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| </h2></div></div></div> |
| <div class="toc"><dl class="toc"> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history">Appendix A: History</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.rationale">Appendix B: Rationale</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.implementation">Appendix C: Implementation |
| Notes</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.faq">Appendix D: FAQ</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.acknowledgements">Appendix E: Acknowledgements</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.tests">Appendix F: Tests</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.tickets">Appendix G: Tickets</a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.todo">Appendix H: Future Plans</a></span></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><span class="section"><a href="appendices.html#ratio.appendices.history.__version_2_1_0__febreary_1__2014___1_56__"><span class="bold"><strong>Version 2.1.0, Febreary 1, 2014 - 1.56</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_2_0_1__febreary_1__2013___1_53__"><span class="bold"><strong>Version 2.0.1, Febreary 1, 2013 - 1.53</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_2_0_0__november_1__2012___1_52__"><span class="bold"><strong>Version 2.0.0, November 1, 2012 - 1.52</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_1_0_3__august_1__2012___1_51__"><span class="bold"><strong>Version 1.0.3, August 1, 2012 - 1.51</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_1_0_2__april_1__2012___1_50__"><span class="bold"><strong>Version 1.0.2, April 1, 2012 - 1.50</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_1_0_1__jan_8__2011___"><span class="bold"><strong>Version 1.0.1, Jan 8, 2011 </strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_1_0_0__jan_2__2011__"><span class="bold"><strong>Version 1.0.0, Jan 2, 2011</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_0_2_1__september_27__2010__"><span class="bold"><strong>Version 0.2.1, September 27, 2010</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_0_2_0__september_22__2010__"><span class="bold"><strong>Version 0.2.0, September 22, 2010</strong></span> </a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.history.__version_0_1_0__september_10__2010__"><span class="bold"><strong>Version 0.1.0, September 10, 2010</strong></span> </a></span></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"><span class="bold"><strong>Version 2.1.0, Febreary 1, 2014 - 1.56</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>New Features:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 2.0.1, Febreary 1, 2013 - 1.53</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 2.0.0, November 1, 2012 - 1.52</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Features:</strong></span> |
| </p> |
| <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> |
| <p> |
| <span class="bold"><strong>Deprecated:</strong></span> |
| </p> |
| <p> |
| 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. |
| </p> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 1.0.3, August 1, 2012 - 1.51</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 1.0.2, April 1, 2012 - 1.50</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 1.0.1, Jan 8, 2011 </strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Features:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 1.0.0, Jan 2, 2011</strong></span> </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"><span class="bold"><strong>Version 0.2.1, September 27, 2010</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 0.2.0, September 22, 2010</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Features:</strong></span> |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| Added ratio_string traits. |
| </li></ul></div> |
| <p> |
| <span class="bold"><strong>Fixes:</strong></span> |
| </p> |
| <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> |
| <p> |
| <span class="bold"><strong>Test:</strong></span> |
| </p> |
| <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"><span class="bold"><strong>Version 0.1.0, September 10, 2010</strong></span> </a> |
| </h4></div></div></div> |
| <p> |
| <span class="bold"><strong>Features:</strong></span> |
| </p> |
| <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> |
| <span class="phrase"><a name="ratio.appendices.rationale.why_ratio_needs_copyconstruction_and_assignment_from_ratios_having_the_same_normalized_form"></a></span><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> |
| <p> |
| Current <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2009/n3000.pdf" target="_top"><span class="bold"><strong>N3000</strong></span></a> doesn't allows to copy-construct or |
| assign ratio instances of ratio classes having the same normalized form. |
| </p> |
| <p> |
| This simple example |
| </p> |
| <pre class="programlisting"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="identifier">r1</span><span class="special">;</span> |
| <a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">3</span><span class="special">,</span><span class="number">9</span><span class="special">></span> <span class="identifier">r2</span><span class="special">;</span> |
| <span class="identifier">r1</span> <span class="special">=</span> <span class="identifier">r2</span><span class="special">;</span> <span class="comment">// (1)</span> |
| </pre> |
| <p> |
| fails to compile in (1). Other example |
| </p> |
| <pre class="programlisting"><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="identifier">r1</span><span class="special">;</span> |
