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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="UTF-8">
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<title> | tait.tech</title>
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<link rel="stylesheet" href="/assets/css/style.css" id="main-stylesheet">
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<meta name="viewport" content="width=device-width, initial-scale=1.0"><link rel="stylesheet" href="/assets/css/katex.css" id="math-stylesheet">
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</head>
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<body>
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<main>
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<div id="wrapper">
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<h1 id="solution">SOLUTION</h1>
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<ul>
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<li>Simon Fraser University</li>
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<li>Computing Science 295</li>
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<li>Fall 2021</li>
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<li>Friday Oct. 15 2021</li>
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<li>Midterm Examination 1</li>
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<li>Time: 45 minutes</li>
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</ul>
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<p>First Name: _________</p>
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<p>Last Name: _________</p>
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<p>Student ID: _________</p>
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<p>This examination has 11 pages.</p>
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<p>Read each question carefully before answering it.</p>
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<ul>
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<li>No textbooks, cheat sheets, calculators, computers, cell phones or other materials may
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be used.</li>
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<li>All assembly code must be x86-64 assembly code.</li>
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<li>Hand in your scrap sheet(s) along with your examination paper. The scrap sheet(s) will
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not be marked.</li>
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<li>The marks for each question are given in [ ]. Use this to manage your time:
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<ul>
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<li>One (1) mark corresponds to one (1) minute of work.</li>
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<li>Do not spend more time on a question than the number of marks assigned to it.</li>
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</ul>
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</li>
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</ul>
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<p>Good luck!</p>
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<h2 id="part-1">Part 1</h2>
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<p>Each question is 2 marks – There are no part marks given!</p>
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<p>Answer the following multiple choice questions on the bubble sheet at the back of this
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examination paper.</p>
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<ol>
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<li>Consider the following syntactically correct C code fragment:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>float aFloat = 3.1415;
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int sum = (int) aFloat + 0xFFFFFFFE;
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</code></pre></div> </div>
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<p>Which value does the variable sum contain when the above C code fragment has
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executed on our target machine?</p>
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<ul>
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<li>a. 1.1415</li>
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<li>b. -1</li>
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<li>c. 0x00000001 (CORRECT)</li>
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</ul>
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<ul>
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<li>aFloat = 3.1415</li>
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<li>0xFFFFFFFE = -2</li>
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<li>(int) aFloat = 3</li>
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<li>sum = 3 + (-2) = 1 or 0x00000001 (int in hex)
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* d. 5
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* e. None of the above</li>
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</ul>
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</li>
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<li>Which step in the compilation process transforms our C code into assembly
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instructions?
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<ul>
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<li>a. The step called the preprocessor</li>
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<li>b. The step called the compiler – See Lecture 8 Slide 7 (CORRECT)</li>
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<li>c. The step called the assembler</li>
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<li>d. The step called the linker</li>
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<li>e. None of the above</li>
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</ul>
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</li>
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<li>Consider the following syntactically correct C function:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>char mystery( char someParam ) {
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char result = 0;
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if ( someParam > 0 ) result = someParam;
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else result = -someParam;
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return result;
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}
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</code></pre></div> </div>
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<p>What will it return once it has executed on our target machine with the parameter
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someParam set to the value -128?</p>
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<ul>
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<li>a. 127</li>
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<li>b. 128</li>
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<li>c. -127</li>
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<li><span id="a3">d. -128 (CORRECT)</span> <a href="#x3">Explaination</a></li>
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<li>e. None of the above</li>
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</ul>
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</li>
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<li>Consider the following syntactically correct C code fragment:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>short count = 0xACE;
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printf( "count = %hhi\n", (char) count );
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</code></pre></div> </div>
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<p>What is printed on the computer monitor screen when the above C code fragment
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has executed on our target machine?</p>
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<ul>
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<li><span id="a4">a. count = -50</span> (CORRECT) <a href="#x4">Explaination</a></li>
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<li>b. count = 0xCE</li>
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<li>c. count = 206</li>
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<li>d. count = 0xACE</li>
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<li>e. None of the above</li>
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</ul>
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</li>
