ceph/src/boost/boost/math/constants/calculate_constants.hpp

   1 //  Copyright John Maddock 2010, 2012.
   2 //  Copyright Paul A. Bristow 2011, 2012.
   3
   4 //  Use, modification and distribution are subject to the
   5 //  Boost Software License, Version 1.0. (See accompanying file
   6 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   7
   8 #ifndef BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
   9 #define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
  10
  11 #include <boost/math/special_functions/trunc.hpp>
  12
  13 namespace boost{ namespace math{ namespace constants{ namespace detail{
  14
  15 template <class T>
  16 template<int N>
  17 inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  18 {
  19    BOOST_MATH_STD_USING
  20
  21    return ldexp(acos(T(0)), 1);
  22
  23    /*
  24    // Although this code works well, it's usually more accurate to just call acos
  25    // and access the number types own representation of PI which is usually calculated
  26    // at slightly higher precision...
  27
  28    T result;
  29    T a = 1;
  30    T b;
  31    T A(a);
  32    T B = 0.5f;
  33    T D = 0.25f;
  34
  35    T lim;
  36    lim = boost::math::tools::epsilon<T>();
  37
  38    unsigned k = 1;
  39
  40    do
  41    {
  42       result = A + B;
  43       result = ldexp(result, -2);
  44       b = sqrt(B);
  45       a += b;
  46       a = ldexp(a, -1);
  47       A = a * a;
  48       B = A - result;
  49       B = ldexp(B, 1);
  50       result = A - B;
  51       bool neg = boost::math::sign(result) < 0;
  52       if(neg)
  53          result = -result;
  54       if(result <= lim)
  55          break;
  56       if(neg)
  57          result = -result;
  58       result = ldexp(result, k - 1);
  59       D -= result;
  60       ++k;
  61       lim = ldexp(lim, 1);
  62    }
  63    while(true);
  64
  65    result = B / D;
  66    return result;
  67    */
  68 }
  69
  70 template <class T>
  71 template<int N>
  72 inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  73 {
  74    return 2 * pi<T, policies::policy<policies::digits2<N> > >();
  75 }
  76
  77 template <class T> // 2 / pi
  78 template<int N>
  79 inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  80 {
  81    return 2 / pi<T, policies::policy<policies::digits2<N> > >();
  82 }
  83
  84 template <class T>  // sqrt(2/pi)
  85 template <int N>
  86 inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  87 {
  88    BOOST_MATH_STD_USING
  89    return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >()));
  90 }
  91
  92 template <class T>
  93 template<int N>
  94 inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  95 {
  96    return 1 / two_pi<T, policies::policy<policies::digits2<N> > >();
  97 }
  98
  99 template <class T>
 100 template<int N>
 101 inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 102 {
 103    BOOST_MATH_STD_USING
 104    return sqrt(pi<T, policies::policy<policies::digits2<N> > >());
 105 }
 106
 107 template <class T>
 108 template<int N>
 109 inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 110 {
 111    BOOST_MATH_STD_USING
 112    return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2);
 113 }
 114
 115 template <class T>
 116 template<int N>
 117 inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 118 {
 119    BOOST_MATH_STD_USING
 120    return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >());
 121 }
 122
 123 template <class T>
 124 template<int N>
 125 inline T constant_log_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 126 {
 127    BOOST_MATH_STD_USING
 128    return log(root_two_pi<T, policies::policy<policies::digits2<N> > >());
 129 }
 130
 131 template <class T>
 132 template<int N>
 133 inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 134 {
 135    BOOST_MATH_STD_USING
 136    return sqrt(log(static_cast<T>(4)));
 137 }
 138
 139 template <class T>
 140 template<int N>
 141 inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 142 {
 143    //
 144    // Although we can clearly calculate this from first principles, this hooks into
 145    // T's own notion of e, which hopefully will more accurate than one calculated to
 146    // a few epsilon:
 147    //
 148    BOOST_MATH_STD_USING
 149    return exp(static_cast<T>(1));
 150 }
 151
 152 template <class T>
 153 template<int N>
 154 inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 155 {
 156    return static_cast<T>(1) / static_cast<T>(2);
 157 }
 158
 159 template <class T>
 160 template<int M>
 161 inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>))
 162 {
 163    BOOST_MATH_STD_USING
 164    //
 165    // This is the method described in:
 166    // "Some New Algorithms for High-Precision Computation of Euler's Constant"
 167    // Richard P Brent and Edwin M McMillan.
