generate all possible math expr of one term

Discussion in 'Perl Misc' started by xahlee@gmail.com, May 15, 2008.

  1. Guest

    Here's a example of Expressiveness of a Language.

    The following is Mathematica code that generates all possible
    equations of one term involving trig function. (tweak the funList and
    nesting level to define what “all possible†means. if nesting level is
    2, it takes about 20 minutes and returns a list of 2876 terms on a
    2004 personal computer.

    << DiscreteMath`Combinatorica`
    funList = {Sin, Tan, Power[#, -1] &};
    Nest[Module[{li = #},
    (Union[#, SameTest -> (Module[{arg1 = #1, arg2 = #2},
    If[(*both expression contains both x and y*)
    And @@ (((((StringMatchQ[#, "*x*"] &&
    StringMatchQ[#, "*y*"]) &)@
    ToString@#) &) /@ {arg1, arg2})
    , SameQ[arg1, arg2 /. {x -> y, y -> x}],
    SameQ[arg1, arg2]]
    ] &)] &)@
    Union@Flatten@(Table[(Times @@ # &) /@ KSubsets[#, i], {i, 1,
    2}] &)@Flatten@{li, Outer[#1@#2 &, funList, li]}
    ] &, {x, y}, 1];
    Select[%, (StringMatchQ[ToString@#, "*x*"] &&
    StringMatchQ[ToString@#, "*y*"]) &]

    The problem is this: generate a list of all possible math expressions
    using the following combination and rules:

    • the math expression involves both x and y. (must have both present)
    • you can use any of the 6 trig functions (you must, since the goal is
    to create all possibilities)
    • The binary operations you can use are multiply, divide. (no addition
    or substraction, since that can make the result 2 terms)
    • a sub-expression (such as x or y) can be replaced by a more
    complicated one. (i.e. you can nest)

    For example, these are first few items from the above code:

    {1/(x^2*y^2), 1/(x*y^2), x/y^2, 1/(x*y), x/y, x^2/y, x*y, x^2*y,
    x^2*y^2, Cos[x]/y^2, Cos[x]/y, Cos[x]/(x*y), (x*Cos[x])/y, y*Cos[x],
    (y*Cos[x])/x, x*y*Cos[x], y^2*Cos[x], Cos[x]*Cos[y], Cot[x]/y^2,
    Cot[x]/(x*y^2)}

    For a gallery of selection plots of these equations, see
    http://xahlee.org/MathGraphicsGallery_dir/dense/dense1.html

    The above i wrote in 2002. If there are requests, i'll translate the
    above code into emacs lisp. The result lisp expression should match
    Mathematica's, token for token. (the code make a lot use of nested
    lambda and or apply and or map) If you are interested, you could
    translate the above into lisp too, it's not difficult (though the
    number of lines will increase maybe 10 fold. And if Common Lisp
    doesn't have combinatorics library providing KSubsets, and also since
    CL doesn't have Outer, so the above in CL might be few hundred lines).
    (see here for a example of how to: http://xahlee.org/UnixResource_dir/writ/notations.html
    )

    PS as a after-thought, i decided to post this to perl, python, and
    java too. This will take about the same number of lines in perl as in
    Common Lisp. Probably llightly more in Python due to syntax. In Java,
    it will be one million lines.

    Gratuitous poem of the day:

    in the climb to geekdom,
    you have few rungs to catch,
    before you see my ass.

    —Xah Lee, 2005

    Xah

    ∑ http://xahlee.org/

    ☄
     
    , May 15, 2008
    #1
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