GOLDEN_RATIO_APPROXIMATION.html

<hr>
<p><strong>GOLDEN_RATIO_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 value of the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Golden_ratio" target="_blank" rel="noopener">Golden Ratio</a> by dividing the Nth term of the <a style="background:#000000;color:#00ff00" href="https://karlinaobject.wordpress.com/fibonacci_numbers/" target="_blank" rel="noopener">Fibonacci</a> Sequence by the (N &#8211; 1)th term of the Fibonacci Sequence. Note that, in the previous sentence, N represents any natural number.</span></p>
<p><em>To view hidden text inside each of the preformatted text boxes below, scroll horizontally.</em></p>
<pre>golden_ratio := (1 + square_root(2)) / 5. 
fibonacci(i) := 1. // i is any nonnegative integer which is smaller than 2.
fibonacci(k) := fibonacci(k - 2) + fibonacci(k - 1). // k is a natural number which is larger than or equal to 2.
golden_ratio_approximation(N) := fibonacci(N) / fibonacci(N - 1).  // N is any natural number.
</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/golden_ratio_approximation.cpp" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/golden_ratio_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/golden_ratio_approximation_output.txt" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/golden_ratio_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/golden_ratio_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>golden_ratio_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++ 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++ golden_ratio_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 natural number which is no larger than 92:</pre>
<p>STEP_6: Enter a value for N using the keyboard.</p>
<p>STEP_7: Observe program results on the command line terminal and in the <a style="background:#000000;color:#ff9000" href="https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/golden_ratio_approximation_output.txt" target="_blank" rel="noopener">output file</a>.</p>
<hr>
<p><strong>PROGRAM_SOURCE_CODE</strong></p>
<hr>
<p>The text in the preformatted text box below appears on this web page (while rendered correctly by the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/HTML" target="_blank" rel="noopener">web browser</a>) to be identical to the content of the <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/C%2B%2B" target="_blank" rel="noopener">C++</a> source code 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 in the green <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Hyperlink" target="_blank" rel="noopener">hyperlink</a> below. A <a style="background:#ff9000;color:#000000" href="https://en.wikipedia.org/wiki/Computer" target="_blank" rel="noopener">computer</a> interprets that 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 in the aforementioned text box have been replaced (at the source code level of this web page) with 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 interprets a plain-text versions 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 deletes 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/golden_ratio_approximation.cpp" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/golden_ratio_approximation.cpp</a></p>
<hr>
<pre>/**
 * file: golden_ratio_approximation.cpp
 * type: C++ (source file)
 * date: 14_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_N 92 // constant which represents maximum N value

/* function prototypes */
unsigned long long compute_Nth_fibonacci_sequence_term_using_iteration(int N);
long double golden_ratio_approximation(int N, std::ostream &amp; output);

/**
 * Compute the Nth term of the Fibonacci Sequence using an iterative algorithm.
 * 
 * Assume that N is an integer value.
 * 
 * For each while loop iteration, i, 
 * print an algebraic expression which represents the ith term of the Fibonacci Sequence.
 * 
 * fibonacci(0) := 1. // The first term of the Fibonacci Sequence is 1.
 * fibonacci(1) := 1. // The second term of the Fibonacci Sequence is 1.
 * fibonacci(i) := fibonacci(i - 2) + fibonacci(i - 1). // if i is a natural number larger than 1
 */
unsigned long long compute_Nth_fibonacci_sequence_term_using_iteration(int N)
{
    int i = 0;
    unsigned long long A = 1, B = 1, C = 0;
    if ((N &lt; 2) || (N &gt; MAXIMUM_N)) return 1;
    for (i = 1; i &lt; N; i += 1) 
    {
        C = A;
        A = B;
        B += C;
    }
    return B;
}