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<li>Consider the following syntactically correct C code fragment:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>short aShort = -2;
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char aChar = aShort;
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short sumS = 0xABBB + (short) aChar + 1;
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</code></pre></div> </div>
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<p>Which statement below is true about the above C code fragment once it has executed
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on our target machine, but has not yet exited the scope of the variables aShort,
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aChar and sumS?</p>
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<ul>
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<li>a. sumS contains the hex value 0xABBA</li>
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<li>b. aChar == aShort</li>
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<li>c. Statements a. and b. are true. (CORRECT) <a href="#x5">Explaination</a></li>
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<li>d. Only the statement b. is true.</li>
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<li>e. None of the above</li>
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</ul>
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</li>
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<li>Consider the following C code fragment:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>char char1 = 101;
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char char2 = _______ ;
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char sumOfChar = char1 + char2;
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</code></pre></div> </div>
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<p>Which value must be assigned to char2 in order for the sum of char1 and char2
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to create a positive overflow?</p>
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<ul>
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<li>a. No numbers would create a positive overflow when added to 101.</li>
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<li>b. <span id="a6">42</span> (CORRECT) <a href="#x6">Explaination</a></li>
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<li>c. 26</li>
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<li>d. -230</li>
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<li>e. None of the above</li>
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</ul>
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</li>
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<li>Consider the following syntactically correct C code fragment:
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>unsigned int x = 0xDECAF000;
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unsigned short y = 0xCAFE;
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if ( x > y ) printf("Caf ");
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if ( x < (signed short) y ) printf("Decaf ");
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if ( (unsigned char) x > y ) printf("Latte ");
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</code></pre></div> </div>
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<p>What is printed on the computer monitor screen when the above C code fragment
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has executed on our target machine?</p>
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<ul>
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<li>a. Caf Decaf Latte</li>
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<li>b. Caf Latte</li>
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<li>c. Caf</li>
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<li>d. Decaf</li>
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<li>e. <span id="a7">None of the above</span> (CORRECT) <a href="#x7">Explaination</a></li>
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</ul>
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</li>
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<li>Which range of values can be stored in the variable y declared in the C fragment
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code of Question 7 above?
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<ul>
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<li>a. [0 .. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>16</mn></msup></mrow><annotation encoding="application/x-tex">2^{16}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">16</span></span></span></span></span></span></span></span></span></span></span></span>]</li>
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<li>b. [-128 .. 127]</li>
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<li>c. [0 .. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>16</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{16}-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">16</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>]</li>
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<li>d. [0 .. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>15</mn></msup><mtext>–</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{15} – 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span><span class="mord">–1</span></span></span></span>]</li>
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<li>e. <span id="a8">None of the above</span> (CORRECT) <a href="#x8">Explaination</a></li>
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</ul>
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</li>
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</ol>
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<h2 id="part-2">Part 2</h2>
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<p>The weight of each question is indicated in [ ] – Write your answer below each question unless instructed otherwise.</p>
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<h3 id="1">1</h3>
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<p>[Total marks: 15] Consider the following function mystery written in x86-64
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assembly code, where the parameter x is in the register %edi and the parameter y is
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in the register %esi:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>.globl power # mystery is crossed out and replaced with power; line 1
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power: # x -> %edi, y -> %esi; mystery replaced with power; line 2
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xorl %eax,%eax # line 3
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movl $1,%r8d # line 4
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loop: # loop replaces .L3 # line 5
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addl $1,%eax # line 6
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imull %edi,%r8d # line 7
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cmpl %eax,%esi # line 8
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jne loop # loop replaces .L3; line 9
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movl %r8d,%eax # line 10
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ret # line 11
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</code></pre></div></div>
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<ul>
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<li>a. [3 marks] If we call this function as follows: mystery( x, y ) where x = 2
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and y = 3, what value will it return?
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<ul>
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<li>Answer: 8</li>
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</ul>
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</li>
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<li>b. [2 marks] Replace the label .L3 with a more descriptive label name. Do this
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replacement in the above code.
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<ul>
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<li>Possible answer: loop</li>
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</ul>
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</li>
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<li>c. [2 marks] Rename this function with a more descriptive name. Do this renaming in
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the above code.
