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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.constants_faq"></a><a class="link" href="constants_faq.html" title="FAQs">FAQs</a>
  28 </h2></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.constants_faq.h0"></a>
  31       <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
  32       <span class="emphasis"><em>these</em></span> Constants Chosen?</a>
  33     </h5>
  34 <p>
  35       It is, of course, impossible to please everyone with a list like this.
  36     </p>
  37 <p>
  38       Some of the criteria we have used are:
  39     </p>
  40 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  41 <li class="listitem">
  42           Used in Boost.Math.
  43         </li>
  44 <li class="listitem">
  45           Commonly used.
  46         </li>
  47 <li class="listitem">
  48           Expensive to compute.
  49         </li>
  50 <li class="listitem">
  51           Requested by users.
  52         </li>
  53 <li class="listitem">
  54           <a href="http://en.wikipedia.org/wiki/Mathematical_constant" target="_top">Used in
  55           science and mathematics.</a>
  56         </li>
  57 <li class="listitem">
  58           No integer values (because so cheap to construct).<br> (You can easily
  59           define your own if found convenient, for example: <code class="computeroutput"><span class="identifier">FPT</span>
  60           <span class="identifier">one</span> <span class="special">=</span><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">FPT</span><span class="special">&gt;(</span><span class="number">42</span><span class="special">);</span></code>).
  61         </li>
  62 </ul></div>
  63 <h5>
  64 <a name="math_toolkit.constants_faq.h1"></a>
  65       <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
  66       are constants named?</a>
  67     </h5>
  68 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  69 <li class="listitem">
  70           Not macros, so no upper case.
  71         </li>
  72 <li class="listitem">
  73           All lower case (following C++ standard names).
  74         </li>
  75 <li class="listitem">
  76           No CamelCase.
  77         </li>
  78 <li class="listitem">
  79           Underscore as _ delimiter between words.
  80         </li>
  81 <li class="listitem">
  82           Numbers spelt as words rather than decimal digits (except following pow).
  83         </li>
  84 <li class="listitem">
  85           Abbreviation conventions:
  86           <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
  87 <li class="listitem">
  88                 root for square root.
  89               </li>
  90 <li class="listitem">
  91                 cbrt for cube root.
  92               </li>
  93 <li class="listitem">
  94                 pow for pow function using decimal digits like pow23 for n<sup>2/3</sup>.
  95               </li>
  96 <li class="listitem">
  97                 div for divided by or operator /.
  98               </li>
  99 <li class="listitem">
 100                 minus for operator -, plus for operator +.
 101               </li>
 102 <li class="listitem">
 103                 sqr for squared.
 104               </li>
 105 <li class="listitem">
 106                 cubed for cubed n<sup>3</sup>.
 107               </li>
 108 <li class="listitem">
 109                 words for greek, like &#960;, &#950; and &#915;.
 110               </li>
 111 <li class="listitem">
 112                 words like half, third, three_quarters, sixth for fractions. (Digit(s)
 113                 can get muddled).
 114               </li>
 115 <li class="listitem">
 116                 log10 for log<sub>10</sub>
 117               </li>
 118 <li class="listitem">
 119                 ln for log<sub>e</sub>
 120               </li>
 121 </ul></div>
 122         </li>
 123 </ul></div>
 124 <h5>
 125 <a name="math_toolkit.constants_faq.h2"></a>
 126       <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
 127       the constants derived?</a>
 128     </h5>
 129 <p>
 130       The constants have all been calculated using high-precision software working
 131       with up to 300-bit precision giving about 100 decimal digits. (The precision
 132       can be arbitrarily chosen and is limited only by compute time).
 133     </p>
 134 <h5>
 135 <a name="math_toolkit.constants_faq.h3"></a>
 136       <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
 137       are the constants?</a>
 138     </h5>
 139 <p>
 140       The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of reasonably-foreseeable
 141       floating-point hardware (256-bit) and should meet most high-precision computations.
 142     </p>
 143 <h5>
 144 <a name="math_toolkit.constants_faq.h4"></a>
 145       <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
 146       constants tested?</a>
 147     </h5>
 148 <div class="orderedlist"><ol class="orderedlist" type="1">
 149 <li class="listitem">
 150           Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double
 151           literals, with at least 35 decimal digits, enough to be accurate for all
 152           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>.
