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<title>Arithmetic underflow</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Arithmetic underflow</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Underflow" redirects here. For the condition of attempting to read an empty buffer, see <a href="Buffer_underrun" title="Buffer underrun">Buffer underrun</a>.</div>

<p>The term <b>arithmetic underflow</b> (also <b>floating-point underflow</b>, or just <b>underflow</b>) is a condition in a <a href="Computer_program" title="Computer program">computer program</a> where the result of a calculation is a number of more precise absolute value than the computer can actually represent in <a href="Computer_memory" title="Computer memory">memory</a> on its <a href="Central_processing_unit" title="Central processing unit">central processing unit</a> (CPU).

</p><p>Arithmetic underflow can occur when the true result of a <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point operation</a> is smaller in magnitude (that is, closer to zero) than the smallest value representable as a <a href="Normal_number_(computing)" title="Normal number (computing)">normal</a> floating-point number in the target <a href="Data_type" title="Data type">datatype</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Underflow can in part be regarded as negative <a href="Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">overflow</a> of the <a href="Exponent" class="mw-redirect" title="Exponent">exponent</a> of the floating-point value. For example, if the exponent part can represent values from −128 to 127, then a result with a value less than −128 may cause underflow.

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<div class="mw-heading mw-heading2"><h2 id="Underflow_gap">Underflow gap</h2></div>

<p>The interval between −<i>fminN</i> and <i>fminN</i>, where <i>fminN</i> is the smallest positive normal floating-point value, is called the underflow gap. This is because the size of this interval is many orders of magnitude larger than the distance between adjacent normal floating-point values just outside the gap. For instance, if the floating-point datatype can represent 20 <a href="Bit" title="Bit">bits</a>, the underflow gap is 2<sup>21</sup> times larger than the absolute distance between adjacent floating-point values just outside the gap.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>In older designs, the underflow gap had just one usable value, zero. When an underflow occurred, the true result was replaced by zero (either directly by the hardware, or by system software handling the primary underflow condition). This replacement is called "flush to zero".

</p><p>The 1984 edition of <a href="IEEE_754" title="IEEE 754">IEEE 754</a> introduced <a href="Subnormal_numbers" class="mw-redirect" title="Subnormal numbers">subnormal numbers</a>. The subnormal numbers (including zero) fill the underflow gap with values where the absolute distance between adjacent values is the same as for adjacent values just outside the underflow gap. This enables "gradual underflow", where a nearest subnormal value is used, just as a nearest normal value is used when possible. Even when using gradual underflow, the nearest value may be zero.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>The absolute distance between adjacent floating-point values just outside the gap is called the <a href="Machine_epsilon" title="Machine epsilon">machine epsilon</a>, typically characterized by the largest value whose sum with the value 1 will result in the answer with value 1 in that floating-point scheme.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This is the maximum value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

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</math></span><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span> that satisfies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {fl} (1+\epsilon )=\operatorname {fl} (1)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {fl} (1+\epsilon )=\operatorname {fl} (1)}</annotation>

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</math></span><img src="./19dcd93764833dcc6385ab27bb0e8b7781734968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.544ex; height:2.843ex;" alt="{\displaystyle \operatorname {fl} (1+\epsilon )=\operatorname {fl} (1)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {fl} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {fl} }</annotation>

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</math></span><img src="./1806d48499c7fbd74889fcbab80466e0264f6aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.359ex; height:2.176ex;" alt="{\displaystyle \operatorname {fl} }" loading="lazy"></span> is a function which converts the real value into the floating-point representation. While the machine epsilon is not to be confused with the underflow level (assuming subnormal numbers), it is closely related. The machine epsilon is dependent on the number of bits which make up the <a href="Significand" title="Significand">significand</a>, whereas the underflow level depends on the number of digits which make up the exponent field. In most floating-point systems, the underflow level is smaller than the machine epsilon.

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<div class="mw-heading mw-heading2"><h2 id="Handling_of_underflow">Handling of underflow</h2></div>

<p>The occurrence of an underflow may set a ("sticky") status bit, raise an exception, at the hardware level generate an interrupt, or may cause some combination of these effects.

</p><p>As specified in <a href="IEEE_754" title="IEEE 754">IEEE 754</a>, the underflow condition is only signaled if there is also a loss of precision. Typically this is determined as the final result being inexact.

However, if the user is <a href="Trap_(computing)" class="mw-redirect" title="Trap (computing)">trapping</a> on underflow, this may happen regardless of consideration for loss of precision. The default handling in IEEE 754 for underflow (as well as other exceptions) is to record as a floating-point status that underflow has occurred. This is specified for the application-programming level, but often also interpreted as how to handle it at the hardware level.

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<ul><li><a href="Denormal_number" class="mw-redirect" title="Denormal number">Denormal number</a></li>

<li><a href="Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point arithmetic</a></li>

<li><a href="Integer_overflow" title="Integer overflow">Integer overflow</a></li>

<li><a href="Logarithmic_number_system" title="Logarithmic number system">Logarithmic number system</a></li>

<li><a href="Machine_epsilon" title="Machine epsilon">Machine epsilon</a></li>

<li><a href="Normal_number_(computing)" title="Normal number (computing)">Normal number (computing)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFCoonen1980" class="citation journal cs1">Coonen, Jerome T. (1980). "An implementation guide to a proposed standard for floating-point arithmetic". <i>Computer</i>. <b>13</b> (1): <span class="nowrap">68–</span>79. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fmc.1980.1653344">10.1109/mc.1980.1653344</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206445847">206445847</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSun_Microsystems2005" class="citation book cs1">Sun Microsystems (2005). <a rel="nofollow" class="external text" href="https://docs.oracle.com/cd/E19422-01/819-3693/"><i>Numerical Computation Guide</i></a>. Oracle<span class="reference-accessdate">. Retrieved <span class="nowrap">21 April</span> 2018</span>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFDemmel1984" class="citation journal cs1">Demmel, James (1984). "Underflow and the Reliability of Numerical Software". <i>SIAM Journal on Scientific and Statistical Computing</i>. <b>5</b> (4): <span class="nowrap">887–</span>919. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0905062">10.1137/0905062</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHeath2002" class="citation book cs1">Heath, Michael T. (2002). <i>Scientific Computing</i> (2nd ed.). New York: McGraw-Hill. p. 20. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-239910-4</bdi>.</cite></span>

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