 168    // Mathematics of Computation, Volume 34, Number 149, Jan 1980, pages 305-312.
 169    // See equation 17 with p = 2.
 170    //
 171    T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4;
 172    T lim = M ? ldexp(T(1), 1 - (std::min)(M, tools::digits<T>())) : tools::epsilon<T>();
 173    T lnn = log(n);
 174    T term = 1;
 175    T N = -lnn;
 176    T D = 1;
 177    T Hk = 0;
 178    T one = 1;
 179
 180    for(unsigned k = 1;; ++k)
 181    {
 182       term *= n * n;
 183       term /= k * k;
 184       Hk += one / k;
 185       N += term * (Hk - lnn);
 186       D += term;
 187
 188       if(term < D * lim)
 189          break;
 190    }
 191    return N / D;
 192 }
 193
 194 template <class T>
 195 template<int N>
 196 inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 197 {
 198   BOOST_MATH_STD_USING
 199   return euler<T, policies::policy<policies::digits2<N> > >()
 200      * euler<T, policies::policy<policies::digits2<N> > >();
 201 }
 202
 203 template <class T>
 204 template<int N>
 205 inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 206 {
 207   BOOST_MATH_STD_USING
 208   return static_cast<T>(1)
 209      / euler<T, policies::policy<policies::digits2<N> > >();
 210 }
 211
 212
 213 template <class T>
 214 template<int N>
 215 inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 216 {
 217    BOOST_MATH_STD_USING
 218    return sqrt(static_cast<T>(2));
 219 }
 220
 221
 222 template <class T>
 223 template<int N>
 224 inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 225 {
 226    BOOST_MATH_STD_USING
 227    return sqrt(static_cast<T>(3));
 228 }
 229
 230 template <class T>
 231 template<int N>
 232 inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 233 {
 234    BOOST_MATH_STD_USING
 235    return sqrt(static_cast<T>(2)) / 2;
 236 }
 237
 238 template <class T>
 239 template<int N>
 240 inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 241 {
 242    //
 243    // Although there are good ways to calculate this from scratch, this hooks into
 244    // T's own notion of log(2) which will hopefully be accurate to the full precision
 245    // of T:
 246    //
 247    BOOST_MATH_STD_USING
 248    return log(static_cast<T>(2));
 249 }
 250
 251 template <class T>
 252 template<int N>
 253 inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 254 {
 255    BOOST_MATH_STD_USING
 256    return log(static_cast<T>(10));
 257 }
 258
 259 template <class T>
 260 template<int N>
 261 inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 262 {
 263    BOOST_MATH_STD_USING
 264    return log(log(static_cast<T>(2)));
 265 }
 266
 267 template <class T>
 268 template<int N>
 269 inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 270 {
 271    BOOST_MATH_STD_USING
 272    return static_cast<T>(1) / static_cast<T>(3);
 273 }
 274
 275 template <class T>
 276 template<int N>
 277 inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 278 {
 279    BOOST_MATH_STD_USING
 280    return static_cast<T>(2) / static_cast<T>(3);
 281 }
 282
 283 template <class T>
 284 template<int N>
 285 inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 286 {
 287    BOOST_MATH_STD_USING
 288    return static_cast<T>(2) / static_cast<T>(3);
 289 }
 290
 291 template <class T>
 292 template<int N>
 293 inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 294 {
 295    BOOST_MATH_STD_USING
 296    return static_cast<T>(3) / static_cast<T>(4);
 297 }
 298
 299 template <class T>
 300 template<int N>
 301 inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 302 {
 303    return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
 304 }
 305
 306 template <class T>
 307 template<int N>
 308 inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 309 {
 310    return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
 311 }
 312
 313 //template <class T>
 314 //template<int N>
 315 //inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 316 //{
 317 //   BOOST_MATH_STD_USING
 318 //   return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
 319 //}
 320
 321 template <class T>
 322 template<int N>
 323 inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 324 {
 325    BOOST_MATH_STD_USING
 326    return exp(static_cast<T>(-0.5));
 327 }
 328
 329 // Pi
 330 template <class T>
 331 template<int N>
