P7_CUBE_ROOT_APPROXIMATION.html

<hr>
<p><strong>CUBE_ROOT_APPROXIMATION</strong></p>
<hr>
<p><span style="background:#ffff00">The <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/C%2B%2B" target="_blank" rel="noopener">C++</a> program featured in this tutorial web page computes the approximate cube root of a real number using an iterative algorithm.</span></p>
<p><em>To view hidden text inside each of the preformatted text boxes below, scroll horizontally.</em></p>
<pre>Y := cube_root(X). // Y = X ^ (1/3).
(Y * Y * Y) = X. // X = Y ^ 3.
</pre>
<hr>
<p><strong>SOFTWARE_APPLICATION_COMPONENTS</strong></p>
<hr>
<p>C++_source_file: <a style="background:#000000;color:#00ff00" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation.cpp" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation.cpp</a></p>
<p>plain-text_file: <a style="background:#000000;color:#ff9000" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation_output.txt" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation_output.txt</a></p>
<hr>
<p><strong>PROGRAM_COMPILATION_AND_EXECUTION</strong></p>
<hr>
<p>STEP_0: Copy and paste the C++ <a style="background:#000000;color:#00ff00" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation.cpp" target="_blank" rel="noopener">source code</a> into a new text editor document and save that document as the following file name:</p>
<pre>cube_root_approximation.cpp</pre>
<p>STEP_1: Open a <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Unix" target="_blank" rel="noopener">Unix</a> command line terminal application and set the current directory to wherever the C++ program file is located on the local machine (e.g. Desktop).</p>
<pre>cd Desktop</pre>
<p>STEP_2: Compile the C++ file into machine-executable instructions (i.e. object file) and then into an executable piece of software named <strong>app</strong> using the following command:</p>
<pre>g++ cube_root_approximation.cpp -o app</pre>
<p>STEP_3: If the program compilation command does not work, then use the following commands (in top-down order) to install the C/C++ compiler (which is part of the <a style="background: #ff9000;color: #000000" href="https://en.wikipedia.org/wiki/GNU_Compiler_Collection" target="_blank" rel="noopener">GNU Compiler Collection (GCC)</a>):</p>
<pre>sudo apt install build-essential</pre>
<pre>sudo apt-get install g++</pre>
<p>STEP_4: After running the <strong>g++</strong> command, run the executable file using the following command:</p>
<pre>./app</pre>
<p>STEP_5: Once the application is running, the following prompt will appear:</p>
<pre>Enter a real number (represented using only base-ten digits with an optional radix and with an optional negative sign), x, which is no larger than 100:</pre>
<p>STEP_6: Enter a value for N using the keyboard.</p>
<p>STEP_7: Statements showing program throughput and the value returned by the cube root function which computes the approximate value of x raised to the power of (1/3) will be printed to the command line terminal and to the file output stream and then the following prompt will appear:</p>
<pre>Would you like to continue inputting program values? (Enter 1 if YES. Enter 0 if NO):</pre>
<p>STEP_8: Enter a value according to your preference until you decide to close the program (and save your program data to the <a style="background:#000000;color:#ff9000" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation_output.txt" target="_blank" rel="noopener">output file</a>).</p>
<hr>
<p><strong>PROGRAM_SOURCE_CODE</strong></p>
<hr>
<p>Note that the text inside of each of the the preformatted text boxes below appears on this web page (while rendered correctly by the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Web_browser" target="_blank" rel="noopener">web browser</a>) to be identical to the content of that preformatted text box text&#8217;s respective <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Plain_text" target="_blank" rel="noopener">plain-text</a> file or <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Source_code" target="_blank" rel="noopener">source code</a> output file (whose <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/URL" target="_blank" rel="noopener">Uniform Resource Locator</a> is displayed as the <strong style="background:#000000;color:#00ff00">green</strong> <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Hyperlink" target="_blank" rel="noopener">hyperlink</a> immediately above that preformatted text box (if that hyperlink points to a <strong>source code file</strong>) or whose Uniform Resource Locator is displayed as the <strong style="background:#000000;color:#ff9000">orange</strong> hyperlink immediately above that preformatted text box (if that hyperlink points to a <strong>plain-text file</strong>)).</p>
<p>A <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Computer" target="_blank" rel="noopener">computer</a> interprets a C++ source code as a series of programmatic instructions (i.e. <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Software" target="_blank" rel="noopener">software</a>) which govern how the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Computer_hardware" target="_blank" rel="noopener">hardware</a> of that computer behaves).</p>
<p><em>(Note that angle brackets which resemble <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/HTML" target="_blank" rel="noopener">HTML</a> tags (i.e. an &#8220;is less than&#8221; symbol (i.e. &#8216;&lt;&#8216;) followed by an &#8220;is greater than&#8221; symbol (i.e. &#8216;&gt;&#8217;)) displayed on this web page have been replaced (at the source code level of this web page) with the Unicode symbols <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Less-than_sign" target="_blank" rel="noopener">U+003C</a> (which is rendered by the web browser as &#8216;&lt;&#8216;) and <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Greater-than_sign" target="_blank" rel="noopener">U+003E</a> (which is rendered by the web browser as &#8216;&gt;&#8217;). That is because the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/WordPress.com" target="_blank" rel="noopener">WordPress</a> web page editor or web browser interprets a plain-text version of an &#8220;is less than&#8221; symbol followed by an &#8220;is greater than&#8221; symbol as being an opening HTML tag (which means that the WordPress web page editor or web browser deletes or fails to display those (plain-text) inequality symbols and the content between those (plain-text) inequality symbols)).</em></p>
<p>C++_source_file: <a style="background:#000000;color:#00ff00" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation.cpp" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation.cpp</a></p>
<hr>
<pre>
/**
 * file: cube_root_approximation.cpp
 * type: C++ (source file)
 * date: 19_JUNE_2023
 * author: karbytes
 * license: PUBLIC_DOMAIN
 */

/* preprocessing directives */
#include &lt;iostream&gt; // standard input (std::cin), standard output (std::cout)
#include &lt;fstream&gt; // file input, file output
#define MAXIMUM_X 1000 // constant which represents maximum absolute value of the program input value
#define E 0.00000001 // constant which represents the degree of accuracy of the square root approximation 

/* function prototypes */
float absolute_value(float x);
long double difference(long double n, long double b);
long double compute_cube_root_of_real_number(float x, std::ostream &amp; output);

/**
 * Return the absolute value of a real number input, x.
 */
float absolute_value(float x)
{
    if (x &lt; 0) return -1 * x;
    return x;
}

/**
 * Return the absolute value of (n - (b *  b * b)).
 */
long double difference(long double n, long double b)
{
    if (n &gt; (b * b * b)) return (n - (b * b * b));
    return ((b * b * b) - n);
}

/**
 * Compute the approximate cube root of a real number, x, using an iterative method.
 * 
 * The cube root of x is x raised to the power of (1/3).
 * 
 * Assume that x is a float type value and that output is an output stream object.
 * 
 * This function returns a value whose data type is long double (which is a floating-point number).
 */
long double compute_cube_root_of_real_number(float x, std::ostream &amp; output)
{
    int i = 0;
    float original_x = x;
    long double A = 0.0, B = 0.0, C = 0.0, epsilon = 0.0;
    x = absolute_value(x);
    x = ((x &gt; MAXIMUM_X) || (x &lt; 1)) ? 0 : x; // If x is out of range, then set x to 0. Also, to avoid an infinite loop as a result of the absolute value of x being too small, set x to 0 if the absolute value of x is smaller than 1.
    C = x;
    output &lt;&lt; &quot;\n\nC = &quot; &lt;&lt; C &lt;&lt; &quot;. // real number to take the cube root of&quot;;
    output &lt;&lt; &quot;\nB = &quot; &lt;&lt; B &lt;&lt; &quot;. // variable for storing the approximate cube root of x&quot;;
    output &lt;&lt; &quot;\nA = &quot; &lt;&lt; A &lt;&lt; &quot;. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i&quot;;
    output &lt;&lt; &quot;\nepsilon = &quot; &lt;&lt; epsilon &lt;&lt; &quot;. // variable for storing the difference between the input value and B raised to the power of 3&quot;;
    while (true)
    {
        output &lt;&lt; &quot;\n\ni := &quot; &lt;&lt; i &lt;&lt; &quot;.&quot;;
        output &lt;&lt; &quot;\nC := &quot; &lt;&lt; C &lt;&lt; &quot;.&quot;;
        output &lt;&lt; &quot;\nA := &quot; &lt;&lt; A &lt;&lt; &quot;.&quot;;
        B = (A + C) / 2;
        epsilon = difference(x, B);
        output &lt;&lt; &quot;\nB := (A + C) / 2 = &quot; &lt;&lt; B &lt;&lt; &quot;.&quot;;
        output &lt;&lt; &quot;\nepsilon = difference(x , B) = &quot; &lt;&lt; epsilon &lt;&lt; &quot;.&quot;;
        if (epsilon &lt;= E) 
        {
            if (original_x &lt; 0) return -1 * B;
            return B;
        }
        if ((B * B * B) &gt; x) C = B;
        else A = B;
        i += 1;
    }
}

/* program entry point */
int main()
{
    // Declare a float type variable and set its initial value to zero.
    float x = 0.0;

    // Declare a double type variable and set its initial value to zero.
    long double S = 0.0;

    // Declare a variable for storing the program user&#039;s answer of whether or not to continue inputting values.
    int input_additional_values = 1;

    // Declare a file output stream object.
    std::ofstream file;