| <a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio_arithmetic" title="ratio Arithmetic"><code class="computeroutput"><span class="identifier">ratio_subtract</span></code></a><span class="special"><</span><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">>,</span><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="special">></span> <span class="identifier">r2</span><span class="special">=</span><span class="identifier">r1</span><span class="special">;</span> <span class="comment">// (2)</span> |
| </pre> |
| <p> |
| 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"><span class="identifier">ratio_subtract</span></code></a><span class="special"><</span><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">>,</span><a class="link" href="reference.html#ratio.reference.std.ratio_hpp.ratio" title="Class Template ratio<>"><code class="computeroutput"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="special">></span></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"><span class="identifier">ratio</span></code></a><span class="special"><</span><span class="number">3</span><span class="special">,</span><span class="number">9</span><span class="special">></span></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"><span class="identifier">ratio</span></code></a><1,3> and the compilation |
| succeeds. |
| </p> |
| <h5> |
| <a name="ratio.appendices.rationale.h1"></a> |
| <span class="phrase"><a name="ratio.appendices.rationale.why_ratio_needs_the_nested_normalizer_typedef_type"></a></span><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> |
| <p> |
| 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. |
| </p> |
| </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> |
| <span class="phrase"><a name="ratio.appendices.implementation.how_does_boost_ratio_try_to_avoid_compile_time_rational_arithmetic_overflow_"></a></span><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> |
| <p> |
| When the result is representable, but a simple application of arithmetic |
| rules would result in overflow, e.g. <code class="computeroutput"><span class="identifier">ratio_multiply</span><span class="special"><</span><span class="identifier">ratio</span><span class="special"><</span><span class="identifier">INTMAX_MAX</span><span class="special">,</span><span class="number">2</span><span class="special">>,</span><span class="identifier">ratio</span><span class="special"><</span><span class="number">2</span><span class="special">,</span><span class="identifier">INTMAX_MAX</span><span class="special">>></span></code> can be reduced to <code class="computeroutput"><span class="identifier">ratio</span><span class="special"><</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">></span></code>, but |
| the direct result of <code class="computeroutput"><span class="identifier">ratio</span><span class="special"><</span><span class="identifier">INTMAX_MAX</span><span class="special">*</span><span class="number">2</span><span class="special">,</span><span class="identifier">INTMAX_MAX</span><span class="special">*</span><span class="number">2</span><span class="special">></span></code> would |
| result in overflow. |
| </p> |
| <p> |
| Boost.Ratio implements some simplifications in order to reduce the possibility |
| of overflow. The general ideas are: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| The <code class="computeroutput"><span class="identifier">num</span></code> and <code class="computeroutput"><span class="identifier">den</span></code> <code class="computeroutput"><span class="identifier">ratio</span><span class="special"><></span></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> |
| <p> |
| The following subsections cover each case in more detail. |
| </p> |
| <p> |
| <span class="bold"><strong>ratio_add</strong></span> |
| </p> |
| <p> |
| In |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)+(</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)=(</span><span class="identifier">n1</span><span class="special">*</span><span class="identifier">d2</span><span class="special">+</span><span class="identifier">n2</span><span class="special">*</span><span class="identifier">d1</span><span class="special">)/(</span><span class="identifier">d1</span><span class="special">*</span><span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| either n1*d2+n2*d1 or d1*d2 can overflow. |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="special">(</span><span class="identifier">n1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">n2</span> <span class="special">*</span> <span class="identifier">d1</span><span class="special">)</span> <span class="special">)</span> |
| <span class="special">--------------------------</span> |
| <span class="special">(</span><span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| Dividing by gcd(d1,d2) on both num and den |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="special">(</span><span class="identifier">n1</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">n2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">)</span> |
| <span class="special">----------------------------------------------------</span> |
| <span class="special">((</span><span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))</span> |
| </pre> |
| <p> |
| Multiplying and diving by gcd(n1,n2) in numerator |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="special">((</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)*(</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> |
| <span class="special">((</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)*(</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> |
