=encoding utf8

=head1 NAME

Math::AnyNum - Arbitrary size precision for integers, rationals, floating-points and complex numbers.

=head1 VERSION

Version 0.25

=head1 SYNOPSIS

Math::AnyNum provides a transparent and easy-to-use interface to L<Math::GMPz>, L<Math::GMPq>, L<Math::MPFR> and L<Math::MPC>, along with a decent number of useful mathematical functions.

    use 5.016;
    use Math::AnyNum qw(:overload factorial);

    # Integers
    say factorial(30);              #=> 265252859812191058636308480000000

    # Floating-point numbers
    say sqrt(1 / factorial(100));   #=> 1.0351378111756264713204945[...]e-79

    # Rational numbers
    my $x = 2/3;
    say ($x * 3);                   #=> 2
    say (2 / $x);                   #=> 3
    say $x;                         #=> 2/3

    # Complex numbers
    say 3 + 4*i;                    #=> 3+4i
    say sqrt(-4);                   #=> 2i
    say log(-1);                    #=> 3.1415926535897932384626433832[...]i

=head1 DESCRIPTION

Math::AnyNum focuses primarily on providing a friendly interface and good performance. In most cases, it can be used as
a drop-in replacement for the L<bignum> and L<bigrat> pragmas.

The philosophy of Math::AnyNum is that mathematics should just work, therefore the support for complex numbers is virtually transparent,
without requiring any explicit conversions. All the conversions are done implicitly, using a fairly sophisticated promotion system,
which tries really hard to do the right thing and as efficiently as possible.

Additionally, each Math::AnyNum object is B<immutable>.

=head1 FUNCTIONS

  :ntheory

    binomial(n,k)             binomial coefficient
    multinomial(@list)        multinomial coefficient
    factorial(n)              product of first n integers: n!
    dfactorial(n)             double-factorial: n!!
    mfactorial(n,k)           multi-factorial: n*(n-k)*(n-2k)*...
    subfactorial(n)           number of derangements: !n
    subfactorial(n,k)         number of derangements with k fixed points
    superfactorial(n)         product of first n factorials
    hyperfactorial(n)         product of k^k for k=1..n
    rising_factorial(n,k)     rising factorial: n^(k)
    falling_factorial(n,k)    falling factorial: (n)_k
    bell(n)                   n-th Bell number
    catalan(n)                n-th Catalan number
    lucas(n)                  n-th Lucas number
    fibonacci(n)              n-th Fibonacci number
    fibonacci(n,k)            n-th Fibonacci number of k-th order
    polygonal(n,k)            n-th k-gonal number
    harmonic(n)               n-th harmonic numbers: 1+1/2+...+1/n
    euler(n)                  n-th Euler number
    bernoulli(n)              n-th Bernoulli number
    euler(n,x)                Euler polynomial
    bernoulli(n,x)            Bernoulli polynomial
    faulhaber_sum(n,k)        sum of powers: 1^k + 2^k + ... + n^k
    geometric_sum(n,r)        geometric sum: r^0 + r^1 + ... + r^n
    kronecker(n,k)            Kronecker (Jacobi) symbol
    gcd(@list)                greatest common divisor
    lcm(@list)                least common multiple
    iadd(a,b)                 integer addition: a+b
    isub(a,b)                 integer subtraction: a-b
    imul(a,b)                 integer multiplication: a*b
    idiv(a,b)                 integer division: int(a/b)
    imod(a,b)                 integer remainder of a/b
    ipow(n,k)                 integer exponentiation: n^k
    ipow2(k)                  integer exponentiation: 2^k
    ipow10(k)                 integer exponentiation: 10^k
    iroot(n,k)                integer k-th root of n
    isqrt(n)                  integer square root of n
    icbrt(n)                  integer cube root of n
    ilog(n,k)                 integer logarithm of n to base k
    ilog2(n)                  integer logarithm of n to base 2
    ilog10(n)                 integer logarithm of n to base 10
    divmod(a,b)               quotient and remainder of a/b
    invmod(n,k)               inverse of n modulo k
    powmod(a,b,n)             modular exponentiation: a ^ b mod n
    isqrtrem(n)               integer sqrt remainder: n - isqrt(n)^2
    irootrem(n,k)             integer root remainder: n - iroot(n,k)^k
    is_square(n)              return 1 if n is a perfect square
    is_power(n)               return 1 if n = c^k for integer c
    is_power(n,k)             return 1 if n = c^k for integer c, k
    is_polygonal(n,k)         return 1 if n is a first k-polygonal number
    is_polygonal2(n,k)        return 1 if n is a second k-polygonal number
    is_coprime(n,k)           return 1 if gcd(n,k) = 1
    is_smooth(n,k)            return 1 if all prime factors of n are <= k
    is_prime(n,[r])           Miller-Rabin primality test
    primorial(n)              product of primes below n
    next_prime(n)             next prime > n
    valuation(n,k)            number of times n is divisible by k
    remdiv(n,k)               n / k^valuation(n,k)
    ipolygonal_root(n,k)      first integer k-gonal root of n
    ipolygonal_root2(n,k)     second integer k-gonal root of n

  :special

    beta(x,y)                    Beta function
    eta(x)                       Dirichlet eta function η(x)
    gamma(x)                     Gamma function Γ(x)
    lgamma(x)                    natural logarithm of abs(Γ(x))
    lngamma(x)                   natural logarithm of Γ(x)
    lnsuperfactorial(n)          natural logarithm of superfactorial(n)
    lnhyperfactorial(n)          natural logarithm of hyperfactorial(n)
    digamma(x)                   Digamma function ψ(x)
    zeta(x)                      Zeta function ζ(x)
    Ai(x)                        Airy function
    Ei(x)                        exponential integral function
    Li(x)                        logarithmic integral function
    Li2(x)                       dilogarithm function
    lgrt(x)                      logarithmic-root: lgrt(x^x) = x
    LambertW(x)                  Lambert W function
    BesselJ(x,n)                 first order Bessel function J_n(x)
    BesselY(x,n)                 second order Bessel function Y_n(x)
    pow(x,y)                     power function: x^y
    sqr(x)                       square function: x^2
    sqrt(x)                      square root function: x^(1/2)
    cbrt(x)                      cube root function: x^(1/3)
    root(x,y)                    root function: x^(1/y)
    exp(x)                       exponential function: e^x
    exp2(x)                      exponential function: 2^x
    exp10(x)                     exponential function: 10^x
    ln(x)                        natural logarithm of x
    log(x,y)                     logarithm of x to base y
    log2(x)                      logarithm of x to base 2
    log10(x)                     logarithm of x to base 10
    mod(x,y)                     remainder of x/y
    polymod(n,@list)             n modulo a list of numbers
    erf(x)                       the Gauss error function
    erfc(x)                      the complementary error function
    abs(x)                       absolute value of x
    agm(x,y)                     arithmetic-geometric mean
    hypot(x,y)                   hypotenuse: sqrt(x^2 + y^2)
    norm(x,y)                    the norm function: x^2 + y^2
    lnbern(n)                    natural logarithm of bernoulli(n)
    bernreal(n)                  Bernoulli number as floating-point
    harmreal(n)                  Harmonic number as floating-point
    polygonal_root(n,k)          first k-gonal root of n
    polygonal_root2(n,k)         second k-gonal root of n

  :trig

    sin(x)              trigonometric sine function
    cos(x)              trigonometric cosine function
    tan(x)              trigonometric tangent function
    csc(x)              trigonometric cosecant function
    sec(x)              trigonometric secant function
    cot(x)              trigonometric cotangent function

    asin(x)             inverse of trigonometric sine
    acos(x)             inverse of trigonometric cosine
    atan(x)             inverse of trigonometric tangent
    acsc(x)             inverse of trigonometric cosecant
    asec(x)             inverse of trigonometric secant
    acot(x)             inverse of trigonometric cotangent

    sinh(x)             hyperbolic sine function
    cosh(x)             hyperbolic cosine function
    tanh(x)             hyperbolic tangent function
    csch(x)             hyperbolic cosecant function
    sech(x)             hyperbolic secant function
    coth(x)             hyperbolic cotangent function

    asinh(x)            inverse of hyperbolic sine
    acosh(x)            inverse of hyperbolic cosine function
    atanh(x)            inverse of hyperbolic tangent
    acsch(x)            inverse of hyperbolic cosecant
    asech(x)            inverse of hyperbolic secant
    acoth(x)            inverse of hyperbolic cotangent

    atan2(x,y)          two-argument variant of arctangent
    rad2deg(x)          convert radians to degrees
    deg2rad(x)          convert degrees to radians

