File:Bird horner rule 6789.pdf

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English: Computation example for Horner's rule from Bird.1989, Lemma 1, p.125 lf. Our example uses integer numbers with binary addition ("") and multiplication (""), of which only their distributivity, and the left-neutrality of wrt. is used. In particular, none of the operations need to be associative. The Horner operation is defined from and .
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Author

Jochen Burghardt

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File:Bird horner rule 456789.pdfFile:Bird horner rule 456789 svg.svgFile:Bird horner rule 56789.pdfFile:Bird horner rule 56789 svg.svgFile:Bird horner rule 6789.pdfFile:Bird horner rule 6789 svg.svg

(General explanations applying to all similar images, including the LaTeX source code of the use file "bird_Preamble.tex" can be found at the description page of File:Bird foldl map fusion.pdf.)

LaTeX source code
\documentclass[12pt]{article}

\setlength{\unitlength}{1mm}

\usepackage[pdftex]{color}
\usepackage{amssymb}
\usepackage[paperwidth=180mm,paperheight=105mm]{geometry}
        
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{180mm}
\setlength{\textheight}{105mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
        
\pagestyle{empty}

\include{bird_Preamble}

\begin{document}
\begin{picture}(175,100)
        %\put(0,0){\makebox(0,0){$+$}}
        %\put(175,100){\makebox(0,0){$+$}}
%
\put(120.000,95.000){\makebox(0.000,0.000)[bl]{{$\List{6\,7\,8\,9}$}}}%
\put(120.000,65.000){\makebox(0.000,0.000)[br]{{$\List{\List{6\,7\,8\,9}\,\List{7\,8\,9}\,\List{8\,9}\,\List{9}\,\Nil}$}}}%
\put(120.000,35.000){\makebox(0.000,0.000)[br]{{$\List{(((1\*6)\*7)\*8)\*9\,((1\*7)\*8)\*9\,(1\*8)\*9\,1\*9\,1}$}}}% 
\put(120.000,5.000){\makebox(0.000,0.000)[br]{{$\A{((((0+\B{(((1\*6)\*7)\*8)\*9})+\B{((1\*7)\*8)\*9})+\B{(1\*8)\*9})+\B{1\*9})+\B{1}}$}}}% 
\put(130.000,5.000){\makebox(0.000,0.000)[bl]{{$\A{(((1\otimes \B{6})\otimes \B{7})\otimes \B{8})\otimes \B{9}}$}}}% 
\put(125.000,6.000){\makebox(0.000,0.000)[b]{{$\mathbf{=}$}}}%
%
\thicklines
%
\textcolor{cFct}{\put(120.000,92.000){\vector(-2,-1){40.000}}}%
\textcolor{cFct}{\put(99.000,82.000){\makebox(0.000,0.000)[br]{$\tails$}}}%
%
\textcolor{cFct}{\put(80.000,62.000){\vector(0,-1){20.000}}}%
\textcolor{cFct}{\put(78.000,52.000){\makebox(0.000,0.000)[br]{$\map \; (\foldl \; (*) \; 1)$}}}%
%
\textcolor{cFct}{\put(80.000,32.000){\vector(0,-1){20.000}}}%
\textcolor{cFct}{\put(78.000,22.000){\makebox(0.000,0.000)[br]{$\foldl \; (+) \; 0$}}}%
%
\textcolor{cFct}{\put(135.000,92.000){\vector(0,-1){80.000}}}%
\textcolor{cFct}{\put(137.000,52.000){\makebox(0.000,0.000)[bl]{$\foldl \; (\otimes) \; 1$}}}%
%
\textcolor{bCmt}{\put(147.000,12.000){\makebox(0.000,0.000)[bl]{\rule{24mm}{16mm}}}}%
\textcolor{cCmt}{\put(148.000,13.000){\makebox(0.000,0.000)[bl]{%
        \shortstack[r]{where $x \otimes y$\\is defined as\\$x \* y + 1$}}}}%
% validate:
% 6*7*8*9 + 7*8*9 + 8*9 + 9 + 1 = 3610
% (((1*6+1)*7+1)*8+1)*9+1 = 3610
\end{picture}
\end{document}

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons

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Date/TimeThumbnailDimensionsUserComment
current13:54, 25 September 2019Thumbnail for version as of 13:54, 25 September 20191,062 × 620 (39 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard

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