    // Set the number of digits of floating-point numbers which are printed to the command line terminal to 100 digits.
    std::cout.precision(100);

    // Set the number of digits of floating-point numbers which are printed to the file output stream to 100 digits.
    file.precision(100);

    /**
     * If cube_root_approximation_output.txt does not already exist in the same directory as cube_root_approximation.cpp, 
     * create a new file named cube_root_approximation_output.txt.
     * 
     * Open the plain-text file named cube_root_approximation_output.txt 
     * and set that file to be overwritten with program data.
     */
    file.open(&quot;cube_root_approximation_output.txt&quot;);

    // Print an opening message to the command line terminal.
    std::cout &lt;&lt; &quot;\n\n--------------------------------&quot;;
    std::cout &lt;&lt; &quot;\nStart Of Program&quot;;
    std::cout &lt;&lt; &quot;\n--------------------------------&quot;;

    // Print an opening message to the file output stream.
    file &lt;&lt; &quot;--------------------------------&quot;;
    file &lt;&lt; &quot;\nStart Of Program&quot;;
    file &lt;&lt; &quot;\n--------------------------------&quot;;

    // Prompt the user to enter an x value as many times as the user chooses to.
    while (input_additional_values != 0)
    {
        // Print &quot;Enter a real number (represented using only base-ten digits with an optional radix and with an optional negative sign), x, which is no larger than {MAXIMUM_X}: &quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\n\nEnter a real number (represented using only base-ten digits with an optional radix and with an optional negative sign), x, which is no larger than &quot; &lt;&lt; MAXIMUM_X &lt;&lt; &quot;: &quot;;

        // Scan the command line terminal for the most recent keyboard input value.
        std::cin &gt;&gt; x;

        // Print &quot;The value which was entered for x is {x}.&quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\nThe value which was entered for x is &quot; &lt;&lt; x &lt;&lt; &quot;.&quot;;

        // Print &quot;The value which was entered for x is {x}.&quot; to the file output stream.
        file &lt;&lt; &quot;\n\nThe value which was entered for x is &quot; &lt;&lt; x &lt;&lt; &quot;.&quot;;

        // Print a horizontal line to the command line terminal.
        std::cout &lt;&lt; &quot;\n\n--------------------------------&quot;;

        // Print a horizontal line to the command line terminal.
        file &lt;&lt; &quot;\n\n--------------------------------&quot;;

        // Print &quot;Computing the approximate cube root of x:&quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\n\nComputing the approximate cube root of x:&quot;;

        // Print &quot;Computing the approximate cube root of x:&quot; to the file output stream.
        file &lt;&lt; &quot;\n\nComputing the approximate cube root of x:&quot;;

        // Compute the approximate cube root of x using the Bijection Method, print the computational steps to the command line terminal, and store the function result in S.
        S = compute_cube_root_of_real_number(x, std::cout);

        // Compute the approximate square root of x using Heron&#039;s Method and print the computational steps to the file output stream.
        compute_cube_root_of_real_number(x, file);

        // Print a horizontal line to the command line terminal.
        std::cout &lt;&lt; &quot;\n\n--------------------------------&quot;;

        // Print a horizontal line to the command line terminal.
        file &lt;&lt; &quot;\n\n--------------------------------&quot;;

        // Print &quot;S = approximate_cube_root({x}) = {S}.&quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\n\nS = approximate_cube_root(&quot; &lt;&lt; x &lt;&lt; &quot;) = &quot; &lt;&lt; S &lt;&lt; &quot;.&quot;;

        // Print &quot;S = approximate_cube_root({x}) = {S}.&quot; to the file output stream.
        file &lt;&lt; &quot;\n\nS = approximate_cube_root(&quot; &lt;&lt; x &lt;&lt; &quot;) = &quot; &lt;&lt; S &lt;&lt; &quot;.&quot;;

        // Print &quot;(S * S * S) = &quot; &lt;&lt; {(S * S * S)} &lt;&lt; &quot;. // the approximate value of x&quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\n\n(S * S * S) = &quot; &lt;&lt; (S * S * S) &lt;&lt; &quot;. // the approximate absolute value of x&quot;;

        // Print &quot;(S * S * S) = &quot; &lt;&lt; {(S * S * S)} &lt;&lt; &quot;. // the approximate value of x&quot; to the command line terminal.
        std::cout &lt;&lt; &quot;\n\n(S * S * S) = &quot; &lt;&lt; (S * S * S) &lt;&lt; &quot;. // the approximate absolute value of x&quot;;

        // Ask the user whether or not to continue inputing values.
        std::cout &lt;&lt; &quot;\n\nWould you like to continue inputing program values? (Enter 1 if YES. Enter 0 if NO): &quot;;

        // Scan the command line terminal for the most recent user input entered via keyboard to store in the variable named input_additional_values.
        std::cin &gt;&gt; input_additional_values;

        // Print a horizontal line to the command line terminal.
        std::cout &lt;&lt; &quot;\n\n--------------------------------&quot;;

        // Print a horizontal line to the command line terminal.
        file &lt;&lt; &quot;\n\n--------------------------------&quot;;
    }

    // Print a closing message to the command line terminal.
    std::cout &lt;&lt; &quot;\nEnd Of Program&quot;;
    std::cout &lt;&lt; &quot;\n--------------------------------\n\n&quot;;

    // Print a closing message to the file output stream.
    file &lt;&lt; &quot;\nEnd Of Program&quot;;
    file &lt;&lt; &quot;\n--------------------------------&quot;;

    // Close the file output stream.
    file.close();

    // Exit the program.
    return 0;
}</pre>
<hr>
<p><strong>SAMPLE_PROGRAM_OUTPUT</strong></p>
<hr>
<p>The text in the preformatted text box below was generated by one use case of the C++ program featured in this <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Computer_programming" target="_blank" rel="noopener">computer programming</a> tutorial web page.</p>
<p>plain-text_file: <a style="background:#000000;color:#ff9000" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation_output.txt" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/cube_root_approximation_output.txt</a></p>
<hr>
<pre>--------------------------------
Start Of Program
--------------------------------

The value which was entered for x is 8.

--------------------------------

Computing the approximate cube root of x:

C = 8. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 8.
A := 0.
B := (A + C) / 2 = 4.
epsilon = difference(x , B) = 56.

i := 1.
C := 4.
A := 0.
B := (A + C) / 2 = 2.
epsilon = difference(x , B) = 0.

--------------------------------

S = approximate_cube_root(8) = 2.

--------------------------------

The value which was entered for x is 125.

--------------------------------

Computing the approximate cube root of x:

C = 125. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 125.
A := 0.
B := (A + C) / 2 = 62.5.
epsilon = difference(x , B) = 244015.625.

i := 1.
C := 62.5.
A := 0.
B := (A + C) / 2 = 31.25.
epsilon = difference(x , B) = 30392.578125.

i := 2.
C := 31.25.
A := 0.
B := (A + C) / 2 = 15.625.
epsilon = difference(x , B) = 3689.697265625.

i := 3.
C := 15.625.
A := 0.
B := (A + C) / 2 = 7.8125.
epsilon = difference(x , B) = 351.837158203125.

i := 4.
C := 7.8125.
A := 0.
B := (A + C) / 2 = 3.90625.
epsilon = difference(x , B) = 65.395355224609375.

i := 5.
C := 7.8125.
A := 3.90625.
B := (A + C) / 2 = 5.859375.
epsilon = difference(x , B) = 76.165676116943359375.

i := 6.
C := 5.859375.
A := 3.90625.
B := (A + C) / 2 = 4.8828125.
epsilon = difference(x , B) = 8.584678173065185546875.

i := 7.
C := 5.859375.
A := 4.8828125.
B := (A + C) / 2 = 5.37109375.
epsilon = difference(x , B) = 29.948793351650238037109375.

i := 8.
C := 5.37109375.
A := 4.8828125.
B := (A + C) / 2 = 5.126953125.
epsilon = difference(x , B) = 9.765286929905414581298828125.

i := 9.
C := 5.126953125.
A := 4.8828125.
B := (A + C) / 2 = 5.0048828125.
epsilon = difference(x , B) = 0.366568681783974170684814453125.

i := 10.
C := 5.0048828125.
A := 4.8828125.
B := (A + C) / 2 = 4.94384765625.
epsilon = difference(x , B) = 4.164306548773311078548431396484375.

i := 11.
C := 5.0048828125.
A := 4.94384765625.
B := (A + C) / 2 = 4.974365234375.
epsilon = difference(x , B) = 1.912767149406136013567447662353515625.

i := 12.
C := 5.0048828125.
A := 4.974365234375.
B := (A + C) / 2 = 4.9896240234375.
epsilon = difference(x , B) = 0.776584445929984212853014469146728515625.

i := 13.
C := 5.0048828125.
A := 4.9896240234375.
B := (A + C) / 2 = 4.99725341796875.
epsilon = difference(x , B) = 0.205880517370360394124872982501983642578125.

i := 14.
C := 5.0048828125.
A := 4.99725341796875.
B := (A + C) / 2 = 5.001068115234375.
epsilon = difference(x , B) = 0.080125756849014351246296428143978118896484375.

i := 15.
C := 5.001068115234375.
A := 4.99725341796875.
B := (A + C) / 2 = 4.9991607666015625.
epsilon = difference(x , B) = 0.062931940783439443976021721027791500091552734375.

i := 16.
C := 5.001068115234375.
A := 4.9991607666015625.
B := (A + C) / 2 = 5.00011444091796875.
epsilon = difference(x , B) = 0.008583265300010634035743350978009402751922607421875.