 332 inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 333 {
 334    return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
 335 }
 336
 337 template <class T>
 338 template<int N>
 339 inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 340 {
 341    return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
 342 }
 343
 344 template <class T>
 345 template<int N>
 346 inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 347 {
 348    return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
 349 }
 350
 351 template <class T>
 352 template<int N>
 353 inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 354 {
 355    BOOST_MATH_STD_USING
 356    return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
 357 }
 358
 359
 360 template <class T>
 361 template<int N>
 362 inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 363 {
 364    BOOST_MATH_STD_USING
 365    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
 366 }
 367
 368 template <class T>
 369 template<int N>
 370 inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 371 {
 372    BOOST_MATH_STD_USING
 373    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(2);
 374 }
 375
 376
 377 template <class T>
 378 template<int N>
 379 inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 380 {
 381    BOOST_MATH_STD_USING
 382    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(3);
 383 }
 384
 385 template <class T>
 386 template<int N>
 387 inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 388 {
 389    BOOST_MATH_STD_USING
 390    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(6);
 391 }
 392
 393 template <class T>
 394 template<int N>
 395 inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 396 {
 397    BOOST_MATH_STD_USING
 398    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
 399 }
 400
 401 template <class T>
 402 template<int N>
 403 inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 404 {
 405    BOOST_MATH_STD_USING
 406    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
 407 }
 408
 409 template <class T>
 410 template<int N>
 411 inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 412 {
 413    BOOST_MATH_STD_USING
 414    return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
 415 }
 416
 417 template <class T>
 418 template<int N>
 419 inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 420 {
 421    BOOST_MATH_STD_USING
 422    return pi<T, policies::policy<policies::digits2<N> > >()
 423    *      pi<T, policies::policy<policies::digits2<N> > >() ; //
 424 }
 425
 426 template <class T>
 427 template<int N>
 428 inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 429 {
 430    BOOST_MATH_STD_USING
 431    return pi<T, policies::policy<policies::digits2<N> > >()
 432    *      pi<T, policies::policy<policies::digits2<N> > >()
 433    / static_cast<T>(6); //
 434 }
 435
 436
 437 template <class T>
 438 template<int N>
 439 inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 440 {
 441    BOOST_MATH_STD_USING
 442    return pi<T, policies::policy<policies::digits2<N> > >()
 443    *      pi<T, policies::policy<policies::digits2<N> > >()
 444    *      pi<T, policies::policy<policies::digits2<N> > >()
 445    ; //
 446 }
 447
 448 template <class T>
 449 template<int N>
 450 inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 451 {
 452    BOOST_MATH_STD_USING
 453    return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
 454 }
 455
 456 template <class T>
 457 template<int N>
 458 inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 459 {
 460    BOOST_MATH_STD_USING
 461    return static_cast<T>(1)
 462    / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
 463 }
 464
 465 // Euler's e
 466
 467 template <class T>
 468 template<int N>
 469 inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 470 {
 471    BOOST_MATH_STD_USING
 472    return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
 473 }
 474
 475 template <class T>
 476 template<int N>
 477 inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 478 {
 479    BOOST_MATH_STD_USING
 480    return sqrt(e<T, policies::policy<policies::digits2<N> > >());
 481 }
 482
 483 template <class T>
 484 template<int N>
 485 inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 486 {
 487    BOOST_MATH_STD_USING
 488    return log10(e<T, policies::policy<policies::digits2<N> > >());