| <span class="special">)</span> |
| <span class="special">--------------------------------------------------</span> |
| <span class="special">(</span> <span class="special">(</span><span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)</span> <span class="special">)</span> |
| </pre> |
| <p> |
| Factorizing gcd(n1,n2) |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)</span> <span class="special">*</span> |
| <span class="special">(</span> <span class="special">((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> <span class="special">((</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">)</span> |
| <span class="special">)</span> |
| <span class="special">-------------------------------------------------------------------------------</span> |
| <span class="special">(</span> <span class="special">(</span><span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)</span> <span class="special">)</span> |
| </pre> |
| <p> |
| Regrouping |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)</span> <span class="special">*</span> |
| <span class="special">(</span> <span class="special">((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> <span class="special">((</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">)</span> |
| <span class="special">)</span> |
| <span class="special">-------------------------------------------------------------------------------</span> |
| <span class="special">(</span> <span class="special">(</span><span class="identifier">d1</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))</span> <span class="special">*</span> <span class="identifier">d2</span> <span class="special">)</span> |
| </pre> |
| <p> |
| Dividing by (d1 / gcd(d1,d2)) |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="special">(</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">d1</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))</span> <span class="special">)</span> <span class="special">*</span> |
| <span class="special">(</span> <span class="special">((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> <span class="special">((</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">)</span> |
| <span class="special">)</span> |
| <span class="special">-------------------------------------------------------------------------------</span> |
| <span class="identifier">d2</span> |
| </pre> |
| <p> |
| Dividing by d2 |
| </p> |
| <pre class="programlisting"><span class="special">(</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">d1</span> <span class="special">/</span> <span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))</span> <span class="special">)</span> <span class="special">*</span> |
| <span class="special">(</span> <span class="special">((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">+</span> <span class="special">((</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">n2</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcd</span><span class="special">(</span><span class="identifier">d1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> <span class="special">/</span> <span class="identifier">d2</span> <span class="special">)</span> |
| </pre> |
| <p> |
| 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. |
| </p> |
| <p> |
| For ratio_subtract the reasoning is the same. |
| </p> |
| <p> |
| <span class="bold"><strong>ratio_multiply</strong></span> |
| </p> |
| <p> |
| In |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)*(</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)=((</span><span class="identifier">n1</span><span class="special">*</span><span class="identifier">n2</span><span class="special">)/(</span><span class="identifier">d1</span><span class="special">*</span><span class="identifier">d2</span><span class="special">))</span> |
| </pre> |
| <p> |
| either n1*n2 or d1*d2 can overflow. |
| </p> |
| <p> |
| Dividing by gcc(n1,d2) numerator and denominator |
| </p> |
| <pre class="programlisting"><span class="special">(((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))*</span><span class="identifier">n2</span><span class="special">)</span> |
| <span class="special">---------------------</span> |
| <span class="special">(</span><span class="identifier">d1</span><span class="special">*(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))))</span> |
| </pre> |
| <p> |
| Dividing by gcc(n2,d1) |
| </p> |
| <pre class="programlisting"><span class="special">((</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">))*(</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n2</span><span class="special">,</span><span class="identifier">d1</span><span class="special">)))</span> |
| <span class="special">---------------------------------</span> |
| <span class="special">((</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n2</span><span class="special">,</span><span class="identifier">d1</span><span class="special">))*(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">gcc</span><span class="special">(</span><span class="identifier">n1</span><span class="special">,</span><span class="identifier">d2</span><span class="special">)))</span> |
| </pre> |
| <p> |
| And now all the initial numerator and denominators have been reduced, avoiding |
| the overflow. |
| </p> |
| <p> |
| For ratio_divide the reasoning is similar. |
| </p> |
| <p> |
| <span class="bold"><strong>ratio_less</strong></span> |
| </p> |
| <p> |
| In order to evaluate |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">n1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)<(</span><span class="identifier">n2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| without moving to floating-point numbers, two techniques are used: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> |