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<ul>
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<li>Possible answer: power or pow; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>y</mi></msup></mrow><annotation encoding="application/x-tex">x^y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span></span></span></span></span></span></span> -> x is raised to the power y</li>
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</ul>
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</li>
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<li>d. [2 marks] What is the data type of the parameters and the return value of this
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function? Express your answer using C data types.
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<ul>
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<li>Answer: int or unsigned int</li>
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</ul>
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</li>
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<li>e. [2 marks] Replace Line 3 with another equivalent x86-64 instruction, i.e., an
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x86-64 instruction that will produce the same result, but is not a xor*
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instruction.
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<ul>
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<li>Answer: Line 3: xorl %eax,%eax</li>
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<li>The purpose of this instruction is to “zero” the register %eax.</li>
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<li>Possible replacement: Line 3: movl $0,%eax</li>
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</ul>
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</li>
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<li>f. [2 marks] Replace Line 6 with another equivalent x86-64 instruction, i.e., an
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x86-64 instruction that will produce the same result, but is not an add*
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instruction.
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<ul>
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<li>Answer: Line 6: addl $1,%eax</li>
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<li>The purpose of this instruction is to increment the value of the
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register %eax by 1.</li>
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<li>Possible replacements:
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<ul>
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<li>Line 6: incl %eax</li>
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<li>Line 6: leal 1(%eax),%eax</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>g. [2 marks] On Line 6 and Line 8, the register %eax is used for a different
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purpose than holding the return value. For what purpose is the register %eax used
|
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on those two lines?</li>
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|
<li>Answer: On Line 6 and Line 8, the register %eax is used as a loop
|
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increment or a loop counter, expressing the number of times the loop
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executes. At every loop iteration, %eax is incremented by 1 (Line 6)
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then compare to y (Line 8). The loop terminates when %eax equals
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the value of y (the power to which we are raising x).</li>
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</ul>
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<h3 id="2">2</h3>
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<p>[Total marks: 14] Consider a floating point number encoding scheme based on the
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IEEE floating point format and defined as follows:</p>
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<ul>
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<li>It uses 7 bits.</li>
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<li>There is one sign bit s.</li>
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<li>The exp can be any number in the range [0 .. 31].</li>
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</ul>
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<p>From the above, we gather:</p>
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<ul>
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<li>the format is: s exp frac</li>
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<li>exp is 5 bits (k = 5) and</li>
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<li>frac is 1 bit1 (1 bit for s + 5 bits for exp + 1 for frac = 7 bits)</li>
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<li>since this floating point number encoding scheme is based on the
|
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IEEE floating point format, the following equations hold:
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|
<ul>
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|
<li>V = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^s M 2^E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></li>
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<li>E= exp – bias (for normalized numbers)</li>
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<li>M = 1 + frac (for normalized numbers)</li>
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|
<li>E= 1 – bias (for denormalized numbers)</li>
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|
<li>M = frac (for denormalized numbers)</li>
|
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|
<li>bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{k-1} – 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span></span></span></span> (for normalized and denormalized numbers)</li>
|
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|
|
</ul>
|
|
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|
|
|
</li>
|
|
|
|
|
|
<li>a. [2 marks] Compute the bias of this IEEE-like floating point number encoding
|
|
|
|
|
|
scheme described above and show your work:
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li>Answer: k = 5</li>
|
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|
|
<li>bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mrow><mn>5</mn><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>4</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>16</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>15</mn></mrow><annotation encoding="application/x-tex">2^{k-1}–1 = 2^{5-1} – 1 = 2^4 – 1 = 16 – 1 = 15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">16–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">15</span></span></span></span></li>
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li>b. [7 marks] Encode the value 24.510 using this IEEE-like floating point number
|
|
|
|
|
|
representation described in this question. Show all your work. Clearly show the
|
|
|
|
|
|
resulting bit pattern and label its three sections s,exp,frac.
