mirror of https://github.com/saitohirga/WSJT-X.git
330 lines
9.9 KiB
C++
330 lines
9.9 KiB
C++
// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#define L22
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//#include "../tools/ntl_rr_lanczos.hpp"
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//#include "../tools/ntl_rr_digamma.hpp"
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#include "multiprecision.hpp"
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#include <boost/math/tools/polynomial.hpp>
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#include <boost/math/special_functions.hpp>
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#include <boost/math/special_functions/zeta.hpp>
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#include <boost/math/special_functions/expint.hpp>
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#include <cmath>
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mp_type f(const mp_type& x, int variant)
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{
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static const mp_type tiny = boost::math::tools::min_value<mp_type>() * 64;
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switch(variant)
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{
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case 0:
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{
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mp_type x_ = sqrt(x == 0 ? 1e-80 : x);
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return boost::math::erf(x_) / x_;
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}
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case 1:
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{
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mp_type x_ = 1 / x;
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return boost::math::erfc(x_) * x_ / exp(-x_ * x_);
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}
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case 2:
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{
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return boost::math::erfc(x) * x / exp(-x * x);
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}
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case 3:
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{
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mp_type y(x);
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if(y == 0)
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y += tiny;
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return boost::math::lgamma(y+2) / y - 0.5;
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}
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case 4:
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//
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// lgamma in the range [2,3], use:
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//
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// lgamma(x) = (x-2) * (x + 1) * (c + R(x - 2))
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//
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// Works well at 80-bit long double precision, but doesn't
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// stretch to 128-bit precision.
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//
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if(x == 0)
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{
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return boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008") / 3;
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}
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return boost::math::lgamma(x+2) / (x * (x+3));
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case 5:
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{
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//
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// lgamma in the range [1,2], use:
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//
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// lgamma(x) = (x - 1) * (x - 2) * (c + R(x - 1))
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//
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// works well over [1, 1.5] but not near 2 :-(
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//
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mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
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mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
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if(x == 0)
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{
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return r1;
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}
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if(x == 1)
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{
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return r2;
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}
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return boost::math::lgamma(x+1) / (x * (x - 1));
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}
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case 6:
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{
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//
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// lgamma in the range [1.5,2], use:
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//
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// lgamma(x) = (2 - x) * (1 - x) * (c + R(2 - x))
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//
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// works well over [1.5, 2] but not near 1 :-(
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//
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mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
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mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
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if(x == 0)
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{
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return r2;
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}
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if(x == 1)
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{
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return r1;
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}
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return boost::math::lgamma(2-x) / (x * (x - 1));
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}
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case 7:
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{
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//
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// erf_inv in range [0, 0.5]
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//
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mp_type y = x;
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if(y == 0)
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y = boost::math::tools::epsilon<mp_type>() / 64;
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return boost::math::erf_inv(y) / (y * (y+10));
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}
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case 8:
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{
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//
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// erfc_inv in range [0.25, 0.5]
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// Use an y-offset of 0.25, and range [0, 0.25]
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// abs error, auto y-offset.
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//
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mp_type y = x;
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if(y == 0)
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y = boost::lexical_cast<mp_type>("1e-5000");
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return sqrt(-2 * log(y)) / boost::math::erfc_inv(y);
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}
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case 9:
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{
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mp_type x2 = x;
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if(x2 == 0)
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x2 = boost::lexical_cast<mp_type>("1e-5000");
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mp_type y = exp(-x2*x2); // sqrt(-log(x2)) - 5;
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return boost::math::erfc_inv(y) / x2;
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}
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case 10:
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{
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//
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// Digamma over the interval [1,2], set x-offset to 1
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// and optimise for absolute error over [0,1].
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//
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int current_precision = get_working_precision();
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if(current_precision < 1000)
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set_working_precision(1000);
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//
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// This value for the root of digamma is calculated using our
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// differentiated lanczos approximation. It agrees with Cody
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// to ~ 25 digits and to Morris to 35 digits. See:
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// TOMS ALGORITHM 708 (Didonato and Morris).
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// and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher.
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//
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//mp_type root = boost::lexical_cast<mp_type>("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871");
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//
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// Actually better to calculate the root on the fly, it appears to be more
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// accurate: convergence is easier with the 1000-bit value, the approximation
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// produced agrees with functions.mathworld.com values to 35 digits even quite
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// near the root.
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//
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static boost::math::tools::eps_tolerance<mp_type> tol(1000);
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static boost::uintmax_t max_iter = 1000;
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mp_type (*pdg)(mp_type) = &boost::math::digamma;
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static const mp_type root = boost::math::tools::bracket_and_solve_root(pdg, mp_type(1.4), mp_type(1.5), true, tol, max_iter).first;
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mp_type x2 = x;
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double lim = 1e-65;
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if(fabs(x2 - root) < lim)
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{
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//
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// This is a problem area:
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// x2-root suffers cancellation error, so does digamma.
