NAME

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

VERSION

Version 0.33

SYNOPSIS

Math::AnyNum provides a transparent and easy-to-use interface to Math::GMPz, Math::GMPq, Math::MPFR and 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

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 bignum and 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 immutable.

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
    catalan(n,k)              C(n,k) entry in Catalan's triangle
    lucas(n)                  n-th Lucas number
    lucasmod(n,m)             n-th Lucas number modulo m
    lucasU(P,Q,n)             Lucas U_n(P, Q) function
    lucasV(P,Q,n)             Lucas V_n(P, Q) function
    lucasUmod(P,Q,n,m)        Lucas U_n(P, Q) modulo m
    lucasVmod(P,Q,n,m)        Lucas V_n(P, Q) modulo m
    fibonacci(n)              n-th Fibonacci number
    fibonacci(n,k)            n-th Fibonacci number of k-th order
    fibmod(n,m)               n-th Fibonacci number modulo m
    polygonal(n,k)            n-th k-gonal number
    harmonic(n)               n-th harmonic number: 1 + 1/2 + ... + 1/n
    secant_number(n)          n-th secant/zig number
    tangent_number(n)         n-th tangent/zag number
    euler(n)                  n-th Euler number: E_n
    euler(n,x)                Euler polynomials: E_n(x)
    bernoulli(n)              n-th Bernoulli number: B_n
    bernoulli(n,x)            Bernoulli polynomials: B_n(x)
    laguerreL(n,x)            Laguerre polynomials: L_n(x)
    legendreP(n,x)            Legendre polynomials: P_n(x)
    hermiteH(n,x)             Physicists' Hermite polynomials: H_n(x)
    hermiteHe(n,x)            Probabilists' Hermite polynomials: He_n(x)
    chebyshevT(n,x)           Chebyshev polynomials of the 1st kind: T_n(x)
    chebyshevU(n,x)           Chebyshev polynomials of the 2nd kind: U_n(x)
    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
    lcm(@list)                least common multiple
    gcd(@list)                greatest common divisor
    gcdext(n,k)               return (u,v,d) where u*n+v*k=d
    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: 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,m)               multiplicative inverse of n modulo m
    powmod(a,b,m)             modular exponentiation: a ^ b mod m
    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 integers c, k > 1
    is_power(n,k)             return 1 if n = c^k for integers c, k
    is_polygonal(n,k)         return 1 if n is a first k-gonal number
    is_polygonal2(n,k)        return 1 if n is a second k-gonal 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_smooth_over_prod(n,k)  return 1 if n is smooth over the primes p|k
    is_prime(n,r=20)          Miller-Rabin primality test
    primorial(n)              product of primes <= 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)                      normalized value of x: abs(x)^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)             pseudorandom floating-point: 0 <= R < a
    rand(a,b)           pseudorandom floating-point: a <= R < b
    irand(a)            pseudorandom integer: 0 <= R <= a
    irand(a,b)          pseudorandom 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 to k places before the decimal point
    round(x,-k)         round x to k places 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 to a rational number
    rat(str)            parse a decimal expansion as an exact fraction
    rat_approx(x)       rational approximation of a real number x
    numerator(r)        numerator of rational number r
    denominator(r)      denominator of rational number r
    nude(r)             numerator and denominator of r
    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
    digits2num(\@d,b)   conversion of digits in base b to an integer
    sumdigits(n,b)      sum of digits of n in base b
    popcount(n)         number of 1's in binary representation of n
    hamdist(n,k)        number of bit-positions where the bits differ
    getbit(n,k)         k-th bit of integer n (1 or 0)
    setbit(n,k)         set the k-th bit of integer n to 1
    clearbit(n,k)       set the k-th bit of integer n to 0
    flipbit(n,k)        flip the k-th bit of integer n
    bit_scan0(n,k)      index of the first 0-bit of n with index >= k
    bit_scan1(n,k)      index of the first 1-bit of n with index >= k

    min(@list)          minimum value from a given list of numbers
    max(@list)          maximum value from a given list of numbers

    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)

    base(n,b)           string-representation of n in base b
    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
    is_congruent(n,k,m) return 1 if n is congruent to k mod m

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 :all keyword, will export all the exportable functions and all the constants.

    use Math::AnyNum qw(:all);

The :overload keyword enables constant overloading, which makes each number a Math::AnyNum object and also exports the i, Inf and 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)

NOTE: :overload is lexical to the current scope only.

