DRAFT: Synopsis 32: Setting Library - Numeric
Rod Adams <rod@rodadams.net> Larry Wall <larry@wall.org> Aaron Sherman <ajs@ajs.com> Mark Stosberg <mark@summersault.com> Carl Mäsak <cmasak@gmail.com> Moritz Lenz <moritz@faui2k3.org> Tim Nelson <wayland@wayland.id.au>
Created: 19 Mar 2009 extracted from S29-functions.pod Last Modified: 16 July 2010 Version: 10
The document is a draft.
If you read the HTML version, it is generated from the Pod in the specs repository under https://github.com/perl6/specs/blob/master/S32-setting-library/Numeric.pod so edit it there in the git repository if you would like to make changes.
This documents Bit, Int, Numeric, Rat, Complex, and Bool.
XXX So where are Bit, Int, and Rat
our Bool multi method succ ( Bool $b: ) is export
Returns Bool::True
.
our Bool multi method pred ( Bool $b: ) is export
Returns Bool::False
.
Numeric
is a role for everything that's a scalar number. So Num
, Int
, Rat
, Complex
and other numeric types do that role. However it is an abstract interface, so $number.WHAT
will never return Numeric
.
Users who provide their own scalar numeric types are encouraged to implement the Numeric
role. It is intended that such types such support the basic arithmetic operators to the extent possible, as well as ==
. In addition, it is hoped that comparison operators will at least return consistent results, even if there is no sensible mathemtical ordering of your type. That allows functions like sort to not choke and die if they are handed a value of your type. (See also the Real
role for scalar numeric types that represent real numbers.)
The following are all defined in the Numeric
role:
Numeric
provides some constants in addition to the basic mathematical functions.
constant pi is export = 3.14159_26535_89793_23846_26433_83279_50288; constant e is export = 2.71828_18284_59045_23536_02874_71352_66249; constant i is export = 1i;
our Real multi method Real ()
If this Numeric
is equivalent to a Real
, return that Real
. (For instance, if this number is a Complex
with a zero imaginary part.) Fail otherwise.
our Int multi method Int ()
If this Numeric
is equivalent to a Real
, return the equivalent of calling truncate
on that Real
to get an Int
.
our Rat multi method Rat ( Real $epsilon = 1.0e-6 )
If this Numeric
is equivalent to a Real
, return a Rat
which is within $epsilon
of that Real
's value.
our Num multi method Num ()
If this Numeric
is equivalent to a Real
, return that Real
as a Num
as accurately as is possible.
our Numeric multi method succ ( Numeric $x: ) is export our Int multi method succ ( Int $x: ) is export
Returns the successor of $x
. This method is used by prefix:<++>
and postfix:<++>
to increment the value in a container.
our Numeric multi method pred ( Numeric $x: ) is export our Int multi method pred ( Int $x: ) is export
Returns the predecessor of $x
. This method is used by prefix:<-->
and postfix:<-->
to decrement the value in a container.
our Numeric multi method abs ( Numeric $x: ) is export
Absolute Value.
our Numeric multi method exp ( Numeric $exponent: Numeric :$base = Num::e ) is export
Performs similar to $base ** $exponent
. $base
defaults to the constant e.
our Numeric multi method log ( Numeric $x: Numeric $base = Num::e ) is export
Logarithm of base $base
, default Natural. Calling with $x == 0
is an error.
our Numeric multi method log10 (Numeric $x:) is export
A base 10
logarithm, otherwise identical to log
.
our Num term:<rand>
Pseudo random number in range 0 ..^ 1
. That is, 0
is theoretically possible, while 1
is not. Note that there is no unary rand
function in Perl 6, but there is a rand
method. For picking a random integer you probably want to use something like (1..6).pick
instead.
our Numeric multi method sqrt ( Numeric $x: ) is export
Returns the principle square root of the parameter.
method roots (Numeric $x: Int $n ) is export
Returns a list of all $n
th (complex) roots of $x
. Returns NaN
if $n <= 0
, itself if $n == 0
, and is free to return a single NaN
if $x
is NaN
or Inf
, or in case of complex numbers if one of the components is.
our Complex multi postfix:<i> ( Numeric $x )
Returns a complex number representing the parameter multiplied by the imaginary unit i
. Note that there is no .i
method. To follow a variable name with the postfix, it's necessary to use a backslash or parentheses:
$land\i ($land)i
our Numeric multi method to-radians ( Numeric $x: TrigBase $base ) is export
Convert from $base
to radians.
our Numeric multi method from-radians ( Numeric $x: TrigBase $base ) is export
Convert from radians to $base
.
role Real does Numeric;
Real
, like Numeric
, is an abstract role that represents the interface of a real scalar number (i.e. neither Complex
nor vector-like). For example Num
, Int
, Bool
and Rat
implement the Real
role.
