<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>Flocq.Core.Fcore_FTZ</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library Flocq.Core.Fcore_FTZ</h1>
<div class="code">
</div>
<div class="doc">
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
<div class="paragraph"> </div>
Copyright (C) 2010-2013 Sylvie Boldo
<br />
Copyright (C) 2010-2013 Guillaume Melquiond
<div class="paragraph"> </div>
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.
<div class="paragraph"> </div>
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
<div class="paragraph"> </div>
<a name="lab7"></a><h1 class="section">Floating-point format with abrupt underflow</h1>
</div>
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_Raux.html#"><span class="id" title="library">Fcore_Raux</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_defs.html#"><span class="id" title="library">Fcore_defs</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_rnd.html#"><span class="id" title="library">Fcore_rnd</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#"><span class="id" title="library">Fcore_generic_fmt</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_float_prop.html#"><span class="id" title="library">Fcore_float_prop</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_ulp.html#"><span class="id" title="library">Fcore_ulp</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_FLX.html#"><span class="id" title="library">Fcore_FLX</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="RND_FTZ"><span class="id" title="section">RND_FTZ</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <span class="id" title="keyword">beta</span> : <a class="idref" href="Flocq.Core.Fcore_Zaux.html#radix"><span class="id" title="record">radix</span></a>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span> := (<a class="idref" href="Flocq.Core.Fcore_Raux.html#bpow"><span class="id" title="definition">bpow</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">e</span>).<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a name="RND_FTZ.prec"><span class="id" title="variable">prec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">prec_gt_0_</span> : <a class="idref" href="Flocq.Core.Fcore_FLX.html#Prec_gt_0"><span class="id" title="class">Prec_gt_0</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="FTZ_format"><span class="id" title="definition">FTZ_format</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">f</span> : <a class="idref" href="Flocq.Core.Fcore_defs.html#float"><span class="id" title="record">float</span></a> <span class="id" title="keyword">beta</span><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a><br/>
<a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R0"><span class="id" title="axiom">R0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#Zabs"><span class="id" title="abbreviation">Zabs</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
(<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#f"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">Z</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> <span class="id" title="var">e</span> := <span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a>) <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <span class="id" title="keyword">then</span> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">else</span> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>.<br/>
<br/>
</div>
<div class="doc">
Properties of the FTZ format
</div>
<div class="code">
<span class="id" title="keyword">Global Instance</span> <a name="FTZ_exp_valid"><span class="id" title="instance">FTZ_exp_valid</span></a> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_exp"><span class="id" title="class">Valid_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="FLXN_format_FTZ"><span class="id" title="lemma">FLXN_format_FTZ</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_format"><span class="id" title="definition">FTZ_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_FLX.html#FLXN_format"><span class="id" title="definition">FLXN_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="generic_format_FTZ"><span class="id" title="lemma">generic_format_FTZ</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_format"><span class="id" title="definition">FTZ_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="FTZ_format_generic"><span class="id" title="lemma">FTZ_format_generic</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_format"><span class="id" title="definition">FTZ_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="FTZ_format_satisfies_any"><span class="id" title="lemma">FTZ_format_satisfies_any</span></a> :<br/>
<a class="idref" href="Flocq.Core.Fcore_rnd.html#satisfies_any"><span class="id" title="inductive">satisfies_any</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_format"><span class="id" title="definition">FTZ_format</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="FTZ_format_FLXN"><span class="id" title="lemma">FTZ_format_FLXN</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>,<br/>
(<a class="idref" href="Flocq.Core.Fcore_FTZ.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="Flocq.Core.Fcore_FLX.html#FLXN_format"><span class="id" title="definition">FLXN_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_format"><span class="id" title="definition">FTZ_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="ulp_FTZ_0"><span class="id" title="lemma">ulp_FTZ_0</span></a>: <a class="idref" href="Flocq.Core.Fcore_ulp.html#ulp"><span class="id" title="definition">ulp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a>1).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="RND_FTZ.FTZ_round"><span class="id" title="section">FTZ_round</span></a>.<br/>
<br/>
</div>
<div class="doc">
Rounding with FTZ
</div>
<div class="code">
<span class="id" title="keyword">Variable</span> <a name="RND_FTZ.FTZ_round.rnd"><span class="id" title="variable">rnd</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">valid_rnd</span> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_rnd"><span class="id" title="class">Valid_rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.FTZ_round.rnd"><span class="id" title="variable">rnd</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Zrnd_FTZ"><span class="id" title="definition">Zrnd_FTZ</span></a> <span class="id" title="var">x</span> :=<br/>
<span class="id" title="keyword">if</span> <a class="idref" href="Flocq.Core.Fcore_Raux.html#Rle_bool"><span class="id" title="definition">Rle_bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R1"><span class="id" title="axiom">R1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>) <span class="id" title="keyword">then</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.FTZ_round.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a>.<br/>
<br/>
<span class="id" title="keyword">Global Instance</span> <a name="valid_rnd_FTZ"><span class="id" title="instance">valid_rnd_FTZ</span></a> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_rnd"><span class="id" title="class">Valid_rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#Zrnd_FTZ"><span class="id" title="definition">Zrnd_FTZ</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_FTZ_FLX"><span class="id" title="lemma">round_FTZ_FLX</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>,<br/>
(<a class="idref" href="Flocq.Core.Fcore_FTZ.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#Zrnd_FTZ"><span class="id" title="definition">Zrnd_FTZ</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="Flocq.Core.Fcore_FLX.html#FLX_exp"><span class="id" title="definition">FLX_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a>) <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.FTZ_round.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_FTZ_small"><span class="id" title="lemma">round_FTZ_small</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>,<br/>
(<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1))%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#FTZ_exp"><span class="id" title="definition">FTZ_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#Zrnd_FTZ"><span class="id" title="definition">Zrnd_FTZ</span></a> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R0"><span class="id" title="axiom">R0</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ.FTZ_round"><span class="id" title="section">FTZ_round</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Core.Fcore_FTZ.html#RND_FTZ"><span class="id" title="section">RND_FTZ</span></a>.<br/>
</div>
</div>
<div id="footer">
<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
</div>
</div>
</body>
</html>
distrib
>
Mageia
>
6
>
armv5tl
>
media
>
core-release
>
by-pkgid
>
3c7997fe0e8405f6d7b5563e63d17a78
>
files
>
86
