Problem 44690. Comparison of floating-point numbers (doubles)

Created by David Verrelli

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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: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 be defined as no more than 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 which</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, 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>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-50\n\n% Output\npracticallyEqual = false]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Explanation: </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 = 2⁻¹⁰⁷⁴, 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 = 2⁻²¹⁹. 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> = 2⁻²¹⁹, and the threshold is 10×2⁻²¹⁹. 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 1×10⁻⁵⁰, and this difference exceeds 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> are</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>not</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/44691\"><w:r><w:t>Problem 44691. Comparison of floating-point numbers (singles)</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 <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 be defined as no more than ten times ε, in which ε is the larger of ε₁ & ε₂, and 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-50
</pre><pre class="language-matlab">% Output
practicallyEqual = false
</pre>Explanation: <tt>A</tt> is represented with a precision of ε₁ = 2⁻¹⁰⁷⁴, whereas <tt>B</tt> is represented with a precision of ε₂ = 2⁻²¹⁹. Thus ε = 2⁻²¹⁹, and the threshold is 10×2⁻²¹⁹. The difference between <tt>A</tt> and <tt>B</tt> is 1×10⁻⁵⁰, and this difference exceeds the threshold. Thus <tt>A</tt> and <tt>B</tt> are not practically equal in this example.RELATED PROBLEMS:<ul><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44691">Problem 44691. Comparison of floating-point numbers (singles)</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>

<https://au.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html Floating-point numbers> 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 <https://au.mathworks.com/help/matlab/ref/double.html double>, namely |A| and |B|, you must return a scalar <https://au.mathworks.com/help/matlab/ref/logical.html logical> output set to <https://au.mathworks.com/help/matlab/ref/true.html |true|> if the difference in magnitude between |A| and |B| is 'small', or otherwise set to <https://au.mathworks.com/help/matlab/ref/false.html |false|>.

For _this_ problem "small" shall be defined as no more than ten times _ε_, in which _ε_ is the larger of _ε_₁ & _ε_₂, and _ε_₁ is the <https://au.mathworks.com/help/matlab/ref/eps.html floating-point precision> with which |A| is represented, and _ε_₂ is the precision with which |B| is represented.

EXAMPLE:

% Input
  A = 0
  B = 1E-50

% Output
  practicallyEqual = false

Explanation:  |A| is represented with a precision of _ε_₁ = 2⁻¹⁰⁷⁴, whereas |B| is represented with a precision of _ε_₂ = 2⁻²¹⁹.  Thus _ε_ = 2⁻²¹⁹, and the threshold is 10×2⁻²¹⁹.  The difference between |A| and |B| is 1×10⁻⁵⁰, and this difference exceeds the threshold.  Thus |A| and |B| are _not_ practically equal in this example.

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Last Solution submitted on Aug 22, 2020

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Rafael S.T. Vieira on 12 Aug 2020

Again, good problem, but it would be better if base-10 or base-2 was adopted as a standard at the description

David Verrelli on 18 Jul 2021

Hello, Rafael. Thank-you for the feedback. Each approach has pros and cons.
Using the same base for all numbers in the problem statement would make comparison easy at a glance, but the values would not all be exact.
The mixture of bases used in the problem statement means that exact values are used for all numbers; although the reader has some inconvenience to perform a conversion in order to compare magnitudes, there may be some more educational value in that, IMHO.
—DIV

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Solution 1565674

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3 Comments

David Verrelli on 22 Jun 2018

"The hard way to go about it, but it worked like a charm!" :-)

James on 22 Jun 2018

If you think this one is ugly (and I agree 100% it is!) you should have seen my first few attempts, which included using "typecast" to get the numbers as uint64s and trying to compare them that way. At least this one worked...eventually.

David Verrelli on 23 Jun 2018

James, actually, it's interesting on Cody to get to see things from another point of view. Kudos for original thinking :-) —DIV

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Problem 44690. Comparison of floating-point numbers (doubles)

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