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Some \ examples:\ \>", "Text"], Cell[BoxData[ \(\(100!\)\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^1000. \)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has always had these, so what is the big deal? " }], "Text"], Cell[TextData[{ "In my opinion, the improvements in ", StyleBox["Mathematica", FontSlant->"Italic"], " 4 are shown off well by the following question which I was asked by a \ representative of the Japanese ", StyleBox["Mathematica", FontSlant->"Italic"], " users group in late 1995:" }], "Text"], Cell[TextData[{ "\"Can ", StyleBox["Mathematica", FontSlant->"Italic"], " compute Pi to 5,000,000 digits in principle? In practice?\"" }], "Text"], Cell["\<\ At the time the question was asked, the answer was basically: in prinicple, \ yes, but in practice, it could take days or weeks. Though I would like to do \ it before your very eyes, it still takes a little long. Nonetheless, the \ computation is practical to do in the time of a coffee break:\ \>", "Text"], Cell[BoxData[{ \(\($MaxPrecision\ = \ Infinity;\)\), "\[IndentingNewLine]", \(Timing[\(pi50k\ = \ N[Pi, 5000000];\)]\)}], "Input"], Cell[BoxData[ \(Timing[\(str\ = \ ToString[pi50k];\)]\)], "Input"], Cell[TextData[{ "Yes, part of the speed improvement has to do with the nice 333MhZ \ processor I have on this machine, but that is actually a small part of the \ story. ", StyleBox["Mathematica", FontSlant->"Italic"], " 3.0 would take days to weeks on the same computation on this machine." }], "Text"], Cell["\<\ The reason I consider this a good example is that there are two distinct \ improvements which went into this functionality. \ \>", "Text"], Cell[TextData[{ "First, multiplication is done with asymptotically efficient algorithms. \ This change effects any computation you do in ", StyleBox["Mathematica", FontSlant->"Italic"], " with more than a few hundred digits." }], "Text"], Cell[TextData[{ "Second, the algorithm for computing \[Pi] was improved to take advantage \ of the faster multiplication. This has been done for many fundamental \ functions in ", StyleBox["Mathematica", FontSlant->"Italic"], ", such as factorials, exponentials, trig functions, and the like." }], "Text"], Cell[TextData[{ "The end result is that, using the same syntax and commands as in ", StyleBox["Mathematica", FontSlant->"Italic"], " 3.0, you may get results an order of magnitude faster with ", StyleBox["Mathematica", FontSlant->"Italic"], " 4. " }], "Text"], Cell[CellGroupData[{ Cell["Asymptotically faster methods for multiplication", "Subtitle"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection"], Cell[BoxData[ \(Square[ll_, \ ur_]\ := \ Polygon[{ll, {ll[\([1]\)], \ ur[\([2]\)]}, ur, {ur[\([1]\)], \ ll[\([2]\)]}}\ ]\)], "Input"], Cell[BoxData[ \(KaratsubaGasket[n_]\ := \ \(ksg[n]\)[{0, 0}, {1, 1}]\)], "Input"], Cell[BoxData[ \(Clear[ksg]\)], "Input"], Cell[BoxData[ \(\(\(ksg[0]\)[c1_, \ c2_]\ := \ Square[c1, c2];\)\)], "Input"], Cell[BoxData[ \(\(ksg[n_]\)[c1_, \ c2_]\ := \ \n With[{mid\ = \ \((c1\ + \ c2)\)/2}, \n\t\t{\(ksg[n - 1]\)[c1, \ mid], \ \(ksg[n - 1]\)[{c1[\([1]\)], \ c2[\([2]\)]}, mid], \ \(ksg[n - 1]\)[{c2[\([1]\)], \ c1[\([2]\)]}, \ mid]}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Karatsuba \"divide and conquer\" method", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(gtab\ = \ Table[KaratsubaGasket[j], {j, 0, 9}];\)\), "\[IndentingNewLine]", \(colors\ = \ Join[Table[ RGBColor[x, \ 0, 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L .05357 .7381 L F .02381 .79762 m .02381 .76786 L .05357 .76786 L .05357 .79762 L F .08333 .7381 m .08333 .76786 L .05357 .76786 L .05357 .7381 L F .02381 .85714 m .02381 .82738 L .05357 .82738 L .05357 .85714 L F .02381 .79762 m .02381 .82738 L .05357 .82738 L .05357 .79762 L F .08333 .85714 m .08333 .82738 L .05357 .82738 L .05357 .85714 L F .14286 .7381 m .14286 .76786 L .1131 .76786 L .1131 .7381 L F .14286 .79762 m .14286 .76786 L .1131 .76786 L .1131 .79762 L F .08333 .7381 m .08333 .76786 L .1131 .76786 L .1131 .7381 L F .2619 .97619 m .2619 .94643 L .23214 .94643 L .23214 .97619 L F .2619 .91667 m .2619 .94643 L .23214 .94643 L .23214 .91667 L F .20238 .97619 m .20238 .94643 L .23214 .94643 L .23214 .97619 L F .2619 .85714 m .2619 .8869 L .23214 .8869 L .23214 .85714 L F .2619 .91667 m .2619 .8869 L .23214 .8869 L .23214 .91667 L F .20238 .85714 m .20238 .8869 L .23214 .8869 L .23214 .85714 L F .14286 .97619 m .14286 .94643 L .17262 .94643 L .17262 .97619 L F .14286 .91667 m .14286 .94643 L .17262 .94643 L .17262 .91667 L F .20238 .97619 m .20238 .94643 L .17262 .94643 L .17262 .97619 L F .02381 .5 m .02381 .52976 L .05357 .52976 L .05357 .5 L F .02381 .55952 m .02381 .52976 L .05357 .52976 L .05357 .55952 L F .08333 .5 m .08333 .52976 L .05357 .52976 L .05357 .5 L F .02381 .61905 m .02381 .58929 L .05357 .58929 L .05357 .61905 L F .02381 .55952 m .02381 .58929 L .05357 .58929 L .05357 .55952 L F .08333 .61905 m .08333 .58929 L .05357 .58929 L .05357 .61905 L F .14286 .5 m .14286 .52976 L .1131 .52976 L .1131 .5 L F .14286 .55952 m .14286 .52976 L .1131 .52976 L .1131 .55952 L F .08333 .5 m .08333 .52976 L .1131 .52976 L .1131 .5 L F .02381 .7381 m .02381 .70833 L .05357 .70833 L .05357 .7381 L F .02381 .67857 m .02381 .70833 L .05357 .70833 L .05357 .67857 L F .08333 .7381 m .08333 .70833 L .05357 .70833 L .05357 .7381 L F .02381 .61905 m .02381 .64881 L .05357 .64881 L .05357 .61905 L F .02381 .67857 m .02381 .64881 L .05357 .64881 L .05357 .67857 L F .08333 .61905 m .08333 .64881 L .05357 .64881 L .05357 .61905 L F .14286 .7381 m .14286 .70833 L .1131 .70833 L .1131 .7381 L F .14286 .67857 m .14286 .70833 L .1131 .70833 L .1131 .67857 L F .08333 .7381 m .08333 .70833 L .1131 .70833 L .1131 .7381 L F .2619 .5 m .2619 .52976 L .23214 .52976 L .23214 .5 L F .2619 .55952 m .2619 .52976 L .23214 .52976 L .23214 .55952 L F .20238 .5 m .20238 .52976 L .23214 .52976 L .23214 .5 L F .2619 .61905 m .2619 .58929 L .23214 .58929 L .23214 .61905 L F .2619 .55952 m .2619 .58929 L .23214 .58929 L .23214 .55952 L F .20238 .61905 m .20238 .58929 L .23214 .58929 L .23214 .61905 L F .14286 .5 m .14286 .52976 L .17262 .52976 L .17262 .5 L F .14286 .55952 m .14286 .52976 L .17262 .52976 L .17262 .55952 L F .20238 .5 m .20238 .52976 L .17262 .52976 L .17262 .5 L F .5 .97619 m .5 .94643 L .47024 .94643 L .47024 .97619 L F .5 .91667 m .5 .94643 L .47024 .94643 L .47024 .91667 L F .44048 .97619 m .44048 .94643 L .47024 .94643 L .47024 .97619 L F .5 .85714 m .5 .8869 L .47024 .8869 L .47024 .85714 L F .5 .91667 m .5 .8869 L .47024 .8869 L .47024 .91667 L F .44048 .85714 m .44048 .8869 L .47024 .8869 L .47024 .85714 L F .38095 .97619 m .38095 .94643 L .41071 .94643 L .41071 .97619 L F .38095 .91667 m .38095 .94643 L .41071 .94643 L .41071 .91667 L F .44048 .97619 m .44048 .94643 L .41071 .94643 L .41071 .97619 L F .5 .7381 m .5 .76786 L .47024 .76786 L .47024 .7381 L F .5 .79762 m .5 .76786 L .47024 .76786 L .47024 .79762 L F .44048 .7381 m .44048 .76786 L .47024 .76786 L .47024 .7381 L F .5 .85714 m .5 .82738 L .47024 .82738 L .47024 .85714 L F .5 .79762 m .5 .82738 L .47024 .82738 L .47024 .79762 L F .44048 .85714 m .44048 .82738 L .47024 .82738 L .47024 .85714 L F .38095 .7381 m .38095 .76786 L .41071 .76786 L .41071 .7381 L F .38095 .79762 m .38095 .76786 L .41071 .76786 L .41071 .79762 L F .44048 .7381 m .44048 .76786 L .41071 .76786 L .41071 .7381 L F .2619 .97619 m .2619 .94643 L .29167 .94643 L .29167 .97619 L F .2619 .91667 m .2619 .94643 L .29167 .94643 L .29167 .91667 L F .32143 .97619 m .32143 .94643 L .29167 .94643 L .29167 .97619 L F .2619 .85714 m .2619 .8869 L .29167 .8869 L .29167 .85714 L F .2619 .91667 m .2619 .8869 L .29167 .8869 L .29167 .91667 L F .32143 .85714 m .32143 .8869 L .29167 .8869 L .29167 .85714 L F .38095 .97619 m .38095 .94643 L .35119 .94643 L .35119 .97619 L F .38095 .91667 m .38095 .94643 L .35119 .94643 L .35119 .91667 L F .32143 .97619 m .32143 .94643 L .35119 .94643 L .35119 .97619 L F .97619 .02381 m .97619 .05357 L .94643 .05357 L .94643 .02381 L F .97619 .08333 m .97619 .05357 L .94643 .05357 L .94643 .08333 L F .91667 .02381 m .91667 .05357 L .94643 .05357 L .94643 .02381 L F .97619 .14286 m .97619 .1131 L .94643 .1131 L .94643 .14286 L F .97619 .08333 m .97619 .1131 L .94643 .1131 L .94643 .08333 L F .91667 .14286 m .91667 .1131 L .94643 .1131 L .94643 .14286 L F .85714 .02381 m .85714 .05357 L .8869 .05357 L .8869 .02381 L F .85714 .08333 m .85714 .05357 L .8869 .05357 L .8869 .08333 L F .91667 .02381 m .91667 .05357 L .8869 .05357 L .8869 .02381 L F .97619 .2619 m .97619 .23214 L .94643 .23214 L .94643 .2619 L F .97619 .20238 m .97619 .23214 L .94643 .23214 L .94643 .20238 L F .91667 .2619 m .91667 .23214 L .94643 .23214 L .94643 .2619 L F .97619 .14286 m .97619 .17262 L .94643 .17262 L .94643 .14286 L F .97619 .20238 m .97619 .17262 L .94643 .17262 L .94643 .20238 L F .91667 .14286 m .91667 .17262 L .94643 .17262 L .94643 .14286 L F .85714 .2619 m .85714 .23214 L .8869 .23214 L .8869 .2619 L F .85714 .20238 m .85714 .23214 L .8869 .23214 L .8869 .20238 L F .91667 .2619 m .91667 .23214 L .8869 .23214 L .8869 .2619 L F .7381 .02381 m .7381 .05357 L .76786 .05357 L .76786 .02381 L F .7381 .08333 m .7381 .05357 L .76786 .05357 L .76786 .08333 L F .79762 .02381 m .79762 .05357 L .76786 .05357 L .76786 .02381 L F .7381 .14286 m .7381 .1131 L .76786 .1131 L .76786 .14286 L F .7381 .08333 m .7381 .1131 L .76786 .1131 L .76786 .08333 L F .79762 .14286 m .79762 .1131 L .76786 .1131 L .76786 .14286 L F .85714 .02381 m .85714 .05357 L .82738 .05357 L .82738 .02381 L F .85714 .08333 m .85714 .05357 L .82738 .05357 L .82738 .08333 L F .79762 .02381 m .79762 .05357 L .82738 .05357 L .82738 .02381 L F .97619 .5 m .97619 .47024 L .94643 .47024 L .94643 .5 L F .97619 .44048 m .97619 .47024 L .94643 .47024 L .94643 .44048 L F .91667 .5 m .91667 .47024 L .94643 .47024 L .94643 .5 L F .97619 .38095 m .97619 .41071 L .94643 .41071 L .94643 .38095 L F .97619 .44048 m .97619 .41071 L .94643 .41071 L .94643 .44048 L F .91667 .38095 m .91667 .41071 L .94643 .41071 L .94643 .38095 L F .85714 .5 m .85714 .47024 L .8869 .47024 L .8869 .5 L F .85714 .44048 m .85714 .47024 L .8869 .47024 L .8869 .44048 L F .91667 .5 m .91667 .47024 L .8869 .47024 L .8869 .5 L F .97619 .2619 m .97619 .29167 L .94643 .29167 L .94643 .2619 L F .97619 .32143 m .97619 .29167 L .94643 .29167 L .94643 .32143 L F .91667 .2619 m .91667 .29167 L .94643 .29167 L .94643 .2619 L F 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.08333 m .5 .05357 L .52976 .05357 L .52976 .08333 L F .55952 .02381 m .55952 .05357 L .52976 .05357 L .52976 .02381 L F .5 .14286 m .5 .1131 L .52976 .1131 L .52976 .14286 L F .5 .08333 m .5 .1131 L .52976 .1131 L .52976 .08333 L F .55952 .14286 m .55952 .1131 L .52976 .1131 L .52976 .14286 L F .61905 .02381 m .61905 .05357 L .58929 .05357 L .58929 .02381 L F .61905 .08333 m .61905 .05357 L .58929 .05357 L .58929 .08333 L F .55952 .02381 m .55952 .05357 L .58929 .05357 L .58929 .02381 L F .5 .2619 m .5 .23214 L .52976 .23214 L .52976 .2619 L F .5 .20238 m .5 .23214 L .52976 .23214 L .52976 .20238 L F .55952 .2619 m .55952 .23214 L .52976 .23214 L .52976 .2619 L F .5 .14286 m .5 .17262 L .52976 .17262 L .52976 .14286 L F .5 .20238 m .5 .17262 L .52976 .17262 L .52976 .20238 L F .55952 .14286 m .55952 .17262 L .52976 .17262 L .52976 .14286 L F .61905 .2619 m .61905 .23214 L .58929 .23214 L .58929 .2619 L F .61905 .20238 m .61905 .23214 L .58929 .23214 L .58929 .20238 L F .55952 .2619 m .55952 .23214 L .58929 .23214 L .58929 .2619 L F .7381 .02381 m .7381 .05357 L .70833 .05357 L .70833 .02381 L F .7381 .08333 m .7381 .05357 L .70833 .05357 L .70833 .08333 L F .67857 .02381 m .67857 .05357 L .70833 .05357 L .70833 .02381 L F .7381 .14286 m .7381 .1131 L .70833 .1131 L .70833 .14286 L F .7381 .08333 m .7381 .1131 L .70833 .1131 L .70833 .08333 L F .67857 .14286 m .67857 .1131 L .70833 .1131 L .70833 .14286 L F .61905 .02381 m .61905 .05357 L .64881 .05357 L .64881 .02381 L F .61905 .08333 m .61905 .05357 L .64881 .05357 L .64881 .08333 L F .67857 .02381 m .67857 .05357 L .64881 .05357 L .64881 .02381 L F .75 0 .25 r .02381 .02381 m .02381 .03869 L .03869 .03869 L .03869 .02381 L F .02381 .05357 m .02381 .03869 L .03869 .03869 L .03869 .05357 L F .05357 .02381 m .05357 .03869 L .03869 .03869 L .03869 .02381 L F .02381 .08333 m .02381 .06845 L .03869 .06845 L .03869 .08333 L F .02381 .05357 m .02381 .06845 L .03869 .06845 L .03869 .05357 L F .05357 .08333 m .05357 .06845 L .03869 .06845 L .03869 .08333 L F .08333 .02381 m .08333 .03869 L .06845 .03869 L .06845 .02381 L F .08333 .05357 m .08333 .03869 L .06845 .03869 L .06845 .05357 L F .05357 .02381 m .05357 .03869 L .06845 .03869 L .06845 .02381 L F .02381 .14286 m .02381 .12798 L .03869 .12798 L .03869 .14286 L F .02381 .1131 m .02381 .12798 L .03869 .12798 L .03869 .1131 L F .05357 .14286 m .05357 .12798 L .03869 .12798 L .03869 .14286 L F .02381 .08333 m .02381 .09821 L .03869 .09821 L .03869 .08333 L F .02381 .1131 m .02381 .09821 L .03869 .09821 L .03869 .1131 L F .05357 .08333 m .05357 .09821 L .03869 .09821 L .03869 .08333 L F .08333 .14286 m .08333 .12798 L .06845 .12798 L .06845 .14286 L F .08333 .1131 m .08333 .12798 L .06845 .12798 L .06845 .1131 L F .05357 .14286 m .05357 .12798 L .06845 .12798 L .06845 .14286 L F .14286 .02381 m .14286 .03869 L .12798 .03869 L .12798 .02381 L F .14286 .05357 m .14286 .03869 L .12798 .03869 L .12798 .05357 L F .1131 .02381 m .1131 .03869 L .12798 .03869 L .12798 .02381 L F .14286 .08333 m .14286 .06845 L .12798 .06845 L .12798 .08333 L F .14286 .05357 m .14286 .06845 L .12798 .06845 L .12798 .05357 L F .1131 .08333 m .1131 .06845 L .12798 .06845 L .12798 .08333 L F .08333 .02381 m .08333 .03869 L .09821 .03869 L .09821 .02381 L F .08333 .05357 m .08333 .03869 L .09821 .03869 L .09821 .05357 L F .1131 .02381 m .1131 .03869 L .09821 .03869 L .09821 .02381 L F .02381 .2619 m .02381 .24702 L .03869 .24702 L .03869 .2619 L F .02381 .23214 m .02381 .24702 L .03869 .24702 L .03869 .23214 L F .05357 .2619 m .05357 .24702 L .03869 .24702 L .03869 .2619 L F .02381 .20238 m .02381 .21726 L .03869 .21726 L .03869 .20238 L F .02381 .23214 m .02381 .21726 L .03869 .21726 L .03869 .23214 L F .05357 .20238 m .05357 .21726 L .03869 .21726 L .03869 .20238 L F .08333 .2619 m .08333 .24702 L .06845 .24702 L .06845 .2619 L F .08333 .23214 m .08333 .24702 L .06845 .24702 L .06845 .23214 L F .05357 .2619 m .05357 .24702 L .06845 .24702 L .06845 .2619 L F .02381 .14286 m .02381 .15774 L .03869 .15774 L .03869 .14286 L F .02381 .17262 m .02381 .15774 L .03869 .15774 L .03869 .17262 L F .05357 .14286 m .05357 .15774 L .03869 .15774 L .03869 .14286 L F .02381 .20238 m .02381 .1875 L .03869 .1875 L .03869 .20238 L F .02381 .17262 m .02381 .1875 L .03869 .1875 L .03869 .17262 L F .05357 .20238 m .05357 .1875 L .03869 .1875 L .03869 .20238 L F .08333 .14286 m .08333 .15774 L .06845 .15774 L .06845 .14286 L F .08333 .17262 m .08333 .15774 L .06845 .15774 L .06845 .17262 L F .05357 .14286 m .05357 .15774 L .06845 .15774 L .06845 .14286 L F .14286 .2619 m .14286 .24702 L .12798 .24702 L .12798 .2619 L F .14286 .23214 m .14286 .24702 L .12798 .24702 L .12798 .23214 L F .1131 .2619 m .1131 .24702 L .12798 .24702 L .12798 .2619 L F .14286 .20238 m .14286 .21726 L .12798 .21726 L .12798 .20238 L F .14286 .23214 m .14286 .21726 L .12798 .21726 L .12798 .23214 L F .1131 .20238 m .1131 .21726 L .12798 .21726 L .12798 .20238 L F .08333 .2619 m .08333 .24702 L .09821 .24702 L .09821 .2619 L F .08333 .23214 m .08333 .24702 L .09821 .24702 L .09821 .23214 L F .1131 .2619 m .1131 .24702 L .09821 .24702 L .09821 .2619 L F .2619 .02381 m .2619 .03869 L .24702 .03869 L .24702 .02381 L F .2619 .05357 m .2619 .03869 L .24702 .03869 L .24702 .05357 L F .23214 .02381 m .23214 .03869 L .24702 .03869 L .24702 .02381 L F .2619 .08333 m .2619 .06845 L .24702 .06845 