Problem 44691. Comparison of floating-point numbers (singles)

Created by David Verrelli

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html\"><w:r><w:t>Floating-point numbers</w:t></w:r></w:hyperlink><w:r><w:t> cannot generally be represented exactly, so it is usually inappropriate to test for 'equality' between two floating-point numbers. Rather, it is generally appropriate to check whether the difference between the two numbers is sufficiently small that they can be considered practically equal. (In other words, so close in value that any differences could be explained by inherent limitations of computations using floating-point numbers.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Based on two scalar inputs of type</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/double.html\"><w:r><w:t>double</w:t></w:r></w:hyperlink><w:r><w:t>, namely</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t>, you must return a scalar</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/logical.html\"><w:r><w:t>logical</w:t></w:r></w:hyperlink><w:r><w:t> output set to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/true.html\"><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>true</w:t></w:r></w:hyperlink><w:r><w:t> if the difference in magnitude between</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>the</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/single.html\"><w:r><w:rPr><w:b/></w:rPr><w:t>single</w:t></w:r></w:hyperlink><w:r><w:rPr><w:b/></w:rPr><w:t>-precision representations</w:t></w:r><w:r><w:t> of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is 'small', or otherwise set to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/false.html\"><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>false</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>this</w:t></w:r><w:r><w:t> problem \"small\" shall mean no more than</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Δ</w:t></w:r><w:r><w:t>, which is defined as the larger of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>δ</w:t></w:r><w:r><w:t> and ten times</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>. In turn,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>δ</w:t></w:r><w:r><w:t> has a fixed value of 1×10⁻¹⁰ and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t> is the larger of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ &amp;</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂, where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ is the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/eps.html\"><w:r><w:t>floating-point precision</w:t></w:r></w:hyperlink><w:r><w:t> with which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> is represented, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂ is the precision with which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is represented.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>EXAMPLE:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[% Input\nA = 0\nB = 1E-20\n\n% Output\npracticallyEqual = true]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Explanation: When represented as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>single</w:t></w:r><w:r><w:t> values,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> is represented with a precision of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ = 2⁻¹⁴⁹, whereas</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is represented with a precision of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂ = 2⁻⁹⁰. Thus</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t> = 2⁻⁹⁰; however the threshold</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Δ</w:t></w:r><w:r><w:t> is 1×10⁻¹⁰ for this situation, because 1×10⁻¹⁰ &gt; 10×2⁻⁹⁰. The difference between</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is 1×10⁻²⁰, and this difference does not exceed the threshold. Thus</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>are</w:t></w:r><w:r><w:t> practically equal in this example.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>RELATED PROBLEMS:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44690\"><w:r><w:t>Problem 44690: Comparison of floating-point numbers (doubles)</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44631\"><w:r><w:t>Problem 44631: Make your own Test Suite (part 0)</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44513\"><w:r><w:t>Problem 44513: Add all the numbers between two limits (inclusive)</w:t></w:r></w:hyperlink></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

<a href = "https://au.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html">Floating-point numbers</a> cannot generally be represented exactly, so it is usually inappropriate to test for 'equality' between two floating-point numbers. Rather, it is generally appropriate to check whether the difference between the two numbers is sufficiently small that they can be considered practically equal. (In other words, so close in value that any differences could be explained by inherent limitations of computations using floating-point numbers.)Based on two scalar inputs of type <a href = "https://au.mathworks.com/help/matlab/ref/double.html">double</a>, namely <tt>A</tt> and <tt>B</tt>, you must return a scalar <a href = "https://au.mathworks.com/help/matlab/ref/logical.html">logical</a> output set to <a href = "https://au.mathworks.com/help/matlab/ref/true.html"><tt>true</tt></a> if the difference in magnitude between the <a href = "https://au.mathworks.com/help/matlab/ref/single.html">single</a>-precision representations of <tt>A</tt> and <tt>B</tt> is 'small', or otherwise set to <a href = "https://au.mathworks.com/help/matlab/ref/false.html"><tt>false</tt></a>.For this problem "small" shall mean no more than Δ, which is defined as the larger of δ and ten times ε. In turn, δ has a fixed value of 1×10⁻¹⁰ and ε is the larger of ε₁ & ε₂, where ε₁ is the <a href = "https://au.mathworks.com/help/matlab/ref/eps.html">floating-point precision</a> with which <tt>A</tt> is represented, and ε₂ is the precision with which <tt>B</tt> is represented.EXAMPLE:<pre class="language-matlab">% Input
A = 0
B = 1E-20
</pre><pre class="language-matlab">% Output
practicallyEqual = true
</pre>Explanation: When represented as <tt>single</tt> values, <tt>A</tt> is represented with a precision of ε₁ = 2⁻¹⁴⁹, whereas <tt>B</tt> is represented with a precision of ε₂ = 2⁻⁹⁰. Thus ε = 2⁻⁹⁰; however the threshold Δ is 1×10⁻¹⁰ for this situation, because 1×10⁻¹⁰ &gt; 10×2⁻⁹⁰. The difference between <tt>A</tt> and <tt>B</tt> is 1×10⁻²⁰, and this difference does not exceed the threshold. Thus <tt>A</tt> and <tt>B</tt> are practically equal in this example.RELATED PROBLEMS:<ul><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44690">Problem 44690: Comparison of floating-point numbers (doubles)</a></li><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44631">Problem 44631: Make your own Test Suite (part 0)</a></li><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44513">Problem 44513: Add all the numbers between two limits (inclusive)</a></li></ul>

Solve

Solution Stats

16 Solutions
5 Solvers

Last Solution submitted on Aug 22, 2020

Last 200 Solutions

Problem Comments

2 Comments

2 Comments

David Verrelli on 12 Aug 2018

Oops. I just noticed that I forgot to rename the function from "compareDoubles" (used in Problem 44690) to "compareSingles". Unfortunately I cannot make the fix now without breaking the two successful submissions. Sorry about any confusion. —DIV

Rafael S.T. Vieira on 12 Aug 2020

Good problem, but it would be better if base-10 or base-2 was adopted as a standard at the description. It is confusing to keep changing them. And the function name should probably be compareSingles and not compareDoubles. You can fix the test suite by using try and catch while accepting both names for the function.

Problem Recent Solvers5

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 44691. Comparison of floating-point numbers (singles)

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers5

Suggested Problems

More from this Author31

Problem Tags

Community Treasure Hunt

Problem 44691. Comparison of floating-point numbers (singles)

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers5

Suggested Problems

More from this Author31

Problem Tags

Community Treasure Hunt

Players