  :misc

    rand(a)             pseudo-random floating-point: 0 <= r < a
    rand(a,b)           pseudo-random floating-point: a <= r < b
    irand(a)            pseudo-random integer: 0 <= r <= a
    irand(a,b)          pseudo-random integer: a <= r <= b
    seed(n)             re-seed the `rand()` function
    iseed(n)            re-seed the `irand()` function
    int(x)              truncate x to an integer
    floor(x)            round x towards -Infinity
    ceil(x)             round x towards +Infinity
    round(x)            round x to the nearest integer
    round(x,+k)         round x before the decimal point
    round(x,-k)         round x after the decimal point
    acmp(a,b)           absolute comparison: abs(a) <=> abs(b)
    approx_cmp(a,b,k)   approximate comparison: round(a,k) <=> round(b,k)
    rat(x)              convert x into a rational number
    rat(str)            parse a decimal expansion as fraction
    rat_approx(x)       rational approximation of real number x
    numerator(q)        numerator of rational number q
    denominator(q)      denominator of rational number q
    nude(q)             numerator and denominator of q
    float(x)            convert x to a floating-point number
    complex(x)          convert x to a floating-point complex number
    complex(a,b)        complex number: a + b*i
    real(z)             real part of complex number z
    imag(z)             imaginary part of complex number z
    reals(z)            real and imaginary part of z as reals
    sgn(x)              -1 if x < 0; 0 if x = 0; 1 if x > 0
    neg(x)              additive inverse of x: -x
    inv(x)              multiplicative inverse of x: 1/x
    conj(x)             complex conjugate of x
    digits(n,b)         digits of n in base b
    sumdigits(n,b)      sum of digits of n in base b
    popcount(n)         number of 1's in binary representation of n
    getbit(n,k)         k-th bit of integer n (1 or 0)
    setbit(n,k)         set the k-th bit of integer to 1
    clearbit(n,k)       set the k-th bit of integer to 0
    flipbit(n,k)        flip the k-th bit of integer

    sum(@list)          sum of a list of numbers
    prod(@list)         product of a list of numbers

    bsearch(n,\&f)      binary search from 0 to n (exact match)
    bsearch(a,b,\&f)    binary search from a to b (exact match)
    bsearch_le(n,\&f)   binary search from 0 to n (less than or equal to)
    bsearch_le(a,b,\&f) binary search from a to b (less than or equal to)
    bsearch_ge(n,\&f)   binary search from 0 to n (greater than or equal to)
    bsearch_ge(a,b,\&f) binary search from a to b (greater than or equal to)

    as_bin(n)           binary string-representation of n
    as_oct(n)           octal string-representation of n
    as_hex(n)           hexadecimal string-representation of n
    as_int(n,b)         integer string-representation of n in base b
    as_rat(n,b)         rational string-representation of n in base b
    as_frac(n,b)        fraction string-representation of n in base b
    as_dec(n,d)         decimal string-expansion of n with d digits

    is_pos(n)           return 1 if n > 0
    is_neg(n)           return 1 if n < 0
    is_int(n)           return 1 if n is an integer
    is_rat(n)           return 1 if n is a rational number
    is_real(n)          return 1 if n is a real number
    is_imag(n)          return 1 if n is an imaginary number
    is_complex(n)       return 1 if n is a complex number
    is_inf(n)           return 1 if n is +Infinity
    is_ninf(n)          return 1 if n is -Infinity
    is_nan(n)           return 1 if n is Not-a-Number (NaN)
    is_zero(n)          return 1 if n = 0
    is_one(n)           return 1 if n = 1
    is_mone(n)          return 1 if n = -1
    is_even(n)          return 1 if n is an integer divisible by 2
    is_odd(n)           return 1 if n is an integer not divisible by 2
    is_div(n,k)         return 1 if n is exactly divisible by k

=head1 IMPORT / EXPORT

Each function can be exported individually, as:

    use Math::AnyNum qw(zeta);

There is also the possibility of exporting an entire group of functions, as:

    use Math::AnyNum qw(:trig);

Additionally, by specifying the C<:all> keyword, will export all the exportable functions and all the constants.

    use Math::AnyNum qw(:all);

The C<:overload> keyword enables constant overloading, which makes
each number a Math::AnyNum object and also exports the C<i>, C<Inf>
and C<NaN> constants:

    use Math::AnyNum qw(:overload);
    say 42;                                       #=> "42"   (as Int)
    say 1/2;                                      #=> "1/2"  (as Rat)
    say 0.5;                                      #=> "0.5"  (as Float)
    say 3 + 4*i;                                  #=> "3+4i" (as Complex)

B<NOTE:> C<:overload> is lexical to the current scope only.

The syntax for disabling the C<:overload> behavior in the current scope, is:

    no Math::AnyNum;        # :overload will be disabled in the current scope

In addition to the exportable functions, Math::AnyNum also provides a list with
useful mathematical constants that can be exported, such as:

    i           # imaginary unit             sqrt(-1)
    e           # e mathematical constant    2.718281828459...
    pi          # PI constant                3.141592653589...
    tau         # TAU constant               6.283185307179...
    ln2         # natural logarithm of 2     0.693147180559...
    phi         # golden ratio               1.618033988749...
    EulerGamma  # Euler-Mascheroni constant  0.577215664901...
    CatalanG    # Catalan G constant         0.915965594177...
    Inf         # positive Infinity
    NaN         # Not-a-Number

The syntax for importing a constant, is:

    use Math::AnyNum qw(pi);
    say pi;                          # 3.141592653589...

Nothing is exported by default.

=head1 HOW IT WORKS

Internally, each Math::AnyNum object holds a reference to an object of type Math::GMPz, Math::GMPq, Math::MPFR or Math::MPC.
Based on the internal types, it decides what functions to call on each operation and does automatic promotion whenever necessarily.

The promotion rules can be summarized as follows:

    (Integer, Integer)   -> Integer  | Rational | Float   | Complex
    (Integer, Rational)  -> Rational | Float    | Complex
    (Integer, Float)     -> Float    | Complex
    (Integer, Complex)   -> Complex

    (Rational, Rational) -> Rational | Float    | Complex
    (Rational, Float)    -> Float    | Complex
    (Rational, Complex)  -> Complex

    (Float, Float)       -> Float    | Complex
    (Float, Complex)     -> Complex

    (Complex, Complex)   -> Complex

For explicit conversions, Math::AnyNum provides the following functions:

    int(x)          # converts x to an integer (NaN if not possible)
    rat(x)          # converts x to a rational (NaN if not possible)
    float(x)        # converts x to a real or complex floating-point number
    complex(x)      # converts x to a complex floating-point number

=head1 PRECISION

The default precision for floating-point numbers is 192 bits, which is equivalent with about
50 digits of precision in base 10.

The precision can be changed by modifying the C<$Math::AnyNum::PREC> variable, such as:

    local $Math::AnyNum::PREC = 1024;

or by specifying the precision at import (this sets the precision globally):

    use Math::AnyNum PREC => 1024;

This precision is used internally whenever a C<Math::MPFR> or a C<Math::MPC> object is created.

For example, if we change the precision to 3 decimal digits (where C<4> is the conversion factor),
we get the following results:

    local $Math::AnyNum::PREC = 3*4;
    say sqrt(2);                                  #=> 1.41
    say 98**7;                                    #=> 86812553324672
    say 1 / 98**7                                 #=> 1/86812553324672

As shown above, integers and rational numbers do not obey this precision, because they can grow and
shrink dynamically, without a specific limit.

Furthermore, a rational number never losses precision and accuracy in rational operations, therefore if we say:

    my $x = 1/3;
    say $x*3;                                     #=> 1
    say 1/$x;                                     #=> 3
    say 3/$x;                                     #=> 9

...the results are exact.

=head1 NOTATIONS

Methods that begin with an B<i> followed by the actual name (e.g.: C<isqrt>),
do integer operations, by first truncating their arguments to integers, if necessary.

The returned types are noted as follows:

    Any         # any type of number
    Int         # an integer value
    Rat         # a rational value
    Float       # a floating-point value
    Complex     # a floating-point complex value
    NaN         # "Not-a-Number" value
    Inf         # +/-Infinity
    Bool        # a Boolean value (1 or 0)
    Scalar      # a Perl scalar

When two or more types are separated with pipe characters (B<|>), it means that the
corresponding function can return any of the specified types.

=head1 INITIALIZATION / CONSTANTS

This section includes methods for creating new B<Math::AnyNum> objects
and some useful mathematical constants.

=head2 new

    Math::AnyNum->new(n)                          #=> Any
    Math::AnyNum->new(n, base)                    #=> Any

Returns a new AnyNum object with the value specified in the first argument,
which can be a Perl numerical value, a string representing a number in a
rational form, such as C<"1/2">, a string holding a decimal expansion number,
such as C<"0.5">, a string holding an integer, such as C<"255"> or a string
holding a complex number, such as C<"3+4i"> or C<"(3 4)">.

    my $z = Math::AnyNum->new('42');        # integer
    my $r = Math::AnyNum->new('3/4');       # rational
    my $f = Math::AnyNum->new('12.34');     # float
    my $c = Math::AnyNum->new('3.1+4i');    # complex

The second argument specifies the base of the number, which must be between 2 and 62.