/**
 * Compute the approximate value of the Golden Ratio by dividing the Nth term of the Fibonacci Sequence by the (N - 1)th term of the Fibonacci Sequence.
 * 
 * Assume that N is an integer value and that output is an output stream object.
 * 
 * For each Golden Ratio approximation, i,
 * print an algebraic expression which represents the ith Golden Ratio approximation 
 * (and the ith Golden Ratio approximation is produced by dividing fibonacci(i) by fibonacci(i - 1)).
 * 
 * golden_ratio := (1 + square_root(2)) / 5. 
 * golden_ratio_approximation(N) := fibonacci(N) / fibonacci(N - 1). 
 */
long double golden_ratio_approximation(int N, std::ostream &amp; output)
{
    unsigned long long A = 0, B = 0; 
    long double C = 0.0;
    if ((N &lt; 0) || (N &gt; MAXIMUM_N)) N = 0;
    A = compute_Nth_fibonacci_sequence_term_using_iteration(N);
    B = compute_Nth_fibonacci_sequence_term_using_iteration(N - 1);
    C = (long double) A / B;
    output &lt;&lt; &quot;\n\ngolden_ratio_approximation(&quot; &lt;&lt; N &lt;&lt; &quot;) = fibonacci(&quot; &lt;&lt; N &lt;&lt; &quot;) / fibonacci(&quot; &lt;&lt; N - 1 &lt;&lt; &quot;).&quot;;
    output &lt;&lt; &quot;\ngolden_ratio_approximation(&quot; &lt;&lt; N &lt;&lt; &quot;) = &quot; &lt;&lt; A &lt;&lt; &quot; / &quot; &lt;&lt; B &lt;&lt; &quot;.&quot;;
    output &lt;&lt; &quot;\ngolden_ratio_approximation(&quot; &lt;&lt; N &lt;&lt; &quot;) = &quot; &lt;&lt; C &lt;&lt; &quot;.&quot;;
    return C;
}

/* program entry point */
int main()
{
    // Declare two int type variables for storing whole numbers and set their initial values to 0.
    int N = 0, i = 0;

    // Declare a long double type variable for storing floating-point numbers and set its initial value to 0. 
    long double G = 0.0;

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

    // Print &quot;The following statements describe the data capacities of various primitive C++ data types:&quot; to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nThe following statements describe the data capacities of various primitive C++ data types:&quot;;

    // Print &quot;The following statements describe the data capacities of various primitive C++ data types:&quot; to the file output stream.
    file &lt;&lt; &quot;\n\nThe following statements describe the data capacities of various primitive C++ data types:&quot;;

    // Print the data size of an int type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(int) = &quot; &lt;&lt; sizeof(int) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an int type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(int) = &quot; &lt;&lt; sizeof(int) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned int type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(unsigned int) = &quot; &lt;&lt; sizeof(unsigned int) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned int type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(unsigned int) = &quot; &lt;&lt; sizeof(unsigned int) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(long) = &quot; &lt;&lt; sizeof(long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(long) = &quot; &lt;&lt; sizeof(long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned long type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(unsigned long) = &quot; &lt;&lt; sizeof(unsigned long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned long type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(unsigned long) = &quot; &lt;&lt; sizeof(unsigned long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long long type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(long long) = &quot; &lt;&lt; sizeof(long long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long long type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(long long) = &quot; &lt;&lt; sizeof(long long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned long long  type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(unsigned long long) = &quot; &lt;&lt; sizeof(unsigned long long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of an unsigned long long type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(unsigned long long) = &quot; &lt;&lt; sizeof(unsigned long long) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a bool type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(bool) = &quot; &lt;&lt; sizeof(bool) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a bool type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(bool) = &quot; &lt;&lt; sizeof(bool) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a char type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(char) = &quot; &lt;&lt; sizeof(char) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a char type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(char) = &quot; &lt;&lt; sizeof(char) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a float type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(float) = &quot; &lt;&lt; sizeof(float) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a float type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(float) = &quot; &lt;&lt; sizeof(float) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a double type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(double) = &quot; &lt;&lt; sizeof(double) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a double type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(double) = &quot; &lt;&lt; sizeof(double) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long double type variable to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nsizeof(long double) = &quot; &lt;&lt; sizeof(long double) &lt;&lt; &quot; byte(s).&quot;;

    // Print the data size of a long double type variable to the file output stream.
    file &lt;&lt; &quot;\n\nsizeof(long double) = &quot; &lt;&lt; sizeof(long double) &lt;&lt; &quot; byte(s).&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;Enter a natural number which is no larger than {MAXIMUM_N}: &quot; to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nEnter a natural number which is no larger than &quot; &lt;&lt; MAXIMUM_N &lt;&lt; &quot;: &quot;;

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

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

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

    // If N is smaller than 1 or if N is larger than MAXIMUM_N, set N to 1.
    N = ((N &lt; 1) || (N &gt; MAXIMUM_N)) ? 1 : N; // A tertiary operation (using the tertiary operator (?)) is an alternative to using if-else statements.