The syntax for disabling the :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 exporting a constant, is:

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

Nothing is exported by default.

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 is necessary.

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

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 $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 Math::MPFR or a Math::MPC object is created.

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

    local $Math::AnyNum::PREC = 3*4;

    # Floating-points
    say sqrt(2);                                  #=> 1.41
    say sqrt(19+13*i);                            #=> 4.58+1.42i

    # Integers
    say 98**7;                                    #=> 86812553324672

    # Rationals
    say 1 / 98**7                                 #=> 1/86812553324672

Notice that 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 or 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.

NOTATIONS

Methods that begin with an i followed by the actual name (e.g.: 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 (|), it means that the corresponding function can return any of the specified types.

INITIALIZATION / CONSTANTS

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

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 "1/2", a string holding a decimal expansion number, such as "0.5", a string holding an integer, such as "255" or a string holding a complex number, such as "3+4i" or "(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);

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

new_si

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

Sets a signed native integer.

Example:

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

new_ui

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

Sets an unsigned native integer.

Example:

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

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);

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

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

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

nan

    Math::AnyNum->nan                             #=> NaN

Returns an object holding the NaN value.

inf

    Math::AnyNum->inf                             #=> Inf

Returns an object representing positive infinity.

ninf

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

Returns an object representing negative infinity.

one

    Math::AnyNum->one                             #=> Int

Returns an object containing the value 1.

mone

    Math::AnyNum->mone                            #=> Int

Returns an object containing the value -1.

zero

    Math::AnyNum->zero                            #=> Int

Returns an object containing the value 0.

i

    Math::AnyNum->i                               #=> Complex

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

e

    Math::AnyNum->e                               #=> Float

Returns the e mathematical constant, which is 2.718....

pi

    Math::AnyNum->pi                              #=> Float

Returns the number PI, which is 3.1415....

tau

    Math::AnyNum->tau                             #=> Float

Returns the number TAU, which is 2*pi.

ln2

    Math::AnyNum->ln2                             #=> Float

Returns the natural logarithm of 2.

phi

    Math::AnyNum->phi                             #=> Float

Returns the value of the golden ratio, which is 1.61803....

EulerGamma

    Math::AnyNum->EulerGamma                      #=> Float

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

CatalanG

    Math::AnyNum->CatalanG                        #=> Float

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

ARITHMETIC OPERATIONS

This section includes basic arithmetic operations.

add

    x + y                                         #=> Any

Adds x and y and returns the result.

sub

    x - y                                         #=> Any

Subtracts y from x and returns the result.

mul

    x * y                                         #=> Any

Multiplies x by y and returns the result.

div

    x / y                                         #=> Any

Divides x by y and returns the result.

mod

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

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

Implemented as:

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

polymod

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

Returns a list of mod results corresponding to the divisors in (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);

conj

    conj(x)                                       #=> Float | Complex

Complex conjugate of x. Return x when x is a real number.

Example:

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

inv

    inv(x)                                        #=> Any

Multiplicative inverse of x. Equivalent with 1/x.

neg

    neg(x)                                        #=> Any

Additive inverse of x. Equivalent with -x.

abs

    abs(x)                                        #=> Any

Absolute value of x.

sqr

    sqr(x)                                        #=> Any

Multiplies x with itself and returns the result. Equivalent with x*x.

norm

    norm(x)                                       #=> Any

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

SPECIAL FUNCTIONS

This section includes the special functions.

sqrt

    sqrt(x)                                       #=> Float | Complex

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

cbrt

    cbrt(x)                                       #=> Float | Complex

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

root

    root(x, y)                                    #=> Float | Complex

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

polygonal_root

    polygonal_root(n, k)                          #=> Float | Complex

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

Example:

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

polygonal_root2

    polygonal_root2(n, k)                         #=> Float | Complex

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

Example:

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

pow

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

Raises x to power y and returns the result.