 489 }
 490
 491 template <class T>
 492 template<int N>
 493 inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 494 {
 495    BOOST_MATH_STD_USING
 496    return  static_cast<T>(1) /
 497      log10(e<T, policies::policy<policies::digits2<N> > >());
 498 }
 499
 500 // Trigonometric
 501
 502 template <class T>
 503 template<int N>
 504 inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 505 {
 506    BOOST_MATH_STD_USING
 507    return pi<T, policies::policy<policies::digits2<N> > >()
 508    / static_cast<T>(180)
 509    ; //
 510 }
 511
 512 template <class T>
 513 template<int N>
 514 inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 515 {
 516    BOOST_MATH_STD_USING
 517    return static_cast<T>(180)
 518    / pi<T, policies::policy<policies::digits2<N> > >()
 519    ; //
 520 }
 521
 522 template <class T>
 523 template<int N>
 524 inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 525 {
 526    BOOST_MATH_STD_USING
 527    return sin(static_cast<T>(1)) ; //
 528 }
 529
 530 template <class T>
 531 template<int N>
 532 inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 533 {
 534    BOOST_MATH_STD_USING
 535    return cos(static_cast<T>(1)) ; //
 536 }
 537
 538 template <class T>
 539 template<int N>
 540 inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 541 {
 542    BOOST_MATH_STD_USING
 543    return sinh(static_cast<T>(1)) ; //
 544 }
 545
 546 template <class T>
 547 template<int N>
 548 inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 549 {
 550    BOOST_MATH_STD_USING
 551    return cosh(static_cast<T>(1)) ; //
 552 }
 553
 554 template <class T>
 555 template<int N>
 556 inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 557 {
 558    BOOST_MATH_STD_USING
 559    return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
 560 }
 561
 562 template <class T>
 563 template<int N>
 564 inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 565 {
 566    BOOST_MATH_STD_USING
 567    //return  log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
 568    return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
 569 }
 570 template <class T>
 571 template<int N>
 572 inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 573 {
 574    BOOST_MATH_STD_USING
 575    return static_cast<T>(1) /
 576      log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
 577 }
 578
 579 // Zeta
 580
 581 template <class T>
 582 template<int N>
 583 inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 584 {
 585    BOOST_MATH_STD_USING
 586
 587      return pi<T, policies::policy<policies::digits2<N> > >()
 588      *  pi<T, policies::policy<policies::digits2<N> > >()
 589      /static_cast<T>(6);
 590 }
 591
 592 template <class T>
 593 template<int N>
 594 inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 595 {
 596    // http://mathworld.wolfram.com/AperysConstant.html
 597    // http://en.wikipedia.org/wiki/Mathematical_constant
 598
 599    // http://oeis.org/A002117/constant
 600    //T zeta3("1.20205690315959428539973816151144999076"
 601    // "4986292340498881792271555341838205786313"
 602    // "09018645587360933525814619915");
 603
 604   //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915"  A002117
 605   // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
 606   //"1.2020569031595942 double
 607   // http://www.spaennare.se/SSPROG/ssnum.pdf  // section 11, Algorithm for Apery's constant zeta(3).
 608   // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
 609
 610   // by Stefan Spannare  September 19, 2007
 611   // zeta(3) = 1/64 * sum
 612    BOOST_MATH_STD_USING
 613    T n_fact=static_cast<T>(1); // build n! for n = 0.
 614    T sum = static_cast<double>(77); // Start with n = 0 case.
 615    // for n = 0, (77/1) /64 = 1.203125
 616    //double lim = std::numeric_limits<double>::epsilon();
 617    T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
 618    for(unsigned int n = 1; n < 40; ++n)
 619    { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
 620       //cout << "n = " << n << endl;
 621       n_fact *= n; // n!
 622       T n_fact_p10 = n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact; // (n!)^10
 623       T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
 624       // int nn = (2 * n + 1);
 625       // T d = factorial(nn); // inline factorial.
