--- /dev/null
+// (C) Copyright John Maddock 2006.
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0. (See accompanying file
+// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+#define L22
+//#include "../tools/ntl_rr_lanczos.hpp"
+//#include "../tools/ntl_rr_digamma.hpp"
+#include "multiprecision.hpp"
+#include <boost/math/tools/polynomial.hpp>
+#include <boost/math/special_functions.hpp>
+#include <boost/math/special_functions/zeta.hpp>
+#include <boost/math/special_functions/expint.hpp>
+#include <boost/math/special_functions/lambert_w.hpp>
+
+#include <cmath>
+
+
+mp_type f(const mp_type& x, int variant)
+{
+ static const mp_type tiny = boost::math::tools::min_value<mp_type>() * 64;
+ switch(variant)
+ {
+ case 0:
+ {
+ mp_type x_ = sqrt(x == 0 ? 1e-80 : x);
+ return boost::math::erf(x_) / x_;
+ }
+ case 1:
+ {
+ mp_type x_ = 1 / x;
+ return boost::math::erfc(x_) * x_ / exp(-x_ * x_);
+ }
+ case 2:
+ {
+ return boost::math::erfc(x) * x / exp(-x * x);
+ }
+ case 3:
+ {
+ mp_type y(x);
+ if(y == 0)
+ y += tiny;
+ return boost::math::lgamma(y+2) / y - 0.5;
+ }
+ case 4:
+ //
+ // lgamma in the range [2,3], use:
+ //
+ // lgamma(x) = (x-2) * (x + 1) * (c + R(x - 2))
+ //
+ // Works well at 80-bit long double precision, but doesn't
+ // stretch to 128-bit precision.
+ //
+ if(x == 0)
+ {
+ return boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008") / 3;
+ }
+ return boost::math::lgamma(x+2) / (x * (x+3));
+ case 5:
+ {
+ //
+ // lgamma in the range [1,2], use:
+ //
+ // lgamma(x) = (x - 1) * (x - 2) * (c + R(x - 1))
+ //
+ // works well over [1, 1.5] but not near 2 :-(
+ //
+ mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
+ mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
+ if(x == 0)
+ {
+ return r1;
+ }
+ if(x == 1)
+ {
+ return r2;
+ }
+ return boost::math::lgamma(x+1) / (x * (x - 1));
+ }
+ case 6:
+ {
+ //
+ // lgamma in the range [1.5,2], use:
+ //
+ // lgamma(x) = (2 - x) * (1 - x) * (c + R(2 - x))
+ //
+ // works well over [1.5, 2] but not near 1 :-(
+ //
+ mp_type r1 = boost::lexical_cast<mp_type>("0.57721566490153286060651209008240243104215933593992");
+ mp_type r2 = boost::lexical_cast<mp_type>("0.42278433509846713939348790991759756895784066406008");
+ if(x == 0)
+ {
+ return r2;
+ }
+ if(x == 1)
+ {
+ return r1;
+ }
+ return boost::math::lgamma(2-x) / (x * (x - 1));
+ }
+ case 7:
+ {
+ //
+ // erf_inv in range [0, 0.5]
+ //
+ mp_type y = x;
+ if(y == 0)
+ y = boost::math::tools::epsilon<mp_type>() / 64;
+ return boost::math::erf_inv(y) / (y * (y+10));
+ }
+ case 8:
+ {
+ //
+ // erfc_inv in range [0.25, 0.5]
+ // Use an y-offset of 0.25, and range [0, 0.25]
+ // abs error, auto y-offset.
+ //
+ mp_type y = x;
+ if(y == 0)
+ y = boost::lexical_cast<mp_type>("1e-5000");
+ return sqrt(-2 * log(y)) / boost::math::erfc_inv(y);
+ }
+ case 9:
+ {
+ mp_type x2 = x;
+ if(x2 == 0)
+ x2 = boost::lexical_cast<mp_type>("1e-5000");
+ mp_type y = exp(-x2*x2); // sqrt(-log(x2)) - 5;
+ return boost::math::erfc_inv(y) / x2;
+ }
+ case 10:
+ {
+ //
+ // Digamma over the interval [1,2], set x-offset to 1
+ // and optimise for absolute error over [0,1].
+ //
+ int current_precision = get_working_precision();
+ if(current_precision < 1000)
+ set_working_precision(1000);
+ //
+ // This value for the root of digamma is calculated using our
+ // differentiated lanczos approximation. It agrees with Cody
+ // to ~ 25 digits and to Morris to 35 digits. See:
+ // TOMS ALGORITHM 708 (Didonato and Morris).