| First compare the sign of the numerators. |
| </li></ul></div> |
| <p> |
| If sign(n1) < sign(n2) the result is true. |
| </p> |
| <p> |
| If sign(n1) == sign(n2) the result depends on the following after making |
| the numerators positive |
| </p> |
| <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> |
| <p> |
| Let call Qi the integer division of ni and di, and Mi the modulo of ni and |
| di. |
| </p> |
| <pre class="programlisting"><span class="identifier">ni</span> <span class="special">=</span> <span class="identifier">Qi</span> <span class="special">*</span> <span class="identifier">di</span> <span class="special">+</span> <span class="identifier">Mi</span> <span class="keyword">and</span> <span class="identifier">Mi</span> <span class="special"><</span> <span class="identifier">di</span> |
| </pre> |
| <p> |
| Form |
| </p> |
| <pre class="programlisting"><span class="special">((</span><span class="identifier">n1</span><span class="special">*</span><span class="identifier">d2</span><span class="special">)<(</span><span class="identifier">d1</span><span class="special">*</span><span class="identifier">n2</span><span class="special">))</span> |
| </pre> |
| <p> |
| we get |
| </p> |
| <pre class="programlisting"><span class="special">(((</span><span class="identifier">Q1</span> <span class="special">*</span> <span class="identifier">d1</span> <span class="special">+</span> <span class="identifier">M1</span><span class="special">)*</span><span class="identifier">d2</span><span class="special">)<(</span><span class="identifier">d1</span><span class="special">*((</span><span class="identifier">Q2</span> <span class="special">*</span> <span class="identifier">d2</span> <span class="special">+</span> <span class="identifier">M2</span><span class="special">))))</span> |
| </pre> |
| <p> |
| Developing |
| </p> |
| <pre class="programlisting"><span class="special">((</span><span class="identifier">Q1</span> <span class="special">*</span> <span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">*</span><span class="identifier">d2</span><span class="special">))<((</span><span class="identifier">d1</span> <span class="special">*</span> <span class="identifier">Q2</span> <span class="special">*</span> <span class="identifier">d2</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">*</span><span class="identifier">M2</span><span class="special">))</span> |
| </pre> |
| <p> |
| Dividing by d1*d2 |
| </p> |
| <pre class="programlisting"><span class="identifier">Q1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">Q2</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| If Q1=Q2 the result depends on |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| If M1==0==M2 the result is false |
| </p> |
| <p> |
| If M1=0 M2!=0 the result is true |
| </p> |
| <p> |
| If M1!=0 M2==0 the result is false |
| </p> |
| <p> |
| If M1!=0 M2!=0 the result depends on |
| </p> |
| <pre class="programlisting"><span class="special">(</span><span class="identifier">d2</span><span class="special">/</span><span class="identifier">M2</span><span class="special">)</span> <span class="special"><</span> <span class="special">(</span><span class="identifier">d1</span><span class="special">/</span><span class="identifier">M1</span><span class="special">)</span> |
| </pre> |
| <p> |
| If Q1!=Q2, the result of |
| </p> |
| <pre class="programlisting"><span class="identifier">Q1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">Q2</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| </pre> |
| <p> |
| depends only on Q1 and Q2 as Qi are integers and (Mi/di) <1 because Mi<di. |
| </p> |
| <p> |
| if Q1>Q2, Q1==Q2+k, k>=1 |
| </p> |
| <pre class="programlisting"><span class="identifier">Q2</span><span class="special">+</span><span class="identifier">k</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">Q2</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| <span class="identifier">k</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| <span class="identifier">k</span> <span class="special"><</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> |
| </pre> |
| <p> |
| but the difference between two numbers between 0 and 1 can not be greater |
| than 1, so the result is false. |
| </p> |
| <p> |
| if Q2>Q1, Q2==Q1+k, k>=1 |
| </p> |
| <pre class="programlisting"><span class="identifier">Q1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">Q1</span><span class="special">+</span><span class="identifier">k</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">k</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> |
| <span class="special">(</span><span class="identifier">M1</span><span class="special">/</span><span class="identifier">d1</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">M2</span><span class="special">/</span><span class="identifier">d2</span><span class="special">)</span> <span class="special"><</span> <span class="identifier">k</span> |
| </pre> |
| <p> |
| which is always true, so the result is true. |
| </p> |
| <p> |
| The following table recapitulates this analisys |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </th> |
| <th> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </th> |
| <th> |
| <p> |
| Q1 |
| </p> |
| </th> |
| <th> |
| <p> |
| Q2 |
| </p> |
| </th> |
| <th> |
| <p> |
| M1 |
| </p> |
| </th> |
| <th> |
| <p> |
| M2 |
| </p> |
| </th> |
| <th> |
| <p> |
| Result |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </td> |
| <td> |
| <p> |
| Q1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Q2 |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Q1 < Q2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| <td> |