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li>Step 1) 24.510 is a positive number so s = 0</li>
|
|
|
|
|
|
<li>Step 2) R2B(24.510)
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li>=> 24 – 16 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">2^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>) = 8 and 0.5 - 0.5 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>) = 0</li>
|
|
|
|
|
|
<li>8-8 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">2^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>) = 0</li>
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|
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|
|
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>24.5</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{24.5}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">24.5</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11000.1</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11000.1}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11000.1</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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|
|
|
|
|
</ul>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li>Step 3) normalize <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11000.1</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11000.1}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11000.1</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1.10001</mtext><mn>2</mn></msub><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\text{1.10001}_{2} \times 2^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></li>
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|
<li>Step 4) Using <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mtext>(-1)</mtext><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = \text{(-1)}^s M 2^E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">(-1)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span>
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<ul>
|
|
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|
|
|
<li>1) E= exp – bias => exp = E + bias
|
|
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|
|
|
<ul>
|
|
|
|
|
|
<li>=> exp = 410 + 1510 = 1910 since E = 410</li>
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<li>=> U2B(<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>19</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{19}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">19</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10011</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10011}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (k = 5)</li>
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</ul>
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</li>
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<li>2) M = 1 + frac => frac = M - 1
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<ul>
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<li>=> frac = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1.10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{1.10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> – 1 => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>0.10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0.10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><msub><mtext>1.10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M = \text{1.10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>=> frac = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (ignoring “0.”)
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but since frac only has 1 bit, 100012
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cannot be stored in frac, so we need to
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round frac <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>.10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{.10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>.1</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{.1}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">.1</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>=> MSBit is the rounding position (in blue)</li>
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<li>=> since the value of the rest of the bits
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(<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>0001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> = 0.03125 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{-5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span>) - see table below) < <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/2</span></span></span></span>
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the worth of rounding position (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/2</span></span></span></span> of 0.5 =
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0.25), then we round down which means we
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only discard the bits <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>0001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>.10001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{.10001}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">.10001</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)</li>
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</ul>
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</li>
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<li>Step 5)
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<ul>
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<li>Using the format: s exp frac
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the resulting bit pattern encoding <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>24.5</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{24.5}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">24.5</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>
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in the IEEE-like floating point number
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representation described in this question is: s=0 exp=10011 frac=1</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>c. [2 marks] Write the “range” (non-contiguous) of real numbers (excluding +/infinity and NAN) that can be encoded using this IEEE-like floating-point
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representation described in this question. Express this range using the bit patterns
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(not the actual real numbers).
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<ul>
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<li>Answer: “range” (non-contiguous) of real numbers (excluding +/- infinity and NAN) that can be encoded
|
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using
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IEEE-like floating-point
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representation
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described
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in
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this
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question
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(expressed using the bit patterns): <code class="language-plaintext highlighter-rouge">[ 1 11110 1 .. 0 11110 1 ]</code></li>
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</ul>
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</li>
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<li>d. [3 marks] Can 6553610 be encoded as a normalized number in this IEEE-like
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floating point representation? Briefly explain why/why not.
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Hint: Use your range in the above question and the table below.
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<ul>