 153         </li>
 154 <li class="listitem">
 155           Comparison with calculation at long double precision. This often requires
 156           a slightly higher tolerance than two epsilon because of computational noise
 157           from round-off etc, especially when trig and other functions are called.
 158         </li>
 159 <li class="listitem">
 160           Comparison with independent published values, for example, using <a href="http://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences (OEIS)</a>
 161           again using at least 35 decimal digits strings.
 162         </li>
 163 <li class="listitem">
 164           Comparison with independely calculated values using arbitrary precision
 165           tools like <a href="http://www.wolfram.com/mathematica/" target="_top">Mathematica</a>,
 166           again using at least 35 decimal digits literal strings.
 167         </li>
 168 </ol></div>
 169 <div class="warning"><table border="0" summary="Warning">
 170 <tr>
 171 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
 172 <th align="left">Warning</th>
 173 </tr>
 174 <tr><td align="left" valign="top"><p>
 175         We have not yet been able to <span class="bold"><strong>check</strong></span> that
 176         <span class="bold"><strong>all</strong></span> constants are accurate at the full arbitrary
 177         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>
 178         appear to be accurate and internal consistencies suggest that others are
 179         this accurate too.
 180       </p></td></tr>
 181 </table></div>
 182 <h5>
 183 <a name="math_toolkit.constants_faq.h5"></a>
 184       <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
 185       important?</a>
 186     </h5>
 187 <p>
 188       Code written using math constants is easily portable even when using different
 189       floating-point types with differing precision.
 190     </p>
 191 <p>
 192       It is a mistake to expect that results of computations will be <span class="bold"><strong>identical</strong></span>,
 193       but you can achieve the <span class="bold"><strong>best accuracy possible for the
 194       floating-point type in use</strong></span>.
 195     </p>
 196 <p>
 197       This has no extra cost to the user, but reduces irritating, and often confusing
 198       and very hard-to-trace effects, caused by the intrinsically limited precision
 199       of floating-point calculations.
 200     </p>
 201 <p>
 202       A harmless symptom of this limit is a spurious least-significant digit; at
 203       worst, slightly inaccurate constants sometimes cause iterating algorithms to
 204       diverge wildly because internal comparisons just fail.
 205     </p>
 206 <h5>
 207 <a name="math_toolkit.constants_faq.h6"></a>
 208       <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
 209       is the Internal Format of the constants, and why?</a>
 210     </h5>
 211 <p>
 212       See <a class="link" href="tutorial.html" title="Tutorial">tutorial</a> above for normal
 213       use, but this FAQ explains the internal details used for the constants.
 214     </p>
 215 <p>
 216       Constants are stored as 100 decimal digit values. However, some compilers do
 217       not accept decimal digits strings as long as this. So the constant is split
 218       into two parts, with the first containing at least 128-bit long double precision
 219       (35 decimal digits), and for consistency should be in scientific format with
 220       a signed exponent.
 221     </p>
 222 <p>
 223       The second part is the value of the constant expressed as a string literal,
 224       accurate to at least 100 decimal digits (in practice that means at least 102
 225       digits). Again for consistency use scientific format with a signed exponent.
 226     </p>
 227 <p>
 228       For types with precision greater than a long double, then if T is constructible
 229       <code class="computeroutput"><span class="identifier">T</span> </code>is constructible from a
 230       <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,
 231       otherwise we fall back on lexical_cast to convert to type <code class="computeroutput"><span class="identifier">T</span></code>.
 232       (Using a string is necessary because you can't use a numeric constant since
 233       even a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
 234       might not have enough digits).
 235     </p>
 236 <p>
 237       So, for example, a constant like pi is internally defined as
 238     </p>
 239 <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>
 240 </pre>
 241 <p>
 242       In this case the significand is 109 decimal digits, ensuring 100 decimal digits
 243       are exact, and exponent is zero.
 244     </p>
 245 <p>
 246       See <a class="link" href="new_const.html" title="Defining New Constants">defining new constants</a> to
 247       calculate new constants.
 248     </p>
 249 <p>
 250       A macro definition like this can be pasted into user code where convenient,
 251       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
 252       is to be added to the Boost.Math library.
 253     </p>
 254 <h5>
 255 <a name="math_toolkit.constants_faq.h7"></a>
 256       <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
 257       Floating-point Types could I use?</a>
 258     </h5>
 259 <p>
 260       Apart from the built-in floating-point types <code class="computeroutput"><span class="keyword">float</span></code>,
 261       <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
 262       <span class="keyword">double</span></code>, there are several arbitrary
 263       precision floating-point classes available, but most are not licensed for commercial
 264       use.