When no base is specified, it defaults to base 10.

For setting an hexadecimal number, we can say:

    my $n = Math::AnyNum->new("deadbeef", 16);

B<NOTE:> no prefix, such as C<"0x"> or C<"0b">, is allowed as part of the number.

=head2 new_si

    Math::AnyNum->new_si(n)                       #=> Int

Sets a I<signed> native integer.

Example:

    my $n = Math::AnyNum->new_si(-42);

=head2 new_ui

    Math::AnyNum->new_ui(n)                       #=> Int

Sets an I<unsigned> native integer.

Example:

    my $n = Math::AnyNum->new_ui(42);

=head2 new_z

    Math::AnyNum->new_z(str)                      #=> Int
    Math::AnyNum->new_z(str, base)                #=> Int

Sets an arbitrary large integer from a given string.

The second argument specifies the base of the number, which must be between 2 and 62.

When no base is specified, it defaults to base 10.

Example:

    my $n = Math::AnyNum->new_z("12345678910111213141516");
    my $m = Math::AnyNum->new_z("fffffffffffffffffff", 16);

=head2 new_q

    Math::AnyNum->new_q(frac)                     #=> Rat
    Math::AnyNum->new_q(num, den)                 #=> Rat
    Math::AnyNum->new_q(num, den, base)           #=> Rat

Sets an arbitrary large rational from a given string.

The third argument specifies the base of the number, which must be between 2 and 62.

When no base is specified, it defaults to base 10.

Example:

    my $n = Math::AnyNum->new_q('12345/67890');         # 823/4526
    my $m = Math::AnyNum->new_q('12345', '67890');      # 823/4526
    my $o = Math::AnyNum->new_q('fffff', 'aaaaa', 16);  # 1048575/699050 = 3/2

=head2 new_f

    Math::AnyNum->new_f(str)                      #=> Float
    Math::AnyNum->new_f(str, base)                #=> Float

Sets a floating-point real number from a given string.

The second argument specifies the base of the number, which must be between 2 and 62.

When no base is specified, it defaults to base 10.

Example:

    my $n = Math::AnyNum->new_f('12.345');          # 12.345
    my $m = Math::AnyNum->new_f('-1.2345e-12');     # -0.0000000000012345
    my $o = Math::AnyNum->new_f('ffffff', 16);      # 16777215

=head2 new_c

    Math::AnyNum->new_c(real)                     #=> Complex
    Math::AnyNum->new_c(real, imag)               #=> Complex
    Math::AnyNum->new_c(real, imag, base)         #=> Complex

Sets a complex number from a given string.

The third argument specifies the base of the number, which must be between 2 and 62.

When no base is specified, it defaults to base 10.

Example:

    my $n = Math::AnyNum->new_c('1.123');           # 1.123
    my $m = Math::AnyNum->new_c('3', '4');          # 3+4i
    my $o = Math::AnyNum->new_c('f', 'a', 16);      # 15+10i

=head2 nan

    Math::AnyNum->nan                             #=> NaN

Returns an object holding the C<NaN> value.

=head2 inf

    Math::AnyNum->inf                             #=> Inf

Returns an object representing positive infinity.

=head2 ninf

    Math::AnyNum->ninf                            #=> -Inf

Returns an object representing negative infinity.

=head2 one

    Math::AnyNum->one                             #=> Int

Returns an object containing the value C<1>.

=head2 mone

    Math::AnyNum->mone                            #=> Int

Returns an object containing the value C<-1>.

=head2 zero

    Math::AnyNum->zero                            #=> Int

Returns an object containing the value C<0>.

=head2 i

    Math::AnyNum->i                               #=> Complex

Returns the imaginary unit, which is C<sqrt(-1)>.

=head2 e

    Math::AnyNum->e                               #=> Float

Returns the I<e> mathematical constant, which is C<2.718...>.

=head2 pi

    Math::AnyNum->pi                              #=> Float

Returns the number PI, which is C<3.1415...>.

=head2 tau

    Math::AnyNum->tau                             #=> Float

Returns the number TAU, which is C<2*pi>.

=head2 ln2

    Math::AnyNum->ln2                             #=> Float

Returns the natural logarithm of C<2>.

=head2 phi

    Math::AnyNum->phi                             #=> Float

Returns the value of the golden ratio, which is C<1.61803...>.

=head2 EulerGamma

    Math::AnyNum->EulerGamma                      #=> Float

Returns the Euler-Mascheroni γ constant, which is C<0.57721...>.

=head2 CatalanG

    Math::AnyNum->CatalanG                        #=> Float

Returns the Catalan G constant, also known as C<Beta(2)>, which is C<0.91596...>.

=head1 ARITHMETIC OPERATIONS

This section includes basic arithmetic operations.

=head2 add

    x + y                                         #=> Any

Adds C<x> and C<y> and returns the result.

=head2 sub

    x - y                                         #=> Any

Subtracts C<y> from C<x> and returns the result.

=head2 mul

    x * y                                         #=> Any

Multiplies C<x> by C<y> and returns the result.

=head2 div

    x / y                                         #=> Any

Divides C<x> by C<y> and returns the result.

=head2 mod

    x % y                                         #=> Any
    mod(x, y)                                     #=> Any

Remainder of C<x> when is divided by C<y>. Returns NaN when C<y> is zero.

Implemented as:

    x % y = x - y*floor(x/y)

=head2 polymod

    polymod(n, a, b, c, ...)                      #=> (Any, Any, ..., Any)

Returns a list of mod results corresponding to the divisors in C<(a, b, c, ...)>. The divisors are
given from smallest "unit" to the largest (e.g. 60 seconds per minute, 60 minutes per hour) and the results
are returned in the same way: from smallest to the largest (5 seconds, 4 minutes).

Example:

    my ($s, $m, $h, $d) = polymod($seconds, 60, 60, 24);

=head2 conj

    conj(x)                                       #=> Any

Complex conjugate of C<x>. Return C<x> when C<x> is a real number.

Example:

    conj("3+4i") = 3-4*i

=head2 inv

    inv(x)                                        #=> Any

Multiplicative inverse of C<x>. Equivalent with C<1/x>.

=head2 neg

    neg(x)                                        #=> Any

Additive inverse of C<x>. Equivalent with C<-x>.

=head2 abs

    abs(x)                                        #=> Any

Absolute value of C<x>.

=head2 sqr

    sqr(x)                                        #=> Any

Multiplies C<x> with itself and returns the result. Equivalent with C<x*x>.

=head2 norm

    norm(x)                                       #=> Any

The square of the absolute value of C<x>. Equivalent with C<abs(x)**2>.

=head1 SPECIAL FUNCTIONS

This section includes the special functions.

=head2 sqrt

    sqrt(x)                                       #=> Any

Square root of C<x>. Returns a complex number when C<x> is negative.

=head2 cbrt

    cbrt(x)                                       #=> Any

Cube root of C<x>. Returns a complex number when C<x> is negative.

=head2 root

    root(x, y)                                    #=> Any

The C<y> root of C<x>. Equivalent with C<x**(1/y)>.

=head2 polygonal_root

    polygonal_root(n, k)                          #=> Any

Returns the k-gonal root of C<n>. Also defined for complex numbers.

Example:

    say polygonal_root($n, 3);      # triangular root
    say polygonal_root($n, 5);      # pentagonal root

=head2 polygonal_root2

    polygonal_root2(n, k)                         #=> Any

Returns the second k-gonal root of C<n>. Also defined for complex numbers.

Example:

    say polygonal_root2($n, 5);      # second pentagonal root

=head2 pow

    x ** y                                        #=> Any
    pow(x, y)                                     #=> Any

Raises C<x> to power C<y> and returns the result.

When C<x> and C<y> are both integers, it does integer exponentiation and returns the exact result.

When C<x> is rational and C<y> is an integer, it does rational exponentiation based on the identity: C<(a/b)**n = a**n / b**n>,
which also computes the exact result.

Otherwise, it does floating-point exponentiation, which is equivalent with C<exp(log(x) * y)>.

=head2 exp

    exp(x)                                        #=> Any

Natural exponentiation of C<x> (i.e.: C<e**x>).

=head2 exp2

    exp2(x)                                       #=> Any

Raises 2 to the power C<x>. (i.e.: C<2**x>)

=head2 exp10

    exp10(x)                                      #=> Any

Raises 10 to the power C<x>. (i.e.: C<10**x>)

=head2 ln / log

    ln(x)                                         #=> Any
    log(x)                                        #=> Any
    log(x, y)                                     #=> Any

Logarithm of C<x> to base C<y> (or base I<e> when C<y> is omitted).

B<NOTE:> C<log(x, y)> is equivalent with C<log(x) / log(y)>.

