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<title>Math Constants FAQs</title>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.constants_faq"></a><a class="link" href="constants_faq.html" title="Math Constants FAQs">Math Constants FAQs</a>
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</h2></div></div></div>
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<h5>
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<a name="math_toolkit.constants_faq.h0"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.why_are_these_constants_chosen"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_are_these_constants_chosen">Why are
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<span class="emphasis"><em>these</em></span> Constants Chosen?</a>
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</h5>
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<p>
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It is, of course, impossible to please everyone with a list like this.
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</p>
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<p>
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Some of the criteria we have used are:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Used in Boost.Math.
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</li>
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<li class="listitem">
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Commonly used.
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</li>
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<li class="listitem">
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Expensive to compute.
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</li>
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<li class="listitem">
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Requested by users.
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</li>
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<li class="listitem">
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<a href="http://en.wikipedia.org/wiki/Mathematical_constant" target="_top">Used in
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science and mathematics.</a>
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</li>
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<li class="listitem">
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No integer values (because so cheap to construct).
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</li>
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</ul></div>
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<p>
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(You can easily define your own if found convenient, for example: <code class="computeroutput"><span class="identifier">FPT</span> <span class="identifier">one</span> <span class="special">=</span><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">FPT</span><span class="special">>(</span><span class="number">42</span><span class="special">);</span></code>).
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</p>
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<h5>
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<a name="math_toolkit.constants_faq.h1"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.how_are_constants_named"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_constants_named">How
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are constants named?</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Not macros, so no upper case.
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</li>
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<li class="listitem">
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All lower case (following C++ standard names).
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</li>
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<li class="listitem">
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No CamelCase.
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</li>
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<li class="listitem">
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Underscore as _ delimiter between words.
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</li>
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<li class="listitem">
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Numbers spelt as words rather than decimal digits (except following pow).
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</li>
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<li class="listitem">
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Abbreviation conventions:
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
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<li class="listitem">
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root for square root.
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</li>
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<li class="listitem">
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cbrt for cube root.
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</li>
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<li class="listitem">
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pow for pow function using decimal digits like pow23 for n<sup>2/3</sup>.
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</li>
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<li class="listitem">
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div for divided by or operator /.
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</li>
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<li class="listitem">
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minus for operator -, plus for operator +.
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</li>
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<li class="listitem">
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sqr for squared.
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</li>
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<li class="listitem">
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cubed for cubed n<sup>3</sup>.
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</li>
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<li class="listitem">
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words for greek, like π, ζ and Γ.
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</li>
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<li class="listitem">
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words like half, third, three_quarters, sixth for fractions. (Digit(s)
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can get muddled).
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</li>
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<li class="listitem">
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log10 for log<sub>10</sub>
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</li>
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<li class="listitem">
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ln for log<sub>e</sub>
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</li>
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</ul></div>
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</li>
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</ul></div>
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<h5>
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<a name="math_toolkit.constants_faq.h2"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_derived"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_derived">How are
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the constants derived?</a>
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</h5>
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<p>
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The constants have all been calculated using high-precision software working
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with up to 300-bit precision giving about 100 decimal digits. (The precision
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can be arbitrarily chosen and is limited only by compute time).
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</p>
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<h5>
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<a name="math_toolkit.constants_faq.h3"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.how_accurate_are_the_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_accurate_are_the_constants">How Accurate
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are the constants?</a>
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</h5>
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<p>
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The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of reasonably-foreseeable
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floating-point hardware (256-bit) and should meet most high-precision computations.
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</p>
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<h5>
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<a name="math_toolkit.constants_faq.h4"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_tested"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_tested">How are the
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constants tested?</a>
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</h5>
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<div class="orderedlist"><ol class="orderedlist" type="1">
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<li class="listitem">
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Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double
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literals, with at least 35 decimal digits, enough to be accurate for all
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long double implementations. The tolerance is usually twice <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">epsilon</span></code>.
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</li>
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<li class="listitem">
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Comparison with calculation at long double precision. This often requires
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a slightly higher tolerance than two epsilon because of computational noise
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from round-off etc, especially when trig and other functions are called.
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</li>
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<li class="listitem">
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Comparison with independent published values, for example, using <a href="http://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences (OEIS)</a>
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again using at least 35 decimal digits strings.
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</li>
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<li class="listitem">
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Comparison with independely calculated values using arbitrary precision
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tools like <a href="http://www.wolfram.com/mathematica/" target="_top">Mathematica</a>,
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again using at least 35 decimal digits literal strings.