 265     </p>
 266 <h6>
 267 <a name="math_toolkit.constants_faq.h8"></a>
 268       <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
 269       by Christopher Kormanyos</a>
 270     </h6>
 271 <p>
 272       This work is based on an earlier work called e-float: Algorithm 910: A Portable
 273       C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS,
 274       {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>
 275       <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
 276       but is now re-factored and available under the Boost license in the Boost-sandbox
 277       at <a href="https://svn.boost.org/svn/boost/sandbox/multiprecision/" target="_top">multiprecision</a>
 278       where it is being refined and prepared for review.
 279     </p>
 280 <h6>
 281 <a name="math_toolkit.constants_faq.h9"></a>
 282       <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
 283       by John Maddock using Expression Templates</a>
 284     </h6>
 285 <p>
 286       <a href="https://svn.boost.org/svn/boost/sandbox/big_number/" target="_top">Big Number</a>
 287       which is a reworking of <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
 288       by Christopher Kormanyos to use expression templates for faster execution.
 289     </p>
 290 <h6>
 291 <a name="math_toolkit.constants_faq.h10"></a>
 292       <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
 293       class quad_float</a>
 294     </h6>
 295 <p>
 296       <a href="http://shoup.net/ntl/" target="_top">NTL</a> by Victor Shoup has fixed and
 297       arbitrary high precision fixed and floating-point types. However none of these
 298       are licenced for commercial use.
 299     </p>
 300 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span> <span class="comment">// quad precision 106-bit, about 32 decimal digits.</span>
 301 <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>
 302 </pre>
 303 <p>
 304       NTL class <code class="computeroutput"><span class="identifier">quad_float</span></code>, which
 305       gives a form of quadruple precision, 106-bit significand (but without an extended
 306       exponent range.) With an IEC559/IEEE 754 compatible processor, for example
 307       Intel X86 family, with 64-bit double, and 53-bit significand, using the significands
 308       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">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">digits10</span></code> is 16, then we get about twice the
 309       precision, so <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">quad_float</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">()</span></code>
 310       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">&lt;</span><span class="identifier">RR</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">()</span></code>
 311       should be about 40). (which seems to agree with experiments). We output constants
 312       (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">&lt;</span><span class="identifier">RR</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">()</span></code>)
 313       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>
 314       <span class="number">3</span><span class="special">);</span></code>
 315     </p>
 316 <p>
 317       Apple Mac/Darwin uses a similar <span class="emphasis"><em>doubledouble</em></span> 106-bit for
 318       its built-in <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
 319       type.
 320     </p>
 321 <div class="note"><table border="0" summary="Note">
 322 <tr>
 323 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
 324 <th align="left">Note</th>
 325 </tr>
 326 <tr><td align="left" valign="top"><p>
 327         The precision of all <code class="computeroutput"><span class="identifier">doubledouble</span></code>
 328         floating-point types is rather odd and values given are only approximate.
 329       </p></td></tr>
 330 </table></div>
 331 <p>
 332       <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
 333     </p>
 334 <h6>
 335 <a name="math_toolkit.constants_faq.h11"></a>
 336       <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
 337       class RR</a>
 338     </h6>
 339 <p>
 340       Arbitrary precision floating point with NTL class RR, default is 150 bit (about
 341       50 decimal digits) used here with 300 bit to output 100 decimal digits, enough
 342       for many practical non-'number-theoretic' C++ applications.
 343     </p>
 344 <p>
 345       <a href="http://www.shoup.net/ntl/" target="_top">NTL A Library for doing Number Theory</a>
 346       is <span class="bold"><strong>not licenced for commercial use</strong></span>.
 347     </p>
 348 <p>
 349       This class is used in Boost.Math and is an option when using big_number projects
 350       to calculate new math constants.
 351     </p>
 352 <p>
 353       <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
 354     </p>
 355 <h6>
 356 <a name="math_toolkit.constants_faq.h12"></a>
 357       <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
 358       and MPFR</a>
 359     </h6>
 360 <p>
 361       <a href="http://gmplib.org" target="_top">GMP</a> and <a href="http://www.mpfr.org/" target="_top">MPFR</a>
 362       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
 363       are <span class="bold"><strong>not licensed for commercial use</strong></span>.