=head2 log2 / log10

    log2(x)                                       #=> Any
    log10(x)                                      #=> Any

Logarithm of C<x> to base 2 and base 10, respectively.

=head2 lgrt

    lgrt(x)                                       #=> Any

Logarithmic-root of C<x>, which is the solution to C<a**a = x>, where C<x> is known.
When the value of C<x> is less than C<e**(-1/e)>, it returns a complex number.

It also accepts a complex number as input.

Example:

     lgrt(100)          # solves for x in `x**x = 100` and returns: `3.59728...`

This function is related to the Lambert-W function via the following identities:

    log(lgrt(exp(x)))     = LambertW(x)
    exp(LambertW(log(x))) = lgrt(x)

=head2 LambertW

    LambertW(x)                                   #=> Any

The Lambert-W function. When the value of C<x> is less than C<-1/e>, it returns a complex number.

It also accepts a complex number as input.

Identities (assumming x>0):

     LambertW(exp(x)*x) = x
     LambertW(log(x)*x) = log(x)

=head2 bernoulli_polynomial

    bernoulli_polynomial(n, x)                    #=> Any

Returns the n-th Bernoulli polynomial.

=head2 euler_polynomial

    euler_polynomial(n, x)                        #=> Any

Returns the n-th Euler polynomial.

=head2 bernoulli / bernfrac

    bernoulli(n)                                  #=> Rat | NaN
    bernoulli(n, x)                               #=> Any

Returns the n-th Bernoulli number C<B_n> as an exact fraction, with C<bernoulli(1) = 1/2>.

When an additional argument is provided, it returns the the n-th Bernoulli polynomial.

=head2 euler

    euler(n)                                      #=> Rat | NaN
    euler(n, x)                                   #=> Any

Returns the n-th Euler number C<E_n>, starting with C<euler(0) = 1>.

When an additional argument is provided, it returns the the n-th Euler polynomial.

=head2 bernreal

    bernreal(n)                                   #=> Float

Returns the n-th Bernoulli number, as a floating-point approximation, with C<bernreal(1) = 0.5>.

=head2 lnbern

    lnbern(n)                                     #=> Float | Complex

Returns the natural logarithm of the n-th Bernoulli number.

=head2 harmonic / harmfrac

    harmonic(n)                                   #=> Rat | NaN

Returns the n-th Harmonic number C<H_n>. The harmonic numbers are the sum of
reciprocals of the first C<n> natural numbers: C<1 + 1/2 + 1/3 + ... + 1/n>.

For values greater than 7000, binary splitting (Fredrik Johansson's elegant formulation) is used.

=head2 harmreal

    harmreal(n)                                   #=> Float

Returns the n-th Harmonic number, as a floating-point approximation, for any real value of C<n>.

Returns NaN for negative integers.

Defined as:

    harmreal(n) = digamma(n+1) + γ

where C<γ> is the Euler-Mascheroni constant.

=head2 agm

    agm(x, y)                                     #=> Any

Arithmetic-geometric mean of C<x> and C<y>. Also defined for complex numbers.

=head2 hypot

    hypot(x, y)                                   #=> Any

The value of the hypotenuse for catheti C<x> and C<y>. Equivalent to C<sqrt(x**2 + y**2)>. Also defined for complex numbers.

=head2 gamma

    gamma(x)                                      #=> Float

The Gamma function on C<x>. Returns Inf when C<x> is zero, and NaN when C<x> is a negative integer.

=head2 lgamma

    lgamma(x)                                     #=> Float

Natural logarithm of the absolute value of the Gamma function.

=head2 lngamma

    lngamma(x)                                    #=> Float

Natural logarithm of the Gamma function on C<x>.

=head2 lnsuperfactorial

    lnsuperfactorial(n)                           #=> Float

Natural logarithm of C<superfactorial(n)>, where C<n> is a non-negative integer.

=head2 lnhyperfactorial

    lnsuperfactorial(n)                           #=> Float

Natural logarithm of C<hyperfactorial(n)>, where C<n> is a non-negative integer.

=head2 digamma

    digamma(x)                                    #=> Float

The Digamma function (sometimes called Psi).
Returns NaN when C<x> is negative, and -Inf when C<x> is 0.

=head2 beta

    beta(x, y)                                    #=> Float

The beta function (also called the Euler integral of the first kind).

Defined as:

    beta(x, y) = gamma(x)*gamma(y) / gamma(x+y)

=head2 zeta

    zeta(x)                                       #=> Float

The Riemann zeta function at C<x>. Returns Inf when C<x> is 1.

=head2 eta

    eta(x)                                        #=> Float

The Dirichlet eta function at C<x>.

Defined as:

    eta(1) = ln(2)
    eta(x) = (1 - 2**(1-x)) * zeta(x)

=head2 BesselJ

    BesselJ(x, n)                                 #=> Float

The first order Bessel function, C<J_n(x)>, where C<n> is a signed integer.

Example:

    BesselJ(x, n)        # represents J_n(x)

=head2 BesselY

    BesselY(x, n)                                 #=> Float

The second order Bessel function, C<Y_n(x)>, where C<n> is a signed integer. Returns NaN for negative values of C<x>.

Example:

    BesselY(x, n)        # represents Y_n(x)

=head2 erf

    erf(x)                                        #=> Float

The error function on C<x>.

=head2 erfc

    erfc(x)                                       #=> Float

Complementary error function on C<x>.

=head2 Ai

    Ai(x)                                         #=> Float

The Airy function on C<x>.

=head2 Ei

    Ei(x)                                         #=> Float

Exponential integral of C<x>. Returns -Inf when C<x> is zero, and NaN when C<x> is negative.

=head2 Li

    Li(x)                                         #=> Float

The logarithmic integral of C<x>, defined as: C<Ei(ln(x))>.
Returns -Inf when C<x> is 1, and NaN when C<x> is less than or equal to 0.

=head2 Li2

    Li2(x)                                        #=> Float

The dilogarithm function, defined as the integral of C<-log(1-t)/t> from 0 to C<x>.

=head1 TRIGONOMETRIC FUNCTIONS

=head2 sin / sinh / asin / asinh

    sin(x)                                        #=> Any
    sinh(x)                                       #=> Any
    asin(x)                                       #=> Any
    asinh(x)                                      #=> Any

Sine, hyperbolic sine, inverse sine and inverse hyperbolic sine.

=head2 cos / cosh / acos / acosh

    cos(x)                                        #=> Any
    cosh(x)                                       #=> Any
    acos(x)                                       #=> Any
    acosh(x)                                      #=> Any

Cosine, hyperbolic cosine, inverse cosine and inverse hyperbolic cosine.

=head2 tan / tanh / atan / atanh

    tan(x)                                        #=> Any
    tanh(x)                                       #=> Any
    atan(x)                                       #=> Any
    atanh(x)                                      #=> Any

Tangent, hyperbolic tangent, inverse tangent and inverse hyperbolic tangent.

=head2 cot / coth / acot / acoth

    cot(x)                                        #=> Any
    coth(x)                                       #=> Any
    acot(x)                                       #=> Any
    acoth(x)                                      #=> Any

Cotangent, hyperbolic cotangent, inverse cotangent and inverse hyperbolic cotangent.

=head2 sec / sech / asec / asech

    sec(x)                                        #=> Any
    sech(x)                                       #=> Any
    asec(x)                                       #=> Any
    asech(x)                                      #=> Any

Secant, hyperbolic secant, inverse secant and inverse hyperbolic secant.

=head2 csc / csch / acsc / acsch

    csc(x)                                        #=> Any
    csch(x)                                       #=> Any
    acsc(x)                                       #=> Any
    acsch(x)                                      #=> Any

Cosecant, hyperbolic cosecant, inverse cosecant and inverse hyperbolic cosecant.

=head2 atan2

    atan2(x, y)                                   #=> Any

The arc tangent of C<x> and C<y>, defined as:

    atan2(x, y) = -i * log((y + x*i) / sqrt(x^2 + y^2))

=head2 deg2rad

    deg2rad(x)                                    #=> Any

Returns the value of C<x> converted from degrees to radians.

Example:

    deg2rad(180) = pi

=head2 rad2deg

    rad2deg(x)                                    #=> Any

Returns the value of C<x> converted from radians to degrees.

Example:

    rad2deg(pi) = 180

=head1 INTEGER FUNCTIONS

All operations in this section are done with integers.

=head2 iadd / isub / imul / idiv / ipow

    iadd(x, y)                                    #=> Int | NaN
    isub(x, y)                                    #=> Int | NaN
    imul(x, y)                                    #=> Int | NaN
    idiv(x, y)                                    #=> Int | NaN
    ipow(x, y)                                    #=> Int | NaN

Integer addition, subtraction, multiplication, division and exponentiation.

=head2 ipow2

    ipow2(n)                                      #=> Int

Raises C<2> to the power C<n>, by first truncating C<n> to an integer.
Returns C<0> when C<n> is negative.