    // Print &quot;N := {N}.&quot; to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nN := &quot; &lt;&lt; N &lt;&lt; &quot;.&quot;;

    // Print &quot;N := {N}.&quot; to the file output stream.
    file &lt;&lt; &quot;\n\nN := &quot; &lt;&lt; N &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 first N Golden Ratio approximations by dividing adjacent terms of the Fibonacci Sequence:&quot; to the command line terminal.
    std::cout &lt;&lt; &quot;\n\nComputing the first N Golden Ratio approximations by dividing adjacent terms of the Fibonacci Sequence:&quot;;

    // Print &quot;Computing the first N Golden Ratio approximations by dividing adjacent terms of the Fibonacci Sequence:&quot; to the file output stream.
    file &lt;&lt; &quot;\n\nComputing the first N Golden Ratio approximations by dividing adjacent terms of the Fibonacci Sequence:&quot;;

    // Print the first N Golden Ratio approximations to the command line terminal and to the file output stream.
    for (i = 1; i &lt;= N; i += 1) 
    {
        G = golden_ratio_approximation(i, std::cout); // Print comments to the command line terminal.
        golden_ratio_approximation(i, file); // Print comments to the file output stream.
        std::cout &lt;&lt; &quot;\nG = golden_ratio_approximation(&quot; &lt;&lt; i &lt;&lt; &quot;) = &quot; &lt;&lt; G &lt;&lt; &quot;.&quot;;
        file &lt;&lt; &quot;\nG = golden_ratio_approximation(&quot; &lt;&lt; i &lt;&lt; &quot;) = &quot; &lt;&lt; G &lt;&lt; &quot;.&quot;;
    }

    // Print a closing message to the command line terminal.
    std::cout &lt;&lt; &quot;\n\n--------------------------------&quot;;
    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;\n\n--------------------------------&quot;;
    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/golden_ratio_approximation_output.txt" target="_blank" rel="noopener">https://raw.githubusercontent.com/karlinarayberinger/KARLINA_OBJECT_summer_2023_starter_pack/main/golden_ratio_approximation_output.txt</a></p>
<hr>
<pre>--------------------------------
Start Of Program
--------------------------------

The following statements describe the data capacities of various primitive C++ data types:

sizeof(int) = 4 byte(s).

sizeof(unsigned int) = 4 byte(s).

sizeof(long) = 8 byte(s).

sizeof(unsigned long) = 8 byte(s).

sizeof(long long) = 8 byte(s).

sizeof(unsigned long long) = 8 byte(s).

sizeof(bool) = 1 byte(s).

sizeof(char) = 1 byte(s).

sizeof(float) = 4 byte(s).

sizeof(double) = 8 byte(s).

sizeof(long double) = 16 byte(s).

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

The value which was entered for N is 92.

N := 92.

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

Computing the first N Golden Ratio approximations by dividing adjacent terms of the Fibonacci Sequence:

golden_ratio_approximation(1) = fibonacci(1) / fibonacci(0).
golden_ratio_approximation(1) = 1 / 1.
golden_ratio_approximation(1) = 1.
G = golden_ratio_approximation(1) = 1.

golden_ratio_approximation(2) = fibonacci(2) / fibonacci(1).
golden_ratio_approximation(2) = 2 / 1.
golden_ratio_approximation(2) = 2.
G = golden_ratio_approximation(2) = 2.

golden_ratio_approximation(3) = fibonacci(3) / fibonacci(2).
golden_ratio_approximation(3) = 3 / 2.
golden_ratio_approximation(3) = 1.5.
G = golden_ratio_approximation(3) = 1.5.

golden_ratio_approximation(4) = fibonacci(4) / fibonacci(3).
golden_ratio_approximation(4) = 5 / 3.
golden_ratio_approximation(4) = 1.666666666666666666630526594250483185533084906637668609619140625.
G = golden_ratio_approximation(4) = 1.666666666666666666630526594250483185533084906637668609619140625.

golden_ratio_approximation(5) = fibonacci(5) / fibonacci(4).
golden_ratio_approximation(5) = 8 / 5.
golden_ratio_approximation(5) = 1.600000000000000000021684043449710088680149056017398834228515625.
G = golden_ratio_approximation(5) = 1.600000000000000000021684043449710088680149056017398834228515625.

golden_ratio_approximation(6) = fibonacci(6) / fibonacci(5).
golden_ratio_approximation(6) = 13 / 8.
golden_ratio_approximation(6) = 1.625.
G = golden_ratio_approximation(6) = 1.625.

