<?xml version="1.0" encoding="UTF-8"?>
<Worksheet>
<Version major="18" minor="1"/>
<Label-Scheme value="2" prefix=""/>
<View-Properties presentation="false" autoexpanding_sections="true">
<Hide name="Group Range"/>
</View-Properties>
<MapleNet-Properties elisiondigitsbefore="100" labelling="true" indentamount="4" elisiontermsthreshold="10000" ansi="false" errorbreak="1" useclientjvm="true" echo="1" imaginaryunit="I" labelwidth="20" unitattributes="&quot;fontweight&quot; = &quot;bold&quot;" contextmenusize="automatic" plotdriver="opengl" elisiondigitsafter="100" plotoutput="terminal" helpbrowser="standard" rtablesize="10" elisiontermsbefore="100" elisiondigitsthreshold="10000" typesetting="standard" plotdevice="inline" verboseproc="1" showassumed="1" quiet="false" errorcursor="false" longdelim="true" plotoptions="" elisiontermsafter="100" screenwidth="79" preplot="" prettyprint="3" displayprecision="-1" screenpixelheight="1050" warnlevel="3" screenheight="25" latexwidth="8.0" postplot="" prompt="&gt; " ShowLabels="true"/>
<Styles>
<Font name="Ordered List 1" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Annotation Text" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 2" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 3" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 4" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 5" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Author" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Annotation Title" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Warning" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Caption Reference" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Input Placeholder" background="[255,255,255]" bold="true" executable="true" family="Courier New" foreground="[200,0,200]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="true"/>
<Font name="Maple Plot" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Code" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Line Printed Output" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Text Output" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Diagnostic" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[40,120,40]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Inert Output" background="[255,255,255]" bold="false" executable="true" family="Times New Roman" foreground="[144,144,144]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Normal" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Hyperlink" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,128,128]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Maple Output" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Dash Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Math" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Input" background="[255,255,255]" bold="true" executable="true" family="Courier New" foreground="[120,0,14]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Output" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Input" background="[255,255,255]" bold="false" executable="true" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="HyperlinkError" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Header and Footer" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="10" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Error" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Title" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 1" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Text" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Bullet Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 4" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Equation Label" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 3" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" opaque="false" readonly="false" size="14" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 2" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="16" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="HyperlinkWarning" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Dictionary Hyperlink" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[147,0,15]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Atomic Variable" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[175,0,175]" italic="true" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Caption Text" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="List Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Layout name="Ordered List 1" alignment="left" bullet="numeric" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Ordered List 2" alignment="left" bullet="alphabetic" firstindent="0" leftmargin="36" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Ordered List 3" alignment="left" bullet="roman" firstindent="0" leftmargin="72" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Ordered List 4" alignment="left" bullet="ALPHABETIC" firstindent="0" leftmargin="108" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Ordered List 5" alignment="left" bullet="ROMAN" firstindent="0" leftmargin="144" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix=""/>
<Layout name="Author" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="8" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Warning" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Annotation Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Maple Plot" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Line Printed Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="any" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Text Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="newline" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Diagnostic" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="any" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Normal" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Maple Output" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.3" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Dash Item" alignment="left" bullet="dash" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="HyperlinkError" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Error" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 1" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="4" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Bullet Item" alignment="left" bullet="dot" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 4" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 3" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 2" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="2" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="HyperlinkWarning" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="List Item" alignment="left" bullet="indent" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Pencil-style name="Pencil 5" pen-color="[255,0,0]" pen-height="5.0" pen-width="5.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 4" pen-color="[0,0,255]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 3" pen-color="[0,0,0]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 2" pen-color="[0,0,255]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 1" pen-color="[0,0,0]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Highlighter-style name="Highlighter 2" pen-color="[255,204,0]" pen-height="14.0" pen-width="14.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 1" pen-color="[255,153,255]" pen-height="12.0" pen-width="8.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 4" pen-color="[0,255,255]" pen-height="32.0" pen-width="32.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 3" pen-color="[51,255,0]" pen-height="24.0" pen-width="24.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 5" pen-color="[255,255,0]" pen-height="48.0" pen-width="48.0" pen-opacity="0.8"/>
</Styles>
<Task-table>
    <Task-category name="&lt;default&gt;"/>
</Task-table>
<Task/><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L5355" drawlabel="true">
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<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" size="16">evalhf, Compile, hfloat and all that</Font><Font size="16">