| <p> |
| false |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| true |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| <td> |
| <p> |
| false |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio<n1,d1> |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio<n2,d2> |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| Q |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| !=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| ratio_less<ratio<d2,M2>, ratio<d1/M1>> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <div class="section"><div class="titlepage"><div><div><h3 class="title"> |
| <a name="ratio.appendices.faq"></a><a class="link" href="appendices.html#ratio.appendices.faq" title="Appendix D: FAQ">Appendix D: FAQ</a> |
| </h3></div></div></div></div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="ratio.appendices.acknowledgements"></a><a class="link" href="appendices.html#ratio.appendices.acknowledgements" title="Appendix E: Acknowledgements">Appendix E: Acknowledgements</a> |
| </h3></div></div></div> |
| <p> |
| The library code was derived from Howard Hinnant's <code class="computeroutput"><span class="identifier">time2_demo</span></code> |
| prototype. Many thanks to Howard for making his code available under the |
| Boost license. The original code was modified by Beman Dawes to conform to |
| Boost conventions. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">time2_demo</span></code> contained this |
| comment: |
| </p> |
| <p> |
| Much thanks to Andrei Alexandrescu, Walter Brown, Peter Dimov, Jeff Garland, |
| Terry Golubiewski, Daniel Krugler, Anthony Williams. |
| </p> |
| <p> |
| Howard Hinnant, who is the real author of the library, has provided valuable |
| feedback and suggestions during the development of the library. In particular, |
| The ratio_io.hpp source has been adapted from the experimental header <code class="computeroutput"><span class="special"><</span><span class="identifier">ratio_io</span><span class="special">></span></code> from Howard Hinnant. |
| </p> |
| <p> |
| The acceptance review of Boost.Ratio took place between October 2nd and 11th |
| 2010. Many thanks to Anthony Williams, the review manager, and to all the |
| reviewers: Bruno Santos, Joel Falcou, Robert Stewart, Roland Bock, Tom Tan |
| and Paul A. Bristol. |
| </p> |
| <p> |
| Thanks to Andrew Chinoff and Paul A. Bristol for his help polishing the documentation. |
| </p> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="ratio.appendices.tests"></a><a class="link" href="appendices.html#ratio.appendices.tests" title="Appendix F: Tests">Appendix F: Tests</a> |
| </h3></div></div></div> |
| <div class="toc"><dl class="toc"> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.tests._ratio_"><code class="computeroutput"><span class="identifier">ratio</span></code></a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.tests._comparison_"><code class="computeroutput"><span class="identifier">comparison</span></code></a></span></dt> |
| <dt><span class="section"><a href="appendices.html#ratio.appendices.tests._arithmetic_"><code class="computeroutput"><span class="identifier">arithmetic</span></code></a></span></dt> |
| </dl></div> |
| <p> |
| In order to test you need to run |
| </p> |
| <pre class="programlisting"><span class="identifier">bjam</span> <span class="identifier">libs</span><span class="special">/</span><span class="identifier">ratio</span><span class="special">/</span><span class="identifier">test</span> |
| </pre> |
| <p> |
| You can also run a specific suite of test by doing |
| </p> |
| <pre class="programlisting"><span class="identifier">cd</span> <span class="identifier">libs</span><span class="special">/</span><span class="identifier">chrono</span><span class="special">/</span><span class="identifier">test</span> |
| <span class="identifier">bjam</span> <span class="identifier">ratio</span> |
| </pre> |
| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="ratio.appendices.tests._ratio_"></a><a class="link" href="appendices.html#ratio.appendices.tests._ratio_" title="ratio"><code class="computeroutput"><span class="identifier">ratio</span></code></a> |
| </h4></div></div></div> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Name |
| </p> |
| </th> |
| <th> |
| <p> |
| kind |
| </p> |
| </th> |
| <th> |
| <p> |
| Description |
| </p> |
| </th> |
| <th> |
| <p> |
| Result |
| </p> |
| </th> |
| <th> |
| <p> |
| Ticket |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| typedefs.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check the num/den are correct for the predefined typedefs |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check the num/den are correctly simplified |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio1.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| The template argument D shall not be zero |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio2.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| the absolute values of the template arguments N and D shall be |
| representable by type intmax_t |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio3.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| the absolute values of the template arguments N and D shall be |
| representable by type intmax_t |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="ratio.appendices.tests._comparison_"></a><a class="link" href="appendices.html#ratio.appendices.tests._comparison_" title="comparison"><code class="computeroutput"><span class="identifier">comparison</span></code></a> |