Users who provide their own scalar real numeric types are encouraged to implement the Real
role. Because real numbers are strictly-totally-ordered and Real
types try to emulate that property, it is desirable that any two Real
types be mutally compatible, even if they are not aware of each other. The current proposal requires you to define a Bridge
method in your Real
type, which converts your type into a neutral Real
type by restating it in terms of the fundamental Perl 6 types and calling Bridge
on them. This then makes the default Real
methods and operators all work with your Real
type. While the name of this method may changed, it is hoped that something like this will remain in the spec.
our Complex multi method Complex ()
Returns a Complex
whose real part is this Real
and whose imaginary part is 0.
our Str multi method Str ()
Returns the Real
as a Str
. All built-in Real
types format it as a decimal number, so for example, the Rat
5/4
is returned as "1.2"
.
our Int multi method floor ( Real $x: ) is export
Returns the highest integer not greater than $x
.
our Int multi method ceiling ( Real $x: ) is export
Returns the lowest integer not less than $x
.
our Int multi method round ( Real $x: $scale = 1) is export
Returns the nearest integer to $x
. The algorithm is:
floor($x / $scale + 0.5) * $scale
(Other rounding algorithms will be given extended names beginning with "round".)
Functions that round to a particular precision may easily be created with currying:
constant &roundcents ::= &round.assuming(:scale(1/100));
our Int multi method truncate ( Real $x: ) is export
Returns the closest integer to $x
whose absolute value is not greater than the absolute value of $x
. (In other words, just chuck any fractional part.) This is the default rounding function used by implicit integer conversions.
You may also truncate using explicit integer casts, either Int()
for an arbitrarily large integers, or int()
for native integers.
our Int multi method sign ( Real $x: ) is export
Returns 1 when $x
is greater than 0, -1 when it is less than 0, 0 when it is equal to 0, or undefined when the value passed is undefined.
multi srand ( Real $seed = default_seed_algorithm())
Seed the generator rand
uses. $seed
defaults to some combination of various platform dependent characteristics to yield a non-deterministic seed. Note that you get one srand()
for free when you start a Perl program, so you must call srand()
yourself if you wish to specify a deterministic seed (or if you wish to be differently nondeterministic).
our Num multi method rand (Real $x:) is export
Pseudo random number in range 0 ..^ $x
. That is, 0
is theoretically possible, while $x
is not. For picking a random integer you probably want to use something like (1..6).pick
instead.
our Complex multi method cis (Real $angle:) is export
Returns 1.unpolar($angle)
our Complex multi method unpolar (Real $mag: Real $angle) is export
Returns a complex number specified in polar coordinates. Angle is in radians.
class Num does Real;
Num
is a machine-precision numeric real value.
Complex
is an immutable type. Each Complex
object stores two numbers, the real and imaginary part. For all practical purposes a Complex
with a NaN
in real or imaginary part may be considered a NaN
itself (and (NaN + 1i) ~~ NaN
is True
).
Coercion of a Complex
to any Real
returns the real part (coerced, if necessary) if the imaginary part is 0, and fails otherwise. Comparison between a Real
number and a Complex
must be smart enough not to coerce the Complex
to a real number blindly.
our Complex multi method new(Real $re, Real $im)
Constructs a Complex
number from real and imaginary part. This is the method form of $re + ($im)i
.
our Seq multi method polar (Complex $nim:) is export
Returns (magnitude, angle) corresponding to the complex number. The magnitude is non-negative, and the angle in the range -π ..^ π
.
our Real multi method re()
Returns the real part of the complex number.
our Real multi method im()
Returns the imaginary part of a complex number.
The following are also defined in Numeric
. The trig functions depend on the current (lexically scoped) trig base:
enum TrigBase is export <Radians Degrees Gradians Circles>; constant $?TRIGBASE = Radians;
Numeric multi method func ( Numeric $x: TrigBase $base = CALLER::<$?TRIGBASE> ) is export
where func is one of: sin, cos, tan, asin, acos, atan, sec, cosec, cotan, asec, acosec, acotan, sinh, cosh, tanh, asinh, acosh, atanh, sech, cosech, cotanh, asech, acosech, acotanh.
Performs the various trigonometric functions.
Option $base
is used to declare how you measure your angles. Given the value of an arc representing a single full revolution.
$base Subdivisions of circle ---- ---------------------- Radians 2*pi Degrees 360 Gradians 400 Circles 1
To change the base within your own lexical scope, it suffices to redefine the compiler constant with the trigbase
pragma:
use trigbase Degrees;
In addition to setting the new lexical $?TRIGBASE
, this also curries a new set of functions into the current lexical scope that assume the new base. (Note that methods calls cannot be curried, so methods must still look up the caller's trigbase. The optimizer may, of course, optimize these into fast function calls.)
our Real multi method atan2 ( Real $y: Real $x = 1, TrigBase $base = CALLER::<$?TRIGBASE> ) our Real multi atan2 ( Real $y, Real $x = 1, TrigBase $base = CALLER::<$?TRIGBASE> )
This second form of atan
computes the arctangent of $y/$x
, and takes the quadrant into account. Otherwise behaves as other trigonometric functions.
An Int
is an immutable, integral number of arbitrary size.
class Rat does Real;
An immutable rational number, represented by two Int
s, a numerator and a denominator. All interface methods return values as if the numerator and denominator were stored in a normal form: both numerator and denominator are minimal in their magnitude, and the denominator is positive. So Rat.new(2, -4).denominator
return 2
, because the normal form is -1/2
.
multi method new(Int $num, Int $denom)
Constructs a Rat
object from the numerator and denominator. Fails if $denom == 0
.
our Seq[Int] multi method nude()
Returns a Seq
of numerator and denominator
our Int multi method denominator()
Returns the denominator
our Int multi method numerator()
Returns the numerator
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