 626       T d = 1;
 627       for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
 628       {
 629         d *= i;
 630       }
 631       T den = d * d * d * d * d; // [(2n+1)!]^5
 632       //cout << "den = " << den << endl;
 633       T term = num/den;
 634       if (n % 2 != 0)
 635       { //term *= -1;
 636         sum -= term;
 637       }
 638       else
 639       {
 640         sum += term;
 641       }
 642       //cout << "term = " << term << endl;
 643       //cout << "sum/64  = " << sum/64 << endl;
 644       if(abs(term) < lim)
 645       {
 646          break;
 647       }
 648    }
 649    return sum / 64;
 650 }
 651
 652 template <class T>
 653 template<int N>
 654 inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 655 { // http://oeis.org/A006752/constant
 656   //T c("0.915965594177219015054603514932384110774"
 657  //"149374281672134266498119621763019776254769479356512926115106248574");
 658
 659   // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
 660
 661    // This is equation (entry) 31 from
 662    // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
 663    // See also http://www.mpfr.org/algorithms.pdf
 664    BOOST_MATH_STD_USING
 665    T k_fact = 1;
 666    T tk_fact = 1;
 667    T sum = 1;
 668    T term;
 669    T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
 670
 671    for(unsigned k = 1;; ++k)
 672    {
 673       k_fact *= k;
 674       tk_fact *= (2 * k) * (2 * k - 1);
 675       term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
 676       sum += term;
 677       if(term < lim)
 678       {
 679          break;
 680       }
 681    }
 682    return boost::math::constants::pi<T, boost::math::policies::policy<> >()
 683       * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
 684        / 8
 685       + 3 * sum / 8;
 686 }
 687
 688 namespace khinchin_detail{
 689
 690 template <class T>
 691 T zeta_polynomial_series(T s, T sc, int digits)
 692 {
 693    BOOST_MATH_STD_USING
 694    //
 695    // This is algorithm 3 from:
 696    //
 697    // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
 698    // Canadian Mathematical Society, Conference Proceedings, 2000.
 699    // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
 700    //
 701    BOOST_MATH_STD_USING
 702    int n = (digits * 19) / 53;
 703    T sum = 0;
 704    T two_n = ldexp(T(1), n);
 705    int ej_sign = 1;
 706    for(int j = 0; j < n; ++j)
 707    {
 708       sum += ej_sign * -two_n / pow(T(j + 1), s);
 709       ej_sign = -ej_sign;
 710    }
 711    T ej_sum = 1;
 712    T ej_term = 1;
 713    for(int j = n; j <= 2 * n - 1; ++j)
 714    {
 715       sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
 716       ej_sign = -ej_sign;
 717       ej_term *= 2 * n - j;
 718       ej_term /= j - n + 1;
 719       ej_sum += ej_term;
 720    }
 721    return -sum / (two_n * (1 - pow(T(2), sc)));
 722 }
 723
 724 template <class T>
 725 T khinchin(int digits)
 726 {
 727    BOOST_MATH_STD_USING
 728    T sum = 0;
 729    T term;
 730    T lim = ldexp(T(1), 1-digits);
 731    T factor = 0;
 732    unsigned last_k = 1;
 733    T num = 1;
 734    for(unsigned n = 1;; ++n)
 735    {
 736       for(unsigned k = last_k; k <= 2 * n - 1; ++k)
 737       {
 738          factor += num / k;
 739          num = -num;
 740       }
 741       last_k = 2 * n;
 742       term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
 743       sum += term;
 744       if(term < lim)
 745          break;
 746    }
 747    return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
 748 }
 749
 750 }
 751
 752 template <class T>
 753 template<int N>
 754 inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 755 {
 756    int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
 757    return khinchin_detail::khinchin<T>(n);
 758 }
 759
 760 template <class T>
 761 template<int N>
 762 inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 763 { // from e_float constants.cpp
 764   // Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
 765    BOOST_MATH_STD_USING
 766    T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
 767     / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
 768
 769 //T ev(
 770 //"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
 771 //"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
 772 //"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
 773 //"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
 774 //"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
 775 //"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
 776 //"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
 777 //"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
 778 //"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
 779 //"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
 780 //"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
 781
 782   return ev;
 783 }
 784
 785 namespace detail{
 786 //
 787 // Calculation of the Glaisher constant depends upon calculating the
 788 // derivative of the zeta function at 2, we can then use the relation:
 789 // zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
 790 // To get the constant A.