 364     </p>
 365 <h5>
 366 <a name="math_toolkit.constants_faq.h13"></a>
 367       <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
 368       happened to a previous collection of constants proposed for Boost?</a>
 369     </h5>
 370 <p>
 371       A review concluded that the way in which the constants were presented did not
 372       meet many peoples needs. None of the methods proposed met many users' essential
 373       requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code>
 374       rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>.
 375       Many science and engineering equations look difficult to read when because
 376       function call brackets can be confused with the many other brackets often needed.
 377       All the methods then proposed of avoiding the brackets failed to meet all needs,
 378       often on grounds of complexity and lack of applicability to various realistic
 379       scenarios.
 380     </p>
 381 <p>
 382       So the simple namespace method, proposed on its own, but rejected at the first
 383       review, has been added to allow users to have convenient access to float, double
 384       and long double values, but combined with template struct and functions to
 385       allow simultaneous use with other non-built-in floating-point types.
 386     </p>
 387 <h5>
 388 <a name="math_toolkit.constants_faq.h14"></a>
 389       <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
 390       the constants (internally) have a struct rather than a simple function?</a>
 391     </h5>
 392 <p>
 393       A function mechanism was provided by in previous versions of Boost.Math.
 394     </p>
 395 <p>
 396       The new mechanism is to permit partial specialization. See Custom Specializing
 397       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>
 398     </p>
 399 <h5>
 400 <a name="math_toolkit.constants_faq.h15"></a>
 401       <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
 402       can I find other high precision constants?</a>
 403     </h5>
 404 <div class="orderedlist"><ol class="orderedlist" type="1">
 405 <li class="listitem">
 406           Constants with very high precision and good accuracy (&gt;40 decimal digits)
 407           from Simon Plouffe's web based collection <a href="http://pi.lacim.uqam.ca/eng/" target="_top">http://pi.lacim.uqam.ca/eng/</a>.
 408         </li>
 409 <li class="listitem">
 410           <a href="https://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences
 411           (OEIS)</a>
 412         </li>
 413 <li class="listitem">
 414           Checks using printed text optically scanned values and converted from:
 415           D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN
 416           0 201 89683 4 (1997)
 417         </li>
 418 <li class="listitem">
 419           M. Abrahamovitz &amp; I. E. Stegun, National Bureau of Standards, Handbook
 420           of Mathematical Functions, a reference source for formulae now superceded
 421           by
 422         </li>
 423 <li class="listitem">
 424           Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark,
 425           NIST Handbook of Mathemetical Functions, Cambridge University Press, ISBN
 426           978-0-521-14063-8, 2010.
 427         </li>
 428 <li class="listitem">
 429           John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7.
 430         </li>
 431 <li class="listitem">
 432           Some values from Cephes Mathematical Library, Stephen L. Moshier and CALC100
 433           100 decimal digit Complex Variable Calculator Program, a DOS utility.
 434         </li>
 435 <li class="listitem">
 436           Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at <a href="http://numbers.computation.free.fr/Constants/constants.html" target="_top">Number,
 437           constants and computation</a>.
 438         </li>
 439 </ol></div>
 440 <h5>
 441 <a name="math_toolkit.constants_faq.h16"></a>
 442       <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
 443       Physical Constants?</a>
 444     </h5>
 445 <p>
 446       Not here in this Boost.Math collection, because physical constants:
 447     </p>
 448 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 449 <li class="listitem">
 450           Are measurements, not truely constants.
 451         </li>
 452 <li class="listitem">
 453           Are not truly constant and keeping changing as mensuration technology improves.
 454         </li>
 455 <li class="listitem">
 456           Have a instrinsic uncertainty.
 457         </li>
 458 <li class="listitem">
 459           Mathematical constants are stored and represented at varying precision,
 460           but should never be inaccurate.
 461         </li>
 462 </ul></div>
 463 <p>
 464       Some physical constants may be available in Boost.Units.
 465     </p>
 466 </div>
 467 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 468 <td align="left"></td>
 469 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 470       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 471       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 472       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 473         Distributed under the Boost Software License, Version 1.0. (See accompanying
 474         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>)
 475       </p>
 476 </div></td>
 477 </tr></table>
 478 <hr>
 479 <div class="spirit-nav">
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