L .24702 .08333 L F .2619 .05357 m .2619 .06845 L .24702 .06845 L .24702 .05357 L F .23214 .08333 m .23214 .06845 L .24702 .06845 L .24702 .08333 L F .20238 .02381 m .20238 .03869 L .21726 .03869 L .21726 .02381 L F .20238 .05357 m .20238 .03869 L .21726 .03869 L .21726 .05357 L F .23214 .02381 m .23214 .03869 L .21726 .03869 L .21726 .02381 L F .2619 .14286 m .2619 .12798 L .24702 .12798 L .24702 .14286 L F .2619 .1131 m .2619 .12798 L .24702 .12798 L .24702 .1131 L F .23214 .14286 m .23214 .12798 L .24702 .12798 L .24702 .14286 L F .2619 .08333 m .2619 .09821 L .24702 .09821 L .24702 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m .08333 .51488 L .06845 .51488 L .06845 .52976 L F .05357 .5 m .05357 .51488 L .06845 .51488 L .06845 .5 L F .02381 .61905 m .02381 .60417 L .03869 .60417 L .03869 .61905 L F .02381 .58929 m .02381 .60417 L .03869 .60417 L .03869 .58929 L F .05357 .61905 m .05357 .60417 L .03869 .60417 L .03869 .61905 L F .02381 .55952 m .02381 .5744 L .03869 .5744 L .03869 .55952 L F .02381 .58929 m .02381 .5744 L .03869 .5744 L .03869 .58929 L F .05357 .55952 m .05357 .5744 L .03869 .5744 L .03869 .55952 L F .08333 .61905 m .08333 .60417 L .06845 .60417 L .06845 .61905 L F .08333 .58929 m .08333 .60417 L .06845 .60417 L .06845 .58929 L F .05357 .61905 m .05357 .60417 L .06845 .60417 L .06845 .61905 L F .14286 .5 m .14286 .51488 L .12798 .51488 L .12798 .5 L F .14286 .52976 m .14286 .51488 L .12798 .51488 L .12798 .52976 L F .1131 .5 m .1131 .51488 L .12798 .51488 L .12798 .5 L F .14286 .55952 m .14286 .54464 L .12798 .54464 L .12798 .55952 L F .14286 .52976 m .14286 .54464 L .12798 .54464 L .12798 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L .21726 .61905 L F .20238 .58929 m .20238 .60417 L .21726 .60417 L .21726 .58929 L F .23214 .61905 m .23214 .60417 L .21726 .60417 L .21726 .61905 L F .14286 .5 m .14286 .51488 L .15774 .51488 L .15774 .5 L F .14286 .52976 m .14286 .51488 L .15774 .51488 L .15774 .52976 L F .17262 .5 m .17262 .51488 L .15774 .51488 L .15774 .5 L F .14286 .55952 m .14286 .54464 L .15774 .54464 L .15774 .55952 L F .14286 .52976 m .14286 .54464 L .15774 .54464 L .15774 .52976 L F .17262 .55952 m .17262 .54464 L .15774 .54464 L .15774 .55952 L F .20238 .5 m .20238 .51488 L .1875 .51488 L .1875 .5 L F .20238 .52976 m .20238 .51488 L .1875 .51488 L .1875 .52976 L F .17262 .5 m .17262 .51488 L .1875 .51488 L .1875 .5 L F .5 .97619 m .5 .96131 L .48512 .96131 L .48512 .97619 L F .5 .94643 m .5 .96131 L .48512 .96131 L .48512 .94643 L F .47024 .97619 m .47024 .96131 L .48512 .96131 L .48512 .97619 L F .5 .91667 m .5 .93155 L .48512 .93155 L .48512 .91667 L F .5 .94643 m .5 .93155 L .48512 .93155 L .48512 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.96131 .05357 L F .94643 .02381 m .94643 .03869 L .96131 .03869 L .96131 .02381 L F .97619 .08333 m .97619 .06845 L .96131 .06845 L .96131 .08333 L F .97619 .05357 m .97619 .06845 L .96131 .06845 L .96131 .05357 L F .94643 .08333 m .94643 .06845 L .96131 .06845 L .96131 .08333 L F .91667 .02381 m .91667 .03869 L .93155 .03869 L .93155 .02381 L F .91667 .05357 m .91667 .03869 L .93155 .03869 L .93155 .05357 L F .94643 .02381 m .94643 .03869 L .93155 .03869 L .93155 .02381 L F .97619 .14286 m .97619 .12798 L .96131 .12798 L .96131 .14286 L F .97619 .1131 m .97619 .12798 L .96131 .12798 L .96131 .1131 L F .94643 .14286 m .94643 .12798 L .96131 .12798 L .96131 .14286 L F .97619 .08333 m .97619 .09821 L .96131 .09821 L .96131 .08333 L F .97619 .1131 m .97619 .09821 L .96131 .09821 L .96131 .1131 L F .94643 .08333 m .94643 .09821 L .96131 .09821 L .96131 .08333 L F .91667 .14286 m .91667 .12798 L .93155 .12798 L .93155 .14286 L F .91667 .1131 m .91667 .12798 L .93155 .12798 L .93155 .1131 L F .94643 .14286 m .94643 .12798 L .93155 .12798 L .93155 .14286 L F .85714 .02381 m .85714 .03869 L .87202 .03869 L .87202 .02381 L F .85714 .05357 m .85714 .03869 L .87202 .03869 L .87202 .05357 L F .8869 .02381 m .8869 .03869 L .87202 .03869 L .87202 .02381 L F .85714 .08333 m .85714 .06845 L .87202 .06845 L .87202 .08333 L F .85714 .05357 m .85714 .06845 L .87202 .06845 L .87202 .05357 L F .8869 .08333 m .8869 .06845 L .87202 .06845 L .87202 .08333 L F .91667 .02381 m .91667 .03869 L .90179 .03869 L .90179 .02381 L F .91667 .05357 m .91667 .03869 L .90179 .03869 L .90179 .05357 L F .8869 .02381 m .8869 .03869 L .90179 .03869 L .90179 .02381 L F .97619 .2619 m .97619 .24702 L .96131 .24702 L .96131 .2619 L F .97619 .23214 m .97619 .24702 L .96131 .24702 L .96131 .23214 L F .94643 .2619 m .94643 .24702 L .96131 .24702 L .96131 .2619 L F .97619 .20238 m .97619 .21726 L .96131 .21726 L .96131 .20238 L F .97619 .23214 m .97619 .21726 L .96131 .21726 L .96131 .23214 L F .94643 .20238 m .94643 .21726 L .96131 .21726 L .96131 .20238 L F .91667 .2619 m .91667 .24702 L .93155 .24702 L .93155 .2619 L F .91667 .23214 m .91667 .24702 L .93155 .24702 L .93155 .23214 L F .94643 .2619 m .94643 .24702 L .93155 .24702 L .93155 .2619 L F .97619 .14286 m .97619 .15774 L .96131 .15774 L .96131 .14286 L F .97619 .17262 m .97619 .15774 L .96131 .15774 L .96131 .17262 L F .94643 .14286 m .94643 .15774 L .96131 .15774 L .96131 .14286 L F .97619 .20238 m .97619 .1875 L .96131 .1875 L .96131 .20238 L F .97619 .17262 m .97619 .1875 L .96131 .1875 L .96131 .17262 L F .94643 .20238 m .94643 .1875 L .96131 .1875 L .96131 .20238 L F .91667 .14286 m .91667 .15774 L .93155 .15774 L .93155 .14286 L F .91667 .17262 m .91667 .15774 L .93155 .15774 L .93155 .17262 L F .94643 .14286 m .94643 .15774 L .93155 .15774 L .93155 .14286 L F .85714 .2619 m .85714 .24702 L .87202 .24702 L .87202 .2619 L F .85714 .23214 m .85714 .24702 L .87202 .24702 L .87202 .23214 L F .8869 .2619 m .8869 .24702 L .87202 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.1131 .25446 L .10565 .25446 L .10565 .24702 L F .09821 .2619 m .09821 .25446 L .10565 .25446 L .10565 .2619 L F .2619 .02381 m .2619 .03125 L .25446 .03125 L .25446 .02381 L F .2619 .03869 m .2619 .03125 L .25446 .03125 L .25446 .03869 L F .24702 .02381 m .24702 .03125 L .25446 .03125 L .25446 .02381 L F .2619 .05357 m .2619 .04613 L .25446 .04613 L .25446 .05357 L F .2619 .03869 m .2619 .04613 L .25446 .04613 L .25446 .03869 L F .24702 .05357 m .24702 .04613 L .25446 .04613 L .25446 .05357 L F .23214 .02381 m .23214 .03125 L .23958 .03125 L .23958 .02381 L F .23214 .03869 m .23214 .03125 L .23958 .03125 L .23958 .03869 L F .24702 .02381 m .24702 .03125 L .23958 .03125 L .23958 .02381 L F .2619 .08333 m .2619 .07589 L .25446 .07589 L .25446 .08333 L F .2619 .06845 m .2619 .07589 L .25446 .07589 L .25446 .06845 L F .24702 .08333 m .24702 .07589 L .25446 .07589 L .25446 .08333 L F .2619 .05357 m .2619 .06101 L .25446 .06101 L .25446 .05357 L F .2619 .06845 m .2619 .06101 L .25446 .06101 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.24702 .14286 m .24702 .13542 L .25446 .13542 L .25446 .14286 L F .2619 .1131 m .2619 .12054 L .25446 .12054 L .25446 .1131 L F .2619 .12798 m .2619 .12054 L .25446 .12054 L .25446 .12798 L F .24702 .1131 m .24702 .12054 L .25446 .12054 L .25446 .1131 L F .23214 .14286 m .23214 .13542 L .23958 .13542 L .23958 .14286 L F .23214 .12798 m .23214 .13542 L .23958 .13542 L .23958 .12798 L F .24702 .14286 m .24702 .13542 L .23958 .13542 L .23958 .14286 L F .2619 .08333 m .2619 .09077 L .25446 .09077 L .25446 .08333 L F .2619 .09821 m .2619 .09077 L .25446 .09077 L .25446 .09821 L F .24702 .08333 m .24702 .09077 L .25446 .09077 L .25446 .08333 L F .2619 .1131 m .2619 .10565 L .25446 .10565 L .25446 .1131 L F .2619 .09821 m .2619 .10565 L .25446 .10565 L .25446 .09821 L F .24702 .1131 m .24702 .10565 L .25446 .10565 L .25446 .1131 L F .23214 .08333 m .23214 .09077 L .23958 .09077 L .23958 .08333 L F .23214 .09821 m .23214 .09077 L .23958 .09077 L .23958 .09821 L F .24702 .08333 m .24702 .09077 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L F .17262 .02381 m .17262 .03125 L .16518 .03125 L .16518 .02381 L F .17262 .03869 m .17262 .03125 L .16518 .03125 L .16518 .03869 L F .15774 .02381 m .15774 .03125 L .16518 .03125 L .16518 .02381 L F .14286 .08333 m .14286 .07589 L .1503 .07589 L .1503 .08333 L F .14286 .06845 m .14286 .07589 L .1503 .07589 L .1503 .06845 L F .15774 .08333 m .15774 .07589 L .1503 .07589 L .1503 .08333 L F .14286 .05357 m .14286 .06101 L .1503 .06101 L .1503 .05357 L F .14286 .06845 m .14286 .06101 L .1503 .06101 L .1503 .06845 L F .15774 .05357 m .15774 .06101 L .1503 .06101 L .1503 .05357 L F .17262 .08333 m .17262 .07589 L .16518 .07589 L .16518 .08333 L F .17262 .06845 m .17262 .07589 L .16518 .07589 L .16518 .06845 L F .15774 .08333 m .15774 .07589 L .16518 .07589 L .16518 .08333 L F .20238 .02381 m .20238 .03125 L .19494 .03125 L .19494 .02381 L F .20238 .03869 m .20238 .03125 L .19494 .03125 L .19494 .03869 L F .1875 .02381 m .1875 .03125 L .19494 .03125 L .19494 .02381 L F .20238 .05357 m .20238 .04613 L .19494 .04613 L .19494 .05357 L F .20238 .03869 m .20238 .04613 L .19494 .04613 L .19494 .03869 L F .1875 .05357 m .1875 .04613 L .19494 .04613 L .19494 .05357 L F .17262 .02381 m .17262 .03125 L .18006 .03125 L .18006 .02381 L F .17262 .03869 m .17262 .03125 L .18006 .03125 L .18006 .03869 L F .1875 .02381 m .1875 .03125 L .18006 .03125 L .18006 .02381 L F .02381 .5 m .02381 .49256 L .03125 .49256 L .03125 .5 L F .02381 .48512 m .02381 .49256 L .03125 .49256 L .03125 .48512 L F .03869 .5 m .03869 .49256 L .03125 .49256 L .03125 .5 L F .02381 .47024 m .02381 .47768 L .03125 .47768 L .03125 .47024 L F .02381 .48512 m .02381 .47768 L .03125 .47768 L .03125 .48512 L F .03869 .47024 m .03869 .47768 L .03125 .47768 L .03125 .47024 L F .05357 .5 m .05357 .49256 L .04613 .49256 L .04613 .5 L F .05357 .48512 m .05357 .49256 L .04613 .49256 L .04613 .48512 L F .03869 .5 m .03869 .49256 L .04613 .49256 L .04613 .5 L F .02381 .44048 m .02381 .44792 L .03125 .44792 L .03125 .44048 L F .02381 .45536 m .02381 .44792 L .03125 .44792 L .03125 .45536 L F .03869 .44048 m .03869 .44792 L .03125 .44792 L .03125 .44048 L F .02381 .47024 m .02381 .4628 L .03125 .4628 L .03125 .47024 L F .02381 .45536 m .02381 .4628 L .03125 .4628 L .03125 .45536 L F .03869 .47024 m .03869 .4628 L .03125 .4628 L .03125 .47024 L F .05357 .44048 m .05357 .44792 L .04613 .44792 L .04613 .44048 L F .05357 .45536 m .05357 .44792 L .04613 .44792 L .04613 .45536 L F .03869 .44048 m .03869 .44792 L .04613 .44792 L .04613 .44048 L F .08333 .5 m .08333 .49256 L .07589 .49256 L .07589 .5 L F .08333 .48512 m .08333 .49256 L .07589 .49256 L .07589 .48512 L F .06845 .5 m .06845 .49256 L .07589 .49256 L .07589 .5 L F .08333 .47024 m .08333 .47768 L .07589 .47768 L .07589 .47024 L F .08333 .48512 m .08333 .47768 L .07589 .47768 L .07589 .48512 L F .06845 .47024 m .06845 .47768 L .07589 .47768 L .07589 .47024 L F .05357 .5 m .05357 .49256 L .06101 .49256 L .06101 .5 L F .05357 .48512 m .05357 .49256 L .06101 .49256 L .06101 .48512 L F .06845 .5 m .06845 .49256 L .06101 .49256 L .06101 .5 L F .02381 .38095 m .02381 .38839 L .03125 .38839 L .03125 .38095 L F .02381 .39583 m .02381 .38839 L .03125 .38839 L .03125 .39583 L F .03869 .38095 m .03869 .38839 L .03125 .38839 L .03125 .38095 L F .02381 .41071 m .02381 .40327 L .03125 .40327 L .03125 .41071 L F .02381 .39583 m .02381 .40327 L .03125 .40327 L .03125 .39583 L F .03869 .41071 m .03869 .40327 L .03125 .40327 L .03125 .41071 L F .05357 .38095 m .05357 .38839 L .04613 .38839 L .04613 .38095 L F .05357 .39583 m .05357 .38839 L .04613 .38839 L .04613 .39583 L F .03869 .38095 m .03869 .38839 L .04613 .38839 L .04613 .38095 L F .02381 .44048 m .02381 .43304 L .03125 .43304 L .03125 .44048 L F .02381 .4256 m .02381 .43304 L .03125 .43304 L .03125 .4256 L F .03869 .44048 m .03869 .43304 L .03125 .43304 L .03125 .44048 L F .02381 .41071 m .02381 .41815 L .03125 .41815 L .03125 .41071 L F .02381 .4256 m .02381 .41815 L .03125 .41815 L .03125 .4256 L F .03869 .41071 m .03869 .41815 L .03125 .41815 L .03125 .41071 L F .05357 .44048 m .05357 .43304 L .04613 .43304 L .04613 .44048 L F .05357 .4256 m .05357 .43304 L .04613 .43304 L .04613 .4256 L F .03869 .44048 m .03869 .43304 L .04613 .43304 L .04613 .44048 L F .08333 .38095 m .08333 .38839 L .07589 .38839 L .07589 .38095 L F .08333 .39583 m .08333 .38839 L .07589 .38839 L .07589 .39583 L F .06845 .38095 m .06845 .38839 L .07589 .38839 L .07589 .38095 L F .08333 .41071 m .08333 .40327 L .07589 .40327 L .07589 .41071 L F .08333 .39583 m .08333 .40327 L .07589 .40327 L .07589 .39583 L F .06845 .41071 m .06845 .40327 L .07589 .40327 L .07589 .41071 L F .05357 .38095 m .05357 .38839 L .06101 .38839 L .06101 .38095 L F .05357 .39583 m .05357 .38839 L .06101 .38839 L .06101 .39583 L F .06845 .38095 m .06845 .38839 L .06101 .38839 L .06101 .38095 L F .14286 .5 m .14286 .49256 L .13542 .49256 L .13542 .5 L F .14286 .48512 m .14286 .49256 L .13542 .49256 L .13542 .48512 L F 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.12054 .44048 L F .08333 .5 m .08333 .49256 L .09077 .49256 L .09077 .5 L F .08333 .48512 m .08333 .49256 L .09077 .49256 L .09077 .48512 L F .09821 .5 m .09821 .49256 L .09077 .49256 L .09077 .5 L F .08333 .47024 m .08333 .47768 L .09077 .47768 L .09077 .47024 L F .08333 .48512 m .08333 .47768 L .09077 .47768 L .09077 .48512 L F .09821 .47024 m .09821 .47768 L .09077 .47768 L .09077 .47024 L F .1131 .5 m .1131 .49256 L .10565 .49256 L .10565 .5 L F .1131 .48512 m .1131 .49256 L .10565 .49256 L .10565 .48512 L F .09821 .5 m .09821 .49256 L .10565 .49256 L .10565 .5 L F .02381 .2619 m .02381 .26935 L .03125 .26935 L .03125 .2619 L F .02381 .27679 m .02381 .26935 L .03125 .26935 L .03125 .27679 L F .03869 .2619 m .03869 .26935 L .03125 .26935 L .03125 .2619 L F .02381 .29167 m .02381 .28423 L .03125 .28423 L .03125 .29167 L F .02381 .27679 m .02381 .28423 L .03125 .28423 L .03125 .27679 L F .03869 .29167 m .03869 .28423 L .03125 .28423 L .03125 .29167 L F .05357 .2619 m .05357 .26935 L .04613 .26935 L .04613 .2619 L F .05357 .27679 m .05357 .26935 L .04613 .26935 L .04613 .27679 L F .03869 .2619 m .03869 .26935 L .04613 .26935 L .04613 .2619 L F .02381 .32143 m .02381 .31399 L .03125 .31399 L .03125 .32143 L F .02381 .30655 m .02381 .31399 L .03125 .31399 L .03125 .30655 L F .03869 .32143 m .03869 .31399 L .03125 .31399 L .03125 .32143 L F .02381 .29167 m .02381 .29911 L .03125 .29911 L .03125 .29167 L F .02381 .30655 m .02381 .29911 L .03125 .29911 L .03125 .30655 L F .03869 .29167 m .03869 .29911 L .03125 .29911 L .03125 .29167 L F .05357 .32143 m .05357 .31399 L .04613 .31399 L .04613 .32143 L F .05357 .30655 m .05357 .31399 L .04613 .31399 L .04613 .30655 L F .03869 .32143 m .03869 .31399 L .04613 .31399 L .04613 .32143 L F .08333 .2619 m .08333 .26935 L .07589 .26935 L .07589 .2619 L F .08333 .27679 m .08333 .26935 L .07589 .26935 L .07589 .27679 L F .06845 .2619 m .06845 .26935 L .07589 .26935 L .07589 .2619 L F .08333 .29167 m .08333 .28423 L .07589 .28423 L .07589 .29167 L F .08333 .27679 m .08333 .28423 L .07589 .28423 L .07589 .27679 L F .06845 .29167 m .06845 .28423 L .07589 .28423 L .07589 .29167 L F .05357 .2619 m .05357 .26935 L .06101 .26935 L .06101 .2619 L F .05357 .27679 m .05357 .26935 L .06101 .26935 L .06101 .27679 L F .06845 .2619 m .06845 .26935 L .06101 .26935 L .06101 .2619 L F .02381 .38095 m .02381 .37351 L .03125 .37351 L .03125 .38095 L F .02381 .36607 m .02381 .37351 L .03125 .37351 L .03125 .36607 L F .03869 .38095 m .03869 .37351 L .03125 .37351 L .03125 .38095 L F .02381 .35119 m .02381 .35863 L .03125 .35863 L .03125 .35119 L F .02381 .36607 m .02381 .35863 L .03125 .35863 L .03125 .36607 L F .03869 .35119 m .03869 .35863 L .03125 .35863 L .03125 .35119 L F .05357 .38095 m .05357 .37351 L .04613 .37351 L .04613 .38095 L F .05357 .36607 m .05357 .37351 L .04613 .37351 L .04613 .36607 L F .03869 .38095 m .03869 .37351 L .04613 .37351 L .04613 .38095 L F .02381 .32143 m .02381 .32887 L .03125 .32887 L .03125 .32143 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