 791 // See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
 792 //
 793 // The derivative of the zeta function is computed by direct differentiation
 794 // of the relation:
 795 // (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s  }
 796 // Which gives us 2 slowly converging but alternating sums to compute,
 797 // for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
 798 // Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
 799 // See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
 800 //
 801 template <class T>
 802 T zeta_series_derivative_2(unsigned digits)
 803 {
 804    // Derivative of the series part, evaluated at 2:
 805    BOOST_MATH_STD_USING
 806    int n = digits * 301 * 13 / 10000;
 807    boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
 808    T d = pow(3 + sqrt(T(8)), n);
 809    d = (d + 1 / d) / 2;
 810    T b = -1;
 811    T c = -d;
 812    T s = 0;
 813    for(int k = 0; k < n; ++k)
 814    {
 815       T a = -log(T(k+1)) / ((k+1) * (k+1));
 816       c = b - c;
 817       s = s + c * a;
 818       b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
 819    }
 820    return s / d;
 821 }
 822
 823 template <class T>
 824 T zeta_series_2(unsigned digits)
 825 {
 826    // Series part of zeta at 2:
 827    BOOST_MATH_STD_USING
 828    int n = digits * 301 * 13 / 10000;
 829    T d = pow(3 + sqrt(T(8)), n);
 830    d = (d + 1 / d) / 2;
 831    T b = -1;
 832    T c = -d;
 833    T s = 0;
 834    for(int k = 0; k < n; ++k)
 835    {
 836       T a = T(1) / ((k + 1) * (k + 1));
 837       c = b - c;
 838       s = s + c * a;
 839       b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
 840    }
 841    return s / d;
 842 }
 843
 844 template <class T>
 845 inline T zeta_series_lead_2()
 846 {
 847    // lead part at 2:
 848    return 2;
 849 }
 850
 851 template <class T>
 852 inline T zeta_series_derivative_lead_2()
 853 {
 854    // derivative of lead part at 2:
 855    return -2 * boost::math::constants::ln_two<T>();
 856 }
 857
 858 template <class T>
 859 inline T zeta_derivative_2(unsigned n)
 860 {
 861    // zeta derivative at 2:
 862    return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
 863       + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
 864 }
 865
 866 }  // namespace detail
 867
 868 template <class T>
 869 template<int N>
 870 inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 871 {
 872
 873    BOOST_MATH_STD_USING
 874    typedef policies::policy<policies::digits2<N> > forwarding_policy;
 875    int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
 876    T v = detail::zeta_derivative_2<T>(n);
 877    v *= 6;
 878    v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
 879    v -= boost::math::constants::euler<T, forwarding_policy>();
 880    v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
 881    v /= -12;
 882    return exp(v);
 883
 884  /*
 885    // from http://mpmath.googlecode.com/svn/data/glaisher.txt
 886      // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
 887      // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
 888   // with Euler-Maclaurin summation for zeta'(2).
 889   T g(
 890   "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
 891   "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
 892   "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
 893   "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
 894   "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
 895   "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
 896   "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
 897   "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
 898   "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
 899   "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
 900   "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
 901   "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
 902   "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
 903
 904   return g;
 905   */
 906 }
 907
 908 template <class T>
 909 template<int N>
 910 inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 911 {  // From e_float
 912   // 1100 digits of the Rayleigh distribution skewness
 913   // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
 914
 915    BOOST_MATH_STD_USING
 916    T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
 917       * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
 918       / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
 919       );
 920  //   6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
 921
 922   //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
 923   //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
 924   //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
 925   //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
 926   //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
 927   //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
 928   //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
 929   //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
 930   //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
 931   //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
 932   //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848");  ;
 933   return rs;
 934 }
 935
 936 template <class T>
 937 template<int N>
 938 inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 939 { // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
 940     // Might provide and calculate this using pi_minus_four.
 941    BOOST_MATH_STD_USING
 942    return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
 943         * pi<T, policies::policy<policies::digits2<N> > >())
 944    - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
 945    /
 946    ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
 947    * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
 948    );
 949 }
 950
 951 template <class T>
 952 template<int N>
 953 inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
 954 { // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
 955   // Might provide and calculate this using pi_minus_four.
 956    BOOST_MATH_STD_USING
 957    return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
 958         * pi<T, policies::policy<policies::digits2<N> > >())
 959    - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
 960    /
 961    ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
 962    * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
 963    );
 964 }
 965
 966 }}}} // namespaces
 967
 968 #endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED