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  27 <a name="math_toolkit.constants_faq"></a><a class="link" href="constants_faq.html" title="Math Constants FAQs">Math Constants 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).
  59         </li>
  60 </ul></div>
  61 <p>
  62       (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">&lt;</span><span class="identifier">FPT</span><span class="special">&gt;(</span><span class="number">42</span><span class="special">);</span></code>).
  63     </p>
  64 <h5>
  65 <a name="math_toolkit.constants_faq.h1"></a>
  66       <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
  67       are constants named?</a>
  68     </h5>
  69 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  70 <li class="listitem">
  71           Not macros, so no upper case.
  72         </li>
  73 <li class="listitem">
  74           All lower case (following C++ standard names).
  75         </li>
  76 <li class="listitem">
  77           No CamelCase.
  78         </li>
  79 <li class="listitem">
  80           Underscore as _ delimiter between words.
  81         </li>
  82 <li class="listitem">
  83           Numbers spelt as words rather than decimal digits (except following pow).
  84         </li>
  85 <li class="listitem">
  86           Abbreviation conventions:
  87           <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
  88 <li class="listitem">
  89                 root for square root.
  90               </li>
  91 <li class="listitem">
  92                 cbrt for cube root.
  93               </li>
  94 <li class="listitem">
  95                 pow for pow function using decimal digits like pow23 for n<sup>2/3</sup>.
  96               </li>
  97 <li class="listitem">
  98                 div for divided by or operator /.
  99               </li>
 100 <li class="listitem">
 101                 minus for operator -, plus for operator +.
 102               </li>
 103 <li class="listitem">
 104                 sqr for squared.
 105               </li>
 106 <li class="listitem">
 107                 cubed for cubed n<sup>3</sup>.
 108               </li>
 109 <li class="listitem">
 110                 words for greek, like &#960;, &#950; and &#915;.
 111               </li>
 112 <li class="listitem">
 113                 words like half, third, three_quarters, sixth for fractions. (Digit(s)
 114                 can get muddled).
 115               </li>
 116 <li class="listitem">
 117                 log10 for log<sub>10</sub>
 118               </li>
 119 <li class="listitem">
 120                 ln for log<sub>e</sub>
 121               </li>
 122 </ul></div>
 123         </li>
 124 </ul></div>
 125 <h5>
 126 <a name="math_toolkit.constants_faq.h2"></a>
 127       <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
 128       the constants derived?</a>
 129     </h5>
 130 <p>
 131       The constants have all been calculated using high-precision software working
 132       with up to 300-bit precision giving about 100 decimal digits. (The precision
 133       can be arbitrarily chosen and is limited only by compute time).
 134     </p>
 135 <h5>
 136 <a name="math_toolkit.constants_faq.h3"></a>
 137       <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
 138       are the constants?</a>
 139     </h5>
 140 <p>
 141       The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of reasonably-foreseeable
 142       floating-point hardware (256-bit) and should meet most high-precision computations.
 143     </p>
 144 <h5>
 145 <a name="math_toolkit.constants_faq.h4"></a>
 146       <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
 147       constants tested?</a>
 148     </h5>
 149 <div class="orderedlist"><ol class="orderedlist" type="1">
 150 <li class="listitem">
 151           Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double
 152           literals, with at least 35 decimal digits, enough to be accurate for all
 153           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>.
 154         </li>
 155 <li class="listitem">
 156           Comparison with calculation at long double precision. This often requires
 157           a slightly higher tolerance than two epsilon because of computational noise
 158           from round-off etc, especially when trig and other functions are called.
 159         </li>
 160 <li class="listitem">
 161           Comparison with independent published values, for example, using <a href="http://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences (OEIS)</a>
 162           again using at least 35 decimal digits strings.
 163         </li>
 164 <li class="listitem">
 165           Comparison with independely calculated values using arbitrary precision
 166           tools like <a href="http://www.wolfram.com/mathematica/" target="_top">Mathematica</a>,
 167           again using at least 35 decimal digits literal strings.