=head2 ipow10

    ipow10(n)                                     #=> Int

Raises C<10> to the power C<n>, by first truncating C<n> to an integer.
Returns C<0> when C<n> is negative.

=head2 imod

    imod(x, y)                                    #=> Int | NaN

The integer modulus operation. Returns NaN when C<y> is zero.

=head2 divmod

    divmod(x, y)                                  #=> (Int, Int) | (NaN, NaN)

Returns the quotient and the remainder from division of C<x> by C<y>,
where both are integers. When C<y> is zero, it returns two NaN values.

=head2 invmod

    invmod(x, y)                                  #=> Int | NaN

Computes the inverse of C<x> modulo C<y> and returns the result.
When no inverse exists, the NaN value is returned.

=head2 powmod

    powmod(x, y, z)                               #=> Int | NaN

Computes C<(x ** y) % z>, where all three values are integers.

Returns NaN when the third argument is 0, or when C<y> is negative and C<gcd(x, z) != 1>.

=head2 isqrt / icbrt

    isqrt(n)                                      #=> Int | NaN
    icbrt(n)                                      #=> Int | NaN

The integer square root of C<n> and the integer cube root of C<n>. Returns NaN when a real root does not exists.

=head2 isqrtrem

    isqrtrem(n)                                   #=> (Int, Int) | (NaN, NaN)

The integer part of the square root of C<n> and the remainder C<n - isqrt(n)**2>, which will be zero when C<n> is a perfect square.

=head2 iroot

    iroot(n, m)                                   #=> Int | NaN

The integer C<m-th> root of C<n>. Returns NaN when a real does not exists.

=head2 irootrem

    irootrem(n, m)                                #=> (Int, Int) | (NaN, NaN)

The integer part of the root of C<n> and the remainder C<n - iroot(n,m)**m>.

Returns C<(NaN,NaN)> when a real root does not exists.

=head2 ilog

    ilog(n)                                       #=> Int | NaN
    ilog(n, m)                                    #=> Int | NaN

The integer part of the logarithm of C<n> to base C<m> or base I<e> when C<m> is not specified.

C<n> must be greater than 0 and C<m> must greater than 1. Returns NaN otherwise.

=head2 ilog2 / ilog10

    ilog2(n)                                      #=> Int | NaN
    ilog10(n)                                     #=> Int | NaN

The integer part of the logarithm of C<n> to base C<2> or base C<10>, respectively.

=head2 and / or / xor / not / lsft / rsft

    x & y                                         #=> Int
    x | y                                         #=> Int
    x ^ y                                         #=> Int
    ~x                                            #=> Int
    x << y                                        #=> Int
    x >> y                                        #=> Int

The bitwise integer operations.

=head2 gcd

    gcd(@list)                                    #=> Int

The greatest common divisor of a list of integers.

=head2 lcm

    lcm(@list)                                    #=> Int

The least common multiple of a list of integers.

=head2 valuation

    valuation(n, k)                               #=> Scalar

Returns the number of times C<n> is divisible by C<k>.

=head2 remdiv

    remdiv(n, k)                                  #=> Int

Removes all occurrences of the divisor C<k> from integer C<n>.

In general, the following statement holds true:

    remdiv(n, k) == n / k**(valuation(n, k))

=head2 kronecker

    kronecker(n, m)                               #=> Scalar

Returns the Kronecker symbol I<(n|m)>, which is a generalization of the Jacobi symbol for all integers I<m>.

=head2 faulhaber_sum

    faulhaber_sum(n, k)                           #=> Int | NaN

Computes the power sum C<1^k + 2^k + 3^k +...+ n^k>, using Faulhaber's formula.

C<k> must be a non-negative integer. Returns NaN otherwise.

Example:

    faulhaber_sum(5, 8) = 1^8 + 2^8 + 3^8 + 4^8 + 5^8 = 462979

=head2 geometric_sum

    geometric_sum(n, r)                           #=> Any

Computes the geometric sum C<1 + r + r^2 + r^3 + ... + r^n>, using the following formula:

    geometric_sum(n, r) = (r^(n+1) - 1) / (r - 1)

Example:

    geometric_sum(5, 8) = 8^0 + 8^1 + 8^2 + 8^3 + 8^4 + 8^5 = 37449

=head2 lucas

    lucas(n)                                      #=> Int | NaN

The n-th Lucas number. Returns NaN when C<n> is negative.

=head2 fibonacci

    fibonacci(n)                                  #=> Int | NaN
    fibonacci(n, k)                               #=> Int | NaN

The n-th Fibonacci number. Returns NaN when C<n> is negative.

When C<k> is specified, it returns the k-th order Fibonacci number.

Example:

    say fibonacci(100, 3);        # 100th Tribonacci number
    say fibonacci(100, 4);        # 100th Tetranacci number
    say fibonacci(100, 5);        # 100th Pentanacci number

=head2 factorial

    factorial(n)                                  #=> Int | NaN

Factorial of C<n> (denoted as C<n!>). Returns NaN when C<n> is negative. (C<1*2*3*...*n>)

=head2 dfactorial

    dfactorial(n)                                 #=> Int | NaN

Double-factorial of C<n> (denoted as C<n!!>). Returns NaN when C<n> is negative. (requires GMP>=5.1.0)

Example:

    dfactorial(7)     # 1*3*5*7 = 105
    dfactorial(8)     # 2*4*6*8 = 384

=head2 mfactorial

    mfactorial(n, m)                              #=> Int | NaN

Generalized m-factorial of C<n>. Returns NaN when C<n> or C<m> is negative. (requires GMP>=5.1.0)

=head2 subfactorial

    subfactorial(n)                               #=> Int | NaN
    subfactorial(n, k)                            #=> Int | NaN

The number of permutations of C<{1, ..., n}> that have exactly C<k> fixed points,
given a positive integer C<n> and an optional integer C<k> (if C<k> is ommited, then C<k=0>).

See also:

=over 4

=item * L<https://en.wikipedia.org/wiki/Rencontres_numbers>

=back

=head2 superfactorial

    superfactorial(n)                               #=> Int | NaN

Product of first C<n> factorials: C<Prod_{k=1..n} k!>.

=head2 hyperfactorial

    hyperfactorial(n)                               #=> Int | NaN

Hyperfactorial of C<n>, defined as: C<Prod_{k=1..n} k^k>.

=head2 bell

    bell(n)                                       #=> Int | NaN

Returns the n-th Bell number.

=head2 catalan

    catalan(n)                                    #=> Int | NaN

Returns the n-th Catalan number.

=head2 binomial

    binomial(n, k)                                #=> Int | NaN

Computes the binomial coefficient C<n> over C<k>, also called the
"choose" function. The result is equivalent to:

                        n!
    binomial(n, k) = -------
                     k!(n-k)!

=head2 multinomial

    multinomial(a, b, c, ...)                     #=> Int | NaN

Computes the multinomial coefficient, given a list of native integers.

Example:

    multinomial(1, 4, 4, 2) = 34650

See also:

=over 4

=item * L<https://en.wikipedia.org/wiki/Multinomial_theorem>

=back

=head2 rising_factorial

    rising_factorial(n, k)                        #=> Int | Rat | NaN

Rising factorial, C<n * (n + 1) * ... * (n + k - 1)>, defined as:

    binomial(n + k - 1, k) * k!

For negative values of C<k>, rising factorial is defined as:

    rising_factorial(n, -k) = 1/rising_factorial(n - k, k)

When the denominator is zero, NaN is returned.

=head2 falling_factorial

    falling_factorial(n, k)                       #=> Int | Rat | NaN

Falling factorial, C<n * (n - 1) * ... * (n - k + 1)>, defined as:

    binomial(n, k) * k!

For negative values of C<k>, falling factorial is defined as:

    falling_factorial(n, -k) = 1/falling_factorial(n + k, k)

When the denominator is zero, NaN is returned.

=head2 primorial

    primorial(n)                                  #=> Int | NaN

Returns the product of all the primes less than or equal to C<n>. (requires GMP>=5.1.0)

=head2 next_prime

    next_prime(n)                                 #=> Int | NaN

Returns the next prime after C<n>.

=head2 is_prime

    is_prime(n)                                   #=> Scalar
    is_prime(n, r)                                #=> Scalar

Returns 2 if C<n> is definitely prime, 1 if C<n> is probably prime (without
being certain), or 0 if C<n> is definitely composite. This method does some
trial divisions, then some Miller-Rabin probabilistic primality tests. It
also accepts an optional argument for specifying the accuracy of the test.
By default, it uses an accuracy value of 20.

Reasonable accuracy values are between 15 and 50.

See also:

=over 4

=item * L<https://en.wikipedia.org/wiki/Miller–Rabin_primality_test>

=item * L<https://gmplib.org/manual/Number-Theoretic-Functions.html>

=back

=head2 is_coprime

    is_coprime(n, k)                              #=> Bool

Returns a true value when C<n> and C<k> are relatively prime to each other. That is, when C<gcd(n, k) == 1>.