i := 17.
C := 5.00011444091796875.
A := 4.9991607666015625.
B := (A + C) / 2 = 4.999637603759765625.
epsilon = difference(x , B) = 0.027177748099647958124336355467676185071468353271484375.

i := 18.
C := 5.00011444091796875.
A := 4.999637603759765625.
B := (A + C) / 2 = 4.9998760223388671875.
epsilon = difference(x , B) = 0.009298094029959631801052211130809155292809009552001953125.

i := 19.
C := 5.00011444091796875.
A := 4.9998760223388671875.
B := (A + C) / 2 = 4.99999523162841796875.
epsilon = difference(x , B) = 0.00035762752759194160745437329751439392566680908203125.

i := 20.
C := 5.00011444091796875.
A := 4.99999523162841796875.
B := (A + C) / 2 = 5.000054836273193359375.
epsilon = difference(x , B) = 0.0041127655949197150508922504741349257528781890869140625.

i := 21.
C := 5.000054836273193359375.
A := 4.99999523162841796875.
B := (A + C) / 2 = 5.0000250339508056640625.
epsilon = difference(x , B) = 0.001877555710920887632742193318335921503603458404541015625.

i := 22.
C := 5.0000250339508056640625.
A := 4.99999523162841796875.
B := (A + C) / 2 = 5.00001013278961181640625.
epsilon = difference(x , B) = 0.00075996076098865106285273895991849713027477264404296875.

i := 23.
C := 5.00001013278961181640625.
A := 4.99999523162841796875.
B := (A + C) / 2 = 5.000002682209014892578125.
epsilon = difference(x , B) = 0.000201165784030642169621927450862131081521511077880859375.

i := 24.
C := 5.000002682209014892578125.
A := 4.99999523162841796875.
B := (A + C) / 2 = 4.9999989569187164306640625.
epsilon = difference(x , B) = 7.823107994742173332269885577261447906494140625e-05.

i := 25.
C := 5.000002682209014892578125.
A := 4.9999989569187164306640625.
B := (A + C) / 2 = 5.00000081956386566162109375.
epsilon = difference(x , B) = 6.14672999998955305045456043444573879241943359375e-05.

i := 26.
C := 5.00000081956386566162109375.
A := 4.9999989569187164306640625.
B := (A + C) / 2 = 4.999999888241291046142578125.
epsilon = difference(x , B) = 8.381902984189171235129833803512156009674072265625e-06.

i := 27.
C := 5.00000081956386566162109375.
A := 4.999999888241291046142578125.
B := (A + C) / 2 = 5.0000003539025783538818359375.
epsilon = difference(x , B) = 2.654269525524666217819458324811421334743499755859375e-05.

i := 28.
C := 5.0000003539025783538818359375.
A := 4.999999888241291046142578125.
B := (A + C) / 2 = 5.00000012107193470001220703125.
epsilon = difference(x , B) = 9.080395322380585554356002830900251865386962890625e-06.

i := 29.
C := 5.00000012107193470001220703125.
A := 4.999999888241291046142578125.
B := (A + C) / 2 = 5.000000004656612873077392578125.
epsilon = difference(x , B) = 3.4924596579999356293910750537179410457611083984375e-07.

i := 30.
C := 5.000000004656612873077392578125.
A := 4.999999888241291046142578125.
B := (A + C) / 2 = 4.9999999464489519596099853515625.
epsilon = difference(x , B) = 4.016328560015047788311903786961920559406280517578125e-06.

i := 31.
C := 5.000000004656612873077392578125.
A := 4.9999999464489519596099853515625.
B := (A + C) / 2 = 4.99999997555278241634368896484375.
epsilon = difference(x , B) = 1.833541309802233509884672457701526582241058349609375e-06.

i := 32.
C := 5.000000004656612873077392578125.
A := 4.99999997555278241634368896484375.
B := (A + C) / 2 = 4.999999990104697644710540771484375.
epsilon = difference(x , B) = 7.42147675182602828414246687316335737705230712890625e-07.

i := 33.
C := 5.000000004656612873077392578125.
A := 4.999999990104697644710540771484375.
B := (A + C) / 2 = 4.9999999973806552588939666748046875.
epsilon = difference(x , B) = 1.96450855478869090831040011835284531116485595703125e-07.

i := 34.
C := 5.000000004656612873077392578125.
A := 4.9999999973806552588939666748046875.
B := (A + C) / 2 = 5.00000000101863406598567962646484375.
epsilon = difference(x , B) = 7.6397554969742653696584966382943093776702880859375e-08.

i := 35.
C := 5.00000000101863406598567962646484375.
A := 4.9999999973806552588939666748046875.
B := (A + C) / 2 = 4.999999999199644662439823150634765625.
epsilon = difference(x , B) = 6.0026650310074369798485349747352302074432373046875e-08.

i := 36.
C := 5.00000000101863406598567962646484375.
A := 4.999999999199644662439823150634765625.
B := (A + C) / 2 = 5.0000000001091393642127513885498046875.
epsilon = difference(x , B) = 8.1854523159563541412353515625e-09.

--------------------------------

S = approximate_cube_root(125) = 5.0000000001091393642127513885498046875.

--------------------------------

The value which was entered for x is -8.

--------------------------------

Computing the approximate cube root of x:

C = 8. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 8.
A := 0.
B := (A + C) / 2 = 4.
epsilon = difference(x , B) = 56.

i := 1.
C := 4.
A := 0.
B := (A + C) / 2 = 2.
epsilon = difference(x , B) = 0.

--------------------------------

S = approximate_cube_root(-8) = -2.

--------------------------------

The value which was entered for x is 1000.

--------------------------------

Computing the approximate cube root of x:

C = 1000. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 1000.
A := 0.
B := (A + C) / 2 = 500.
epsilon = difference(x , B) = 124999000.

i := 1.
C := 500.
A := 0.
B := (A + C) / 2 = 250.
epsilon = difference(x , B) = 15624000.

i := 2.
C := 250.
A := 0.
B := (A + C) / 2 = 125.
epsilon = difference(x , B) = 1952125.

i := 3.
C := 125.
A := 0.
B := (A + C) / 2 = 62.5.
epsilon = difference(x , B) = 243140.625.

i := 4.
C := 62.5.
A := 0.
B := (A + C) / 2 = 31.25.
epsilon = difference(x , B) = 29517.578125.

i := 5.
C := 31.25.
A := 0.
B := (A + C) / 2 = 15.625.
epsilon = difference(x , B) = 2814.697265625.

i := 6.
C := 15.625.
A := 0.
B := (A + C) / 2 = 7.8125.
epsilon = difference(x , B) = 523.162841796875.

i := 7.
C := 15.625.
A := 7.8125.
B := (A + C) / 2 = 11.71875.
epsilon = difference(x , B) = 609.325408935546875.

i := 8.
C := 11.71875.
A := 7.8125.
B := (A + C) / 2 = 9.765625.
epsilon = difference(x , B) = 68.677425384521484375.

i := 9.
C := 11.71875.
A := 9.765625.
B := (A + C) / 2 = 10.7421875.
epsilon = difference(x , B) = 239.590346813201904296875.

i := 10.
C := 10.7421875.
A := 9.765625.
B := (A + C) / 2 = 10.25390625.
epsilon = difference(x , B) = 78.122295439243316650390625.

i := 11.
C := 10.25390625.
A := 9.765625.
B := (A + C) / 2 = 10.009765625.
epsilon = difference(x , B) = 2.932549454271793365478515625.

i := 12.
C := 10.009765625.
A := 9.765625.
B := (A + C) / 2 = 9.8876953125.
epsilon = difference(x , B) = 33.314452390186488628387451171875.

i := 13.
C := 10.009765625.
A := 9.8876953125.
B := (A + C) / 2 = 9.94873046875.
epsilon = difference(x , B) = 15.302137195249088108539581298828125.

i := 14.
C := 10.009765625.
A := 9.94873046875.
B := (A + C) / 2 = 9.979248046875.
epsilon = difference(x , B) = 6.212675567439873702824115753173828125.

i := 15.
C := 10.009765625.
A := 9.979248046875.
B := (A + C) / 2 = 9.9945068359375.
epsilon = difference(x , B) = 1.647044138962883152998983860015869140625.

i := 16.
C := 10.009765625.
A := 9.9945068359375.
B := (A + C) / 2 = 10.00213623046875.
epsilon = difference(x , B) = 0.641006054792114809970371425151824951171875.

i := 17.
C := 10.00213623046875.
A := 9.9945068359375.
B := (A + C) / 2 = 9.998321533203125.
epsilon = difference(x , B) = 0.503455526267515551808173768222332000732421875.

i := 18.
C := 10.00213623046875.
A := 9.998321533203125.
B := (A + C) / 2 = 10.0002288818359375.
epsilon = difference(x , B) = 0.068666122400085072285946807824075222015380859375.

i := 19.
C := 10.0002288818359375.
A := 9.998321533203125.
B := (A + C) / 2 = 9.99927520751953125.
epsilon = difference(x , B) = 0.217421984797183664994690843741409480571746826171875.

i := 20.
C := 10.0002288818359375.
A := 9.99927520751953125.
B := (A + C) / 2 = 9.999752044677734375.
epsilon = difference(x , B) = 0.074384752239677054408417689046473242342472076416015625.

i := 21.
C := 10.0002288818359375.
A := 9.999752044677734375.
B := (A + C) / 2 = 9.9999904632568359375.
epsilon = difference(x , B) = 0.00286102022073553285963498638011515140533447265625.

i := 22.
C := 10.0002288818359375.
A := 9.9999904632568359375.
B := (A + C) / 2 = 10.00010967254638671875.
epsilon = difference(x , B) = 0.0329021247593577204071380037930794060230255126953125.