 168         </li>
 169 </ol></div>
 170 <div class="warning"><table border="0" summary="Warning">
 171 <tr>
 172 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
 173 <th align="left">Warning</th>
 174 </tr>
 175 <tr><td align="left" valign="top"><p>
 176         We have not yet been able to <span class="bold"><strong>check</strong></span> that
 177         <span class="bold"><strong>all</strong></span> constants are accurate at the full arbitrary
 178         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>
 179         appear to be accurate and internal consistencies suggest that others are
 180         this accurate too.
 181       </p></td></tr>
 182 </table></div>
 183 <h5>
 184 <a name="math_toolkit.constants_faq.h5"></a>
 185       <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
 186       important?</a>
 187     </h5>
 188 <p>
 189       Code written using math constants is easily portable even when using different
 190       floating-point types with differing precision.
 191     </p>
 192 <p>
 193       It is a mistake to expect that results of computations will be <span class="bold"><strong>identical</strong></span>,
 194       but you can achieve the <span class="bold"><strong>best accuracy possible for the
 195       floating-point type in use</strong></span>.
 196     </p>
 197 <p>
 198       This has no extra cost to the user, but reduces irritating, and often confusing
 199       and very hard-to-trace effects, caused by the intrinsically limited precision
 200       of floating-point calculations.
 201     </p>
 202 <p>
 203       A harmless symptom of this limit is a spurious least-significant digit; at
 204       worst, slightly inaccurate constants sometimes cause iterating algorithms to
 205       diverge wildly because internal comparisons just fail.
 206     </p>
 207 <h5>
 208 <a name="math_toolkit.constants_faq.h6"></a>
 209       <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
 210       is the Internal Format of the constants, and why?</a>
 211     </h5>
 212 <p>
 213       See <a class="link" href="tutorial.html" title="Tutorial">tutorial</a> above for normal
 214       use, but this FAQ explains the internal details used for the constants.
 215     </p>
 216 <p>
 217       Constants are stored as 100 decimal digit values. However, some compilers do
 218       not accept decimal digits strings as long as this. So the constant is split
 219       into two parts, with the first containing at least 128-bit long double precision
 220       (35 decimal digits), and for consistency should be in scientific format with
 221       a signed exponent.
 222     </p>
 223 <p>
 224       The second part is the value of the constant expressed as a string literal,
 225       accurate to at least 100 decimal digits (in practice that means at least 102
 226       digits). Again for consistency use scientific format with a signed exponent.
 227     </p>
 228 <p>
 229       For types with precision greater than a long double, then if T is constructible
 230       <code class="computeroutput"><span class="identifier">T</span> </code>is constructible from a
 231       <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,
 232       otherwise we fall back on lexical_cast to convert to type <code class="computeroutput"><span class="identifier">T</span></code>.
 233       (Using a string is necessary because you can't use a numeric constant since
 234       even a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
 235       might not have enough digits).
 236     </p>
 237 <p>
 238       So, for example, a constant like pi is internally defined as
 239     </p>
 240 <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>
 241 </pre>
 242 <p>
 243       In this case the significand is 109 decimal digits, ensuring 100 decimal digits
 244       are exact, and exponent is zero.
 245     </p>
 246 <p>
 247       See <a class="link" href="new_const.html" title="Defining New Constants">defining new constants</a> to
 248       calculate new constants.
 249     </p>
 250 <p>
 251       A macro definition like this can be pasted into user code where convenient,
 252       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
 253       is to be added to the Boost.Math library.
 254     </p>
 255 <h5>
 256 <a name="math_toolkit.constants_faq.h7"></a>
 257       <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
 258       Floating-point Types could I use?</a>
 259     </h5>
 260 <p>
 261       Apart from the built-in floating-point types <code class="computeroutput"><span class="keyword">float</span></code>,
 262       <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
 263       <span class="keyword">double</span></code>, there are several arbitrary
 264       precision floating-point classes available, but most are not licensed for commercial
 265       use.