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

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

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

exp

    exp(x)                                        #=> Float | Complex

Natural exponentiation of x (i.e.: e**x).

exp2

    exp2(x)                                       #=> Any

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

exp10

    exp10(x)                                      #=> Any

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

ln | log

    ln(x)                                         #=> Float | Complex
    log(x)                                        #=> Float | Complex
    log(x, y)                                     #=> Float | Complex

Logarithm of x to base y (or base e when y is omitted).

NOTE: log(x, y) is equivalent with log(x) / log(y).

log2 | log10

    log2(x)                                       #=> Float | Complex
    log10(x)                                      #=> Float | Complex

Logarithm of x to base 2 and base 10, respectively.

lgrt

    lgrt(x)                                       #=> Float | Complex

Logarithmic-root of x, which is the solution to a**a = x, where x is known. When the value of x is less than 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)

LambertW

    LambertW(x)                                   #=> Float | Complex

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

It also accepts a complex number as input.

Identities (assuming x>0):

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

bernreal

    bernreal(n)                                   #=> Float

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

lnbern

    lnbern(n)                                     #=> Float | Complex

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

harmreal

    harmreal(n)                                   #=> Float

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

Returns NaN for negative integers.

Defined as:

    harmreal(n) = digamma(n+1) + γ

where γ is the Euler-Mascheroni constant.

agm

    agm(x, y)                                     #=> Float | Complex

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

hypot

    hypot(x, y)                                   #=> Float | Complex

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

gamma

    gamma(x)                                      #=> Float

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

lgamma

    lgamma(x)                                     #=> Float

Natural logarithm of the absolute value of the Gamma function.

lngamma

    lngamma(x)                                    #=> Float

Natural logarithm of the Gamma function for which the logarithm is a real number. Returns NaN otherwise.

lnsuperfactorial

    lnsuperfactorial(n)                           #=> Float

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

lnhyperfactorial

    lnsuperfactorial(n)                           #=> Float

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

digamma

    digamma(x)                                    #=> Float

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

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)

zeta

    zeta(x)                                       #=> Float

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

eta

    eta(x)                                        #=> Float

The Dirichlet eta function at x.

Defined as:

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

BesselJ

    BesselJ(x, n)                                 #=> Float

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

Example:

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

BesselY

    BesselY(x, n)                                 #=> Float

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

Example:

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

erf

    erf(x)                                        #=> Float

The error function on x.

erfc

    erfc(x)                                       #=> Float

Complementary error function on x.

Ai

    Ai(x)                                         #=> Float

The Airy function on x.

Ei

    Ei(x)                                         #=> Float

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

Li

    Li(x)                                         #=> Float

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

Li2

    Li2(x)                                        #=> Float

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

TRIGONOMETRIC FUNCTIONS

sin | sinh | asin | asinh

    sin(x)                                        #=> Float | Complex
    sinh(x)                                       #=> Float | Complex
    asin(x)                                       #=> Float | Complex
    asinh(x)                                      #=> Float | Complex

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

cos | cosh | acos | acosh

    cos(x)                                        #=> Float | Complex
    cosh(x)                                       #=> Float | Complex
    acos(x)                                       #=> Float | Complex
    acosh(x)                                      #=> Float | Complex

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

tan | tanh | atan | atanh

    tan(x)                                        #=> Float | Complex
    tanh(x)                                       #=> Float | Complex
    atan(x)                                       #=> Float | Complex
    atanh(x)                                      #=> Float | Complex

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

cot | coth | acot | acoth

    cot(x)                                        #=> Float | Complex
    coth(x)                                       #=> Float | Complex
    acot(x)                                       #=> Float | Complex
    acoth(x)                                      #=> Float | Complex

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

sec | sech | asec | asech

    sec(x)                                        #=> Float | Complex
    sech(x)                                       #=> Float | Complex
    asec(x)                                       #=> Float | Complex
    asech(x)                                      #=> Float | Complex

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

csc | csch | acsc | acsch

    csc(x)                                        #=> Float | Complex
    csch(x)                                       #=> Float | Complex
    acsc(x)                                       #=> Float | Complex
    acsch(x)                                      #=> Float | Complex

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

atan2

    atan2(x, y)                                   #=> Float | Complex

The arc tangent of x and y, defined as:

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

deg2rad

    deg2rad(x)                                    #=> Float | Complex

Returns the value of x converted from degrees to radians.

Example:

    deg2rad(180) = pi

rad2deg

    rad2deg(x)                                    #=> Float | Complex

Returns the value of x converted from radians to degrees.

Example:

    rad2deg(pi) = 180

INTEGER FUNCTIONS

All operations in this section are done with integers.

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.

ipow2

    ipow2(n)                                      #=> Int

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

ipow10

    ipow10(n)                                     #=> Int

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

imod

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

The integer modulus operation. Returns NaN when y is zero.

divmod

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

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

invmod

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

Computes the multiplicative inverse of x modulo y and returns the result.

When no inverse exists (i.e.: gcd(x, y) != 1), the NaN value is returned.

powmod

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

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

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

isqrt | icbrt

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

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

isqrtrem

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

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

iroot

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

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

irootrem

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

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

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

ilog

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

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

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

ilog2 | ilog10

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

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

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.

lcm

    lcm(@list)                                    #=> Int

The least common multiple of a list of integers.

gcd

    gcd(@list)                                    #=> Int

The greatest common divisor of a list of integers.

gcdext

    gcdext(n, k)                                  #=> (Int, Int, Int)

The extended greatest common divisor of n and k, returning (u, v, d), where d is the greatest common divisor of n and k, while u and v are the coefficients satisfying u*n + v*k = d. The value of d is always non-negative.

valuation

    valuation(n, k)                               #=> Scalar

Returns the number of times n is divisible by k.

remdiv

    remdiv(n, k)                                  #=> Int

Removes all occurrences of the divisor k from integer n.

In general, the following statement holds true:

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

kronecker

    kronecker(n, m)                               #=> Scalar

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

faulhaber_sum

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

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

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

geometric_sum

    geometric_sum(n, r)                           #=> Any

Computes the geometric sum 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

harmonic | harmfrac

    harmonic(n)                                   #=> Rat | NaN

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

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

secant_number

    secant_number(n)                              #=> Int

Returns the n-th secant number (A000364), starting with secant_number(0) = 1.

tangent_number

    tangent_number(n)                             #=> Int

Returns the n-th tangent number (A000182), starting with tangent_number(1) = 1.

bernoulli_polynomial

    bernoulli_polynomial(n, x)                    #=> Any

Returns the n-th Bernoulli polynomial: B_n(x).

euler_polynomial

    euler_polynomial(n, x)                        #=> Any

Returns the n-th Euler polynomial: E_n(x).

bernoulli | bernfrac

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

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

When an additional argument is provided, it returns the n-th Bernoulli polynomial: B_n(x).

euler

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

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

When an additional argument is provided, it returns the n-th Euler polynomial: E_n(x).

lucas

    lucas(n)                                      #=> Int | NaN

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

lucasU

    lucasU(P, Q, n)                               #=> Int | NaN

The Lucas U_n(P, Q) function.

Example:

    lucasU(1, -1, $n)       # the Fibonacci numbers
    lucasU(2, -1, $n)       # the Pell numbers
    lucasU(1, -2, $n)       # the Jacobsthal numbers

lucasV

    lucasV(P, Q, n)                               #=> Int | NaN

The Lucas V_n(P, Q) function.

Example:

    lucasV(1, -1, $n)       # the Lucas numbers
    lucasV(2, -1, $n)       # the Pell-Lucas numbers
    lucasV(1, -2, $n)       # the Jacobsthal-Lucas numbers

lucasmod

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

Efficiently compute the n-th Lucas number modulo m.

lucasUmod

    lucasUmod(P, Q, n, m)                         #=> Int | NaN

Efficiently compute the Lucas U_n(P, Q) function modulo m.

lucasVmod

    lucasVmod(P, Q, n, m)                         #=> Int | NaN

Efficiently compute the Lucas V_n(P, Q) function modulo m.

fibonacci

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

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

When 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

fibmod

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

Efficiently compute the n-th Fibonacci number modulo m.

chebyshevT

    chebyshevT(n, x)                              #=> Any

Compute the Chebyshev polynomials of the first kind: T_n(x), where n must be a native integer.

Defined as:

    T(0, x) = 1
    T(1, x) = x
    T(n, x) = 2*x*T(n-1, x) - T(n-2, x)

chebyshevU

    chebyshevU(n, x)                              #=> Any

Compute the Chebyshev polynomials of the second kind: U_n(x), where n must be a native integer.

Defined as:

    U(0, x) = 1
    U(1, x) = 2*x
    U(n, x) = 2*x*U(n-1, x) - U(n-2, x)

laguerreL

    laguerreL(n, x)                               #=> Any

Compute the Laguerre polynomials: L_n(x), where n must be a non-negative native integer.

legendreP

    legendreP(n, x)                               #=> Any

Compute the Legendre polynomials: P_n(x), where n must be a non-negative native integer.

hermiteH

    hermiteH(n, x)                                #=> Any

Compute the physicists' Hermite polynomials: H_n(x), where n must be a non-negative native integer.

hermiteHe

    hermiteHe(n, x)                               #=> Any

Compute the probabilists' Hermite polynomials: He_n(x), where n must be a non-negative native integer.

factorial

    factorial(n)                                  #=> Int | NaN

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

dfactorial

    dfactorial(n)                                 #=> Int | NaN

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

Example:

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

mfactorial

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

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

subfactorial

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

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

See also:

superfactorial

    superfactorial(n)                             #=> Int | NaN

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

hyperfactorial

    hyperfactorial(n)                             #=> Int | NaN

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

bell

    bell(n)                                       #=> Int | NaN

Returns the n-th Bell number.

catalan

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

Returns the n-th Catalan number.

If two arguments are provided, it returns the C(n,k) entry in Catalan's triangle.

binomial

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

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

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

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:

rising_factorial

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

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

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

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

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

When the denominator is zero, NaN is returned.

falling_factorial

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

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

    binomial(n, k) * k!

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

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

When the denominator is zero, NaN is returned.

primorial

    primorial(n)                                  #=> Int | NaN

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

next_prime

    next_prime(n)                                 #=> Int | NaN

Returns the next prime after n.

is_prime

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

Returns 2 if n is definitely prime, 1 if n is probably prime (without being certain), or 0 if 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:

is_coprime

    is_coprime(n, k)                              #=> Bool

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

is_smooth

    is_smooth(n, k)                               #=> Bool

Returns a true value when all the prime factors of n are less than or equal to k, where n and 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

is_smooth_over_prod

    is_smooth_over_prod(n, k)                     #=> Bool

Returns a true value when n is the product of a subset of powers of prime factors of k, where k is a square-free integer, whose prime factors represent a set A, such that n is said to be smooth over a set A if there exists a factorization of n where the factors are powers of elements in A.

Example:

    is_smooth_over_prod($n, 3*5*7*11)   # true for odd 11-smooth numbers $n

is_power

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

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

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

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

When k is not specified, it returns true if n can be expressed as a**b for some integers a and b, with 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

is_square

    is_square(n)                                  #=> Bool

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

polygonal

    polygonal(n, k)                               #=> Int

Returns the nth k-gonal number. When 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

ipolygonal_root

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

Integer k-gonal root of 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"

ipolygonal_root2

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

Second integer k-gonal root of 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"

is_polygonal

    is_polygonal(n, k)                            #=> Bool

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

The values of n and 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

is_polygonal2

    is_polygonal2(n, k)                           #=> Bool

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

The values of n and 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)

MISCELLANEOUS

This section includes various useful methods.

min | max

    min(@list)                                    #=> Any
    max(@list)                                    #=> Any

Smallest and greatest value, respectively, from a given list of numbers.

Returns undef if the list contains NaN or if the list is empty.

sum | prod

    sum(@list)                                    #=> Any
    prod(@list)                                   #=> Any

Sum and product of a given list of numbers.

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 k such that f(k) = 0. Returns undef otherwise.

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

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 k such that f(k) <= 0 and f(k+1) > 0. Returns undef otherwise.

    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)

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 k such that f(k-1) < 0 and f(k) >= 0. Returns 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)

floor

    floor(x)                                      #=> Any

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

Example:

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

ceil

    ceil(x)                                       #=> Any

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

Example:

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

round

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

Rounds 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

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 x (inclusive) and y (exclusive). Otherwise, returns a number between 0 (inclusive) and x (exclusive).

If x is greater than y, the returned value will be in the range [y, x).

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

Example:

    rand(10)        # a pseudorandom floating-point in the interval [0, 10)
    rand(10, 20)    # a pseudorandom floating-point in the interval [10, 20)

irand

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

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

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

If x is greater than y, the returned integer will be in the range [y, x].

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

Example:

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

seed | iseed

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

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

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

sgn

    sgn(x)                                        #=> Scalar | Complex

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

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

    sgn(x) = x / abs(x)

length

    $n->length                                    #=> Scalar
    $n->length($base)                             #=> Scalar

Returns the number of digits of the integer part of n in a given base (default 10).

Example:

    5040->length        # size in base 10
    5040->length(2)     # size in base 2

Returns undef when n cannot be truncated to an integer.

inc

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

Returns x + 1.

dec

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

Return x - 1.

copy

    $x->copy                                      #=> Any

Returns a deep-copy of the self-object.

popcount

    popcount(n)                                   #=> Scalar

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

Return undef when n cannot be truncated to an integer.

This value is also known as the Hamming weight value.

Example:

    popcount(0b1011) = 3

hamdist

    hamdist(n, k)                                 #=> Scalar

Returns the Hamming distance (number of bit-positions where the bits differ) between integers n and k.

Return undef when n or k cannot be truncated to an integer.

getbit

    getbit(n, k)                                  #=> Bool

Returns 1 if bit k of n is set, and 0 if it is not set.

Return undef when n cannot be truncated to an integer or when k is negative.

Example:

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

setbit

    setbit(n, k)                                  #=> Int

Returns a copy of n with bit k set to 1.

Example:

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

flipbit

    flipbit(n, k)                                 #=> Int

Returns a copy of n with bit k inverted.

Example:

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

clearbit

    clearbit(n, k)                                #=> Int

Returns a copy of n with bit k set to 0.

Example:

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

bit_scan0 | bit_scan1

    bit_scan0(n, k)                               #=> Scalar
    bit_scan1(n, k)                               #=> Scalar

Scan n, starting from bit index k, towards more significant bits, until 0 or 1 bit (respectively) is found.

When k is omitted, k=0 is assumed.

Returns undef if n cannot be truncated to an integer or if k is negative.

* Introspection

is_int

    is_int(x)                                     #=> Bool

Returns a true value when x is an integer.

is_rat

    is_rat(x)                                     #=> Bool

Returns a true value when x is a rational number.

is_real

    is_real(x)                                    #=> Bool

Returns a true value when 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)

is_imag

Returns a true value when 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)

is_complex

    is_complex(x)                                 #=> Bool

Returns a true value when 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

is_even

    is_even(n)                                    #=> Bool

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

is_odd

    is_odd(n)                                     #=> Bool

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

is_div

    is_div(n, k)                                  #=> Bool

Returns a true value when n is exactly divisible by k (i.e.: when the remainder n % k is zero).

Also defined for rationals, floats and complex numbers.

is_congruent

    is_congruent(n, k, m)                         #=> Bool

Returns a true value when n is congruent to k modulo m (i.e.: when the remainder n % m equals k % m).

Also defined for rationals, floats and complex numbers.

is_pos

    is_pos(x)                                     #=> Bool

Returns a true value when x is positive.

is_neg

    is_neg(x)                                     #=> Bool

Returns a true value when x is negative.

is_zero

    is_zero(n)                                    #=> Bool

Returns a true value when n equals 0.

is_one

    is_one(n)                                     #=> Bool

Returns a true value when n equals 1.

is_mone

    is_mone(n)                                    #=> Bool

Returns a true value when n equals -1.

is_inf

    is_inf(x)                                     #=> Bool

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

is_ninf

    is_ninf(x)                                    #=> Bool

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

is_nan

    is_nan(x)                                     #=> Bool

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

* Conversions

int

    int(x)                                        #=> Int | NaN

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

rat

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

Converts 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 x is a floating-point real number, consider using rat_approx() instead.