i := 23.
C := 10.00010967254638671875.
A := 9.9999904632568359375.
B := (A + C) / 2 = 10.000050067901611328125.
epsilon = difference(x , B) = 0.015020445687367101061937546546687372028827667236328125.

i := 24.
C := 10.000050067901611328125.
A := 9.9999904632568359375.
B := (A + C) / 2 = 10.0000202655792236328125.
epsilon = difference(x , B) = 0.00607968608790920850282191167934797704219818115234375.

i := 25.
C := 10.0000202655792236328125.
A := 9.9999904632568359375.
B := (A + C) / 2 = 10.00000536441802978515625.
epsilon = difference(x , B) = 0.001609326272245137356975419606897048652172088623046875.

i := 26.
C := 10.00000536441802978515625.
A := 9.9999904632568359375.
B := (A + C) / 2 = 9.999997913837432861328125.
epsilon = difference(x , B) = 0.00062584863957937386658159084618091583251953125.

i := 27.
C := 10.00000536441802978515625.
A := 9.999997913837432861328125.
B := (A + C) / 2 = 10.0000016391277313232421875.
epsilon = difference(x , B) = 0.0004917383999991642440363648347556591033935546875.

i := 28.
C := 10.0000016391277313232421875.
A := 9.999997913837432861328125.
B := (A + C) / 2 = 9.99999977648258209228515625.
epsilon = difference(x , B) = 6.7055223873513369881038670428097248077392578125e-05.

i := 29.
C := 10.0000016391277313232421875.
A := 9.99999977648258209228515625.
B := (A + C) / 2 = 10.000000707805156707763671875.
epsilon = difference(x , B) = 0.00021234156204197329742555666598491370677947998046875.

i := 30.
C := 10.000000707805156707763671875.
A := 9.99999977648258209228515625.
B := (A + C) / 2 = 10.0000002421438694000244140625.
epsilon = difference(x , B) = 7.2643162579044684434848022647202014923095703125e-05.

i := 31.
C := 10.0000002421438694000244140625.
A := 9.99999977648258209228515625.
B := (A + C) / 2 = 10.00000000931322574615478515625.
epsilon = difference(x , B) = 2.79396772639994850351286004297435283660888671875e-06.

i := 32.
C := 10.00000000931322574615478515625.
A := 9.99999977648258209228515625.
B := (A + C) / 2 = 9.999999892897903919219970703125.
epsilon = difference(x , B) = 3.2130628480120382306495230295695364475250244140625e-05.

i := 33.
C := 10.00000000931322574615478515625.
A := 9.999999892897903919219970703125.
B := (A + C) / 2 = 9.9999999511055648326873779296875.
epsilon = difference(x , B) = 1.4668330478417868079077379661612212657928466796875e-05.

i := 34.
C := 10.00000000931322574615478515625.
A := 9.9999999511055648326873779296875.
B := (A + C) / 2 = 9.99999998020939528942108154296875.
epsilon = difference(x , B) = 5.937181401460822627313973498530685901641845703125e-06.

i := 35.
C := 10.00000000931322574615478515625.
A := 9.99999998020939528942108154296875.
B := (A + C) / 2 = 9.999999994761310517787933349609375.
epsilon = difference(x , B) = 1.571606843830952726648320094682276248931884765625e-06.

i := 36.
C := 10.00000000931322574615478515625.
A := 9.999999994761310517787933349609375.
B := (A + C) / 2 = 10.0000000020372681319713592529296875.
epsilon = difference(x , B) = 6.11180439757941229572679731063544750213623046875e-07.

i := 37.
C := 10.0000000020372681319713592529296875.
A := 9.999999994761310517787933349609375.
B := (A + C) / 2 = 9.99999999839928932487964630126953125.
epsilon = difference(x , B) = 4.80213202480594958387882797978818416595458984375e-07.

i := 38.
C := 10.0000000020372681319713592529296875.
A := 9.99999999839928932487964630126953125.
B := (A + C) / 2 = 10.000000000218278728425502777099609375.
epsilon = difference(x , B) = 6.54836185276508331298828125e-08.

i := 39.
C := 10.000000000218278728425502777099609375.
A := 9.99999999839928932487964630126953125.
B := (A + C) / 2 = 9.9999999993087840266525745391845703125.
epsilon = difference(x , B) = 2.0736479200422763824462890625e-07.

i := 40.
C := 10.000000000218278728425502777099609375.
A := 9.9999999993087840266525745391845703125.
B := (A + C) / 2 = 9.99999999976353137753903865814208984375.
epsilon = difference(x , B) = 7.0940586738288402557373046875e-08.

i := 41.
C := 10.000000000218278728425502777099609375.
A := 9.99999999976353137753903865814208984375.
B := (A + C) / 2 = 9.999999999990905052982270717620849609375.
epsilon = difference(x , B) = 2.7284841053187847137451171875e-09.

--------------------------------

S = approximate_cube_root(1000) = 9.999999999990905052982270717620849609375.

--------------------------------

The value which was entered for x is 3.1400001049041748046875.

--------------------------------

Computing the approximate cube root of x:

C = 3.1400001049041748046875. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 3.1400001049041748046875.
A := 0.
B := (A + C) / 2 = 1.57000005245208740234375.
epsilon = difference(x , B) = 0.72989328296328886773979005564427779972902499139308929443359375.

i := 1.
C := 1.57000005245208740234375.
A := 0.
B := (A + C) / 2 = 0.785000026226043701171875.
epsilon = difference(x , B) = 2.65626343142074184560698368873232766418368555605411529541015625.

i := 2.
C := 1.57000005245208740234375.
A := 0.785000026226043701171875.
B := (A + C) / 2 = 1.1775000393390655517578125.
epsilon = difference(x , B) = 1.5073888318975885679262827210322939208708703517913818359375.

i := 3.
C := 1.57000005245208740234375.
A := 1.1775000393390655517578125.
B := (A + C) / 2 = 1.37375004589557647705078125.
epsilon = difference(x , B) = 0.5474738704539012898626915148980742742423899471759796142578125.

i := 4.
C := 1.57000005245208740234375.
A := 1.37375004589557647705078125.
B := (A + C) / 2 = 1.471875049173831939697265625.
epsilon = difference(x , B) = 0.04869378768681393893254238935952571409870870411396026611328125.

i := 5.
C := 1.471875049173831939697265625.
A := 1.37375004589557647705078125.
B := (A + C) / 2 = 1.4228125475347042083740234375.
epsilon = difference(x , B) = 0.25966472170411463902227333644390228073461912572383880615234375.

i := 6.
C := 1.471875049173831939697265625.
A := 1.4228125475347042083740234375.
B := (A + C) / 2 = 1.44734379835426807403564453125.
epsilon = difference(x , B) = 0.10809842450396796591401138432075867967796511948108673095703125.

i := 7.
C := 1.471875049173831939697265625.
A := 1.44734379835426807403564453125.
B := (A + C) / 2 = 1.459609423764050006866455078125.
epsilon = difference(x , B) = 0.0303610937093032767254696668857150143594481050968170166015625.

i := 8.
C := 1.471875049173831939697265625.
A := 1.459609423764050006866455078125.
B := (A + C) / 2 = 1.4657422364689409732818603515625.
epsilon = difference(x , B) = 0.0090009611727116579406315910460989471175707876682281494140625.

i := 9.
C := 1.4657422364689409732818603515625.
A := 1.459609423764050006866455078125.
B := (A + C) / 2 = 1.46267583011649549007415771484375.
epsilon = difference(x , B) = 0.0107213262234489643993928797982562173274345695972442626953125.

i := 10.
C := 1.4657422364689409732818603515625.
A := 1.46267583011649549007415771484375.
B := (A + C) / 2 = 1.464209033292718231678009033203125.
epsilon = difference(x , B) = 0.0008705083265141623678762261562269486603327095508575439453125.

i := 11.
C := 1.4657422364689409732818603515625.
A := 1.464209033292718231678009033203125.
B := (A + C) / 2 = 1.4649756348808296024799346923828125.
epsilon = difference(x , B) = 0.0040626436212677177230168101829121951595880091190338134765625.

i := 12.
C := 1.4649756348808296024799346923828125.
A := 1.464209033292718231678009033203125.
B := (A + C) / 2 = 1.46459233408677391707897186279296875.
epsilon = difference(x , B) = 0.00159542211586210179972977751816642921767197549343109130859375.

i := 13.
C := 1.46459233408677391707897186279296875.
A := 1.464209033292718231678009033203125.
B := (A + C) / 2 = 1.464400683689746074378490447998046875.
epsilon = difference(x , B) = 0.00036229553291318612739946303236138192005455493927001953125.

i := 14.
C := 1.464400683689746074378490447998046875.
A := 1.464209033292718231678009033203125.
B := (A + C) / 2 = 1.4643048584912321530282497406005859375.
epsilon = difference(x , B) = 0.00025414673460094866323799589480358918081037700176239013671875.

i := 15.
C := 1.464400683689746074378490447998046875.
A := 1.4643048584912321530282497406005859375.
B := (A + C) / 2 = 1.46435277109048911370337009429931640625.
epsilon = difference(x , B) = 5.406431437603685663528807481270632706582546234130859375e-05.

i := 16.
C := 1.46435277109048911370337009429931640625.
A := 1.4643048584912321530282497406005859375.
B := (A + C) / 2 = 1.464328814790860633365809917449951171875.
epsilon = difference(x , B) = 0.00010004373126623069233109841746909296489320695400238037109375.