 266     </p>
 267 <h6>
 268 <a name="math_toolkit.constants_faq.h8"></a>
 269       <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
 270       by Christopher Kormanyos</a>
 271     </h6>
 272 <p>
 273       This work is based on an earlier work called e-float: Algorithm 910: A Portable
 274       C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS,
 275       {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>
 276       <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
 277       but is now re-factored and available under the Boost license in the Boost-sandbox
 278       at <a href="https://svn.boost.org/svn/boost/sandbox/multiprecision/" target="_top">multiprecision</a>
 279       where it is being refined and prepared for review.
 280     </p>
 281 <h6>
 282 <a name="math_toolkit.constants_faq.h9"></a>
 283       <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
 284       by John Maddock using Expression Templates</a>
 285     </h6>
 286 <p>
 287       <a href="https://svn.boost.org/svn/boost/sandbox/big_number/" target="_top">Big Number</a>
 288       which is a reworking of <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a>
 289       by Christopher Kormanyos to use expression templates for faster execution.
 290     </p>
 291 <h6>
 292 <a name="math_toolkit.constants_faq.h10"></a>
 293       <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
 294       class quad_float</a>
 295     </h6>
 296 <p>
 297       <a href="http://shoup.net/ntl/" target="_top">NTL</a> by Victor Shoup has fixed and
 298       arbitrary high precision fixed and floating-point types. However none of these
 299       are licenced for commercial use.
 300     </p>
 301 <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>
 302 <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>
 303 </pre>
 304 <p>
 305       NTL class <code class="computeroutput"><span class="identifier">quad_float</span></code>, which
 306       gives a form of quadruple precision, 106-bit significand (but without an extended
 307       exponent range.) With an IEC559/IEEE 754 compatible processor, for example
 308       Intel X86 family, with 64-bit double, and 53-bit significand, using the significands
 309       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
 310       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>
 311       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>
 312       should be about 40). (which seems to agree with experiments). We output constants
 313       (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>)
 314       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>
 315       <span class="number">3</span><span class="special">);</span></code>
 316     </p>
 317 <p>
 318       Apple Mac/Darwin uses a similar <span class="emphasis"><em>doubledouble</em></span> 106-bit for
 319       its built-in <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
 320       type.
 321     </p>
 322 <div class="note"><table border="0" summary="Note">
 323 <tr>
 324 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
 325 <th align="left">Note</th>
 326 </tr>
 327 <tr><td align="left" valign="top"><p>
 328         The precision of all <code class="computeroutput"><span class="identifier">doubledouble</span></code>
 329         floating-point types is rather odd and values given are only approximate.
 330       </p></td></tr>
 331 </table></div>
 332 <p>
 333       <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
 334     </p>
 335 <h6>
 336 <a name="math_toolkit.constants_faq.h11"></a>
 337       <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
 338       class RR</a>
 339     </h6>
 340 <p>
 341       Arbitrary precision floating point with NTL class RR, default is 150 bit (about
 342       50 decimal digits) used here with 300 bit to output 100 decimal digits, enough
 343       for many practical non-'number-theoretic' C++ applications.
 344     </p>
 345 <p>
 346       <a href="http://www.shoup.net/ntl/" target="_top">NTL A Library for doing Number Theory</a>
 347       is <span class="bold"><strong>not licenced for commercial use</strong></span>.
 348     </p>
 349 <p>
 350       This class is used in Boost.Math and is an option when using big_number projects
 351       to calculate new math constants.
 352     </p>
 353 <p>
 354       <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
 355     </p>
 356 <h6>
 357 <a name="math_toolkit.constants_faq.h12"></a>
 358       <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
 359       and MPFR</a>
 360     </h6>
 361 <p>
 362       <a href="http://gmplib.org" target="_top">GMP</a> and <a href="http://www.mpfr.org/" target="_top">MPFR</a>
 363       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
 364       are <span class="bold"><strong>not licensed for commercial use</strong></span>.
 365     </p>
 366 <h5>
 367 <a name="math_toolkit.constants_faq.h13"></a>
 368       <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
 369       happened to a previous collection of constants proposed for Boost?</a>
 370     </h5>
 371 <p>
 372       A review concluded that the way in which the constants were presented did not
 373       meet many peoples needs. None of the methods proposed met many users' essential
 374       requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code>
 375       rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>.