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</li>
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</ol></div>
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<div class="warning"><table border="0" summary="Warning">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
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<th align="left">Warning</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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We have not yet been able to <span class="bold"><strong>check</strong></span> that
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<span class="bold"><strong>all</strong></span> constants are accurate at the full arbitrary
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precision, at present 100 decimal digits. But certain key values like <code class="computeroutput"><span class="identifier">e</span></code> and <code class="computeroutput"><span class="identifier">pi</span></code>
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appear to be accurate and internal consistencies suggest that others are
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this accurate too.
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</p></td></tr>
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</table></div>
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<h5>
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<a name="math_toolkit.constants_faq.h5"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.why_is_portability_important"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_is_portability_important">Why is Portability
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important?</a>
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</h5>
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<p>
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Code written using math constants is easily portable even when using different
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floating-point types with differing precision.
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</p>
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<p>
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It is a mistake to expect that results of computations will be <span class="bold"><strong>identical</strong></span>,
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but you can achieve the <span class="bold"><strong>best accuracy possible for the
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floating-point type in use</strong></span>.
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</p>
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<p>
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This has no extra cost to the user, but reduces irritating, and often confusing
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and very hard-to-trace effects, caused by the intrinsically limited precision
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of floating-point calculations.
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</p>
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<p>
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A harmless symptom of this limit is a spurious least-significant digit; at
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worst, slightly inaccurate constants sometimes cause iterating algorithms to
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diverge wildly because internal comparisons just fail.
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</p>
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<h5>
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<a name="math_toolkit.constants_faq.h6"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.what_is_the_internal_format_of_t"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_is_the_internal_format_of_t">What
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is the Internal Format of the constants, and why?</a>
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</h5>
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<p>
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See <a class="link" href="tutorial.html" title="Tutorial">tutorial</a> above for normal
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use, but this FAQ explains the internal details used for the constants.
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</p>
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<p>
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Constants are stored as 100 decimal digit values. However, some compilers do
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not accept decimal digits strings as long as this. So the constant is split
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into two parts, with the first containing at least 128-bit long double precision
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(35 decimal digits), and for consistency should be in scientific format with
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a signed exponent.
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</p>
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<p>
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The second part is the value of the constant expressed as a string literal,
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accurate to at least 100 decimal digits (in practice that means at least 102
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digits). Again for consistency use scientific format with a signed exponent.
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</p>
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<p>
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For types with precision greater than a long double, then if T is constructible
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<code class="computeroutput"><span class="identifier">T</span> </code>is constructible from a
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<code class="computeroutput"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span></code> then it's directly constructed from the string,
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otherwise we fall back on lexical_cast to convert to type <code class="computeroutput"><span class="identifier">T</span></code>.
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(Using a string is necessary because you can't use a numeric constant since
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even a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
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might not have enough digits).
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</p>
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<p>
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So, for example, a constant like pi is internally defined as
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</p>
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<pre class="programlisting"><span class="identifier">BOOST_DEFINE_MATH_CONSTANT</span><span class="special">(</span><span class="identifier">pi</span><span class="special">,</span> <span class="number">3.141592653589793238462643383279502884e+00</span><span class="special">,</span> <span class="string">"3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00"</span><span class="special">);</span>
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</pre>
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<p>
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In this case the significand is 109 decimal digits, ensuring 100 decimal digits
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are exact, and exponent is zero.
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</p>
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<p>
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See <a class="link" href="new_const.html" title="Defining New Constants">defining new constants</a> to
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calculate new constants.
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</p>
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<p>
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A macro definition like this can be pasted into user code where convenient,
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or into <code class="computeroutput"><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">constants</span><span class="special">.</span><span class="identifier">hpp</span></code> if it
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is to be added to the Boost.Math library.
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</p>
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<h5>
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<a name="math_toolkit.constants_faq.h7"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.what_floating_point_types_could_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_floating_point_types_could_">What
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Floating-point Types could I use?</a>
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</h5>
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<p>
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Apart from the built-in floating-point types <code class="computeroutput"><span class="keyword">float</span></code>,
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<code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
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<span class="keyword">double</span></code>, there are several arbitrary
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precision floating-point classes available, but most are not licensed for commercial
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use.