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<li>Answer:
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>65536</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{65536}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">65536</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> cannot be encoded as a normalized number
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in this IEEE-like floating point representation
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because expressed as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mtext>(-1)</mtext><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = \text{(-1)}^s M 2^E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">(-1)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span>
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6553610 is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mtext>(-1)</mtext><mn>0</mn></msup><mn>1.0</mn><msup><mn>2</mn><mn>16</mn></msup><mo>=</mo><msup><mn>2</mn><mn>16</mn></msup></mrow><annotation encoding="application/x-tex">V = \text{(-1)}^0 1.0 2^{16} = 2^{16}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.204008em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">(-1)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span><span class="mord">1.0</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">16</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">16</span></span></span></span></span></span></span></span></span></span></span></span> (see table below)
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<ul>
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<li>E= exp – bias => exp = E + bias</li>
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<li>exp = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>16</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{16}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">16</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>15</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{15}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">15</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>31</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{31}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">31</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> since E = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>16</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{16}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">16</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></sp
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and U2B(<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>31</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{31}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">31</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11111}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11111</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>
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(k = 5) which is outside
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the range for exp as indicated in the range given
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as the answer to the above question:
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<code class="language-plaintext highlighter-rouge">[ 1 11110 1 .. 0 11110 1 ]</code></li>
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<li>When <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>exp</mtext><mo>=</mo><msub><mtext>11111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{exp} = \text{11111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11111</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for k = 5, it indicates overflow, i.e., one of the special cases.</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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<p>Table Powers of 2:</p>
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<table>
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<thead>
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<tr>
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<th>Power of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">2^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></th>
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<th>Value</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>-10</td>
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<td>1/1024 = 0.0009765625</td>
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</tr>
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<tr>
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<td>-9</td>
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<td>1/512 = 0.001953125</td>
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</tr>
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<tr>
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<td>-8</td>
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<td>1/256 = 0.00390625</td>
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</tr>
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<tr>
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<td>-7</td>
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<td>1/128 = 0.0078125</td>
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</tr>
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<tr>
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<td>-6</td>
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<td>1/64 = 0.015625</td>
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</tr>
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<tr>
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<td>-5</td>
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<td>1/32 = 0.03125</td>
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</tr>
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<tr>
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<td>-4</td>
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<td>1/16 = 0.0625</td>
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</tr>
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<tr>
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<td>-3</td>
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<td>1/8 = 0.125</td>
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</tr>
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<tr>
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<td>-2</td>
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<td>1/4 = 0.25</td>
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</tr>
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<tr>
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<td>-1</td>
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<td>1/2 = 0.5</td>
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</tr>
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<tr>
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<td>0</td>
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<td>1</td>
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</tr>
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<tr>
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<td>1</td>
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<td>2</td>
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</tr>
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<tr>
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<td>2</td>
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<td>4</td>
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</tr>
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<tr>
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<td>3</td>
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<td>8</td>
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</tr>
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<tr>
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<td>4</td>