 376       Many science and engineering equations look difficult to read when because
 377       function call brackets can be confused with the many other brackets often needed.
 378       All the methods then proposed of avoiding the brackets failed to meet all needs,
 379       often on grounds of complexity and lack of applicability to various realistic
 380       scenarios.
 381     </p>
 382 <p>
 383       So the simple namespace method, proposed on its own, but rejected at the first
 384       review, has been added to allow users to have convenient access to float, double
 385       and long double values, but combined with template struct and functions to
 386       allow simultaneous use with other non-built-in floating-point types.
 387     </p>
 388 <h5>
 389 <a name="math_toolkit.constants_faq.h14"></a>
 390       <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
 391       the constants (internally) have a struct rather than a simple function?</a>
 392     </h5>
 393 <p>
 394       A function mechanism was provided by in previous versions of Boost.Math.
 395     </p>
 396 <p>
 397       The new mechanism is to permit partial specialization. See Custom Specializing
 398       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>
 399     </p>
 400 <h5>
 401 <a name="math_toolkit.constants_faq.h15"></a>
 402       <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
 403       can I find other high precision constants?</a>
 404     </h5>
 405 <div class="orderedlist"><ol class="orderedlist" type="1">
 406 <li class="listitem">
 407           Constants with very high precision and good accuracy (&gt;40 decimal digits)
 408           from Simon Plouffe's web based collection <a href="http://pi.lacim.uqam.ca/eng/" target="_top">http://pi.lacim.uqam.ca/eng/</a>.
 409         </li>
 410 <li class="listitem">
 411           <a href="https://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences
 412           (OEIS)</a>
 413         </li>
 414 <li class="listitem">
 415           Checks using printed text optically scanned values and converted from:
 416           D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN
 417           0 201 89683 4 (1997)
 418         </li>
 419 <li class="listitem">
 420           M. Abrahamovitz &amp; I. E. Stegun, National Bureau of Standards, Handbook
 421           of Mathematical Functions, a reference source for formulae now superceded
 422           by
 423         </li>
 424 <li class="listitem">
 425           Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark,
 426           NIST Handbook of Mathemetical Functions, Cambridge University Press, ISBN
 427           978-0-521-14063-8, 2010.
 428         </li>
 429 <li class="listitem">
 430           John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7.
 431         </li>
 432 <li class="listitem">
 433           Some values from Cephes Mathematical Library, Stephen L. Moshier and CALC100
 434           100 decimal digit Complex Variable Calculator Program, a DOS utility.
 435         </li>
 436 <li class="listitem">
 437           Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at <a href="http://numbers.computation.free.fr/Constants/constants.html" target="_top">Number,
 438           constants and computation</a>.
 439         </li>
 440 </ol></div>
 441 <h5>
 442 <a name="math_toolkit.constants_faq.h16"></a>
 443       <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
 444       Physical Constants?</a>
 445     </h5>
 446 <p>
 447       Not here in this Boost.Math collection, because physical constants:
 448     </p>
 449 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 450 <li class="listitem">
 451           Are measurements, not truely constants.
 452         </li>
 453 <li class="listitem">
 454           Are not truly constant and keeping changing as mensuration technology improves.
 455         </li>
 456 <li class="listitem">
 457           Have a instrinsic uncertainty.
 458         </li>
 459 <li class="listitem">
 460           Mathematical constants are stored and represented at varying precision,
 461           but should never be inaccurate.
 462         </li>
 463 </ul></div>
 464 <p>
 465       Some physical constants may be available in Boost.Units.
 466     </p>
 467 </div>
 468 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 469 <td align="left"></td>
 470 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 471       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 472       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 473       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 474       Daryle Walker and Xiaogang Zhang<p>
 475         Distributed under the Boost Software License, Version 1.0. (See accompanying
 476         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>)
 477       </p>
 478 </div></td>
 479 </tr></table>
 480 <hr>
 481 <div class="spirit-nav">
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