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</p>
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<h6>
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<a name="math_toolkit.constants_faq.h8"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.boost_multiprecision_by_christop"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_multiprecision_by_christop">Boost.Multiprecision
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by Christopher Kormanyos</a>
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</h6>
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<p>
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This work is based on an earlier work called e-float: Algorithm 910: A Portable
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C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS,
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{VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. <a href="http://doi.acm.org/10.1145/1916461.1916469" target="_top">http://doi.acm.org/10.1145/1916461.1916469</a>
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<a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
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but is now re-factored and available under the Boost license in the Boost-sandbox
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at <a href="https://svn.boost.org/svn/boost/sandbox/multiprecision/" target="_top">multiprecision</a>
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where it is being refined and prepared for review.
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</p>
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<h6>
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<a name="math_toolkit.constants_faq.h9"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_">Boost.cpp_float
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by John Maddock using Expression Templates</a>
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</h6>
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<p>
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<a href="https://svn.boost.org/svn/boost/sandbox/big_number/" target="_top">Big Number</a>
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which is a reworking of <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
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by Christopher Kormanyos to use expression templates for faster execution.
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</p>
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<h6>
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<a name="math_toolkit.constants_faq.h10"></a>
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<span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_quad_float"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_quad_float">NTL
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class quad_float</a>
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</h6>
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<p>
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<a href="http://shoup.net/ntl/" target="_top">NTL</a> by Victor Shoup has fixed and
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arbitrary high precision fixed and floating-point types. However none of these
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are licenced for commercial use.
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">NTL</span><span class="special">/</span><span class="identifier">quad_float</span><span class="special">.</span><span class="identifier">h</span><span class="special">></span> <span class="comment">// quad precision 106-bit, about 32 decimal digits.</span>
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<span class="keyword">using</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">to_quad_float</span><span class="special">;</span> <span class="comment">// Less precise than arbitrary precision NTL::RR.</span>
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</pre>
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<p>
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NTL class <code class="computeroutput"><span class="identifier">quad_float</span></code>, which
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gives a form of quadruple precision, 106-bit significand (but without an extended
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exponent range.) With an IEC559/IEEE 754 compatible processor, for example
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Intel X86 family, with 64-bit double, and 53-bit significand, using the significands
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of <span class="bold"><strong>two</strong></span> 64-bit doubles, if <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span></code> is 16, then we get about twice the
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precision, so <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">quad_float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">()</span></code>
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should be 32. (the default <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">RR</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">()</span></code>
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should be about 40). (which seems to agree with experiments). We output constants
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(including some noisy bits, an approximation to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">RR</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">()</span></code>)
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by adding 2 or 3 extra decimal digits, so using <code class="computeroutput"><span class="identifier">quad_float</span><span class="special">::</span><span class="identifier">SetOutputPrecision</span><span class="special">(</span><span class="number">32</span> <span class="special">+</span>
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<span class="number">3</span><span class="special">);</span></code>
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</p>
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<p>
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Apple Mac/Darwin uses a similar <span class="emphasis"><em>doubledouble</em></span> 106-bit for
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its built-in <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
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type.
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</p>
|
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
|
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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The precision of all <code class="computeroutput"><span class="identifier">doubledouble</span></code>
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floating-point types is rather odd and values given are only approximate.
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</p></td></tr>
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</table></div>
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<p>
|
|
<span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
|
|
</p>
|
|
<h6>
|
|
<a name="math_toolkit.constants_faq.h11"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_rr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_rr">NTL
|
|
class RR</a>
|
|
</h6>
|
|
<p>
|
|
Arbitrary precision floating point with NTL class RR, default is 150 bit (about
|
|
50 decimal digits) used here with 300 bit to output 100 decimal digits, enough
|
|
for many practical non-'number-theoretic' C++ applications.
|
|
</p>
|
|
<p>
|
|
<a href="http://www.shoup.net/ntl/" target="_top">NTL A Library for doing Number Theory</a>
|
|
is <span class="bold"><strong>not licenced for commercial use</strong></span>.
|
|
</p>
|
|
<p>
|
|
This class is used in Boost.Math and is an option when using big_number projects
|
|
to calculate new math constants.