</Font><Font style="Normal">Dave Linder</Font></Text-field>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L5351" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal" alignment="centred">Software Architect, Mathematical Software</Text-field>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L5352" drawlabel="true">
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<Text-field style="Normal" layout="Normal" alignment="centred">Maplesoft</Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Questions that I see asked quite often by users involve how to make floating-point (numeric) computations perform faster in Maple. Answers often include references to special terms such as <Hyperlink linktarget="Help:evalhf" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">evalhf</Font></Hyperlink>, the <Hyperlink linktarget="Help:Compiler" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Compiler</Font></Hyperlink>, and <Hyperlink linktarget="Help:option_hfloat" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">option hfloat</Font></Hyperlink>. A difficulty for the non-expert lies in knowing which of these can be used and when. So let's try and clear up some of the mystery of these terms, by discussion and functionality comparison.  This document is aimed at the user with an intermediate level of Maple experience and knowledge.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">These special techniques can often improve performance dramatically, sometimes improving computation time by a factor of 20 or more. A follow-up article will cover relative performance in detail, as that is example dependent.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Let's review a few basics and technical terms.Maple is an interpreted language. The Maple kernel interprets and executes code. Normally all floating-point numbers (<Font italic="true">floats</Font>) are represented and stored by the kernel as <Font italic="true">software floats</Font>, which means that internally they are stored in memory in the general data structure also used for symbolic objects, known as a <Font italic="true">DAG</Font> (Directed Acyclic Graph). We don't need to understand the details of DAGs here, but Maple's basic structures are discussed in the <Hyperlink linktarget="http://www.maplesoft.com/support/help/Maple/view.aspx?path=ProgrammingGuide/Appendix1" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Programming Manual</Font></Hyperlink>.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">The acceleration techniques under discussion all operate with some representation of <Font italic="true">hardware double precision</Font> floating-point numbers. The key to the speed of these mechanisms is in the internal use of hardware floats rather than software floats. Software floats can have arbitrary precision, but hardware floats have a <Hyperlink linktarget="http://en.wikipedia.org/wiki/Double_precision" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">double precision</Font></Hyperlink> internal representation.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">A good place to start is by using <Hyperlink linktarget="Help:procedure" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">procedures</Font></Hyperlink> so that computations can be easily performed for various input values. Examples below use procedures, and some of the performance acceleration techniques require use of procedures.</Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">evalhf</Text-field></Title>
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<Text-field style="Text" layout="Normal">The <Font bold="true">evalhf</Font> command provides an alternate interpreter in which all numeric quantities are stored (temporarily in the case of integers) as hardware double-precision floating-point numbers. Unassigned symbolic names cannot be handled directly in this special environment. Indeed, under <Font bold="true">evalhf</Font> there are a number of restrictions which allow much faster execution of numeric code. Function calls, and even basic arithmetic, can be computed much faster because the restrictions entail that a great deal of internal machinery is not needed.There is a help page on the <Hyperlink linktarget="Help:evalhf,procedure" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">restrictions for procedures</Font></Hyperlink> under <Font bold="true">evalhf</Font>. Another help page documents what commands can be used directly from within a procedure running under <Font bold="true">evalhf</Font>.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">One reason that <Font bold="true">evalhf</Font> can compute so quickly is that it does not create temporary DAG structures which must be handled as collectible garbage by Maple's internal memory management system.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Typically a call to evalhf is wrapped around either an individual expression or a function call. For example,</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">evalhf( 7 + 1/3 );</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIzLkxMTExMTEx0ISM8</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p := proc(x)
   x^2 + sin(x);
end proc:

evalhf( p(2.0) );</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIzJD5vRG9VKEg0XCEjPA==</Equation></Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Sometimes it is difficult to divide up a computation into the parts that can run under <Font bold="true">evalhf</Font> and the parts that cannot. It may be that the most natural way to program a procedure entails parts that run into the restrictions of the evalhf environment, in which case, other options can be used.</Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">option hfloat</Text-field></Title>
<Text-field style="Text" layout="Normal">By providing a procedure with <Font bold="true">option hfloat</Font> all floats are stored internally in a very lightweight DAG known as an <Hyperlink linktarget="Help:HFloat" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">HFloat</Font></Hyperlink>. which is a light wrapper around a hardware float. Such HFloats temporarily created during computation are still collectible garbage and entail memory management. But arithmetic can be done with them much faster, and many commands such as trigonometric and elementary functions can dispatch to fast methods more quickly.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">An advantage of <Font bold="true">option hfloat</Font> is that it does not put the same kind of restrictions on a procedure that <Font bold="true">evalhf</Font> does. Amongst other things, this means that it allows a procedure to use global variables, lexical scoping, unassigned symbol names and operations, direct calls to non-evalhfable commands, create Matrices, etc.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Here is the second example above, using <Font bold="true">option hfloat</Font>,</Text-field>
<Group labelreference="L5367" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ph := proc( x )
   option hfloat;
   x^2 + sin( x );
end proc:

ph( 2.0 );</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFExNC45MDkyOTc0MjY4MjU2OEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==">XTQwMTNBMzFFREQ1QTM2ODk=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L5368" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Notice that both results above compute at something close to 15 decimal digits, as a consequence of computing with double precision floats. In contrast, under Maple's normal interpreter the float computations are done with &quot;software floats&quot;, respect the Digits setting of precision, and return results with 10 decimal digits.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L5366" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p( 2.0 ); </Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrRnVINFwhIio=</Equation></Text-field>
</Output>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Compiler</Text-field></Title>
<Text-field style="Text" layout="Normal">The <Font bold="true">Compiler</Font> provides another fast computational environment in a Maple session. The <Hyperlink linktarget="Help:Compiler,Compile" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Compile</Font></Hyperlink> command internally translates a numeric procedure to the C language, compiles it, and links to the resulting object. It returns a new procedure which can be called like the original, but which computes faster.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">In <Hyperlink linktarget="http://www.maplesoft.com/support/help/Maple/view.aspx?path=updates/Maple18/Performance" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Maple 18</Font></Hyperlink> the C compiler is included with the Maple installation, and the compiled object code is stored directly in memory. The runtime of the Compiler is also <Hyperlink linktarget="http://www.maplesoft.com/support/help/Maple/view.aspx?path=updates%2fMaple18%2fLanguage#Compiler" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">thread-safe in Maple 18</Font></Hyperlink>, in consequence of which the resulting compiled procedures can run in parallel under the <Hyperlink linktarget="Help:Theads" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Threads</Font></Hyperlink> package.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">There is overhead in performing the compilation of a procedure, and of loading its runtime into memory the first time its compiled output procedure is called.Excluding such overhead it is generally the fastest floating-point computation environment within Maple. There are restriction of its use, however.Its runtime provides fast internal versions of key <Hyperlink linktarget="Help:CodeGeneration,General,TranslationDetails" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">commands</Font></Hyperlink> which may be translated.The <Hyperlink linktarget="Help:Compiler" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Compiler</Font></Hyperlink> help page details the restrictions on use. Several of the restrictions are similar to those of <Font bold="true">evalhf</Font>, for example Matrices must be passed in as arguments to the procedure.An important restriction is that nested procedures cannot be compiled.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">Here is the simple procedure example from above, compiled.</Text-field>
<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pr := proc( x::float )
   x^2 + sin(x);
end proc:

pc := Compiler:-Compile( pr ):

pc( 2.0 );</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIzJD5vRG9VKEg0XCEjPA==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L5379" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Summary</Text-field></Title>
<Text-field style="Text" layout="Normal">Below is a table which outlines some of the key functionality aspects of the three acceleration techniques discussed above.</Text-field><Table interior="group" showinput="true" alignment="left" exterior="all" width="100.0%" labelreference="L5371" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="237" separator="true"></Table-Column><Table-Column weight="238" separator="true"></Table-Column><Table-Column weight="237" separator="true"></Table-Column><Table-Column weight="237" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true">option hfloat</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true">evalhf</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true">Compile</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">nested procedures</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">no compilation overhead</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">variable typing not required</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">Matrix/Vector/Array creation</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">symbolic expressions (names, lists)</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">global variables</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">parallelizable</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred">no collectible garbage</Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[255,0,0]">no</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]">
<Text-field style="Text" layout="Normal" alignment="centred"><Font bold="true" foreground="[0,128,0]">yes</Font></Text-field></Table-Cell></Table-Row></Table>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">As can be seen, <Font bold="true">option hfloat</Font> is the least restricted. In general <Font bold="true">Compile</Font> attains the best speed performance if the time for procedure compilation is excluded, with <Font bold="true">evalhf</Font> as second best. The relative performance depends on the particular example.The overall pattern is a trade-off between flexibility and performance.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">In a follow-up article the topic of parallelism and running under the <Hyperlink linktarget="Help:Threads" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Threads</Font></Hyperlink> environment will be discussed, with a closer look at serial and parallel performance for more involved examples.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
</Section>
<Group labelreference="L5357" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8" italic="true" size="10">Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2014. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>