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copied!<p>When comparing floating point numbers for closeness you usually want to measure their relative difference, which is defined as</p> <pre><code>if (abs(x) != 0 || abs(y) != 0) rel_diff (x, y) = abs((x - y) / max(abs(x),abs(y)) else rel_diff(x,y) = max(abs(x),abs(y)) </code></pre> <p>For example, </p> <pre><code>rel_diff(1.12345, 1.12367) = 0.000195787019 rel_diff(112345.0, 112367.0) = 0.000195787019 rel_diff(112345E100, 112367E100) = 0.000195787019 </code></pre> <p>The idea is to measure the number of leading significant digits the numbers have in common; if you take the -log10 of 0.000195787019 you get 3.70821611, which is about the number of leading base 10 digits all the examples have in common.</p> <p>If you need to determine if two floating point numbers are equal you should do something like</p> <pre><code>if (rel_diff(x,y) < error_factor * machine_epsilon()) then print "equal\n"; </code></pre> <p>where machine epsilon is the smallest number that can be held in the mantissa of the floating point hardware being used. Most computer languages have a function call to get this value. error_factor should be based on the number of significant digits you think will be consumed by rounding errors (and others) in the calculations of the numbers x and y. For example, if I knew that x and y were the result of about 1000 summations and did not know any bounds on the numbers being summed, I would set error_factor to about 100.</p> <p>Tried to add these as links but couldn't since this is my first post:</p> <ul> <li>en.wikipedia.org/wiki/Relative_difference</li> <li>en.wikipedia.org/wiki/Machine_epsilon</li> <li>en.wikipedia.org/wiki/Significand (mantissa) </li> <li>en.wikipedia.org/wiki/Rounding_error</li> </ul>

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