This is a Raku symbolic math module. It parses, manipulates, evaluates, and
outputs mathematical expressions and relations. This module is PRE-ALPHA
symbolic --m=1 --b=0 'y=m*x+b' x
on the command line, or
say Math::Symbolic.new("y=m*x+b").isolate("x").evaluate(:m(1), :b(0));
in Raku code, will print
A basic command line program named 'symbolic' is provided, and may be installed
in your PATH. It takes at least one positional argument: the relation or
expression to work with. If a second positional is passed, it is the name of
the variable to isolate in the given relation (non-relation expressions are
unsupported for isolation). If "0" is passed instead of a variable name, the
relation is arranged with 0 on the right side and terms grouped by variable
(when possible) on the left side.
If any named args are passed, they are substituted into the expression for the
variables they name. Each named argument's value is parsed as an expression
itself, so it doesn't just have to be a numeric value to give the variable.
The resulting expression will be printed after applying all requested
transformations, and attempting to simplify.
For development of Math::Symbolic itself, there is also a 'symbolic' bash
script in the module's root directory. This will use the module and command
line script in the wrapper's directory, instead of the installed versions. It
also intentionally does not include blib, since precompilation isn't ordinarily
done during development.
At the time of this writing, most of the API is too unstable to document yet.
To minimize exposure to the internal chaos and to provide a starting point for
thinking about what functionality needs to exist in a more formal future API, a
minimal temporary public interface is implemented as a small collection of
simple methods in the main Math::Symbolic class. Note that where the actual
signatures differ from what is documented here, the undocumented differences
are considered "private", and may not do what is expected.
Creates a new Math::Symbolic object, initialized with the tree resulting from a
parse of $expression (which may also be a relation; currently only "=" equality
is supported for relations).
Returns a clone of the object with an independent copy of the tree structure.
This is important because all manipulations (below) are done in place, and
cloning avoids the parsing overhead of .new().
Arranges a relation with $var on the left and everything else on the right. Or
attempts to. It supports simple operation inversion when only one instance of a
variable exists, as well as attempting to combine instances via distributive
property and/or factoring of polynomial expressions, if necessary. Calling
.isolate on a non-relation expression is not supported.
Replaces all instances of variables named by the keys of the hash, with the
expressions in the values of the hash, and simplifies the result as much as
possible (see .simplify below). If the resulting expression has no variables,
this means it can be fully evaluated down to a single value.
Note that fully evaluating an equation with valid values would result in
something mostly unhelpful like "0=0" if the simplifier is smart enough.
Though in the future, when such a relation can be evaluated for truth, that
will become useful.
Makes a childish attempt to reduce the complexity of the expression by
evaluating operations on constants, removing operations on identity values (and
eventually other special cases like 0, inverse identity, etc). Also already
does a very small number of rearrangements of combinations of operations, like
transforming a+-b into a-b.
.simplify is sometimes called after certain manipulations like .isolate and
.poly which might otherwise leave messy expressions e.g. operations with
identity values and awkward forms of negation and inversion. It is also called
at the end of the command line tool for output of the final result.
Attempts to arrange the equation/expression in polynomial form. If $var is
given, terms are grouped for that specific variable. Otherwise terms are
grouped separately according to the set of all variables in a term. For example
"x²+x²*y" will be unchanged by .poly(), but .poly('x') will rearrange it to
something like "x²*(1+y)". Unlike the formal definition of a polynomial, this
function may accept and return any expression for coefficients, and allows for
exponents of any constant value.
If .poly is called on a relation, it is first arranged so that the right side
is equal to zero, before grouping terms on the left. An attempt is made to
guess which side should be subtracted from the other to avoid ending up with an
excessive amount of negation.
Creates a new Math::Symbolic object from the expression on the right-hand side
of a relation after isolating $var on the left. Note that unlike the above
transformations, no changes are made to the original object.
.compile($positional = False, $defaults?)
Returns a Raku subroutine which is mathematically equivalent to the
Math::Symbolic expression. Not all operations are currently supported.
Compiling relations is also undefined behavior at this time.
All arguments are named by default. If $positional is True, all arguments are
positional instead, sorted in default Raku sort order. If $positional is itself
a Positional class then only the listed variables will be taken positionally,
in the specified order.
All arguments are also required by default. If $defaults is an Associative
object, it is taken as a map of variable names to default values, and the
listed variables will be optional. If $defaults is any other defined value,
that value is taken as the default for all arguments.
.routine($positional = False, $defaults?)
Identical to .compile (above), but returns the code as a string without
compiling via EVAL, for instance to embed the code into another module or
Returns the number of nodes in the expression's tree. This could be useful to
determine if an expression has been fully evaluated, or used as a crude
Prints out the tree structure of the expression. This really should return the
string instead, and perhaps be renamed.
Returns the expression re-assembled into a string by the same syntactic
constructs which govern parsing. As with all Raku objects, this is also the
method which gets called when the object is coerced to a string by other means,
e.g. interpolation, context, or the ~ prefix. The .gist() method is also
handled by this routine, for easy printing of a readable result.
Passing the result of .Str() back in to .new() should always yield a
mathematically equivalent structure (exact representation may vary by some
auto-simplification), giving the same type of round-trip characteristics to
expressions that .raku() and EVAL() provide for Raku objects. This allows a
user to, for instance, isolate a variable in an equation, then plug the result
in to .evaluate() for that variable in a different equation, all with the
simplicity of strings; no additional classes or APIs for the user to worry
about (albeit at a steep performance penalty).
Returns the expression coerced first to a string (see above), then to a number.
This will fail if the expression hasn't already been evaluated/simplified (see
further above) to a single constant value. As with all Raku objects, this is
also the method which gets called when the object is coerced to a number by
other means, e.g. context or the + prefix.
Syntax and Operations
All whitespace is optional. Implicit operations, e.g. multiplication by putting
two variables in a row with no infix operator, are not supported, and likely
never will be. It leads to far too many consequences, compromises, complexities
The available operations and syntax in order of precedence are currently:
- valid characters for variable names are alphanumerics and underscore
- first character must not be a number, mainly to avoid collisions with
E notation (see below)
- optional sign (only "-" for now)
- E notation supported
- case insensitive
- no restriction on value of exponent, with sign and decimal
- subexpressions not supported (e is numeric syntax, not an op)
- leading zeros before decimals not required
- imaginary, complex, quaternion, etc NYI
- vector, matrix, set, etc NYI
- () Grouping Parens
- || Absolute Value
- cannot invert this op for solving/manipulating, ± NYI
- ! Factorial
- syntax only, no functional implementation
- ² Square (same as ^ 2)
- √ Square Root (same as ^ .5)
- Infix Operation
- ^ Exponentiation, like raku's **
- mathematical convention dictates that this operation be chained
right-to-left, which is NYI
- √ Root, n√x is the same as x^(1/n)
- evaluates to positive only (± NYI)
- imaginary numbers NYI
- ^/ is a Texan variant of √ with the operands reversed
- x^/n is the same as x^(1/n) or n√x
- Logarithms are NYI, so no solving for variables in exponents yet
- Infix Relation
- = Equality is currently the only relation supported
- this is really because proper relations are more-or-less NYI
- note that relations are optional in the input string, it automatically
detects whether it is working with an expression or a relation
- really it just breaks if you call 'solve' on an expression ATM
Many, in all likelihood. Please report them to email@example.com.
Patches graciously accepted.