=head2 is_smooth

    is_smooth(n, k)                               #=> Bool

Returns a true value when all the prime factors of C<n> are less than or equal to C<k>, where C<n> and C<k> are positive integers.

Example:

    is_smooth(36, 3)    # true  : 36 = 2^2 * 3^2
    is_smooth(39, 6)    # false : 39 = 3 * 13, where 13 > 6

=head2 is_power

    is_power(n)
    is_power(n, k)                                #=> Bool

Return a true value when C<n> is a perfect power of a given integer C<k>.

When C<n> is not an integer, it always returns false. On the other hand, when C<k> is not an integer,
it will implicitly be truncated to an integer. If C<k> is not positive after truncation, C<0> is returned.

A true value is returned iff there exists some integer C<a> satisfying the equation: C<a**k = n>.

When C<k> is not specified, it returns true if C<n> can be expressed as C<a**b> for some integers C<a> and C<b>, with C<b> greater than 1.

Example:

    is_power(100, 2)       # true: 100 is a square (10**2)
    is_power(125, 3)       # true: 125 is a cube   ( 5**3)
    is_power(279841)       # true: 279841 is 23**4

=head2 is_square

    is_square(n)                                  #=> Bool

Returns a true value when C<n> is a perfect square.
When C<n> is not an integer, a false value is returned.

=head2 polygonal

    polygonal(n, k)                               #=> Int

Returns the nth k-gonal number. When C<n> is negative, it returns the second k-gonal number.

Example:

    say join(' ', map { polygonal( $_, 3) } 1..10);  # triangular numbers
    say join(' ', map { polygonal( $_, 5) } 1..10);  # pentagonal numbers
    say join(' ', map { polygonal(-$_, 5) } 1..10);  # second pentagonal numbers

=head2 ipolygonal_root

    ipolygonal_root(n, k)                         #=> Int | NaN

Integer k-gonal root of C<n>. Returns NaN when a real root does not exists.

Example:

    say ipolygonal_root($n, 5);                  # integer pentagonal root
    say ipolygonal_root(polygonal(10, 5), 5);    # prints: "10"

=head2 ipolygonal_root2

    ipolygonal_root2(n, k)                        #=> Int | NaN

Second integer k-gonal root of C<n>. Returns NaN when a real root does not exists.

Example:

    say ipolygonal_root2($n, 5);                   # second integer pentagonal root
    say ipolygonal_root2(polygonal(-10, 5), 5);    # prints: "-10"

=head2 is_polygonal

    is_polygonal(n, k)                            #=> Bool

Returns a true value when C<n> is a k-gonal number.

The values of C<n> and C<k> can be any arbitrary large integers.

Example:

    say is_polygonal(145, 5);      #=> 1 ("145" is a pentagonal number)
    say is_polygonal(155, 5);      #=> 0

=head2 is_polygonal2

    is_polygonal2(n, k)                           #=> Bool

Returns a true value when C<n> is a second k-gonal number.

The values of C<n> and C<k> can be any arbitrary large integers.

Example:

    say is_polygonal2(145, 5);      #=> 0
    say is_polygonal2(155, 5);      #=> 1 ("155" is a second-pentagonal number)

=head1 MISCELLANEOUS

This section includes various useful methods.

=head2 sum

    sum(@list)                                    #=>  Any

Sum of a list of numbers.

=head2 prod

    prod(@list)                                   #=> Any

Product of a list of numbers.

=head2 bsearch

    bsearch(n, \&f)                               #=> Int | undef
    bsearch(a, b, \&f)                            #=> Int | undef

Binary search from to 0 to n, or from a to b, which can be any arbitrary large integers.

The last argument is a subroutine reference which does the comparisons.

This function finds a value C<k> such that f(k) = 0. Returns C<undef> otherwise.

Example:

    bsearch(20,      sub { $_*$_  <=> 49   });   #=> 7   (7*7  = 49)
    bsearch(3, 1000, sub { $_**$_ <=> 3125 });   #=> 5   (5**5 = 3125)

=head2 bsearch_le

    bsearch_le(n, \&f)                               #=> Int | undef
    bsearch_le(a, b, \&f)                            #=> Int | undef

Binary search from to 0 to n, or from a to b, which can be any arbitrary large integers.

The last argument is a subroutine reference which does the comparisons.

This function finds a value C<k> such that f(k) <= 0 and f(k+1) > 0. Returns C<undef> otherwise.

Example:

    bsearch_le(10**6,         sub { exp($_) <=> 1e+9 });  #=>  20   (exp( 20) <= 1e+9)
    bsearch_le(-10**6, 10**6, sub { exp($_) <=> 1e-9 });  #=> -21   (exp(-21) <= 1e-9)

=head2 bsearch_ge

    bsearch_ge(n, \&f)                               #=> Int | undef
    bsearch_ge(a, b, \&f)                            #=> Int | undef

Binary search from to 0 to n, or from a to b, which can be any arbitrary large integers.

The last argument is a subroutine reference which does the comparisons.

This function finds a value C<k> such that f(k-1) < 0 and f(k) >= 0. Returns C<undef> otherwise.

    bsearch_ge(10**6,         sub { exp($_) <=> 1e+9 });  #=>  21   (exp( 21) >= 1e+9)
    bsearch_ge(-10**6, 10**6, sub { exp($_) <=> 1e-9 });  #=> -20   (exp(-20) >= 1e-9)

=head2 floor

    floor(x)                                      #=> Any

Returns C<x> if C<x> is an integer, otherwise it rounds C<x> towards -Infinity.

Example:

    floor( 2.5) =  2
    floor(-2.5) = -3

=head2 ceil

    ceil(x)                                       #=> Any

Returns C<x> if C<x> is an integer, otherwise it rounds C<x> towards +Infinity.

Example:

    ceil( 2.5) =  3
    ceil(-2.5) = -2

=head2 round

    round(x)                                      #=> Any
    round(x, p)                                   #=> Any

Rounds C<x> to the nth place. A negative argument rounds that many digits
after the decimal point, while a positive argument rounds that many digits
before the decimal point.

Example:

    round('1234.567')         = 1235
    round('1234.567', 2)      = 1200
    round('3.123+4.567i', -2) = 3.12+4.57*i

=head2 rand

    rand(x)                                       #=> Float
    rand(x, y)                                    #=> Float

Returns a pseudorandom floating-point value. When an additional argument is provided,
it returns a number between C<x> (inclusive) and C<y> (exclusive). Otherwise, returns a number between
C<0> (inclusive) and C<x> (exclusive).

The PRNG behind this function is called the "Mersenne Twister". Although it generates pseudorandom
numbers of very good quality, B<it is NOT cryptographically secure>. You should not rely on it in
security-sensitive situations.

Example:

    rand(10)        # a random number in the interval [0, 10)
    rand(10, 20)    # a random number in the interval [10, 20)

=head2 irand

    irand(x)                                      #=> Int
    irand(x, y)                                   #=> Int

Returns a pseudorandom integer. Unlike the C<rand()> function, C<irand()> is inclusive in both sides.

When an additional argument is provided, it returns an integer between C<x> (inclusive) and C<y> (inclusive),
otherwise returns an integer between C<0> (inclusive) and C<x> (inclusive).

If C<x> is greater that C<y>, the returned result will be in the range C<[y, x]>.

The PRNG behind this function is called the "Mersenne Twister". Although it generates high-quality
pseudorandom integers, B<it is NOT cryptographically secure>. You should not rely on it in
security-sensitive situations.

Example:

    irand(10)        # a random integer in the interval [0, 10]
    irand(10, 20)    # a random integer in the interval [10, 20]

=head2 seed / iseed

    seed(n)                                       #=> Int
    iseed(n)                                      #=> Int

Reseeds the C<rand()> and the C<irand()> function, respectively, with the value of C<n>, which can be any arbitrary large integer.

Returns back the integer part of C<n>. If C<n> cannot be truncated to an integer,
the method dies with an appropriate error message.

=head2 sgn

    sgn(x)                                        #=> Scalar | Complex

Returns C<-1> when C<x> is negative, C<1> when C<x> is positive, and C<0> when C<x> is zero.

When C<x> is a complex number, it computes the sign using the identity:

    sgn(x) = x / abs(x)

=head2 length

    $x->length                                    #=> Scalar

Returns the number of digits in the integer part of C<x>.
Returns C<-1> when C<x> cannot be truncated to an integer.

For C<-1234.56>, it returns C<4>.

=head2 inc

    ++$x                                          #=> Any
    $x++                                          #=> Any

Returns C<x + 1>.

=head2 dec

    --$x                                          #=> Any
    $x--                                          #=> Any

Return C<x - 1>.