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<td>16</td>
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</tr>
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<tr>
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<td>5</td>
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<td>32</td>
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</tr>
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<tr>
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<td>6</td>
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<td>64</td>
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</tr>
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<tr>
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<td>7</td>
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<td>128</td>
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</tr>
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<tr>
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<td>8</td>
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<td>256</td>
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</tr>
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<tr>
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<td>9</td>
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<td>512</td>
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</tr>
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<tr>
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<td>10</td>
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<td>1024</td>
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</tr>
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<tr>
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<td>11</td>
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<td>2048</td>
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</tr>
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<tr>
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<td>12</td>
|
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<td>4096</td>
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</tr>
|
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<tr>
|
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<td>13</td>
|
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<td>8192</td>
|
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</tr>
|
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<tr>
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<td>14</td>
|
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<td>16384</td>
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</tr>
|
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<tr>
|
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<td>15</td>
|
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<td>32768</td>
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</tr>
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<tr>
|
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<td>16</td>
|
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<td>65536</td>
|
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</tr>
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</tbody>
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</table>
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|
<p>Table of x86-64 Jumps:</p>
|
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<table>
|
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<thead>
|
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<tr>
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<th>Instruction</th>
|
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<th>Condition</th>
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<th>Description</th>
|
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</tr>
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</thead>
|
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<tbody>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jmp</code></td>
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<td>always</td>
|
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<td>Unconditional jump</td>
|
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">je</code>/<code class="language-plaintext highlighter-rouge">jz</code></td>
|
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<td>ZF</td>
|
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<td>Jump if equal / zero</td>
|
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</tr>
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<tr>
|
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<td><code class="language-plaintext highlighter-rouge">jne</code>/<code class="language-plaintext highlighter-rouge">jnz</code></td>
|
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<td>~ZF</td>
|
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<td>Jump if not equal / not zero</td>
|
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</tr>
|
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<tr>
|
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<td><code class="language-plaintext highlighter-rouge">js</code></td>
|
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<td>SF</td>
|
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<td>Jump if negative</td>
|
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|
</tr>
|
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<tr>
|
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<td><code class="language-plaintext highlighter-rouge">jns</code></td>
|
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<td>~SF</td>
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<td>Jump if nonnegative</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jo</code></td>
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<td>OF</td>
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<td>Jump if overflow</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jno</code></td>
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<td>~OF</td>
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<td>Jump if not overflow</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jq</code>/<code class="language-plaintext highlighter-rouge">jnle</code></td>
|
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<td>~(SF ^ OF) & ~ZF</td>
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<td>Jump if greater (signed >)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jge</code>/<code class="language-plaintext highlighter-rouge">jnl</code></td>
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<td>~(SF ^ OF)</td>
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<td>Jump if greater or equal (signed >=)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jl</code>/<code class="language-plaintext highlighter-rouge">jnge</code></td>
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<td>SF ^ OF</td>
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<td>Jump if less (signed <)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jle</code>/<code class="language-plaintext highlighter-rouge">jng</code></td>
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<td>(SF ^ OF) | ZF</td>
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<td>Jump if less or equal (signed <=)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">ja</code>/<code class="language-plaintext highlighter-rouge">jnbe</code></td>
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<td>~CF & ~ZF</td>
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<td>JumP if greater (unsigned >)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jae</code>/<code class="language-plaintext highlighter-rouge">jnb</code></td>
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<td>~CF</td>
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<td>Jump if greater or equal (unsigned >=)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jb</code>/<code class="language-plaintext highlighter-rouge">jnae</code>/<code class="language-plaintext highlighter-rouge">jc</code></td>
