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<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Appendices</title> <link rel="stylesheet" href="../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> <link rel="home" href="../index.html" title="The Boost C++ Libraries BoostBook Documentation Subset"> <link rel="up" href="../ratio.html" title="Chapter 25. Boost.Ratio 2.1.0"> <link rel="prev" href="reference.html" title="Reference"> <link rel="next" href="../signals.html" title="Chapter 26. Boost.Signals"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../boost.png"></td> <td align="center"><a href="../../../index.html">Home</a></td> <td align="center"><a href="../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="reference.html"><img src="../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ratio.html"><img src="../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../signals.html"><img src="../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h2 class="title" style="clear: both"> <a name="ratio.appendices"></a><a class="link" href="appendices.html" title="Appendices">Appendices</a> </h2></div></div></div> <div class="toc"><dl class="toc"> <dt><a href="appendices.html#ratio.appendices.history">Appendix A: History</a></dt> <dt><a href="appendices.html#ratio.appendices.rationale">Appendix B: Rationale</a></dt> <dt><a href="appendices.html#ratio.appendices.implementation">Appendix C: Implementation Notes</a></dt> <dt><a href="appendices.html#ratio.appendices.faq">Appendix D: FAQ</a></dt> <dt><a href="appendices.html#ratio.appendices.acknowledgements">Appendix E: Acknowledgements</a></dt> <dt><a href="appendices.html#ratio.appendices.tests">Appendix F: Tests</a></dt> <dt><a href="appendices.html#ratio.appendices.tickets">Appendix G: Tickets</a></dt> <dt><a href="appendices.html#ratio.appendices.todo">Appendix H: Future Plans</a></dt> </dl></div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="ratio.appendices.history"></a><a class="link" href="appendices.html#ratio.appendices.history" title="Appendix A: History">Appendix A: History</a> </h3></div></div></div> <div class="toc"><dl class="toc"> <dt><a href="appendices.html#ratio.appendices.history.__version_2_1_0__febreary_1__2014___1_56__">Version 2.1.0, Febreary 1, 2014 - 1.56 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_2_0_1__febreary_1__2013___1_53__">Version 2.0.1, Febreary 1, 2013 - 1.53 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_2_0_0__november_1__2012___1_52__">Version 2.0.0, November 1, 2012 - 1.52 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_1_0_3__august_1__2012___1_51__">Version 1.0.3, August 1, 2012 - 1.51 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_1_0_2__april_1__2012___1_50__">Version 1.0.2, April 1, 2012 - 1.50 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_1_0_1__jan_8__2011___">Version 1.0.1, Jan 8, 2011 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_1_0_0__jan_2__2011__">Version 1.0.0, Jan 2, 2011 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_0_2_1__september_27__2010__">Version 0.2.1, September 27, 2010 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_0_2_0__september_22__2010__">Version 0.2.0, September 22, 2010 </a></dt> <dt><a href="appendices.html#ratio.appendices.history.__version_0_1_0__september_10__2010__">Version 0.1.0, September 10, 2010 </a></dt> </dl></div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_2_1_0__febreary_1__2014___1_56__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_2_1_0__febreary_1__2014___1_56__" title="Version 2.1.0, Febreary 1, 2014 - 1.56">Version 2.1.0, Febreary 1, 2014 - 1.56 </a> </h4></div></div></div> New Features: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/XXXX" target="_top">#XXXX</a> Add ratio_power. </li> <li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/XXXX" target="_top">#XXXX</a> Add IEC binary prefixes. </li> </ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_2_0_1__febreary_1__2013___1_53__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_2_0_1__febreary_1__2013___1_53__" title="Version 2.0.1, Febreary 1, 2013 - 1.53">Version 2.0.1, Febreary 1, 2013 - 1.53 </a> </h4></div></div></div> Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/7616" target="_top">#7616</a> br_mul::nan - warning C4293: '<<' : shift count negative or too big, undefined behavior`. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_2_0_0__november_1__2012___1_52__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_2_0_0__november_1__2012___1_52__" title="Version 2.0.0, November 1, 2012 - 1.52">Version 2.0.0, November 1, 2012 - 1.52 </a> </h4></div></div></div> Features: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> Replace the short_name and long_name functions by symbol and prefix functions respectively. </li></ul></div> Deprecated: The ratio_string<>::short_name and ratio_string<>::long_name are deprecated. Use ratio_string<>::symbol and ratio_string<>::prefix respectively. These functions be removed in 1.55. Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/7478" target="_top">#7478</a> Compiles fails with compilers supporting char16_t and char32_t fails if the library doesn't provides std::u16string and std::u32string. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_1_0_3__august_1__2012___1_51__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_1_0_3__august_1__2012___1_51__" title="Version 1.0.3, August 1, 2012 - 1.51">Version 1.0.3, August 1, 2012 - 1.51 </a> </h4></div></div></div> Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/7075" target="_top">#7075</a> Workaround for error: the type of partial specialization template parameter constant "n1" depends on another template parameter. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_1_0_2__april_1__2012___1_50__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_1_0_2__april_1__2012___1_50__" title="Version 1.0.2, April 1, 2012 - 1.50">Version 1.0.2, April 1, 2012 - 1.50 </a> </h4></div></div></div> Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> <a href="http://svn.boost.org/trac/boost/ticket/6498" target="_top">#6498</a> boost::ratio won't compile with default settings. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_1_0_1__jan_8__2011___"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_1_0_1__jan_8__2011___" title="Version 1.0.1, Jan 8, 2011">Version 1.0.1, Jan 8, 2011 </a> </h4></div></div></div> Features: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> Added MPL Rational Constant and the associated numeric metafunction specializations. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_1_0_0__jan_2__2011__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_1_0_0__jan_2__2011__" title="Version 1.0.0, Jan 2, 2011">Version 1.0.0, Jan 2, 2011 </a> </h4></div></div></div> <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> Moved ratio to trunk. </li> <li class="listitem"> Documentation revision. </li> </ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_0_2_1__september_27__2010__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_0_2_1__september_27__2010__" title="Version 0.2.1, September 27, 2010">Version 0.2.1, September 27, 2010 </a> </h4></div></div></div> Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> Removal of LLVM adapted files due to incompatible License issue. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_0_2_0__september_22__2010__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_0_2_0__september_22__2010__" title="Version 0.2.0, September 22, 2010">Version 0.2.0, September 22, 2010 </a> </h4></div></div></div> Features: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> Added ratio_string traits. </li></ul></div> Fixes: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> ratio_less overflow avoided following the algorithm from libc++. </li></ul></div> Test: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> A more complete test has been included adapted from the test of from libc++/ratio. </li></ul></div> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> <a name="ratio.appendices.history.__version_0_1_0__september_10__2010__"></a><a class="link" href="appendices.html#ratio.appendices.history.