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<center><b><big><big>An Epsilon Accurate Exponential Approximation</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>#&#xA0;include&#xA0;&quot;exp_eps.hpp&quot;</span></font></code>

<code><span style='white-space: nowrap'><br/>
</span></code>
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;exp_eps(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>



<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
This is a an example algorithm that is used to demonstrate 
how Algorithmic Differentiation works with loops and  
boolean decision variables
(see <a href="exp_2.xml" target="_top"><span style='white-space: nowrap'>exp_2</span></a>
 for a simpler example). 

<br/>
<br/>
<b><big><a name="Mathematical Function" id="Mathematical Function">Mathematical Function</a></big></b>
<br/>
The exponential function can be defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>1</mn>
<mo stretchy="false">!</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>2</mn>
<mo stretchy="false">!</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
</mrow></math>

We define 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 as the smallest
non-negative integer such that 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">&#x02265;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
</mrow></math>

; i.e.,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi>min</mi>
<mo stretchy="false">{</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">&#x02208;</mo>
<msub><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>Z</mi>
</mstyle></mrow>
<mo stretchy="false">+</mo>
</msub>
<mspace width='.3em'/>
<mo stretchy="false">|</mo>
<mspace width='.3em'/>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">&#x02265;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>k</mi>
</msup>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">!</mo>
<mo stretchy="false">}</mo>
</mrow></math>

The mathematical form for our approximation of the exponential function is

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>exp</mi>
<mo stretchy="false">_</mo>
<mi mathvariant='normal'>eps</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>exp</mi>
<mo stretchy="false">_</mo>
<mi mathvariant='normal'>eps</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">&lt;</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mn>1</mn>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>1</mn>
</msup>
<mo stretchy="false">/</mo>
<mn>1</mn>
<mo stretchy="false">!</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">!</mo>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><big><a name="include" id="include">include</a></big></b>
<br/>
The include command in the syntax is relative to 

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;cppad-</span></font><i><font color="black"><span style='white-space: nowrap'>yyyymmdd</span></font></i><font color="blue"><span style='white-space: nowrap'>/introduction/exp_apx<br/>
</span></font></code>
where 
<code><font color="blue"><span style='white-space: nowrap'>cppad-</span></font><i><font color="black"><span style='white-space: nowrap'>yyyymmdd</span></font></i></code>
 is the distribution directory
created during the beginning steps of the
<a href="install.xml" target="_top"><span style='white-space: nowrap'>installation</span></a>
 of CppAD.


<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 below).
It specifies the point at which to evaluate the 
approximation for the exponential function.

<br/>
<br/>
<b><big><a name="epsilon" id="epsilon">epsilon</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It specifies the accuracy with which
to approximate the exponential function value; i.e.,
it is the value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
</mrow></math>

 in the 
exponential function approximation defined above.

<br/>
<br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The result 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is the value of the exponential function 
approximation defined above.

<br/>
<br/>
<b><big><a name="Type" id="Type">Type</a></big></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>
 are 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 objects and 
<code><i><font color="black"><span style='white-space: nowrap'>i</span></font></i></code>

is an <code><font color="blue">int</font></code>: 

<table><tr><td align='left'  valign='top'>

<b>Operation</b>  </td><td align='left'  valign='top'>
 <b>Result Type</b> </td><td align='left'  valign='top'>
 <b>Description</b>
</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
 
	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 object with value equal to 
<code><i><font color="black"><span style='white-space: nowrap'>i</span></font></i></code>
 
</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>Type&#xA0;u&#xA0;</span></font></i><font color="blue"><span style='white-space: nowrap'>=</span></font><i><font color="black"><span style='white-space: nowrap'>&#xA0;v</span></font></i></code>
 
	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 construct 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i></code>
 with value equal to 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>
 
</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

	</td><td align='left'  valign='top'>
 <code><font color="blue">bool</font></code>
	</td><td align='left'  valign='top'>
 true,
	if 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i></code>
 greater than 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>
, an false otherwise
</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>
 
	</td><td align='left'  valign='top'>
 new 
<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i></code>
 (and result) is value of 
<code><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 result is value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>v</mi>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;/&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 result is value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>v</mi>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>


<code><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>v</span></font></i></code>

	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 result is value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>v</mi>
</mrow></math>


</td></tr><tr><td align='left'  valign='top'>


<code><font color="blue"><span style='white-space: nowrap'>-</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i></code>

	</td><td align='left'  valign='top'>
 
<code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code>

	</td><td align='left'  valign='top'>
 result is value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>u</mi>
</mrow></math>


</td></tr>
</table>
<br/>
<b><big><a name="Implementation" id="Implementation">Implementation</a></big></b>
<br/>
The file <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>

contains a C++ implementation of this function.

<br/>
<br/>
<b><big><a name="Test" id="Test">Test</a></big></b>
<br/>
The file <a href="exp_eps.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.cpp</span></a>
 
contains a test of this implementation.
It returns true for success and false for failure.

<br/>
<br/>
<b><big><a name="Exercises" id="Exercises">Exercises</a></big></b>

<ol type="1"><li>
Using the definition of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 above,
what is the value of 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mn>.5</mn>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

, 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mn>.5</mn>
<mo stretchy="false">,</mo>
<mn>.1</mn>
<mo stretchy="false">)</mo>
</mrow></math>

, and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">(</mo>
<mn>.5</mn>
<mo stretchy="false">,</mo>
<mn>.01</mn>
<mo stretchy="false">)</mo>
</mrow></math>

 ?
</li><li>

Suppose that we make the following call to <code><font color="blue">exp_eps</font></code>:
<code><font color='blue'><pre style='display:inline'> 
	double x       = 1.;
	double epsilon = .01;
	double y = exp_eps(x, epsilon);
</pre></font></code>

What is the value assigned to 
<code><font color="blue">k</font></code>, <code><font color="blue">temp</font></code>, <code><font color="blue">term</font></code>, and <code><font color="blue">sum</font></code>
the first time through the <code><font color="blue">while</font></code> loop in <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
 ?
</li><li>

Continuing the previous exercise, 
what is the value assigned to 
<code><font color="blue">k</font></code>, <code><font color="blue">temp</font></code>, <code><font color="blue">term</font></code>, and <code><font color="blue">sum</font></code>
the second time through the <code><font color="blue">while</font></code> loop in <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
 ?
</li></ol>




<hr/>Input File: introduction/exp_apx/exp_eps.hpp

</body>
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