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<td>CF</td>
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<td>Jump if less (unsigned <)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">jbe</code>/<code class="language-plaintext highlighter-rouge">jna</code>/<code class="language-plaintext highlighter-rouge">jnc</code></td>
|
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<td>CF | ZF</td>
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<td>Jump if less or equal (unsigned <=)</td>
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</tr>
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</tbody>
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</table>
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|
<p>Table of x86-64 Registers</p>
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<table>
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<thead>
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<tr>
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<th>64-bit (quad)</th>
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<th>32-bit (double)</th>
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<th>16-bit (word)</th>
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<th>8-bit (byte)</th>
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<th>8-bit (byte)</th>
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<th> </th>
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</tr>
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<tr>
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<th>63..0</th>
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|
<th>31..0</th>
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<th>15..0</th>
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<th>15..8</th>
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<th>7..0</th>
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<th> </th>
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</tr>
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|
</thead>
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|
|
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|
<tbody>
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|
|
|
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<tr>
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|
<td>rax</td>
|
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|
<td>eax</td>
|
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|
<td>ax</td>
|
|
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|
|
<td>ah</td>
|
|
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|
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|
<td>al</td>
|
|
|
|
|
|
<td>Return value</td>
|
|
|
|
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|
</tr>
|
|
|
|
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<tr>
|
|
|
|
|
|
<td>rbx</td>
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|
<td>ebx</td>
|
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|
<td>bx</td>
|
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|
<td>bh</td>
|
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|
|
|
<td>bl</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rcx</td>
|
|
|
|
|
|
<td>ecx</td>
|
|
|
|
|
|
<td>cx</td>
|
|
|
|
|
|
<td>ch</td>
|
|
|
|
|
|
<td>cl</td>
|
|
|
|
|
|
<td>4th arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rdx</td>
|
|
|
|
|
|
<td>edx</td>
|
|
|
|
|
|
<td>dx</td>
|
|
|
|
|
|
<td>dh</td>
|
|
|
|
|
|
<td>dl</td>
|
|
|
|
|
|
<td>3rd arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rsi</td>
|
|
|
|
|
|
<td>esi</td>
|
|
|
|
|
|
<td>si</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>sil</td>
|
|
|
|
|
|
<td>2nd arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rdi</td>
|
|
|
|
|
|
<td>edi</td>
|
|
|
|
|
|
<td>di</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>dil</td>
|
|
|
|
|
|
<td>1st arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rbp</td>
|
|
|
|
|
|
<td>ebp</td>
|
|
|
|
|
|
<td>bp</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>bpl</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>rsp</td>
|
|
|
|
|
|
<td>esp</td>
|
|
|
|
|
|
<td>sp</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>spl</td>
|
|
|
|
|
|
<td>Stack position</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r8</td>
|
|
|
|
|
|
<td>r8d</td>
|
|
|
|
|
|
<td>r8w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r8b</td>
|
|
|
|
|
|
<td>5th arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r9</td>
|
|
|
|
|
|
<td>r9d</td>
|
|
|
|
|
|
<td>r9w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r9b</td>
|
|
|
|
|
|
<td>6th arg</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r10</td>
|
|
|
|
|
|
<td>r10d</td>
|
|
|
|
|
|
<td>r10w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r10b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r11</td>
|
|
|
|
|
|
<td>r11d</td>
|
|
|
|
|
|
<td>r11w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r11b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r12</td>
|
|
|
|
|
|
<td>r12d</td>
|
|
|
|
|
|
<td>r12w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r12b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r13</td>
|
|
|
|
|
|
<td>r13d</td>
|
|
|
|
|
|
<td>r13w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r13b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r14</td>
|
|
|
|
|
|
<td>r14d</td>
|
|
|
|
|
|
<td>r14w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r14b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
|
|
|
|
<td>r15</td>
|
|
|
|
|
|
<td>r15d</td>
|
|
|
|
|
|
<td>r15w</td>
|
|
|
|
|
|
<td> </td>
|
|
|
|
|
|
<td>r15b</td>
|
|
|
|
|
|
<td>Callee served</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
</tbody>
|
|
|
|
|
|
</table>
|
|
|
|
|
|
|
|
|
|
|
|
<h2 id="explainations">Explainations:</h2>
|
|
|
|
|
|
|
|
|
|
|
|
<h3 id="x3">3</h3>
|
|
|
|
|
|
|
|
|
|
|
|
<p><a href="#a3">Back to answer</a></p>
|
|
|
|
|
|
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>char mystery( -128 ) {
|
|
|
|
|
|
char result = 0;
|
|
|
|
|
|
if ( -128 > 0 ) result = someParam;
|
|
|
|
|
|
else result = -(-128);
|
|
|
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
|
|
|
|
|
|
<p>So, it seems that result = 128
|
|
|
|
|
|
What is the bit pattern of 128?
|
|
|
|
|
|
Interpreting 128 as an unsigned char we get:
|
|
|
|
|
|
B2U(10000000) -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">2^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> -> 128
|
|
|
|
|
|
but we cannot interpret 128 as a signed char
|
|
|
|
|
|
because 128 is outside the range of signed char
|
|
|
|
|
|
-> [-128 .. 127], so the bit pattern 10000000
|
|
|
|
|
|
interpreted as a signed char is -128
|
|
|
|
|
|
Therefore even though it seems that
|
|
|
|
|
|
result = 128 (10000000)
|
|
|
|
|
|
It is actually the case that
|
|
|
|
|
|
result = -128
|
|
|
|
|
|
return result i.e., -128</p>
|
|
|
|
|
|
|
|
|
|
|
|
<h4 id="extra">EXTRA</h4>
|
|
|
|
|
|
|
|
|
|
|
|
<p>What is -128 as a bit pattern?</p>
|
|
|
|
|
|
|
|
|
|
|
|
<p>-128 -> T2B(X) -> (~(U2B(|X|)))+1 and X = -128</p>
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li>See Lecture 3 Slide 11 Method 1</li>
|
|
|
|
|
|
<li>(~(U2B(|-128|)))+1</li>
|
|
|
|
|
|
<li>(~(U2B(128)))+1</li>
|
|
|
|
|
|
<li>(~(10000000))+1</li>
|
|
|
|
|
|
<li>-> someParam is a char -> w = 8 bits</li>
|
|
|
|
|
|
<li>(01111111) + 1 = 10000000</li>
|
|
|
|
|
|
<li>Check: B2T(10000000) -> -27 -> -128</li>
|
|
|
|
|
|
<li>-128 -> T2B(X) -> U2B(X + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">2^w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span>)</li>
|
|
|
|
|
|
<li>See Lecture 3 Slide 11 Method 2</li>
|
|
|
|
|
|
<li>U2B(-128 + 2w) -> U2B(-128 + 28)</li>
|
|
|
|
|
|
<li>-> U2B(-128 + 256)</li>
|
|
|
|
|
|
<li>-> U2B(128) -> 10000000</li>
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
|
|
|
|
|
|
<h3 id="x4">4</h3>
|
|
|
|
|
|
|
|
|
|
|
|
<p><a href="#a4">Back to answer</a></p>
|
|
|
|
|
|