=head2 copy

    $x->copy                                      #=> Any

Returns a deep-copy of the self-object.

=head2 popcount

    popcount(n)                                   #=> Scalar

Returns the population count of the positive integer part of C<x>, which is the number of 1's in
its binary representation.

This value is also known as the Hamming weight.

Example:

    popcount(0b1011) = 3

=head2 getbit

    getbit(n, k)                                  #=> Bool

Verifies C<n>'s k-th bit and returns a true value if the bit is set to 1. False otherwise.

Example:

    getbit(0b1001, 0) = 1
    getbit(0b1000, 0) = 0

=head2 setbit

    setbit(n, k)                                  #=> Int

Sets C<n>'s k-th bit to 1 (without modifying C<n> in-place).

Example:

    setbit(0b1000, 0) = 0b1001
    setbit(0b1000, 2) = 0b1100

=head2 flipbit

    flipbit(n, k)                                 #=> Int

Flips C<n>'s k-th bit from 1 to 0, and viceversa (without modifying C<n> in-place).

Example:

    flipbit(0b1000, 0) = 0b1001
    flipbit(0b1001, 0) = 0b1000

=head2 clearbit

    clearbit(n, k)                                #=> Int

Clears C<n>'s k-th bit, by setting it to 0 (without modifying C<n> in-place).

Example:

    clearbit(0b1001, 0) = 0b1000
    clearbit(0b1100, 2) = 0b1000

=head1 * Introspection

=head2 is_int

    is_int(x)                                     #=> Bool

Returns a true value when C<x> is an integer.

=head2 is_rat

    is_rat(x)                                     #=> Bool

Returns a true value when C<x> is a rational number.

=head2 is_real

    is_real(x)                                    #=> Bool

Returns a true value when C<x> is a real number (i.e.: when the imaginary part is zero and it holds a real value in the real part).

Example:

    is_real(complex('4'))           # true
    is_real(complex('4i'))          # false (is imaginary)
    is_real(complex('3+4i'))        # false (is complex)

=head2 is_imag

Returns a true value when C<x> is an imaginary number (i.e.: when the real part is zero and it has a non-zero imaginary part).

Example:

    is_imag(complex('4'))           # false (is real)
    is_imag(complex('4i'))          # true
    is_imag(complex('3+4i'))        # false (is complex)

=head2 is_complex

    is_complex(x)                                 #=> Bool

Returns a true value when C<x> is a complex number (i.e.: when the real part and the imaginary part are non-zero).

Example:

    is_complex(complex('4'))        # false (is real)
    is_complex(complex('4i'))       # false (is imaginary)
    is_complex(complex('3+4i'))     # true

=head2 is_even

    is_even(n)                                    #=> Bool

Returns a true value when C<n> is a real integer divisible by 2.

=head2 is_odd

    is_odd(n)                                     #=> Bool

Returns a true value when C<n> is a real integer not divisible by 2.

=head2 is_div

    is_div(x, y)                                  #=> Bool

Returns a true value when C<x> is exactly divisible by C<y> (i.e.: when the remainder C<x % y> is zero).

This is a generalized version of the above two methods and it also works with complex numbers.

=head2 is_pos

    is_pos(x)                                     #=> Bool

Returns a true value when C<x> is positive.

=head2 is_neg

    is_neg(x)                                     #=> Bool

Returns a true value when C<x> is negative.

=head2 is_zero

    is_zero(n)                                    #=> Bool

Returns a true value when C<n> equals 0.

=head2 is_one

    is_one(n)                                     #=> Bool

Returns a true value when C<n> equals 1.

=head2 is_mone

    is_mone(n)                                    #=> Bool

Returns a true value when C<n> equals -1.

=head2 is_inf

    is_inf(x)                                     #=> Bool

Returns a true value when C<x> holds the positive Infinity special value.

=head2 is_ninf

    is_ninf(x)                                    #=> Bool

Returns a true value when C<x> holds the negative Infinity special value.

=head2 is_nan

    is_nan(x)                                     #=> Bool

Returns a true value when C<x> holds the Not-a-Number special value.

=head1 * Conversions

=head2 int

    int(x)                                        #=> Int | NaN

Returns the integer part of C<x>. Returns NaN when C<x> cannot be truncated to an integer.

=head2 rat

    rat(x)                                        #=> Rat | NaN
    rat(str)                                      #=> Rat | NaN

Converts C<x> to a rational number. Returns NaN when this conversion is not possible.

When the given argument is a decimal expansion string, it will be specially parsed as an exact fraction.

If C<x> is a floating-point real number, consider using C<rat_approx()> instead.

Example:

    rat('0.5')       = 1/2
    rat('1234/5678') = 617/2839

=head2 rat_approx

    rat_approx(n)                                 #=> Rat | NaN

Given a real number C<n>, it returns a very good (sometimes exact) rational approximation to C<n>, computed with continued fractions.

Example:

    rat_approx(3.14)     = 22/7
    rat_approx(zeta(-5)) = -1/252

Return NaN when C<n> is not a real number.

=head2 float

    float(x)                                      #=> Float | Complex
    float(str)                                    #=> Float | Complex

Converts C<x> to a real or a complex floating-point number (in this order).

Example:

    float(3.1415926) = 3.1415926   (as Float)
    float('777/222') = 3.5         (as Float)
    float('123+45i') = 123 + 45*i  (as Complex)

=head2 complex

    complex(x)                                    #=> Complex
    complex(str)                                  #=> Complex
    complex(x, y)                                 #=> Complex

Converts C<x> to a complex number. When a second argument is given, it sets C<x> as the real part and C<y> as the imaginary part.

If C<x> or C<y> are complex numbers, the function returns the result of C<x + y*i>.

Example:

    complex("3+4i")        = 3+4*i
    complex(3, 4)          = 3+4*i
    complex("5+2i", "-4i") = 9+2*i

=head2 stringify

    "$x"                                          #=> Scalar

Returns a string representing the value of C<x>, in base 10.

=head2 boolify

    !!$x                                          #=> Bool

Returns a false value when the number is zero. True otherwise.

=head2 numify

    $x->numify                                    #=> Scalar

Returns a Perl numerical scalar containing the value of C<x>, truncated if necessary.

If C<x> is an integer that fits inside a native signed or unsigned integer, the returned result will be exact,
otherwise the result is returned as a double, with possible truncation.

If C<x> is a complex number, only the real part is considered.

=head2 as_bin

    as_bin(n)                                     #=> Scalar

Returns a string representing the integer part of C<n> in binary (base 2).

Example:

    as_bin(42) = "101010"

Returns C<undef> when C<n> cannot be converted to an integer.

=head2 as_oct

    as_oct(n)                                     #=> Scalar

Returns a string representing the integer part of C<n> in octal (base 8).

Example:

    as_oct(42) = "52"

Returns C<undef> when C<n> cannot be converted to an integer.

=head2 as_hex

    as_hex(n)                                     #=> Scalar

Returns a string representing the integer part of C<n> in hexadecimal (base 16).

Example:

    as_hex(42) = "2a"

Returns C<undef> when C<n> cannot be converted to an integer.

=head2 as_int

    as_int(n)                                     #=> Scalar
    as_int(n, b)                                  #=> Scalar

Returns the integer part of C<n> as a string, in a given base, where the base must be between 2 and 62.

When the base is omitted, it defaults to base 10.

Example:

    as_int(255)     = "255"
    as_int(255, 16) = "ff"

Returns C<undef> when C<n> cannot be converted to an integer.

=head2 as_rat

    as_rat(n)                                     #=> Scalar
    as_rat(n, b)                                  #=> Scalar

Returns C<n> as a rational string-representation in a given base, where the base must be between 2 and 62.

When the base is omitted, it defaults to base 10.

Example:

    as_rat(42)      = "42"
    as_rat("2/4")   = "1/2"
    as_rat(255, 16) = "ff"

Returns C<undef> when C<n> cannot be converted to a rational number.

=head2 as_frac

    as_frac(n)                                    #=> Scalar | undef
    as_frac(n, b)                                 #=> Scalar | undef

Returns C<n> as a fraction in a given base, where the base must be between 2 and 62.

When the base is omitted, it defaults to base 10.

Example:

    as_frac(42)      = "42/1"
    as_frac("2/4")   = "1/2"
    as_frac(255, 16) = "ff/1"

Returns C<undef> when C<n> cannot be converted to a rational number.

=head2 as_dec

    as_dec(n)                                     #=> Scalar
    as_dec(n, digits)                             #=> Scalar

Returns C<n> as a decimal expansion string, with an optional number of digits.

When the second argument is undefined, it uses the default precision.

The value of C<n> can also be a complex number.

Example:

    as_dec(1/2)        = "0.5"
    as_dec(sqrt(2), 3) = "1.41"

=head1 * Dissections

=head2 numerator

    numerator(x)                                  #=> Int | NaN

Returns the numerator of C<x> as a signed C<Math::AnyNum> object. When C<x> is not a rational number,
it tries to convert it to a rational. Returns NaN when this conversion is not possible.