|
|
</p>
|
|
<p>
|
|
<span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
|
|
</p>
|
|
<h6>
|
|
<a name="math_toolkit.constants_faq.h12"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.gmp_and_mpfr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.gmp_and_mpfr">GMP
|
|
and MPFR</a>
|
|
</h6>
|
|
<p>
|
|
<a href="http://gmplib.org" target="_top">GMP</a> and <a href="http://www.mpfr.org/" target="_top">MPFR</a>
|
|
have also been used to compute constants, but are licensed under the <a href="http://www.gnu.org/copyleft/lesser.html" target="_top">Lesser GPL license</a> and
|
|
are <span class="bold"><strong>not licensed for commercial use</strong></span>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.constants_faq.h13"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.what_happened_to_a_previous_coll"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_happened_to_a_previous_coll">What
|
|
happened to a previous collection of constants proposed for Boost?</a>
|
|
</h5>
|
|
<p>
|
|
A review concluded that the way in which the constants were presented did not
|
|
meet many peoples needs. None of the methods proposed met many users' essential
|
|
requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code>
|
|
rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>.
|
|
Many science and engineering equations look difficult to read when because
|
|
function call brackets can be confused with the many other brackets often needed.
|
|
All the methods then proposed of avoiding the brackets failed to meet all needs,
|
|
often on grounds of complexity and lack of applicability to various realistic
|
|
scenarios.
|
|
</p>
|
|
<p>
|
|
So the simple namespace method, proposed on its own, but rejected at the first
|
|
review, has been added to allow users to have convenient access to float, double
|
|
and long double values, but combined with template struct and functions to
|
|
allow simultaneous use with other non-built-in floating-point types.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.constants_faq.h14"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.why_do_the_constants_internally_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_do_the_constants_internally_">Why do
|
|
the constants (internally) have a struct rather than a simple function?</a>
|
|
</h5>
|
|
<p>
|
|
A function mechanism was provided by in previous versions of Boost.Math.
|
|
</p>
|
|
<p>
|
|
The new mechanism is to permit partial specialization. See Custom Specializing
|
|
a constant above. It should also allow use with other packages like <a href="http://www.ttmath.org/" target="_top">ttmath Bignum C++ library.</a>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.constants_faq.h15"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.where_can_i_find_other_high_prec"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_can_i_find_other_high_prec">Where
|
|
can I find other high precision constants?</a>
|
|
</h5>
|
|
<div class="orderedlist"><ol class="orderedlist" type="1">
|
|
<li class="listitem">
|
|
Constants with very high precision and good accuracy (>40 decimal digits)
|
|
from Simon Plouffe's web based collection <a href="http://pi.lacim.uqam.ca/eng/" target="_top">http://pi.lacim.uqam.ca/eng/</a>.
|
|
</li>
|
|
<li class="listitem">
|
|
<a href="https://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences
|
|
(OEIS)</a>
|
|
</li>
|
|
<li class="listitem">
|
|
Checks using printed text optically scanned values and converted from:
|
|
D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN
|
|
0 201 89683 4 (1997)
|
|
</li>
|
|
<li class="listitem">
|
|
M. Abrahamovitz & I. E. Stegun, National Bureau of Standards, Handbook
|
|
of Mathematical Functions, a reference source for formulae now superceded
|
|
by
|
|
</li>
|
|
<li class="listitem">
|
|
Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark,
|
|
NIST Handbook of Mathemetical Functions, Cambridge University Press, ISBN
|
|
978-0-521-14063-8, 2010.
|
|
</li>
|
|
<li class="listitem">
|
|
John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7.
|
|
</li>
|
|
<li class="listitem">
|
|
Some values from Cephes Mathematical Library, Stephen L. Moshier and CALC100
|
|
100 decimal digit Complex Variable Calculator Program, a DOS utility.
|
|
</li>
|
|
<li class="listitem">
|
|
Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at <a href="http://numbers.computation.free.fr/Constants/constants.html" target="_top">Number,
|
|
constants and computation</a>.
|
|
</li>
|
|
</ol></div>
|
|
<h5>
|
|
<a name="math_toolkit.constants_faq.h16"></a>
|
|
<span class="phrase"><a name="math_toolkit.constants_faq.where_are_physical_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_are_physical_constants">Where are
|
|
Physical Constants?</a>
|
|
</h5>
|
|
<p>
|
|
Not here in this Boost.Math collection, because physical constants:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Are measurements, not truely constants.
|
|
</li>
|
|
<li class="listitem">
|
|
Are not truly constant and keeping changing as mensuration technology improves.
|
|
</li>
|
|
<li class="listitem">
|
|
Have a instrinsic uncertainty.
|
|
</li>
|
|
<li class="listitem">
|
|
Mathematical constants are stored and represented at varying precision,
|
|
but should never be inaccurate.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
Some physical constants may be available in Boost.Units.
|
|
</p>
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<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>
|
|
<hr>
|
|
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|
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