+ // and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher.
+ //
+ //mp_type root = boost::lexical_cast<mp_type>("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871");
+ //
+ // Actually better to calculate the root on the fly, it appears to be more
+ // accurate: convergence is easier with the 1000-bit value, the approximation
+ // produced agrees with functions.mathworld.com values to 35 digits even quite
+ // near the root.
+ //
+ static boost::math::tools::eps_tolerance<mp_type> tol(1000);
+ static std::uintmax_t max_iter = 1000;
+ mp_type (*pdg)(mp_type) = &boost::math::digamma;
+ 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;
+
+ mp_type x2 = x;
+ double lim = 1e-65;
+ if(fabs(x2 - root) < lim)
+ {
+ //
+ // This is a problem area:
+ // x2-root suffers cancellation error, so does digamma.
+ // That gets compounded again when Remez calculates the error
+ // function. This cludge seems to stop the worst of the problems:
+ //
+ static const mp_type a = boost::math::digamma(root - lim) / -lim;
+ static const mp_type b = boost::math::digamma(root + lim) / lim;
+ mp_type fract = (x2 - root + lim) / (2*lim);
+ mp_type r = (1-fract) * a + fract * b;
+ std::cout << "In root area: " << r;
+ return r;
+ }
+ mp_type result = boost::math::digamma(x2) / (x2 - root);
+ if(current_precision < 1000)
+ set_working_precision(current_precision);
+ return result;
+ }
+ case 11:
+ // expm1:
+ if(x == 0)
+ {
+ static mp_type lim = 1e-80;
+ static mp_type a = boost::math::expm1(-lim);
+ static mp_type b = boost::math::expm1(lim);
+ static mp_type l = (b-a) / (2 * lim);
+ return l;
+ }
+ return boost::math::expm1(x) / x;
+ case 12:
+ // demo, and test case:
+ return exp(x);
+ case 13:
+ // K(k):
+ {
+ return boost::math::ellint_1(x);
+ }
+ case 14:
+ // K(k)
+ {
+ return boost::math::ellint_1(1-x) / log(x);
+ }
+ case 15:
+ // E(k)
+ {
+ // x = 1-k^2
+ mp_type z = 1 - x * log(x);
+ return boost::math::ellint_2(sqrt(1-x)) / z;
+ }
+ case 16:
+ // Bessel I0(x) over [0,16]:
+ {
+ return boost::math::cyl_bessel_i(0, sqrt(x));
+ }
+ case 17:
+ // Bessel I0(x) over [16,INF]
+ {
+ mp_type z = 1 / (mp_type(1)/16 - x);
+ return boost::math::cyl_bessel_i(0, z) * sqrt(z) / exp(z);
+ }
+ case 18:
+ // Zeta over [0, 1]
+ {
+ return boost::math::zeta(1 - x) * x - x;
+ }
+ case 19:
+ // Zeta over [1, n]
+ {
+ return boost::math::zeta(x) - 1 / (x - 1);
+ }
+ case 20:
+ // Zeta over [a, b] : a >> 1
+ {
+ return log(boost::math::zeta(x) - 1);
+ }
+ case 21:
+ // expint[1] over [0,1]:
+ {
+ mp_type tiny = boost::lexical_cast<mp_type>("1e-5000");
+ mp_type z = (x <= tiny) ? tiny : x;
+ return boost::math::expint(1, z) - z + log(z);
+ }
+ case 22:
+ // expint[1] over [1,N],
+ // Note that x varies from [0,1]:
+ {
+ mp_type z = 1 / x;
+ return boost::math::expint(1, z) * exp(z) * z;
+ }
+ case 23:
+ // expin Ei over [0,R]
+ {
+ static const mp_type root =
+ boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
+ mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::min)() : x;
+ return (boost::math::expint(z) - log(z / root)) / (z - root);
+ }
+ case 24:
+ // Expint Ei for large x:
+ {
+ static const mp_type root =