Example:

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

rat_approx

    rat_approx(n)                                 #=> Rat | NaN

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

Example:

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

Return NaN when n is not a real number.

float

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

Converts 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)

complex

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

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

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

Example:

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

stringify

    "$x"                                          #=> Scalar

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

boolify

    !!$x                                          #=> Bool

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

numify

    $x->numify                                    #=> Scalar

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

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

base

    base(n, b)                                    #=> Scalar

Returns a string-representation of n in a given base b (between 2 and 62), where n can be any type of number, including a floating-point or a complex number.

Example:

    base(42, 2)         = "101010"
    base(17.5, 36)      = "h.i"
    base("99/43", 16)   = "63/2b"
    base("17.5+5i", 36) = "(h.i 5)"

The output of this function can be passed to new(), along with the base-number, which converts it back to the original number:

    say Math::AnyNum->new("101010", 2);    #=> 42
    say Math::AnyNum->new("h.i", 36);      #=> 17.5

as_bin

    as_bin(n)                                     #=> Scalar

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

Example:

    as_bin(42) = "101010"

Returns undef when n cannot be converted to an integer.

as_oct

    as_oct(n)                                     #=> Scalar

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

Example:

    as_oct(42) = "52"

Returns undef when n cannot be converted to an integer.

as_hex

    as_hex(n)                                     #=> Scalar

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

Example:

    as_hex(42) = "2a"

Returns undef when n cannot be converted to an integer.

as_int

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

Returns the integer part of 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 undef when n cannot be converted to an integer.

as_rat

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

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

as_frac

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

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

as_dec

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

Returns 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 n can also be a complex number.

Example:

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

* Dissections

numerator

    numerator(x)                                  #=> Int | NaN

Returns the numerator of x as a signed Math::AnyNum object. When 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

denominator

    denominator(x)                                #=> Int | NaN

Returns the denominator of x as an unsigned Math::AnyNum object. When 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

nude

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

Returns the numerator and the denominator of x.

Example:

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

real

    real(x)                                       #=> Any

Returns the real part of x.

Example:

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

imag

    imag(x)                                       #=> Any

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

Example:

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

reals

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

Return the real and the imaginary part of x as real numbers.

Example:

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

digits

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

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

Only the absolute integer part of n is considered.

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

Example:

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

digits2num

    digits2num([...], b=10)                       #=> Int | NaN

Takes an array-ref of digits (in reverse order) and an optional base (default 10), converting the digits to an integer in the given base.

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

Example:

    digits2num([5, 4, 3, 2, 1])  = 12345
    digits2num([45, 23, 1], 100) = 12345

sumdigits

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

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

Only the absolute integer part of n is considered.

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

Example:

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

* Comparisons

eq

    x == y                                        #=> Bool

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

ne

    x != y                                        #=> Bool

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

gt

    x > y                                         #=> Bool

Returns a true value when x is greater than y.

ge

    x >= y                                        #=> Bool

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

lt

    x < y                                         #=> Bool

Returns a true value when x is less than y.

le

    x <= y                                        #=> Bool

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

cmp

    x <=> y                                       #=> Scalar

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

Complex numbers are compared as:

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

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

acmp

    acmp(x, y)                                    #=> Scalar

Absolute comparison of x and y.

Defined as:

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

approx_cmp

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

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

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

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

PERFORMANCE

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

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

  • minimal overhead in object creations and conversions.

  • minimal Perl code is executed per operation.

  • the GMP, MPFR and MPC libraries are extremely efficient.

To achieve the best performance, try to:

  • use the i* functions/methods wherever applicable.

  • use floating-point numbers when accuracy is not important.

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

MOTIVATION

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

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

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

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

SEE ALSO

  • Fast math libraries

    Math::GMP - High speed arbitrary size integer math.

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

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

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

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

  • Portable math libraries

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

    Math::BigFloat - Arbitrary size floating point math package.

    Math::BigRat - Arbitrary big rational numbers.

  • Math utilities

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

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

REPOSITORY

https://github.com/trizen/Math-AnyNum

AUTHOR

Daniel Șuteu, <trizen at cpan.org>

COPYRIGHT AND LICENSE

Copyright (C) 2017-2019 Daniel Șuteu

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.22.0 or, at your option, any later version of Perl 5 you may have available.