| </h4></div></div></div> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Name |
| </p> |
| </th> |
| <th> |
| <p> |
| kind |
| </p> |
| </th> |
| <th> |
| <p> |
| Description |
| </p> |
| </th> |
| <th> |
| <p> |
| Result |
| </p> |
| </th> |
| <th> |
| <p> |
| Ticket |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| ratio_equal.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_equal metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_not_equal.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_not_equal metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_less.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_less metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_less_equal.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_less_equal metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_greater.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_greater metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_greater_equal.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_greater_equal metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="ratio.appendices.tests._arithmetic_"></a><a class="link" href="appendices.html#ratio.appendices.tests._arithmetic_" title="arithmetic"><code class="computeroutput"><span class="identifier">arithmetic</span></code></a> |
| </h4></div></div></div> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Name |
| </p> |
| </th> |
| <th> |
| <p> |
| kind |
| </p> |
| </th> |
| <th> |
| <p> |
| Description |
| </p> |
| </th> |
| <th> |
| <p> |
| Result |
| </p> |
| </th> |
| <th> |
| <p> |
| Ticket |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| ratio_add.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_add metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_subtract.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_subtract metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_multiply.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_multiply metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_divide.pass |
| </p> |
| </td> |
| <td> |
| <p> |
| run |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_divide metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_add.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_add overflow metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_subtract.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_subtract underflow metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_multiply.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_multiply overflow metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| ratio_divide.fail |
| </p> |
| </td> |
| <td> |
| <p> |
| compile-fails |
| </p> |
| </td> |
| <td> |
| <p> |
| check ratio_divide overflow metafunction class |
| </p> |
| </td> |
| <td> |
| <p> |
| Pass |
| </p> |
| </td> |
| <td> |
| <p> |
| # |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="ratio.appendices.tickets"></a><a class="link" href="appendices.html#ratio.appendices.tickets" title="Appendix G: Tickets">Appendix G: Tickets</a> |
| </h3></div></div></div> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Ticket |
| </p> |
| </th> |
| <th> |
| <p> |
| Description |
| </p> |
| </th> |
| <th> |
| <p> |
| Resolution |
| </p> |
| </th> |
| <th> |
| <p> |
| State |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 1 |
| </p> |
| </td> |
| <td> |
| <p> |
| result of metafunctions ratio_multiply and ratio_divide were not |
| normalized ratios. |
| </p> |
| </td> |
| <td> |
| <p> |
| Use of the nested ratio typedef type on ratio arithmetic operations. |
| </p> |
| </td> |
| <td> |
| <p> |
| Closed |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 2 |
| </p> |
| </td> |
| <td> |
| <p> |
| INTMAX_C is not always defined. |
| </p> |
| </td> |
| <td> |
| <p> |
| Replace INTMAX_C by BOOST_INTMAX_C until boost/cstdint.hpp ensures |
| INTMAX_C is always defined. |
| </p> |
| </td> |
| <td> |
| <p> |
| Closed |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 3 |
| </p> |
| </td> |
| <td> |
| <p> |
| MSVC reports a warning instead of an error when there is an integral |
| constant overflow. |
| </p> |
| </td> |
| <td> |
| <p> |
| manage with MSVC reporting a warning instead of an error when there |
| is an integral constant overflow. |
| </p> |
| </td> |
| <td> |
| <p> |
| Closed |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 4 |
| </p> |
| </td> |
| <td> |
| <p> |
| ration_less overflow on cases where it can be avoided. |
| </p> |
| </td> |
| <td> |
| <p> |
| Change the algorithm as implemented in libc++. |
| </p> |
| </td> |
| <td> |
| <p> |
| Closed |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="ratio.appendices.todo"></a><a class="link" href="appendices.html#ratio.appendices.todo" title="Appendix H: Future Plans">Appendix H: Future Plans</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="ratio.appendices.todo.h0"></a> |
| <span class="phrase"><a name="ratio.appendices.todo.for_later_releases"></a></span><a class="link" href="appendices.html#ratio.appendices.todo.for_later_releases">For |
| later releases</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Use template aliases on compiler providing it. |
| </li> |
| <li class="listitem"> |
| Implement <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2010/n3135.html#3135" target="_top">multiple |
| arguments</a> ratio arithmetic. |
| </li> |
| </ul></div> |
| </div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2008 Howard Hinnant<br>Copyright © 2006, 2008 Beman Dawes<br>Copyright © 2009-2012 Vicente |
| J. Botet Escriba<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
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