+ boost::lexical_cast<mp_type>("0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392");
+ mp_type z = x < (std::numeric_limits<long double>::min)() ? (std::numeric_limits<long double>::max)() : mp_type(1 / x);
+ return (boost::math::expint(z) - z) * z * exp(-z);
+ //return (boost::math::expint(z) - log(z)) * z * exp(-z);
+ }
+ case 25:
+ // Expint Ei for large x:
+ {
+ return (boost::math::expint(x) - x) * x * exp(-x);
+ }
+ case 26:
+ {
+ //
+ // erf_inv in range [0, 0.5]
+ //
+ mp_type y = x;
+ if(y == 0)
+ y = boost::math::tools::epsilon<mp_type>() / 64;
+ y = sqrt(y);
+ return boost::math::erf_inv(y) / (y);
+ }
+ case 28:
+ {
+ // log1p over [-0.5,0.5]
+ mp_type y = x;
+ if(fabs(y) < 1e-100)
+ y = (y == 0) ? 1e-100 : boost::math::sign(y) * 1e-100;
+ return (boost::math::log1p(y) - y + y * y / 2) / (y);
+ }
+ case 29:
+ {
+ // cbrt over [0.5, 1]
+ return boost::math::cbrt(x);
+ }
+ case 30:
+ {
+ // trigamma over [x,y]
+ mp_type y = x;
+ y = sqrt(y);
+ return boost::math::trigamma(x) * (x * x);
+ }
+ case 31:
+ {
+ // trigamma over [x, INF]
+ if(x == 0) return 1;
+ mp_type y = (x == 0) ? (std::numeric_limits<double>::max)() / 2 : mp_type(1/x);
+ return boost::math::trigamma(y) * y;
+ }
+ case 32:
+ {
+ // I0 over [N, INF]
+ // Don't need to go past x = 1/1000 = 1e-3 for double, or
+ // 1/15000 = 0.0006 for long double, start at 1/7.75=0.13
+ mp_type arg = 1 / x;
+ return sqrt(arg) * exp(-arg) * boost::math::cyl_bessel_i(0, arg);
+ }
+ case 33:
+ {
+ // I0 over [0, N]
+ mp_type xx = sqrt(x) * 2;
+ return (boost::math::cyl_bessel_i(0, xx) - 1) / x;
+ }
+ case 34:
+ {
+ // I1 over [0, N]
+ mp_type xx = sqrt(x) * 2;
+ return (boost::math::cyl_bessel_i(1, xx) * 2 / xx - 1 - x / 2) / (x * x);
+ }
+ case 35:
+ {
+ // I1 over [N, INF]
+ mp_type xx = 1 / x;
+ return boost::math::cyl_bessel_i(1, xx) * sqrt(xx) * exp(-xx);
+ }
+ case 36:
+ {
+ // K0 over [0, 1]
+ mp_type xx = sqrt(x);
+ return boost::math::cyl_bessel_k(0, xx) + log(xx) * boost::math::cyl_bessel_i(0, xx);
+ }
+ case 37:
+ {
+ // K0 over [1, INF]
+ mp_type xx = 1 / x;
+ return boost::math::cyl_bessel_k(0, xx) * exp(xx) * sqrt(xx);
+ }
+ case 38:
+ {
+ // K1 over [0, 1]
+ mp_type xx = sqrt(x);
+ return (boost::math::cyl_bessel_k(1, xx) - log(xx) * boost::math::cyl_bessel_i(1, xx) - 1 / xx) / xx;
+ }
+ case 39:
+ {
+ // K1 over [1, INF]
+ mp_type xx = 1 / x;
+ return boost::math::cyl_bessel_k(1, xx) * sqrt(xx) * exp(xx);
+ }
+ // Lambert W0
+ case 40:
+ return boost::math::lambert_w0(x);
+ case 41:
+ {
+ if (x == 0)
+ return 1;
+ return boost::math::lambert_w0(x) / x;
+ }
+ case 42:
+ {
+ static const mp_type e1 = exp(mp_type(-1));
+ return x / -boost::math::lambert_w0(-e1 + x);
+ }
+ case 43:
+ {
+ mp_type xx = 1 / x;
+ return 1 / boost::math::lambert_w0(xx);
+ }
+ case 44:
+ {
+ mp_type ex = exp(x);
+ return boost::math::lambert_w0(ex) - x;
+ }
+ }
+ return 0;
+}
+
+void show_extra(
+ const boost::math::tools::polynomial<mp_type>& n,
+ const boost::math::tools::polynomial<mp_type>& d,
+ const mp_type& x_offset,
+ const mp_type& y_offset,
+ int variant)
+{
+ switch(variant)
+ {
+ default:
+ // do nothing here...
+ ;
+ }
+}
+
projects / ceph.git / blobdiff