i := 17.
C := 1.46435277109048911370337009429931640625.
A := 1.464328814790860633365809917449951171875.
B := (A + C) / 2 = 1.4643407929406748735345900058746337890625.
epsilon = difference(x , B) = 2.2990338738696457221433178119696094654500484466552734375e-05.

i := 18.
C := 1.46435277109048911370337009429931640625.
A := 1.4643407929406748735345900058746337890625.
B := (A + C) / 2 = 1.46434678201558199361898005008697509765625.
epsilon = difference(x , B) = 1.55368302446261090377088720515530440025031566619873046875e-05.

i := 19.
C := 1.46434678201558199361898005008697509765625.
A := 1.4643407929406748735345900058746337890625.
B := (A + C) / 2 = 1.464343787478128433576785027980804443359375.
epsilon = difference(x , B) = 3.72679364046553211753387557791938888840377330780029296875e-06.

i := 20.
C := 1.46434678201558199361898005008697509765625.
A := 1.464343787478128433576785027980804443359375.
B := (A + C) / 2 = 1.4643452847468552135978825390338897705078125.
epsilon = difference(x , B) = 5.90500845371239903303095530873179086484014987945556640625e-06.

i := 21.
C := 1.4643452847468552135978825390338897705078125.
A := 1.464343787478128433576785027980804443359375.
B := (A + C) / 2 = 1.46434453611249182358733378350734710693359375.
epsilon = difference(x , B) = 1.08910494453257240821120177542979945428669452667236328125e-06.

i := 22.
C := 1.46434453611249182358733378350734710693359375.
A := 1.464343787478128433576785027980804443359375.
B := (A + C) / 2 = 1.464344161795310128582059405744075775146484375.
epsilon = difference(x , B) = 1.318844963489086696828422873295494355261325836181640625e-06.

i := 23.
C := 1.46434453611249182358733378350734710693359375.
A := 1.464344161795310128582059405744075775146484375.
B := (A + C) / 2 = 1.4643443489539009760846965946257114410400390625.
epsilon = difference(x , B) = 1.1487016335874622452450921628042124211788177490234375e-07.

i := 24.
C := 1.46434453611249182358733378350734710693359375.
A := 1.4643443489539009760846965946257114410400390625.
B := (A + C) / 2 = 1.46434444253319639983601518906652927398681640625.
epsilon = difference(x , B) = 4.8711735211692634706093230079204658977687358856201171875e-07.

i := 25.
C := 1.46434444253319639983601518906652927398681640625.
A := 1.4643443489539009760846965946257114410400390625.
B := (A + C) / 2 = 1.464344395743548687960355891846120357513427734375.
epsilon = difference(x , B) = 1.8612358476167469023554446039270260371267795562744140625e-07.

i := 26.
C := 1.464344395743548687960355891846120357513427734375.
A := 1.4643443489539009760846965946257114410400390625.
B := (A + C) / 2 = 1.4643443723487248320225262432359158992767333984375.
epsilon = difference(x , B) = 3.562670829702907493441443875781260430812835693359375e-08.

i := 27.
C := 1.4643443723487248320225262432359158992767333984375.
A := 1.4643443489539009760846965946257114410400390625.
B := (A + C) / 2 = 1.46434436065131290405361141893081367015838623046875.
epsilon = difference(x , B) = 3.9621728131940259221011046975036151707172393798828125e-08.

i := 28.
C := 1.4643443723487248320225262432359158992767333984375.
A := 1.46434436065131290405361141893081367015838623046875.
B := (A + C) / 2 = 1.464344366500018868038068831083364784717559814453125.
epsilon = difference(x , B) = 1.997510067942853684286319548846222460269927978515625e-09.

--------------------------------

S = approximate_cube_root(3.1400001049041748046875) = 1.464344366500018868038068831083364784717559814453125.

--------------------------------

The value which was entered for x is 2.

--------------------------------

Computing the approximate cube root of x:

C = 2. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 2.
A := 0.
B := (A + C) / 2 = 1.
epsilon = difference(x , B) = 1.

i := 1.
C := 2.
A := 1.
B := (A + C) / 2 = 1.5.
epsilon = difference(x , B) = 1.375.

i := 2.
C := 1.5.
A := 1.
B := (A + C) / 2 = 1.25.
epsilon = difference(x , B) = 0.046875.

i := 3.
C := 1.5.
A := 1.25.
B := (A + C) / 2 = 1.375.
epsilon = difference(x , B) = 0.599609375.

i := 4.
C := 1.375.
A := 1.25.
B := (A + C) / 2 = 1.3125.
epsilon = difference(x , B) = 0.260986328125.

i := 5.
C := 1.3125.
A := 1.25.
B := (A + C) / 2 = 1.28125.
epsilon = difference(x , B) = 0.103302001953125.

i := 6.
C := 1.28125.
A := 1.25.
B := (A + C) / 2 = 1.265625.
epsilon = difference(x , B) = 0.027286529541015625.

i := 7.
C := 1.265625.
A := 1.25.
B := (A + C) / 2 = 1.2578125.
epsilon = difference(x , B) = 0.010024547576904296875.

i := 8.
C := 1.265625.
A := 1.2578125.
B := (A + C) / 2 = 1.26171875.
epsilon = difference(x , B) = 0.008573234081268310546875.

i := 9.
C := 1.26171875.
A := 1.2578125.
B := (A + C) / 2 = 1.259765625.
epsilon = difference(x , B) = 0.000740073621273040771484375.

i := 10.
C := 1.26171875.
A := 1.259765625.
B := (A + C) / 2 = 1.2607421875.
epsilon = difference(x , B) = 0.003912973217666149139404296875.

i := 11.
C := 1.2607421875.
A := 1.259765625.
B := (A + C) / 2 = 1.26025390625.
epsilon = difference(x , B) = 0.001585548394359648227691650390625.

i := 12.
C := 1.26025390625.
A := 1.259765625.
B := (A + C) / 2 = 1.260009765625.
epsilon = difference(x , B) = 0.000422512079239822924137115478515625.

i := 13.
C := 1.260009765625.
A := 1.259765625.
B := (A + C) / 2 = 1.2598876953125.
epsilon = difference(x , B) = 0.000158837092385510914027690887451171875.

i := 14.
C := 1.260009765625.
A := 1.2598876953125.
B := (A + C) / 2 = 1.25994873046875.
epsilon = difference(x , B) = 0.000131823412402809481136500835418701171875.

i := 15.
C := 1.25994873046875.
A := 1.2598876953125.
B := (A + C) / 2 = 1.259918212890625.
epsilon = difference(x , B) = 1.3510360162172219133935868740081787109375e-05.

i := 16.
C := 1.25994873046875.
A := 1.259918212890625.
B := (A + C) / 2 = 1.2599334716796875.
epsilon = difference(x , B) = 5.9155646066955114292795769870281219482421875e-05.

i := 17.
C := 1.2599334716796875.
A := 1.259918212890625.
B := (A + C) / 2 = 1.25992584228515625.
epsilon = difference(x , B) = 2.2822422940382836031858460046350955963134765625e-05.

i := 18.
C := 1.25992584228515625.
A := 1.259918212890625.
B := (A + C) / 2 = 1.259922027587890625.
epsilon = difference(x , B) = 4.655976386269689015762196504510939121246337890625e-06.

i := 19.
C := 1.259922027587890625.
A := 1.259918212890625.
B := (A + C) / 2 = 1.2599201202392578125.
epsilon = difference(x , B) = 4.427205638639353235674889219808392226696014404296875e-06.

i := 20.
C := 1.259922027587890625.
A := 1.2599201202392578125.
B := (A + C) / 2 = 1.25992107391357421875.
epsilon = difference(x , B) = 1.14381936140543760682675156203913502395153045654296875e-07.

i := 21.
C := 1.25992107391357421875.
A := 1.2599201202392578125.
B := (A + C) / 2 = 1.259920597076416015625.
epsilon = difference(x , B) = 2.156412710667735509184606002008877112530171871185302734375e-06.

i := 22.
C := 1.25992107391357421875.
A := 1.259920597076416015625.
B := (A + C) / 2 = 1.2599208354949951171875.
epsilon = difference(x , B) = 1.02101560211825988233602657828669180162250995635986328125e-06.

i := 23.
C := 1.25992107391357421875.
A := 1.2599208354949951171875.
B := (A + C) / 2 = 1.25992095470428466796875.
epsilon = difference(x , B) = 4.5331688670251051032078493108201655559241771697998046875e-07.

i := 24.
C := 1.25992107391357421875.
A := 1.25992095470428466796875.
B := (A + C) / 2 = 1.259921014308929443359375.
epsilon = difference(x , B) = 1.6946748870936938213827005483835819177329540252685546875e-07.

i := 25.
C := 1.25992107391357421875.
A := 1.259921014308929443359375.
B := (A + C) / 2 = 1.2599210441112518310546875.
epsilon = difference(x , B) = 2.7542779641536417611913378777899197302758693695068359375e-08.

i := 26.
C := 1.25992107391357421875.
A := 1.2599210441112518310546875.
B := (A + C) / 2 = 1.25992105901241302490234375.
epsilon = difference(x , B) = 4.341957741027697992297618156953831203281879425048828125e-08.

i := 27.
C := 1.25992105901241302490234375.
A := 1.2599210441112518310546875.
B := (A + C) / 2 = 1.259921051561832427978515625.
epsilon = difference(x , B) = 7.93839867452295067096201819367706775665283203125e-09.

--------------------------------

S = approximate_cube_root(2) = 1.259921051561832427978515625.

--------------------------------

The value which was entered for x is -10.

--------------------------------

Computing the approximate cube root of x:

C = 10. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 10.
A := 0.
B := (A + C) / 2 = 5.
epsilon = difference(x , B) = 115.

i := 1.
C := 5.
A := 0.
B := (A + C) / 2 = 2.5.
epsilon = difference(x , B) = 5.625.

i := 2.
C := 2.5.
A := 0.
B := (A + C) / 2 = 1.25.
epsilon = difference(x , B) = 8.046875.

i := 3.
C := 2.5.
A := 1.25.
B := (A + C) / 2 = 1.875.
epsilon = difference(x , B) = 3.408203125.

i := 4.
C := 2.5.
A := 1.875.
B := (A + C) / 2 = 2.1875.
epsilon = difference(x , B) = 0.467529296875.

i := 5.
C := 2.1875.
A := 1.875.
B := (A + C) / 2 = 2.03125.
epsilon = difference(x , B) = 1.619110107421875.

i := 6.
C := 2.1875.
A := 2.03125.
B := (A + C) / 2 = 2.109375.
epsilon = difference(x , B) = 0.614414215087890625.

i := 7.
C := 2.1875.
A := 2.109375.
B := (A + C) / 2 = 2.1484375.
epsilon = difference(x , B) = 0.083277225494384765625.

i := 8.
C := 2.1875.
A := 2.1484375.
B := (A + C) / 2 = 2.16796875.
epsilon = difference(x , B) = 0.189644992351531982421875.

i := 9.
C := 2.16796875.
A := 2.1484375.
B := (A + C) / 2 = 2.158203125.
epsilon = difference(x , B) = 0.052566416561603546142578125.

i := 10.
C := 2.158203125.
A := 2.1484375.
B := (A + C) / 2 = 2.1533203125.
epsilon = difference(x , B) = 0.015509421937167644500732421875.

i := 11.
C := 2.158203125.
A := 2.1533203125.
B := (A + C) / 2 = 2.15576171875.
epsilon = difference(x , B) = 0.018489949288778007030487060546875.

i := 12.
C := 2.15576171875.
A := 2.1533203125.
B := (A + C) / 2 = 2.154541015625.
epsilon = difference(x , B) = 0.001480632126913405954837799072265625.

i := 13.
C := 2.154541015625.
A := 2.1533203125.
B := (A + C) / 2 = 2.1539306640625.
epsilon = difference(x , B) = 0.007016802110229036770761013031005859375.

i := 14.
C := 2.154541015625.
A := 2.1539306640625.
B := (A + C) / 2 = 2.15423583984375.
epsilon = difference(x , B) = 0.002768686878198423073627054691314697265625.

i := 15.
C := 2.154541015625.
A := 2.15423583984375.
B := (A + C) / 2 = 2.154388427734375.
epsilon = difference(x , B) = 0.000644177857935801512212492525577545166015625.

i := 16.
C := 2.154541015625.
A := 2.154388427734375.
B := (A + C) / 2 = 2.1544647216796875.
epsilon = difference(x , B) = 0.000418189512583211353557999245822429656982421875.

i := 17.
C := 2.1544647216796875.
A := 2.154388427734375.
B := (A + C) / 2 = 2.15442657470703125.
epsilon = difference(x , B) = 0.000113003577986159342572136665694415569305419921875.

i := 18.
C := 2.1544647216796875.
A := 2.15442657470703125.
B := (A + C) / 2 = 2.154445648193359375.
epsilon = difference(x , B) = 0.000152590615950243257969987098476849496364593505859375.

i := 19.
C := 2.154445648193359375.
A := 2.15442657470703125.
B := (A + C) / 2 = 2.1544361114501953125.
epsilon = difference(x , B) = 1.9792931147573356032154379136045463383197784423828125e-05.

i := 20.
C := 2.1544361114501953125.
A := 2.15442657470703125.
B := (A + C) / 2 = 2.15443134307861328125.
epsilon = difference(x , B) = 4.6605470377584883034938201262775692157447338104248046875e-05.

i := 21.
C := 2.1544361114501953125.
A := 2.15443134307861328125.
B := (A + C) / 2 = 2.154433727264404296875.
epsilon = difference(x , B) = 1.34063063546192851038796334250946529209613800048828125e-05.

i := 22.
C := 2.1544361114501953125.
A := 2.154433727264404296875.
B := (A + C) / 2 = 2.1544349193572998046875.
epsilon = difference(x , B) = 3.193303211568125632435766192429582588374614715576171875e-06.

i := 23.
C := 2.1544349193572998046875.
A := 2.154433727264404296875.
B := (A + C) / 2 = 2.15443432331085205078125.
epsilon = difference(x , B) = 5.106503867751722991474849777659983374178409576416015625e-06.

i := 24.
C := 2.1544349193572998046875.
A := 2.15443432331085205078125.
B := (A + C) / 2 = 2.154434621334075927734375.
epsilon = difference(x , B) = 9.56600902148226073240522282503661699593067169189453125e-07.

i := 25.
C := 2.1544349193572998046875.
A := 2.154434621334075927734375.
B := (A + C) / 2 = 2.1544347703456878662109375.
epsilon = difference(x , B) = 1.118351011196276612036371034264448098838329315185546875e-06.

i := 26.
C := 2.1544347703456878662109375.
A := 2.154434621334075927734375.
B := (A + C) / 2 = 2.15443469583988189697265625.
epsilon = difference(x , B) = 8.0875018644739615769623242158559150993824005126953125e-08.

i := 27.
C := 2.15443469583988189697265625.
A := 2.154434621334075927734375.
B := (A + C) / 2 = 2.154434658586978912353515625.
epsilon = difference(x , B) = 4.37862950721130961273530601829406805336475372314453125e-07.

i := 28.
C := 2.15443469583988189697265625.
A := 2.154434658586978912353515625.
B := (A + C) / 2 = 2.1544346772134304046630859375.
epsilon = difference(x , B) = 1.78493968280325765451976849362836219370365142822265625e-07.

i := 29.
C := 2.15443469583988189697265625.
A := 2.1544346772134304046630859375.
B := (A + C) / 2 = 2.15443468652665615081787109375.
epsilon = difference(x , B) = 4.88094753781087575816854950971901416778564453125e-08.

i := 30.
C := 2.15443469583988189697265625.
A := 2.15443468652665615081787109375.
B := (A + C) / 2 = 2.154434691183269023895263671875.
epsilon = difference(x , B) = 1.6032771493236508408841700656921602785587310791015625e-08.

i := 31.
C := 2.154434691183269023895263671875.
A := 2.15443468652665615081787109375.
B := (A + C) / 2 = 2.1544346888549625873565673828125.
epsilon = difference(x , B) = 1.63883519775642749749522408819757401943206787109375e-08.

i := 32.
C := 2.154434691183269023895263671875.
A := 2.1544346888549625873565673828125.
B := (A + C) / 2 = 2.15443469001911580562591552734375.
epsilon = difference(x , B) = 1.777902512711815319335073581896722316741943359375e-10.

--------------------------------

S = approximate_cube_root(-10) = -2.15443469001911580562591552734375.

--------------------------------

The value which was entered for x is -1.

--------------------------------

Computing the approximate cube root of x:

C = 1. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 1.
A := 0.
B := (A + C) / 2 = 0.5.
epsilon = difference(x , B) = 0.875.

i := 1.
C := 1.
A := 0.5.
B := (A + C) / 2 = 0.75.
epsilon = difference(x , B) = 0.578125.

i := 2.
C := 1.
A := 0.75.
B := (A + C) / 2 = 0.875.
epsilon = difference(x , B) = 0.330078125.

i := 3.
C := 1.
A := 0.875.
B := (A + C) / 2 = 0.9375.
epsilon = difference(x , B) = 0.176025390625.

i := 4.
C := 1.
A := 0.9375.
B := (A + C) / 2 = 0.96875.
epsilon = difference(x , B) = 0.090850830078125.

i := 5.
C := 1.
A := 0.96875.
B := (A + C) / 2 = 0.984375.
epsilon = difference(x , B) = 0.046146392822265625.

i := 6.
C := 1.
A := 0.984375.
B := (A + C) / 2 = 0.9921875.
epsilon = difference(x , B) = 0.023254871368408203125.

i := 7.
C := 1.
A := 0.9921875.
B := (A + C) / 2 = 0.99609375.
epsilon = difference(x , B) = 0.011673033237457275390625.

i := 8.
C := 1.
A := 0.99609375.
B := (A + C) / 2 = 0.998046875.
epsilon = difference(x , B) = 0.005847938358783721923828125.

i := 9.
C := 1.
A := 0.998046875.
B := (A + C) / 2 = 0.9990234375.
epsilon = difference(x , B) = 0.002926827408373355865478515625.

i := 10.
C := 1.
A := 0.9990234375.
B := (A + C) / 2 = 0.99951171875.
epsilon = difference(x , B) = 0.001464128610678017139434814453125.

i := 11.
C := 1.
A := 0.99951171875.
B := (A + C) / 2 = 0.999755859375.
epsilon = difference(x , B) = 0.000732243075617589056491851806640625.

i := 12.
C := 1.
A := 0.999755859375.
B := (A + C) / 2 = 0.9998779296875.
epsilon = difference(x , B) = 0.000366166235835407860577106475830078125.

i := 13.
C := 1.
A := 0.9998779296875.
B := (A + C) / 2 = 0.99993896484375.
epsilon = difference(x , B) = 0.000183094293106478289701044559478759765625.

i := 14.
C := 1.
A := 0.99993896484375.
B := (A + C) / 2 = 0.999969482421875.
epsilon = difference(x , B) = 9.1549940435697862994857132434844970703125e-05.

i := 15.
C := 1.
A := 0.999969482421875.
B := (A + C) / 2 = 0.9999847412109375.
epsilon = difference(x , B) = 4.5775668699121752069913782179355621337890625e-05.

i := 16.
C := 1.
A := 0.9999847412109375.
B := (A + C) / 2 = 0.99999237060546875.
epsilon = difference(x , B) = 2.2888008971211348807628382928669452667236328125e-05.

i := 17.
C := 1.
A := 0.99999237060546875.
B := (A + C) / 2 = 0.999996185302734375.
epsilon = difference(x , B) = 1.1444048141184826050675837905146181583404541015625e-05.

i := 18.
C := 1.
A := 0.999996185302734375.
B := (A + C) / 2 = 0.9999980926513671875.
epsilon = difference(x , B) = 5.722034984508017618765052247908897697925567626953125e-06.

i := 19.
C := 1.
A := 0.9999980926513671875.
B := (A + C) / 2 = 0.99999904632568359375.
epsilon = difference(x , B) = 2.861020220735512042953274658430018462240695953369140625e-06.

i := 20.
C := 1.
A := 0.99999904632568359375.
B := (A + C) / 2 = 0.999999523162841796875.
epsilon = difference(x , B) = 1.430510792488457090521070114164103870280086994171142578125e-06.

i := 21.
C := 1.
A := 0.999999523162841796875.
B := (A + C) / 2 = 0.9999997615814208984375.
epsilon = difference(x , B) = 7.1525556677443091757595539093017578125e-07.

i := 22.
C := 1.
A := 0.9999997615814208984375.
B := (A + C) / 2 = 0.99999988079071044921875.
epsilon = difference(x , B) = 3.576278260197796043939888477325439453125e-07.

i := 23.
C := 1.
A := 0.99999988079071044921875.
B := (A + C) / 2 = 0.999999940395355224609375.
epsilon = difference(x , B) = 1.78813923668030838598497211933135986328125e-07.

i := 24.
C := 1.
A := 0.999999940395355224609375.
B := (A + C) / 2 = 0.9999999701976776123046875.
epsilon = difference(x , B) = 8.940696449855067839962430298328399658203125e-08.

i := 25.
C := 1.
A := 0.9999999701976776123046875.
B := (A + C) / 2 = 0.99999998509883880615234375.
epsilon = difference(x , B) = 4.47034829154091539749060757458209991455078125e-08.

i := 26.
C := 1.
A := 0.99999998509883880615234375.
B := (A + C) / 2 = 0.999999992549419403076171875.
epsilon = difference(x , B) = 2.2351741624238030681226518936455249786376953125e-08.

i := 27.
C := 1.
A := 0.999999992549419403076171875.
B := (A + C) / 2 = 0.9999999962747097015380859375.
epsilon = difference(x , B) = 1.117587085375237876405662973411381244659423828125e-08.

i := 28.
C := 1.
A := 0.9999999962747097015380859375.
B := (A + C) / 2 = 0.99999999813735485076904296875.
epsilon = difference(x , B) = 5.5879354372845302378891574335284531116485595703125e-09.

--------------------------------

S = approximate_cube_root(-1) = -0.99999999813735485076904296875.

--------------------------------

The value which was entered for x is 0.

--------------------------------

Computing the approximate cube root of x:

C = 0. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 0.
A := 0.
B := (A + C) / 2 = 0.
epsilon = difference(x , B) = 0.

--------------------------------

S = approximate_cube_root(0) = 0.

--------------------------------

The value which was entered for x is 0.5.

--------------------------------

Computing the approximate cube root of x:

C = 0. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 0.
A := 0.
B := (A + C) / 2 = 0.
epsilon = difference(x , B) = 0.

--------------------------------

S = approximate_cube_root(0.5) = 0.

--------------------------------

The value which was entered for x is -81.

--------------------------------

Computing the approximate cube root of x:

C = 81. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 81.
A := 0.
B := (A + C) / 2 = 40.5.
epsilon = difference(x , B) = 66349.125.

i := 1.
C := 40.5.
A := 0.
B := (A + C) / 2 = 20.25.
epsilon = difference(x , B) = 8222.765625.

i := 2.
C := 20.25.
A := 0.
B := (A + C) / 2 = 10.125.
epsilon = difference(x , B) = 956.970703125.

i := 3.
C := 10.125.
A := 0.
B := (A + C) / 2 = 5.0625.
epsilon = difference(x , B) = 48.746337890625.

i := 4.
C := 5.0625.
A := 0.
B := (A + C) / 2 = 2.53125.
epsilon = difference(x , B) = 64.781707763671875.

i := 5.
C := 5.0625.
A := 2.53125.
B := (A + C) / 2 = 3.796875.
epsilon = difference(x , B) = 26.263263702392578125.

i := 6.
C := 5.0625.
A := 3.796875.
B := (A + C) / 2 = 4.4296875.
epsilon = difference(x , B) = 5.919909954071044921875.

i := 7.
C := 4.4296875.
A := 3.796875.
B := (A + C) / 2 = 4.11328125.
epsilon = difference(x , B) = 11.407054603099822998046875.

i := 8.
C := 4.4296875.
A := 4.11328125.
B := (A + C) / 2 = 4.271484375.
epsilon = difference(x , B) = 3.064295388758182525634765625.

i := 9.
C := 4.4296875.
A := 4.271484375.
B := (A + C) / 2 = 4.3505859375.
epsilon = difference(x , B) = 1.346141687594354152679443359375.

i := 10.
C := 4.3505859375.
A := 4.271484375.
B := (A + C) / 2 = 4.31103515625.
epsilon = difference(x , B) = 0.879307645722292363643646240234375.

i := 11.
C := 4.3505859375.
A := 4.31103515625.
B := (A + C) / 2 = 4.330810546875.
epsilon = difference(x , B) = 0.228336121697793714702129364013671875.

i := 12.
C := 4.330810546875.
A := 4.31103515625.
B := (A + C) / 2 = 4.3209228515625.
epsilon = difference(x , B) = 0.326753086765165789984166622161865234375.

i := 13.
C := 4.330810546875.
A := 4.3209228515625.
B := (A + C) / 2 = 4.32586669921875.
epsilon = difference(x , B) = 0.049525676228995507699437439441680908203125.

i := 14.
C := 4.330810546875.
A := 4.32586669921875.
B := (A + C) / 2 = 4.328338623046875.
epsilon = difference(x , B) = 0.089325878997186691776732914149761199951171875.

i := 15.
C := 4.328338623046875.
A := 4.32586669921875.
B := (A + C) / 2 = 4.3271026611328125.
epsilon = difference(x , B) = 0.019880271113965619633745518513023853302001953125.

i := 16.
C := 4.3271026611328125.
A := 4.32586669921875.
B := (A + C) / 2 = 4.32648468017578125.
epsilon = difference(x , B) = 0.014827659417025795818290134775452315807342529296875.

i := 17.
C := 4.3271026611328125.
A := 4.32648468017578125.
B := (A + C) / 2 = 4.326793670654296875.
epsilon = difference(x , B) = 0.002525066545089493796893975741113536059856414794921875.

i := 18.
C := 4.326793670654296875.
A := 4.32648468017578125.
B := (A + C) / 2 = 4.3266391754150390625.
epsilon = difference(x , B) = 0.006151606250750417392847424480351037345826625823974609375.

i := 19.
C := 4.326793670654296875.
A := 4.3266391754150390625.
B := (A + C) / 2 = 4.32671642303466796875.
epsilon = difference(x , B) = 0.001813347307908885763794160084216855466365814208984375.

i := 20.
C := 4.326793670654296875.
A := 4.32671642303466796875.
B := (A + C) / 2 = 4.326755046844482421875.
epsilon = difference(x , B) = 0.00035584025464784063697010196847259066998958587646484375.

i := 21.
C := 4.326755046844482421875.
A := 4.32671642303466796875.
B := (A + C) / 2 = 4.3267357349395751953125.
epsilon = difference(x , B) = 0.000728758367594527223243261460083886049687862396240234375.

i := 22.
C := 4.326755046844482421875.
A := 4.3267357349395751953125.
B := (A + C) / 2 = 4.32674539089202880859375.
epsilon = difference(x , B) = 0.000186460266717043687823007758197491057217121124267578125.

i := 23.
C := 4.326755046844482421875.
A := 4.32674539089202880859375.
B := (A + C) / 2 = 4.326750218868255615234375.
epsilon = difference(x , B) = 8.4689691404134237462386636252631433308124542236328125e-05.

i := 24.
C := 4.326750218868255615234375.
A := 4.32674539089202880859375.
B := (A + C) / 2 = 4.3267478048801422119140625.
epsilon = difference(x , B) = 5.0885363296726549009463269612751901149749755859375e-05.

i := 25.
C := 4.326750218868255615234375.
A := 4.3267478048801422119140625.
B := (A + C) / 2 = 4.32674901187419891357421875.
epsilon = difference(x , B) = 1.69021451436324188222215525456704199314117431640625e-05.

i := 26.
C := 4.32674901187419891357421875.
A := 4.3267478048801422119140625.
B := (A + C) / 2 = 4.326748408377170562744140625.
epsilon = difference(x , B) = 1.6991613804064054082942902823560871183872222900390625e-05.

i := 27.
C := 4.32674901187419891357421875.
A := 4.326748408377170562744140625.
B := (A + C) / 2 = 4.3267487101256847381591796875.
epsilon = difference(x , B) = 4.47355121002690481191166327334940433502197265625e-08.