__version_0_1_0__september_10__2010__" title="Version 0.1.0, September 10, 2010">Version 0.1.0, September 10, 2010 </a> </h4></div></div></div> Features: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> Ratio has been extracted from Boost.Chrono. </li></ul></div> </div> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="ratio.appendices.rationale"></a><a class="link" href="appendices.html#ratio.appendices.rationale" title="Appendix B: Rationale">Appendix B: Rationale</a> </h3></div></div></div> <h5> <a name="ratio.appendices.rationale.h0"></a> <a name="ratio.appendices.rationale.why_ratio_needs_copyconstruction_and_assignment_from_ratios_having_the_same_normalized_form"></a><a class="link" href="appendices.html#ratio.appendices.rationale.why_ratio_needs_copyconstruction_and_assignment_from_ratios_having_the_same_normalized_form">Why ratio needs CopyConstruction and Assignment from ratios having the same normalized form</a> </h5> Current <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2009/n3000.pdf" target="_top">N3000</a> doesn't allows to copy-construct or assign ratio instances of ratio classes having the same normalized form. This simple example <pre class="programlisting"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><1,3> r1; <a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><3,9> r2; r1 = r2; // (1) </pre> fails to compile in (1). Other example <pre class="programlisting"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><1,3> r1; <a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio_arithmetic" title="ratio Arithmetic"><code class="computeroutput">ratio_subtract</code></a><<a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><2,3>,<a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><1,3> > r2=r1; // (2) </pre> The type of <code class="computeroutput"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio_arithmetic" title="ratio Arithmetic"><code class="computeroutput">ratio_subtract</code></a><<a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><2,3>,<a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><1,3> ></code> could be <code class="computeroutput"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><3,9></code> so the compilation could fail in (2). It could also be <a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput">ratio</code></a><1,3> and the compilation succeeds. <h5> <a name="ratio.appendices.rationale.h1"></a> <a name="ratio.appendices.rationale.why_ratio_needs_the_nested_normalizer_typedef_type"></a><a class="link" href="appendices.html#ratio.appendices.rationale.why_ratio_needs_the_nested_normalizer_typedef_type">Why ratio needs the nested normalizer typedef type</a> </h5> The current resolution of issue LWG 1281 acknowledges the need for a nested type typedef, so Boost.Ratio is tracking the likely final version of std::ratio. </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="ratio.appendices.implementation"></a><a class="link" href="appendices.html#ratio.appendices.implementation" title="Appendix C: Implementation Notes">Appendix C: Implementation Notes</a> </h3></div></div></div> <h5> <a name="ratio.appendices.implementation.h0"></a> <a name="ratio.appendices.implementation.how_does_boost_ratio_try_to_avoid_compile_time_rational_arithmetic_overflow_"></a><a class="link" href="appendices.html#ratio.appendices.implementation.how_does_boost_ratio_try_to_avoid_compile_time_rational_arithmetic_overflow_">How does Boost.Ratio try to avoid compile-time rational arithmetic overflow?</a> </h5> When the result is representable, but a simple application of arithmetic rules would result in overflow, e.g. <code class="computeroutput">ratio_multiply<ratio<INTMAX_MAX,2>,ratio<2,INTMAX_MAX>></code> can be reduced to <code class="computeroutput">ratio<1,1></code>, but the direct result of <code class="computeroutput">ratio<INTMAX_MAX*2,INTMAX_MAX*2></code> would result in overflow. Boost.Ratio implements some simplifications in order to reduce the possibility of overflow. The general ideas are: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> The <code class="computeroutput">num</code> and <code class="computeroutput">den</code> <code class="computeroutput">ratio<></code> fields are normalized. </li> <li class="listitem"> Use the gcd of some of the possible products that can overflow, and simplify before doing the product. </li> <li class="listitem"> Use some equivalences relations that avoid addition or subtraction that can overflow or underflow. </li> </ul></div> The following subsections cover each case in more detail. ratio_add In <pre class="programlisting">(n1/d1)+(n2/d2)=(n1*d2+n2*d1)/(d1*d2) </pre> either n1*d2+n2*d1 or d1*d2 can overflow. <pre