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// That gets compounded again when Remez calculates the error
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// function. This cludge seems to stop the worst of the problems:
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//
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static const mp_type a = boost::math::digamma(root - lim) / -lim;
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static const mp_type b = boost::math::digamma(root + lim) / lim;
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mp_type fract = (x2 - root + lim) / (2*lim);
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mp_type r = (1-fract) * a + fract * b;
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std::cout << "In root area: " << r;
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return r;
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}
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mp_type result = boost::math::digamma(x2) / (x2 - root);
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if(current_precision < 1000)
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set_working_precision(current_precision);
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return result;
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}
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case 11:
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// expm1:
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if(x == 0)
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{
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static mp_type lim = 1e-80;
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static mp_type a = boost::math::expm1(-lim);
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static mp_type b = boost::math::expm1(lim);
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static mp_type l = (b-a) / (2 * lim);
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return l;
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}
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return boost::math::expm1(x) / x;
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case 12:
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// demo, and test case:
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return exp(x);
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case 13:
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// K(k):
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{
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return boost::math::ellint_1(x);
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}
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case 14:
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// K(k)
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{
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return boost::math::ellint_1(1-x) / log(x);
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}
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case 15:
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// E(k)
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{
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// x = 1-k^2
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mp_type z = 1 - x * log(x);
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return boost::math::ellint_2(sqrt(1-x)) / z;
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}
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case 16:
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// Bessel I0(x) over [0,16]:
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{
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return boost::math::cyl_bessel_i(0, sqrt(x));
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}
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case 17:
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// Bessel I0(x) over [16,INF]
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{
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mp_type z = 1 / (mp_type(1)/16 - x);
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return boost::math::cyl_bessel_i(0, z) * sqrt(z) / exp(z);
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}
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case 18:
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// Zeta over [0, 1]
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{
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return boost::math::zeta(1 - x) * x - x;
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}
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case 19:
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// Zeta over [1, n]
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{
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return boost::math::zeta(x) - 1 / (x - 1);
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}
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case 20:
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// Zeta over [a, b] : a >> 1
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{
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return log(boost::math::zeta(x) - 1);
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}
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case 21:
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// expint[1] over [0,1]:
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{
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mp_type tiny = boost::lexical_cast<mp_type>("1e-5000");
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mp_type z = (x <= tiny) ? tiny : x;
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return boost::math::expint(1, z) - z + log(z);
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}
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case 22:
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// expint[1] over [1,N],
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// Note that x varies from [0,1]:
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{
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mp_type z = 1 / x;
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return boost::math::expint(1, z) * exp(z) * z;
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}
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case 23:
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// expin Ei over [0,R]
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{
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static const mp_type root =
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boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
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mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::min)() : x;
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return (boost::math::expint(z) - log(z / root)) / (z - root);
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}
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case 24:
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// Expint Ei for large x:
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{
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static const mp_type root =
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boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
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mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::max)() : mp_type(1 / x);
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return (boost::math::expint(z) - z) * z * exp(-z);
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//return (boost::math::expint(z) - log(z)) * z * exp(-z);
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}
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case 25:
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// Expint Ei for large x:
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{
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return (boost::math::expint(x) - x) * x * exp(-x);
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}
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case 26:
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{
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//
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// erf_inv in range [0, 0.5]
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//
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mp_type y = x;
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if(y == 0)
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y = boost::math::tools::epsilon<mp_type>() / 64;
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y = sqrt(y);
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return boost::math::erf_inv(y) / (y);
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}
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case 28:
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{
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// log1p over [-0.5,0.5]
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mp_type y = x;
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if(fabs(y) < 1e-100)
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y = (y == 0) ? 1e-100 : boost::math::sign(y) * 1e-100;
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return (boost::math::log1p(y) - y + y * y / 2) / (y);
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}
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case 29:
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{
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// cbrt over [0.5, 1]
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return boost::math::cbrt(x);
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}
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case 30:
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{
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// trigamma over [x,y]
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mp_type y = x;
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y = sqrt(y);
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return boost::math::trigamma(x) * (x * x);
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}
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case 31:
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{
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// trigamma over [x, INF]
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if(x == 0) return 1;
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mp_type y = (x == 0) ? (std::numeric_limits<double>::max)() / 2 : mp_type(1/x);
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return boost::math::trigamma(y) * y;
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}
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}
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return 0;
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}
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void show_extra(
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const boost::math::tools::polynomial<mp_type>& n,
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const boost::math::tools::polynomial<mp_type>& d,
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const mp_type& x_offset,
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const mp_type& y_offset,
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int variant)
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{
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switch(variant)
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{
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default:
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// do nothing here...
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;
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}
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}
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