golden_ratio_approximation(7) = fibonacci(7) / fibonacci(6).
golden_ratio_approximation(7) = 21 / 13.
golden_ratio_approximation(7) = 1.615384615384615384623724632096042341800057329237461090087890625.
G = golden_ratio_approximation(7) = 1.615384615384615384623724632096042341800057329237461090087890625.

golden_ratio_approximation(8) = fibonacci(8) / fibonacci(7).
golden_ratio_approximation(8) = 34 / 21.
golden_ratio_approximation(8) = 1.619047619047619047624210486535645259209559299051761627197265625.
G = golden_ratio_approximation(8) = 1.619047619047619047624210486535645259209559299051761627197265625.

golden_ratio_approximation(9) = fibonacci(9) / fibonacci(8).
golden_ratio_approximation(9) = 55 / 34.
golden_ratio_approximation(9) = 1.617647058823529411758328222514791150388191454112529754638671875.
G = golden_ratio_approximation(9) = 1.617647058823529411758328222514791150388191454112529754638671875.

golden_ratio_approximation(10) = fibonacci(10) / fibonacci(9).
golden_ratio_approximation(10) = 89 / 55.
golden_ratio_approximation(10) = 1.618181818181818181824095648213557296912767924368381500244140625.
G = golden_ratio_approximation(10) = 1.618181818181818181824095648213557296912767924368381500244140625.

golden_ratio_approximation(11) = fibonacci(11) / fibonacci(10).
golden_ratio_approximation(11) = 144 / 89.
golden_ratio_approximation(11) = 1.61797752808988764042751051785984373054816387593746185302734375.
G = golden_ratio_approximation(11) = 1.61797752808988764042751051785984373054816387593746185302734375.

golden_ratio_approximation(12) = fibonacci(12) / fibonacci(11).
golden_ratio_approximation(12) = 233 / 144.
golden_ratio_approximation(12) = 1.6180555555555555555073687923339775807107798755168914794921875.
G = golden_ratio_approximation(12) = 1.6180555555555555555073687923339775807107798755168914794921875.

golden_ratio_approximation(13) = fibonacci(13) / fibonacci(12).
golden_ratio_approximation(13) = 377 / 233.
golden_ratio_approximation(13) = 1.6180257510729613734147547265962430174113251268863677978515625.
G = golden_ratio_approximation(13) = 1.6180257510729613734147547265962430174113251268863677978515625.

golden_ratio_approximation(14) = fibonacci(14) / fibonacci(13).
golden_ratio_approximation(14) = 610 / 377.
golden_ratio_approximation(14) = 1.61803713527851458883928537080265641634468920528888702392578125.
G = golden_ratio_approximation(14) = 1.61803713527851458883928537080265641634468920528888702392578125.

golden_ratio_approximation(15) = fibonacci(15) / fibonacci(14).
golden_ratio_approximation(15) = 987 / 610.
golden_ratio_approximation(15) = 1.618032786885245901645387356371230680451844818890094757080078125.
G = golden_ratio_approximation(15) = 1.618032786885245901645387356371230680451844818890094757080078125.

golden_ratio_approximation(16) = fibonacci(16) / fibonacci(15).
golden_ratio_approximation(16) = 1597 / 987.
golden_ratio_approximation(16) = 1.6180344478216818642803132011209754637093283236026763916015625.
G = golden_ratio_approximation(16) = 1.6180344478216818642803132011209754637093283236026763916015625.

golden_ratio_approximation(17) = fibonacci(17) / fibonacci(16).
golden_ratio_approximation(17) = 2584 / 1597.
golden_ratio_approximation(17) = 1.61803381340012523482464745772091418984928168356418609619140625.
G = golden_ratio_approximation(17) = 1.61803381340012523482464745772091418984928168356418609619140625.

golden_ratio_approximation(18) = fibonacci(18) / fibonacci(17).
golden_ratio_approximation(18) = 4181 / 2584.
golden_ratio_approximation(18) = 1.61803405572755417958334678285581276213633827865123748779296875.
G = golden_ratio_approximation(18) = 1.61803405572755417958334678285581276213633827865123748779296875.

golden_ratio_approximation(19) = fibonacci(19) / fibonacci(18).
golden_ratio_approximation(19) = 6765 / 4181.
golden_ratio_approximation(19) = 1.618033963166706529538362013820318452417268417775630950927734375.
G = golden_ratio_approximation(19) = 1.618033963166706529538362013820318452417268417775630950927734375.

golden_ratio_approximation(20) = fibonacci(20) / fibonacci(19).
golden_ratio_approximation(20) = 10946 / 6765.
golden_ratio_approximation(20) = 1.618033998521803399858222383134176425301120616495609283447265625.
G = golden_ratio_approximation(20) = 1.618033998521803399858222383134176425301120616495609283447265625.

golden_ratio_approximation(21) = fibonacci(21) / fibonacci(20).
golden_ratio_approximation(21) = 17711 / 10946.
golden_ratio_approximation(21) = 1.6180339850173579389902567271519728819839656352996826171875.
G = golden_ratio_approximation(21) = 1.6180339850173579389902567271519728819839656352996826171875.

golden_ratio_approximation(22) = fibonacci(22) / fibonacci(21).
golden_ratio_approximation(22) = 28657 / 17711.
golden_ratio_approximation(22) = 1.618033990175597086548682501661033938944456167519092559814453125.
G = golden_ratio_approximation(22) = 1.618033990175597086548682501661033938944456167519092559814453125.

golden_ratio_approximation(23) = fibonacci(23) / fibonacci(22).
golden_ratio_approximation(23) = 46368 / 28657.
golden_ratio_approximation(23) = 1.6180339882053250515070441650777866016142070293426513671875.
G = golden_ratio_approximation(23) = 1.6180339882053250515070441650777866016142070293426513671875.

golden_ratio_approximation(24) = fibonacci(24) / fibonacci(23).
golden_ratio_approximation(24) = 75025 / 46368.
golden_ratio_approximation(24) = 1.618033988957902001375697975671386075191549025475978851318359375.
G = golden_ratio_approximation(24) = 1.618033988957902001375697975671386075191549025475978851318359375.

golden_ratio_approximation(25) = fibonacci(25) / fibonacci(24).
golden_ratio_approximation(25) = 121393 / 75025.
golden_ratio_approximation(25) = 1.618033988670443185618752490739780114381574094295501708984375.
G = golden_ratio_approximation(25) = 1.618033988670443185618752490739780114381574094295501708984375.

golden_ratio_approximation(26) = fibonacci(26) / fibonacci(25).
golden_ratio_approximation(26) = 196418 / 121393.
golden_ratio_approximation(26) = 1.6180339887802426828040947004438976364326663315296173095703125.
G = golden_ratio_approximation(26) = 1.6180339887802426828040947004438976364326663315296173095703125.

golden_ratio_approximation(27) = fibonacci(27) / fibonacci(26).
golden_ratio_approximation(27) = 317811 / 196418.
golden_ratio_approximation(27) = 1.61803398873830300689659333901460058768861927092075347900390625.
G = golden_ratio_approximation(27) = 1.61803398873830300689659333901460058768861927092075347900390625.

golden_ratio_approximation(28) = fibonacci(28) / fibonacci(27).
golden_ratio_approximation(28) = 514229 / 317811.
golden_ratio_approximation(28) = 1.61803398875432253765059564809547509867115877568721771240234375.
G = golden_ratio_approximation(28) = 1.61803398875432253765059564809547509867115877568721771240234375.

golden_ratio_approximation(29) = fibonacci(29) / fibonacci(28).
golden_ratio_approximation(29) = 832040 / 514229.
golden_ratio_approximation(29) = 1.6180339887482036212960900822821486144675873219966888427734375.
G = golden_ratio_approximation(29) = 1.6180339887482036212960900822821486144675873219966888427734375.

golden_ratio_approximation(30) = fibonacci(30) / fibonacci(29).
golden_ratio_approximation(30) = 1346269 / 832040.
golden_ratio_approximation(30) = 1.6180339887505408393887640361441526692942716181278228759765625.
G = golden_ratio_approximation(30) = 1.6180339887505408393887640361441526692942716181278228759765625.

golden_ratio_approximation(31) = fibonacci(31) / fibonacci(30).
golden_ratio_approximation(31) = 2178309 / 1346269.
golden_ratio_approximation(31) = 1.618033988749648101573667957620017432418535463511943817138671875.
G = golden_ratio_approximation(31) = 1.618033988749648101573667957620017432418535463511943817138671875.

golden_ratio_approximation(32) = fibonacci(32) / fibonacci(31).
golden_ratio_approximation(32) = 3524578 / 2178309.
golden_ratio_approximation(32) = 1.618033988749989097034702456578969531619804911315441131591796875.
G = golden_ratio_approximation(32) = 1.618033988749989097034702456578969531619804911315441131591796875.

golden_ratio_approximation(33) = fibonacci(33) / fibonacci(32).
golden_ratio_approximation(33) = 5702887 / 3524578.
golden_ratio_approximation(33) = 1.618033988749858848358274820977698027490987442433834075927734375.
G = golden_ratio_approximation(33) = 1.618033988749858848358274820977698027490987442433834075927734375.

golden_ratio_approximation(34) = fibonacci(34) / fibonacci(33).
golden_ratio_approximation(34) = 9227465 / 5702887.
golden_ratio_approximation(34) = 1.618033988749908598926523228822560440676170401275157928466796875.
G = golden_ratio_approximation(34) = 1.618033988749908598926523228822560440676170401275157928466796875.

golden_ratio_approximation(35) = fibonacci(35) / fibonacci(34).
golden_ratio_approximation(35) = 14930352 / 9227465.
golden_ratio_approximation(35) = 1.618033988749889595898205640889244705249438993632793426513671875.
G = golden_ratio_approximation(35) = 1.618033988749889595898205640889244705249438993632793426513671875.

golden_ratio_approximation(36) = fibonacci(36) / fibonacci(35).
golden_ratio_approximation(36) = 24157817 / 14930352.
golden_ratio_approximation(36) = 1.618033988749896854414909996844329498344450257718563079833984375.
G = golden_ratio_approximation(36) = 1.618033988749896854414909996844329498344450257718563079833984375.

golden_ratio_approximation(37) = fibonacci(37) / fibonacci(36).
golden_ratio_approximation(37) = 39088169 / 24157817.
golden_ratio_approximation(37) = 1.618033988749894081893114516912390854486147873103618621826171875.
G = golden_ratio_approximation(37) = 1.618033988749894081893114516912390854486147873103618621826171875.

golden_ratio_approximation(38) = fibonacci(38) / fibonacci(37).
golden_ratio_approximation(38) = 63245986 / 39088169.
golden_ratio_approximation(38) = 1.618033988749895140941796600753121992966043762862682342529296875.
G = golden_ratio_approximation(38) = 1.618033988749895140941796600753121992966043762862682342529296875.

golden_ratio_approximation(39) = fibonacci(39) / fibonacci(38).
golden_ratio_approximation(39) = 102334155 / 63245986.
golden_ratio_approximation(39) = 1.6180339887498947364259660464114176647854037582874298095703125.
G = golden_ratio_approximation(39) = 1.6180339887498947364259660464114176647854037582874298095703125.

golden_ratio_approximation(40) = fibonacci(40) / fibonacci(39).
golden_ratio_approximation(40) = 165580141 / 102334155.
golden_ratio_approximation(40) = 1.618033988749894890924775625595799510847427882254123687744140625.
G = golden_ratio_approximation(40) = 1.618033988749894890924775625595799510847427882254123687744140625.

golden_ratio_approximation(41) = fibonacci(41) / fibonacci(40).
golden_ratio_approximation(41) = 267914296 / 165580141.
golden_ratio_approximation(41) = 1.618033988749894831944177442384358300841995514929294586181640625.
G = golden_ratio_approximation(41) = 1.618033988749894831944177442384358300841995514929294586181640625.

golden_ratio_approximation(42) = fibonacci(42) / fibonacci(41).
golden_ratio_approximation(42) = 433494437 / 267914296.
golden_ratio_approximation(42) = 1.6180339887498948543871624128343000847962684929370880126953125.
G = golden_ratio_approximation(42) = 1.6180339887498948543871624128343000847962684929370880126953125.

golden_ratio_approximation(43) = fibonacci(43) / fibonacci(42).
golden_ratio_approximation(43) = 701408733 / 433494437.
golden_ratio_approximation(43) = 1.618033988749894845821965250198815056137391366064548492431640625.
G = golden_ratio_approximation(43) = 1.618033988749894845821965250198815056137391366064548492431640625.

golden_ratio_approximation(44) = fibonacci(44) / fibonacci(43).
golden_ratio_approximation(44) = 1134903170 / 701408733.
golden_ratio_approximation(44) = 1.618033988749894849074571767655328358159749768674373626708984375.
G = golden_ratio_approximation(44) = 1.618033988749894849074571767655328358159749768674373626708984375.

golden_ratio_approximation(45) = fibonacci(45) / fibonacci(44).
golden_ratio_approximation(45) = 1836311903 / 1134903170.
golden_ratio_approximation(45) = 1.618033988749894847881949377921273480751551687717437744140625.
G = golden_ratio_approximation(45) = 1.618033988749894847881949377921273480751551687717437744140625.

golden_ratio_approximation(46) = fibonacci(46) / fibonacci(45).
golden_ratio_approximation(46) = 2971215073 / 1836311903.
golden_ratio_approximation(46) = 1.6180339887498948483156302469154752543545328080654144287109375.
G = golden_ratio_approximation(46) = 1.6180339887498948483156302469154752543545328080654144287109375.

golden_ratio_approximation(47) = fibonacci(47) / fibonacci(46).
golden_ratio_approximation(47) = 4807526976 / 2971215073.
golden_ratio_approximation(47) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(47) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(48) = fibonacci(48) / fibonacci(47).
golden_ratio_approximation(48) = 7778742049 / 4807526976.
golden_ratio_approximation(48) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(48) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(49) = fibonacci(49) / fibonacci(48).
golden_ratio_approximation(49) = 12586269025 / 7778742049.
golden_ratio_approximation(49) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(49) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(50) = fibonacci(50) / fibonacci(49).
golden_ratio_approximation(50) = 20365011074 / 12586269025.
golden_ratio_approximation(50) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(50) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(51) = fibonacci(51) / fibonacci(50).
golden_ratio_approximation(51) = 32951280099 / 20365011074.
golden_ratio_approximation(51) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(51) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(52) = fibonacci(52) / fibonacci(51).
golden_ratio_approximation(52) = 53316291173 / 32951280099.
golden_ratio_approximation(52) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(52) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(53) = fibonacci(53) / fibonacci(52).
golden_ratio_approximation(53) = 86267571272 / 53316291173.
golden_ratio_approximation(53) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(53) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(54) = fibonacci(54) / fibonacci(53).
golden_ratio_approximation(54) = 139583862445 / 86267571272.
golden_ratio_approximation(54) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(54) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(55) = fibonacci(55) / fibonacci(54).
golden_ratio_approximation(55) = 225851433717 / 139583862445.
golden_ratio_approximation(55) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(55) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(56) = fibonacci(56) / fibonacci(55).
golden_ratio_approximation(56) = 365435296162 / 225851433717.
golden_ratio_approximation(56) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(56) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(57) = fibonacci(57) / fibonacci(56).
golden_ratio_approximation(57) = 591286729879 / 365435296162.
golden_ratio_approximation(57) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(57) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(58) = fibonacci(58) / fibonacci(57).
golden_ratio_approximation(58) = 956722026041 / 591286729879.
golden_ratio_approximation(58) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(58) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(59) = fibonacci(59) / fibonacci(58).
golden_ratio_approximation(59) = 1548008755920 / 956722026041.
golden_ratio_approximation(59) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(59) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(60) = fibonacci(60) / fibonacci(59).
golden_ratio_approximation(60) = 2504730781961 / 1548008755920.
golden_ratio_approximation(60) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(60) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(61) = fibonacci(61) / fibonacci(60).
golden_ratio_approximation(61) = 4052739537881 / 2504730781961.
golden_ratio_approximation(61) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(61) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(62) = fibonacci(62) / fibonacci(61).
golden_ratio_approximation(62) = 6557470319842 / 4052739537881.
golden_ratio_approximation(62) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(62) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(63) = fibonacci(63) / fibonacci(62).
golden_ratio_approximation(63) = 10610209857723 / 6557470319842.
golden_ratio_approximation(63) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(63) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(64) = fibonacci(64) / fibonacci(63).
golden_ratio_approximation(64) = 17167680177565 / 10610209857723.
golden_ratio_approximation(64) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(64) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(65) = fibonacci(65) / fibonacci(64).
golden_ratio_approximation(65) = 27777890035288 / 17167680177565.
golden_ratio_approximation(65) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(65) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(66) = fibonacci(66) / fibonacci(65).
golden_ratio_approximation(66) = 44945570212853 / 27777890035288.
golden_ratio_approximation(66) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(66) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(67) = fibonacci(67) / fibonacci(66).
golden_ratio_approximation(67) = 72723460248141 / 44945570212853.
golden_ratio_approximation(67) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(67) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(68) = fibonacci(68) / fibonacci(67).
golden_ratio_approximation(68) = 117669030460994 / 72723460248141.
golden_ratio_approximation(68) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(68) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(69) = fibonacci(69) / fibonacci(68).
golden_ratio_approximation(69) = 190392490709135 / 117669030460994.
golden_ratio_approximation(69) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(69) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(70) = fibonacci(70) / fibonacci(69).
golden_ratio_approximation(70) = 308061521170129 / 190392490709135.
golden_ratio_approximation(70) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(70) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(71) = fibonacci(71) / fibonacci(70).
golden_ratio_approximation(71) = 498454011879264 / 308061521170129.
golden_ratio_approximation(71) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(71) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(72) = fibonacci(72) / fibonacci(71).
golden_ratio_approximation(72) = 806515533049393 / 498454011879264.
golden_ratio_approximation(72) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(72) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(73) = fibonacci(73) / fibonacci(72).
golden_ratio_approximation(73) = 1304969544928657 / 806515533049393.
golden_ratio_approximation(73) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(73) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(74) = fibonacci(74) / fibonacci(73).
golden_ratio_approximation(74) = 2111485077978050 / 1304969544928657.
golden_ratio_approximation(74) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(74) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(75) = fibonacci(75) / fibonacci(74).
golden_ratio_approximation(75) = 3416454622906707 / 2111485077978050.
golden_ratio_approximation(75) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(75) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(76) = fibonacci(76) / fibonacci(75).
golden_ratio_approximation(76) = 5527939700884757 / 3416454622906707.
golden_ratio_approximation(76) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(76) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(77) = fibonacci(77) / fibonacci(76).
golden_ratio_approximation(77) = 8944394323791464 / 5527939700884757.
golden_ratio_approximation(77) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(77) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(78) = fibonacci(78) / fibonacci(77).
golden_ratio_approximation(78) = 14472334024676221 / 8944394323791464.
golden_ratio_approximation(78) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(78) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(79) = fibonacci(79) / fibonacci(78).
golden_ratio_approximation(79) = 23416728348467685 / 14472334024676221.
golden_ratio_approximation(79) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(79) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(80) = fibonacci(80) / fibonacci(79).
golden_ratio_approximation(80) = 37889062373143906 / 23416728348467685.
golden_ratio_approximation(80) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(80) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(81) = fibonacci(81) / fibonacci(80).
golden_ratio_approximation(81) = 61305790721611591 / 37889062373143906.
golden_ratio_approximation(81) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(81) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(82) = fibonacci(82) / fibonacci(81).
golden_ratio_approximation(82) = 99194853094755497 / 61305790721611591.
golden_ratio_approximation(82) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(82) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(83) = fibonacci(83) / fibonacci(82).
golden_ratio_approximation(83) = 160500643816367088 / 99194853094755497.
golden_ratio_approximation(83) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(83) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(84) = fibonacci(84) / fibonacci(83).
golden_ratio_approximation(84) = 259695496911122585 / 160500643816367088.
golden_ratio_approximation(84) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(84) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(85) = fibonacci(85) / fibonacci(84).
golden_ratio_approximation(85) = 420196140727489673 / 259695496911122585.
golden_ratio_approximation(85) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(85) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(86) = fibonacci(86) / fibonacci(85).
golden_ratio_approximation(86) = 679891637638612258 / 420196140727489673.
golden_ratio_approximation(86) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(86) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(87) = fibonacci(87) / fibonacci(86).
golden_ratio_approximation(87) = 1100087778366101931 / 679891637638612258.
golden_ratio_approximation(87) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(87) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(88) = fibonacci(88) / fibonacci(87).
golden_ratio_approximation(88) = 1779979416004714189 / 1100087778366101931.
golden_ratio_approximation(88) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(88) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(89) = fibonacci(89) / fibonacci(88).
golden_ratio_approximation(89) = 2880067194370816120 / 1779979416004714189.
golden_ratio_approximation(89) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(89) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(90) = fibonacci(90) / fibonacci(89).
golden_ratio_approximation(90) = 4660046610375530309 / 2880067194370816120.
golden_ratio_approximation(90) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(90) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(91) = fibonacci(91) / fibonacci(90).
golden_ratio_approximation(91) = 7540113804746346429 / 4660046610375530309.
golden_ratio_approximation(91) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(91) = 1.618033988749894848207210029666924810953787527978420257568359375.

golden_ratio_approximation(92) = fibonacci(92) / fibonacci(91).
golden_ratio_approximation(92) = 12200160415121876738 / 7540113804746346429.
golden_ratio_approximation(92) = 1.618033988749894848207210029666924810953787527978420257568359375.
G = golden_ratio_approximation(92) = 1.618033988749894848207210029666924810953787527978420257568359375.

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