class="programlisting">( (n1 * d2) + (n2 * d1) ) -------------------------- (d1 * d2) </pre> Dividing by gcd(d1,d2) on both num and den <pre class="programlisting">( (n1 * (d2/gcd(d1,d2))) + (n2 * (d1/gcd(d1,d2))) ) ---------------------------------------------------- ((d1 * d2) / gcd(d1,d2)) </pre> Multiplying and diving by gcd(n1,n2) in numerator <pre class="programlisting">( ((gcd(n1,n2)*(n1/gcd(n1,n2))) * (d2/gcd(d1,d2))) + ((gcd(n1,n2)*(n2/gcd(n1,n2))) * (d1/gcd(d1,d2))) ) -------------------------------------------------- ( (d1 * d2) / gcd(d1,d2) ) </pre> Factorizing gcd(n1,n2) <pre class="programlisting">( gcd(n1,n2) * ( ((n1/gcd(n1,n2)) * (d2/gcd(d1,d2))) + ((n2/gcd(n1,n2)) * (d1/gcd(d1,d2))) ) ) ------------------------------------------------------------------------------- ( (d1 * d2) / gcd(d1,d2) ) </pre> Regrouping <pre class="programlisting">( gcd(n1,n2) * ( ((n1/gcd(n1,n2)) * (d2/gcd(d1,d2))) + ((n2/gcd(n1,n2)) * (d1/gcd(d1,d2))) ) ) ------------------------------------------------------------------------------- ( (d1 / gcd(d1,d2)) * d2 ) </pre> Dividing by (d1 / gcd(d1,d2)) <pre class="programlisting">( ( gcd(n1,n2) / (d1 / gcd(d1,d2)) ) * ( ((n1/gcd(n1,n2)) * (d2/gcd(d1,d2))) + ((n2/gcd(n1,n2)) * (d1/gcd(d1,d2))) ) ) ------------------------------------------------------------------------------- d2 </pre> Dividing by d2 <pre class="programlisting">( gcd(n1,n2) / (d1 / gcd(d1,d2)) ) * ( ((n1/gcd(n1,n2)) * (d2/gcd(d1,d2))) + ((n2/gcd(n1,n2)) * (d1/gcd(d1,d2))) / d2 ) </pre> This expression correspond to the multiply of two ratios that have less risk of overflow as the initial numerators and denominators appear now in most of the cases divided by a gcd. For ratio_subtract the reasoning is the same. ratio_multiply In <pre class="programlisting">(n1/d1)*(n2/d2)=((n1*n2)/(d1*d2)) </pre> either n1*n2 or d1*d2 can overflow. Dividing by gcc(n1,d2) numerator and denominator <pre class="programlisting">(((n1/gcc(n1,d2))*n2) --------------------- (d1*(d2/gcc(n1,d2)))) </pre> Dividing by gcc(n2,d1) <pre class="programlisting">((n1/gcc(n1,d2))*(n2/gcc(n2,d1))) --------------------------------- ((d1/gcc(n2,d1))*(d2/gcc(n1,d2))) </pre> And now all the initial numerator and denominators have been reduced, avoiding the overflow. For ratio_divide the reasoning is similar. ratio_less In order to evaluate <pre class="programlisting">(n1/d1)<(n2/d2) </pre> without moving to floating-point numbers, two techniques are used: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> First compare the sign of the numerators. </li></ul></div> If sign(n1) < sign(n2) the result is true. If sign(n1) == sign(n2) the result depends on the following after making the numerators positive <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> When the sign is equal the technique used is to work with integer division and modulo when the signs are equal. </li></ul></div> Let call Qi the integer division of ni and di, and Mi the modulo of ni and di. <pre class="programlisting">ni = Qi * di + Mi and Mi < di </pre> Form <pre class="programlisting">((n1*d2)<(d1*n2)) </pre> we get <pre class="programlisting">(((Q1 * d1 + M1)*d2)<(d1*((Q2 * d2 + M2)))) </pre> Developing <pre class="programlisting">((Q1 * d1 * d2)+ (M1*d2))<((d1 * Q2 * d2) + (d1*M2)) </pre> Dividing by d1*d2 <pre class="programlisting">Q1 + (M1/d1) < Q2 + (M2/d2) </pre> If Q1=Q2 the result depends on <pre class="programlisting">(M1/d1) < (M2/d2) </pre> If M1==0==M2 the result is false If M1=0 M2!=0 the result is true If M1!=0 M2==0 the result is false If M1!=0 M2!=0 the result depends on <pre class="programlisting">(d2/M2) < (d1/M1) </pre> If Q1!=Q2, the result of <pre class="programlisting">Q1 + (M1/d1) < Q2 + (M2/d2) </pre> depends only on Q1 and Q2 as Qi are integers and (Mi/di) <1 because Mi<di. if Q1>Q2, Q1==Q2+k, k>=1 <pre class="programlisting">Q2+k + (M1/d1) < Q2 + (M2/d2) k + (M1/d1) < (M2/d2) k < (M2/d2) - (M1/d1) </pre> but the difference between two numbers between 0 and 1 can not be greater than 1, so the result is false. if Q2>Q1, Q2==Q1+k, k>=1 <pre class="programlisting">Q1 + (M1/d1) < Q1+k + (M2/d2) (M1/d1) < k + (M2/d2) (M1/d1) - (M2/d2) < k </pre> which is always true, so the result is true. The following table recapitulates this analisys <div class="informaltable"><table class="ta