i := 28.
C := 4.32674901187419891357421875.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.32674886099994182586669921875.
epsilon = difference(x , B) = 8.42870452029409467087361917947418987751007080078125e-06.

i := 29.
C := 4.32674886099994182586669921875.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.326748785562813282012939453125.
epsilon = difference(x , B) = 4.191984430225448310380897964932955801486968994140625e-06.

i := 30.
C := 4.326748785562813282012939453125.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.3267487478442490100860595703125.
epsilon = difference(x , B) = 2.073624440601662399785709567368030548095703125e-06.

i := 31.
C := 4.3267487478442490100860595703125.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.32674872898496687412261962890625.
epsilon = difference(x , B) = 1.014444459636332229734989596181549131870269775390625e-06.

i := 32.
C := 4.32674872898496687412261962890625.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.326748719555325806140899658203125.
epsilon = difference(x , B) = 4.8485447261270575580738295684568583965301513671875e-07.

i := 33.
C := 4.326748719555325806140899658203125.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.3267487148405052721500396728515625.
epsilon = difference(x , B) = 2.20059479971723703783936798572540283203125e-07.

i := 34.
C := 4.3267487148405052721500396728515625.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.32674871248309500515460968017578125.
epsilon = difference(x , B) = 8.76619838663383887933377991430461406707763671875e-08.

i := 35.
C := 4.32674871248309500515460968017578125.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.326748711304389871656894683837890625.
epsilon = difference(x , B) = 2.1463235862217988625388898071832954883575439453125e-08.

i := 36.
C := 4.326748711304389871656894683837890625.
A := 4.3267487101256847381591796875.
B := (A + C) / 2 = 4.3267487107150373049080371856689453125.
epsilon = difference(x , B) = 1.1636138115556082794910253142006695270538330078125e-08.

i := 37.
C := 4.326748711304389871656894683837890625.
A := 4.3267487107150373049080371856689453125.
B := (A + C) / 2 = 4.32674871100971358828246593475341796875.
epsilon = difference(x , B) = 4.9135488733309529152393224649131298065185546875e-09.

--------------------------------

S = approximate_cube_root(-81) = -4.32674871100971358828246593475341796875.

--------------------------------

The value which was entered for x is 27.

--------------------------------

Computing the approximate cube root of x:

C = 27. // real number to take the cube root of
B = 0. // variable for storing the approximate cube root of x
A = 0. // number to add to C before dividing the sum of A and C by 2 for each while loop iteration, i
epsilon = 0. // variable for storing the difference between the input value and B raised to the power of 3

i := 0.
C := 27.
A := 0.
B := (A + C) / 2 = 13.5.
epsilon = difference(x , B) = 2433.375.

i := 1.
C := 13.5.
A := 0.
B := (A + C) / 2 = 6.75.
epsilon = difference(x , B) = 280.546875.

i := 2.
C := 6.75.
A := 0.
B := (A + C) / 2 = 3.375.
epsilon = difference(x , B) = 11.443359375.

i := 3.
C := 3.375.
A := 0.
B := (A + C) / 2 = 1.6875.
epsilon = difference(x , B) = 22.194580078125.

i := 4.
C := 3.375.
A := 1.6875.
B := (A + C) / 2 = 2.53125.
epsilon = difference(x , B) = 10.781707763671875.

i := 5.
C := 3.375.
A := 2.53125.
B := (A + C) / 2 = 2.953125.
epsilon = difference(x , B) = 1.245952606201171875.

i := 6.
C := 3.375.
A := 2.953125.
B := (A + C) / 2 = 3.1640625.
epsilon = difference(x , B) = 4.676352024078369140625.

i := 7.
C := 3.1640625.
A := 2.953125.
B := (A + C) / 2 = 3.05859375.
epsilon = difference(x , B) = 1.613131463527679443359375.

i := 8.
C := 3.05859375.
A := 2.953125.
B := (A + C) / 2 = 3.005859375.
epsilon = difference(x , B) = 0.158512316644191741943359375.

i := 9.
C := 3.005859375.
A := 2.953125.
B := (A + C) / 2 = 2.9794921875.
epsilon = difference(x , B) = 0.549934429116547107696533203125.

i := 10.
C := 3.005859375.
A := 2.9794921875.
B := (A + C) / 2 = 2.99267578125.
epsilon = difference(x , B) = 0.197271501529030501842498779296875.

i := 11.
C := 3.005859375.
A := 2.99267578125.
B := (A + C) / 2 = 2.999267578125.
epsilon = difference(x , B) = 0.019770563041674904525279998779296875.

i := 12.
C := 3.005859375.
A := 2.999267578125.
B := (A + C) / 2 = 3.0025634765625.
epsilon = difference(x , B) = 0.069273026741939247585833072662353515625.

i := 13.
C := 3.0025634765625.
A := 2.999267578125.
B := (A + C) / 2 = 3.00091552734375.
epsilon = difference(x , B) = 0.024726782761490539996884763240814208984375.

i := 14.
C := 3.00091552734375.
A := 2.999267578125.
B := (A + C) / 2 = 3.000091552734375.
epsilon = difference(x , B) = 0.002471999266020930008380673825740814208984375.

i := 15.
C := 3.000091552734375.
A := 2.999267578125.
B := (A + C) / 2 = 2.9996795654296875.
epsilon = difference(x , B) = 0.008650809326514519170814310200512409210205078125.

i := 16.
C := 3.000091552734375.
A := 2.9996795654296875.
B := (A + C) / 2 = 2.99988555908203125.
epsilon = difference(x , B) = 0.003089786916141701311744327540509402751922607421875.

i := 17.
C := 3.000091552734375.
A := 2.99988555908203125.
B := (A + C) / 2 = 2.999988555908203125.
epsilon = difference(x , B) = 0.000308989299811990303368247623438946902751922607421875.

i := 18.
C := 3.000091552734375.
A := 2.999988555908203125.
B := (A + C) / 2 = 3.0000400543212890625.
epsilon = difference(x , B) = 0.001081481114006833943452789981165551580488681793212890625.

i := 19.
C := 3.0000400543212890625.
A := 2.999988555908203125.
B := (A + C) / 2 = 3.00001430511474609375.
epsilon = difference(x , B) = 0.00038623993987423055340713062832946889102458953857421875.

i := 20.
C := 3.00001430511474609375.
A := 2.999988555908203125.
B := (A + C) / 2 = 3.000001430511474609375.
epsilon = difference(x , B) = 3.86238282317243053487487713937298394739627838134765625e-05.

i := 21.
C := 3.000001430511474609375.
A := 2.999988555908203125.
B := (A + C) / 2 = 2.9999949932098388671875.
epsilon = difference(x , B) = 0.00013518310873918137904325931231142021715641021728515625.

i := 22.
C := 3.000001430511474609375.
A := 2.9999949932098388671875.
B := (A + C) / 2 = 2.99999821186065673828125.
epsilon = difference(x , B) = 4.827973349109081213637040264075039885938167572021484375e-05.

i := 23.
C := 3.000001430511474609375.
A := 2.99999821186065673828125.
B := (A + C) / 2 = 2.999999821186065673828125.
epsilon = difference(x , B) = 4.827975939036832642159424722194671630859375e-06.

i := 24.
C := 3.000001430511474609375.
A := 2.999999821186065673828125.
B := (A + C) / 2 = 3.0000006258487701416015625.
epsilon = difference(x , B) = 1.689792031900338997729704715311527252197265625e-05.

i := 25.
C := 3.0000006258487701416015625.
A := 2.999999821186065673828125.
B := (A + C) / 2 = 3.00000022351741790771484375.
epsilon = difference(x , B) = 6.0349707331486257544383988715708255767822265625e-06.

i := 26.
C := 3.00000022351741790771484375.
A := 2.999999821186065673828125.
B := (A + C) / 2 = 3.000000022351741790771484375.
epsilon = difference(x , B) = 6.03497032847233327856883988715708255767822265625e-07.

i := 27.
C := 3.000000022351741790771484375.
A := 2.999999821186065673828125.
B := (A + C) / 2 = 2.9999999217689037322998046875.
epsilon = difference(x , B) = 2.11223954414696546422192113823257386684417724609375e-06.

i := 28.
C := 3.000000022351741790771484375.
A := 2.9999999217689037322998046875.
B := (A + C) / 2 = 2.99999997206032276153564453125.
epsilon = difference(x , B) = 7.543712784129075199501812676317058503627777099609375e-07.

i := 29.
C := 3.000000022351741790771484375.
A := 2.99999997206032276153564453125.
B := (A + C) / 2 = 2.999999997206032276153564453125.
epsilon = difference(x , B) = 7.543712847446482072655271622352302074432373046875e-08.

i := 30.
C := 3.000000022351741790771484375.
A := 2.999999997206032276153564453125.
B := (A + C) / 2 = 3.0000000097788870334625244140625.
epsilon = difference(x , B) = 2.640299507639110032641838188283145427703857421875e-07.

i := 31.
C := 3.0000000097788870334625244140625.
A := 2.999999997206032276153564453125.
B := (A + C) / 2 = 3.00000000349245965480804443359375.
epsilon = difference(x , B) = 9.429641078910477869357009694795124232769012451171875e-08.

i := 32.
C := 3.00000000349245965480804443359375.
A := 2.999999997206032276153564453125.
B := (A + C) / 2 = 3.000000000349245965480804443359375.
epsilon = difference(x , B) = 9.429641067981719970703125e-09.

--------------------------------

S = approximate_cube_root(27) = 3.000000000349245965480804443359375.

--------------------------------
End Of Program
--------------------------------</pre>
<hr>
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