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li>count = 0xACE = 0000 1010 1100 1110</li>
|
|
|
|
|
|
<li>(char) count = 0xCE = 1100 1110</li>
|
|
|
|
|
|
<li>hhi -> signed char numerical output
|
|
|
|
|
|
<ul>
|
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<li>-> 1100 1110 = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">-2^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">2^6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">2^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">2^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span> = -128 + 64 + 8 + 4
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</ul>
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</li>
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<li>See Lecture_4_Demo.c</li>
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</ul>
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<h3 id="x5">5</h3>
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<p><a href="#a5">Back to answer</a></p>
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<ul>
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<li>aShort = -2 = 0xFFFE</li>
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<li>aChar = aShort -> aChar = -2 = 0xFE -> 1111 1110</li>
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<li>1111 1110 = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">-2^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">2^6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">2^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">2^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">2^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> + <span class="katex"><span cl
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<li>So, Statement a. => aChar (-2) == aShort (-2) is TRUE</li>
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<li>(short) aChar = 0xFFFE -> 1111 1111 1111 1110 = -2</li>
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<li>(short) aChar + 1 = -2 + 1 = -1 -> 0xFFFF</li>
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<li>0xFFFF = 1111 1111 1111 1111</li>
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<li>0xABBB (1010 1011 1011 1011) + 0xFFFF -> 0xABBA</li>
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<li>short sumS = 0xABBB + (short) aChar + 1;</li>
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<li>So, Statement b. => sumS contains the hex value 0xABBA is TRUE</li>
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<li>So, Statements a. and b. are true.</li>
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</ul>
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<h3 id="x6">6</h3>
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<p><a href="#a6">Back to answer</a></p>
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<ul>
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<li>char1 = 101;</li>
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<li>range of char -> [-128 .. 127]</li>
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<li>For char1 + char2 > 127 i.e., create a positive overflow, char2 > 127 – char1 (101) -> char2 > 26</li>
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<li>So char2 = 42 satisfies the above condition</li>
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<li>char1 (101) + char2 (42) = 143 > 127</li>
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<li>char1 (101) + char2 (26) = 127 -> still within the range (on positive side of range)</li>
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<li>char1 (101) + char2 (-203) = -102 -> still within the range (on negative side of range)</li>
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</ul>
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<h3 id="x7">7</h3>
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<p><a href="#a7">Back to answer</a></p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>unsigned int x = 0xDECAF000;
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unsigned short y = 0xCAFE;
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if ( x > y ) printf("Caf ");
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</code></pre></div></div>
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<p>Promoting y to 32 bits as an unsigned i.e. padding with 0’s: 0xDECAF000 > 0x0000CAFE without a calculator, we can see that these 32
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bits 0xDECAF000, interpreted as an unsigned
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value, will be > than 0x0000CAFE, also
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interpreted as an unsigned value.
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So, Caf is printed on the computer monitor screen
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of target machine.</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>if ( x < (signed short) y ) printf("Decaf ");
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</code></pre></div></div>
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<p>Casting y to 16 bits as a signed i.e. interpreting 0xCAFE as a signed value and promoting it to 32 bits still as a signed i.e.
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padding with 1’s: 0xFFFFCAFE
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0xDECAF000 < 0xFFFFCAFE
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without a calculator, we can see that these 32
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bits 0xDECAF000, interpreted as a signed value,
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will represent a larger negative value than
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0xFFFFCAFE, also interpreted as a signed value
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Remember from previous questions that 0xFE =
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0xFFFE = 0xFFFFFFFE = -2
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So, Decaf is printed on the computer monitor
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screen of target machine.</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>if ( (unsigned char) x > y ) printf("Latte ");
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</code></pre></div></div>
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<p>Casting x (0xDECAF000) to a char -> 8 bits, we
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get 0x00 = 0 (unsigned value)
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promoting it to 16 bits still gives us 0
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0x0000 > 0xCAFE
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without a calculator, we can see that this is not
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the case.
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So, Latte is NOT printed on the computer monitor
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screen of target machine.</p>
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<p>The answer (Caf Decaf) is not one of the options.</p>
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<h3 id="x8">8</h3>
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<p><a href="#a8">Return to answer</a></p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>unsigned short y = 0xCAFE;
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</code></pre></div></div>
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<p>y is declared as an unsigned short -> 16 bits
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so range of unsigned short is [0 .. 216 -1]
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which is not one of the options sbove.</p>
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<footer>
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</footer>
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</div>
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</main>
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