Example:

    numerator("-42")  = -42
    numerator("-3/4") = -3

=head2 denominator

    denominator(x)                                #=> Int | NaN

Returns the denominator of C<x> as an unsigned C<Math::AnyNum> object. When C<x> is not a rational number,
it tries to convert it to a rational. Returns NaN when this conversion is not possible.

Example:

    denominator("-42")  = 1
    denominator("-3/4") = 4

=head2 nude

    nude(x)                                       #=> (Int | NaN, Int | NaN)

Returns the numerator and the denominator of C<x>.

Example:

    nude("42")   = (42, 1)
    nude("-3/4") = (-3, 4)

=head2 real

    real(x)                                       #=> Any

Returns the real part of C<x>.

Example:

    real("42")   = 42
    real("42i")  = 0
    real("3-4i") = 3

=head2 imag

    imag(x)                                       #=> Any

Returns the imaginary part of C<x>, if any. Otherwise, returns zero.

Example:

    imag("42")   =  0
    imag("42i")  = 42
    imag("3-4i") = -4

=head2 reals

    reals(x)                                      #=> (Any, Any)

Return the real and the imaginary part of C<x> as real numbers.

Example:

    reals("42")   = (42, 0)
    reals("42i")  = (0, 42)
    reals("3-4i") = (3, -4)

=head2 digits

    digits(n)                                     #=> (Scalar, Scalar, ...)
    digits(n, b)                                  #=> (Scalar | Int, Scalar | Int, ...)

Returns a list with the digits of C<n> in a given base. When no base is specified, it defaults to base 10.

Only the absolute integer part of C<n> is considered.

The value of C<b> must be greater than C<1>. Returns an empty list otherwise.

Example:

    digits(12345)      = (5, 4, 3, 2, 1)
    digits(12345, 100) = (45, 23, 1)

=head2 sumdigits

    sumdigits(n)                                  #=> Int | NaN
    sumdigits(n, b)                               #=> Int | NaN

Sum the digits of C<n> in a given base. When no base is specified, it defaults to base 10.

Only the absolute integer part of C<n> is considered.

The value of C<b> must be greater than C<1>. Returns NaN otherwise.

Example:

    sumdigits(12345)      = 15
    sumdigits(12345, 100) = 69

=head1 * Comparisons

=head2 eq

    x == y                                        #=> Bool

Equality check: returns a true value when C<x> and C<y> are equal.

=head2 ne

    x != y                                        #=> Bool

Inequality check: returns a true value when C<x> and C<y> are not equal.

=head2 gt

    x > y                                         #=> Bool

Returns a true value when C<x> is greater than C<y>.

=head2 ge

    x >= y                                        #=> Bool

Returns a true value when C<x> is equal or greater than C<y>.

=head2 lt

    x < y                                         #=> Bool

Returns a true value when C<x> is less than C<y>.

=head2 le

    x <= y                                        #=> Bool

Returns a true value when C<x> is equal or less than C<y>.

=head2 cmp

    x <=> y                                       #=> Scalar

Compares C<x> to C<y> and returns a negative value when C<x> is less than C<y>,
0 when C<x> and C<y> are equal, and a positive value when C<x> is greater than C<y>.

Complex numbers are compared as:

    (real(x) <=> real(y)) ||
    (imag(x) <=> imag(y))

Comparing anything to NaN (including NaN itself), returns C<undef>.

=head2 acmp

    acmp(x, y)                                    #=> Scalar

Absolute comparison of C<x> and C<y>.

Defined as:

    acmp(x, y) = abs(x) <=> abs(y)

=head2 approx_cmp

    approx_cmp(x, y)                              #=> Scalar
    approx_cmp(x, y, k)                           #=> Scalar

Approximate comparison, by rounding the values of C<x> and C<y> at a given number of decimal places.

A negative value for C<k> rounds that many digits after the decimal point, while a positive value rounds before the decimal point.

When no value is given for C<k>, it uses the default precision - 1.

=head1 PERFORMANCE

The performance varies greatly, but, in most cases, Math::AnyNum is between 2x up to 10x
faster than L<Math::BigFloat> with the B<GMP> backend, and about 100x faster than L<Math::BigFloat>
without the B<GMP> backend (to be modest).

Math::AnyNum is fast because of the following facts:

=over 4

=item *

minimal overhead in object creations and conversions.

=item *

minimal Perl code is executed per operation.

=item *

the B<GMP>, B<MPFR> and B<MPC> libraries are extremely efficient.

=back

To achieve the best performance, try to:

=over 4

=item *

use the B<i*> functions/methods wherever applicable.

=item *

use floating-point numbers when accuracy is not important.

=item *

pass Perl integers as arguments to methods, if you can.

=back

=head1 MOTIVATION

This module came into existence as a response to Dana Jacobsen's request for a transparent
interface to L<Math::GMPz> and L<Math::MPFR>, which he talked about at the YAPC NA, in 2015.

See his great presentation at: L<https://www.youtube.com/watch?v=Dhl4_Chvm_g>.

The main aim of this module is to provide a fast and correct alternative to L<Math::BigInt>,
L<Maht::BigFloat> and L<Math::BigRat>, as well as to L<bigint>, L<bignum> and L<bigrat> pragmas.

The original project was called C<Math::BigNum>, but because of some design flaws, the project
was abandoned and much of its code ended up in this module.

=head1 AUTHOR

Daniel Șuteu, C<< <trizen@protonmail.com> >>

=head1 BUGS

Please report any bugs or feature requests to L<https://github.com/trizen/Math-AnyNum/issues>.
I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

=head1 SUPPORT

You can find documentation for this module with the perldoc command.

    perldoc Math::AnyNum

You can also look for information at:

=over 4

=item * GitHub

L<https://github.com/trizen/Math-AnyNum>

=item * AnnoCPAN: Annotated CPAN documentation

L<http://annocpan.org/dist/Math-AnyNum>

=item * CPAN Ratings

L<http://cpanratings.perl.org/d/Math-AnyNum>

=item * Search CPAN

L<http://search.cpan.org/dist/Math-AnyNum/>

=back

=head1 SEE ALSO

=over 4

=item * Fast math libraries

L<Math::GMP> - High speed arbitrary size integer math.

L<Math::GMPz> - perl interface to the GMP library's integer (mpz) functions.

L<Math::GMPq> - perl interface to the GMP library's rational (mpq) functions.

L<Math::MPFR> - perl interface to the MPFR (floating point) library.

L<Math::MPC> - perl interface to the MPC (multi precision complex) library.

=item * Portable math libraries

L<Math::BigInt> - Arbitrary size integer/float math package.

L<Math::BigFloat> - Arbitrary size floating point math package.

L<Math::BigRat> - Arbitrary big rational numbers.

=item * Math utilities

L<Math::Prime::Util> - Utilities related to prime numbers, including fast sieves and factoring.

L<Math::GComplex> - Generic library for complex number operations, with support for Gaussian integers.

=back

=head1 LICENSE AND COPYRIGHT

Copyright 2017-2018 Daniel Șuteu.

This program is free software; you can redistribute it and/or modify it
under the terms of the the Artistic License (2.0). You may obtain a
copy of the full license at:

L<http://www.perlfoundation.org/artistic_license_2_0>

Any use, modification, and distribution of the Standard or Modified
Versions is governed by this Artistic License. By using, modifying or
distributing the Package, you accept this license. Do not use, modify,
or distribute the Package, if you do not accept this license.

If your Modified Version has been derived from a Modified Version made
by someone other than you, you are nevertheless required to ensure that
your Modified Version complies with the requirements of this license.

This license does not grant you the right to use any trademark, service
mark, tradename, or logo of the Copyright Holder.

This license includes the non-exclusive, worldwide, free-of-charge
patent license to make, have made, use, offer to sell, sell, import and
otherwise transfer the Package with respect to any patent claims
licensable by the Copyright Holder that are necessarily infringed by the
Package. If you institute patent litigation (including a cross-claim or
counterclaim) against any party alleging that the Package constitutes
direct or contributory patent infringement, then this Artistic License
to you shall terminate on the date that such litigation is filed.

Disclaimer of Warranty: THE PACKAGE IS PROVIDED BY THE COPYRIGHT HOLDER
AND CONTRIBUTORS "AS IS' AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES.
THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR
PURPOSE, OR NON-INFRINGEMENT ARE DISCLAIMED TO THE EXTENT PERMITTED BY
YOUR LOCAL LAW. UNLESS REQUIRED BY LAW, NO COPYRIGHT HOLDER OR
CONTRIBUTOR WILL BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, OR
CONSEQUENTIAL DAMAGES ARISING IN ANY WAY OUT OF THE USE OF THE PACKAGE,
EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.


=cut
