` is greater than `b`, and `-1` if `a` is smaller than `b`.

    val eq_big_int : big_int -> big_int -> bool

    val le_big_int : big_int -> big_int -> bool

    val ge_big_int : big_int -> big_int -> bool

    val lt_big_int : big_int -> big_int -> bool

    val gt_big_int : big_int -> big_int -> bool

Usual boolean comparisons between two big integers.

    val max_big_int : big_int -> big_int -> big_int

Return the greater of its two arguments.

    val min_big_int : big_int -> big_int -> big_int

Return the smaller of its two arguments.

    val num_digits_big_int : big_int -> int

Return the number of machine words used to store the given big integer.

###### Conversions to and from strings

    val to_string : big_int -> string

    val string_of_big_int : big_int -> string

Return the string representation of the given big integer, in decimal (base 10).

    val of_string : string -> big_int

    val big_int_of_string : string -> big_int

Convert a string to a big integer, in decimal. The string consists of an optional `-` or `+` sign, followed by one or several decimal digits.

    val to_string_in_binary : big_int -> string

as `string_of_big_int`, but in base 2

    val to_string_in_octal : big_int -> string

as `string_of_big_int`, but in base 8

    val to_string_in_hexa : big_int -> string

as `string_of_big_int`, but in base 16

    val to_string_in_base : int -> big_int -> string

`to_string_in_base b n` returns the string representation in base `b` of the given big integer `n`. Should you have advanced needs (arbitrarily large bases, or custom digits instead of the usual `0,1,...9,a,b,...,z`), use `to_string_in_custom_base` instead.
Raises `Invalid_argument` if b is not in `2 .. 36`.

    val to_string_in_custom_base : string -> int -> big_int -> string

First argument, called `symbols`, is the vector of the symbols used to represent the digits in base `b`. `to_string_in_base` is almost equivalent to `to_string_in_custom_base big_int_base_default_symbols`, the difference being that `to_string_in_custom_base` allows the base to be arbitrarily large, provided that `symbols` can accommodate it. Concretely, the base `b` must be at least `2`, and `symbols` must be of size at least `b`. The default value of `big_int_base_default_symbols` contains 62 symbols, as it uses lowercase and uppercase letters both. See below for more information.
Raises `Invalid_argument` if `b` is incorrect.

    val big_int_base_default_symbols : string

Default vector of symbols used by `to_string_in_base` and its fixed-base derivatives `to_string_in_binary`, `to_string_in_octal` and `to_string_in_hexa` to represent digits. The symbol at position `p` encodes the value `p`. The original value of this vector is, schematically, `['0'..'9' 'a' 'b'..'z' 'A' 'B'..'Z']`, which is sufficient for bases up to and including 62. The basic `to_string_in_base` function is capped to base 36 to avoid unexpected behaviours do to the case-sensitivity of the output in bases 37 to 62. You technically can mutate the vector, for instance if you prefer to exchange lower- and upper-case symbols program-wide. As usual where mutability is concerned, discretion is advised. Most of the time, it is better to build custom functions using `to_string_in_custom_base`.

###### Conversions to and from other numerical types

    val of_int : int -> big_int

    val big_int_of_int : int -> big_int

Convert a small integer to a big integer.

    val is_int_big_int : big_int -> bool

Test whether the given big integer is small enough to be representable as a small integer (type `int`) without loss of precision. On a 32-bit platform, `is_int_big_int a` returns `true` if and only if `a` is between -230 and 230-1. On a 64-bit platform, `is_int_big_int a` returns `true` if and only if `a` is between -262 and 262-1.

    val to_int : big_int -> int

    val int_of_big_int : big_int -> int

Convert a big integer to a small integer (type `int`).
Raises `Failure` if the big integer is not representable as a small integer.

    val big_int_of_int32 : int32 -> big_int

Convert a 32-bit integer to a big integer.

    val big_int_of_nativeint : nativeint -> big_int

Convert a native integer to a big integer.

    val big_int_of_int64 : int64 -> big_int

Convert a 64-bit integer to a big integer.

    val int32_of_big_int : big_int -> int32

Convert a big integer to a 32-bit integer.
Raises `Failure` if the big integer is outside the range `[-2{^31}, 2{^31}-1]`.

    val nativeint_of_big_int : big_int -> nativeint

Convert a big integer to a native integer.
Raises `Failure` if the big integer is outside the range `[Nativeint.min_int, Nativeint.max_int]`.

    val int64_of_big_int : big_int -> int64

Convert a big integer to a 64-bit integer.
Raises `Failure` if the big integer is outside the range `[-2{^63}, 2{^63}-1]`.

    val float_of_big_int : big_int -> float

Returns a floating-point number approximating the given big integer.

    val of_float : float -> big_int

rounds to the nearest integer
Raises `Invalid_argument` when given NaN or +/-infinity

    val to_float : big_int -> float

###### Bit-oriented operations

    val and_big_int : big_int -> big_int -> big_int

Bitwise logical ``and''. The arguments must be positive or zero.

    val or_big_int : big_int -> big_int -> big_int

Bitwise logical ``or''. The arguments must be positive or zero.

    val xor_big_int : big_int -> big_int -> big_int

Bitwise logical ``exclusive or''. The arguments must be positive or zero.

    val shift_left_big_int : big_int -> int -> big_int

`shift_left_big_int b n` returns `b` shifted left by `n` bits. Equivalent to multiplication by `2^n`.

    val shift_right_big_int : big_int -> int -> big_int

`shift_right_big_int b n` returns `b` shifted right by `n` bits. Equivalent to division by `2^n` with the result being rounded towards minus infinity.

    val shift_right_towards_zero_big_int : big_int -> int -> big_int

`shift_right_towards_zero_big_int b n` returns `b` shifted right by `n` bits. The shift is performed on the absolute value of `b`, and the result has the same sign as `b`. Equivalent to division by `2^n` with the result being rounded towards zero.

    val extract_big_int : big_int -> int -> int -> big_int

`extract_big_int bi ofs n` returns a nonnegative number corresponding to bits `ofs` to `ofs + n - 1` of the binary representation of `bi`. If `bi` is negative, a two's complement representation is used.

###### Submodules grouping all infix operators

    module Infix: BatNumber.Infix  with type bat__infix_t = t

    module Compare: BatNumber.Compare  with type bat__compare_t = t

###### Boilerplate code

Printing

    val print : 'a BatIO.output -> t -> unit