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214 lines
8.6 KiB
C++
214 lines
8.6 KiB
C++
// Copyright Paul A. Bristow 2014, 2015.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// Note that this file contains Quickbook mark-up as well as code
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// and comments, don't change any of the special comment mark-ups!
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// Example of finding nth root using 1st and 2nd derivatives of x^n.
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#include <boost/math/tools/roots.hpp>
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//using boost::math::policies::policy;
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//using boost::math::tools::newton_raphson_iterate;
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//using boost::math::tools::halley_iterate;
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//using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits.
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//using boost::math::tools::bracket_and_solve_root;
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//using boost::math::tools::toms748_solve;
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#include <boost/math/special_functions/next.hpp>
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#include <boost/multiprecision/cpp_dec_float.hpp>
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#include <boost/math/special_functions/pow.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/multiprecision/cpp_dec_float.hpp> // For cpp_dec_float_50.
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#include <boost/multiprecision/cpp_bin_float.hpp> // using boost::multiprecision::cpp_bin_float_50;
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#ifndef _MSC_VER // float128 is not yet supported by Microsoft compiler at 2013.
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# include <boost/multiprecision/float128.hpp>
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#endif
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#include <iostream>
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// using std::cout; using std::endl;
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#include <iomanip>
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// using std::setw; using std::setprecision;
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#include <limits>
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using std::numeric_limits;
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#include <tuple>
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#include <utility> // pair, make_pair
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//[root_finding_nth_functor_2deriv
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template <int N, class T = double>
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struct nth_functor_2deriv
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{ // Functor returning both 1st and 2nd derivatives.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
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nth_functor_2deriv(T const& to_find_root_of) : a(to_find_root_of)
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{ /* Constructor stores value a to find root of, for example: */ }
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// using boost::math::tuple; // to return three values.
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std::tuple<T, T, T> operator()(T const& x)
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{
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// Return f(x), f'(x) and f''(x).
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using boost::math::pow;
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T fx = pow<N>(x) - a; // Difference (estimate x^n - a).
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T dx = N * pow<N - 1>(x); // 1st derivative f'(x).
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T d2x = N * (N - 1) * pow<N - 2 >(x); // 2nd derivative f''(x).
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return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
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}
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private:
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T a; // to be 'nth_rooted'.
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};
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//] [/root_finding_nth_functor_2deriv]
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/*
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To show the progress, one might use this before the return statement above?
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#ifdef BOOST_MATH_ROOT_DIAGNOSTIC
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std::cout << " x = " << x << ", fx = " << fx << ", dx = " << dx << ", dx2 = " << d2x << std::endl;
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#endif
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*/
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// If T is a floating-point type, might be quicker to compute the guess using a built-in type,
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// probably quickest using double, but perhaps with float or long double, T.
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// If T is a type for which frexp and ldexp are not defined,
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// then it is necessary to compute the guess using a built-in type,
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// probably quickest (but limited range) using double,
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// but perhaps with float or long double, or a multiprecision T for the full range of T.
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// typedef double guess_type; is used to specify the this.
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//[root_finding_nth_function_2deriv
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template <int N, class T = double>
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T nth_2deriv(T x)
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{ // return nth root of x using 1st and 2nd derivatives and Halley.
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using namespace std; // Help ADL of std functions.
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using namespace boost::math::tools; // For halley_iterate.
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BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!");
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BOOST_STATIC_ASSERT_MSG((N > 0) == true, "root N must be > 0!");
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BOOST_STATIC_ASSERT_MSG((N > 1000) == false, "root N is too big!");
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typedef double guess_type; // double may restrict (exponent) range for a multiprecision T?
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int exponent;
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frexp(static_cast<guess_type>(x), &exponent); // Get exponent of z (ignore mantissa).
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T guess = ldexp(static_cast<guess_type>(1.), exponent / N); // Rough guess is to divide the exponent by n.
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T min = ldexp(static_cast<guess_type>(1.) / 2, exponent / N); // Minimum possible value is half our guess.
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T max = ldexp(static_cast<guess_type>(2.), exponent / N); // Maximum possible value is twice our guess.
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int digits = std::numeric_limits<T>::digits * 0.4; // Accuracy triples with each step, so stop when
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// slightly more than one third of the digits are correct.
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const boost::uintmax_t maxit = 20;
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boost::uintmax_t it = maxit;
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T result = halley_iterate(nth_functor_2deriv<N, T>(x), guess, min, max, digits, it);
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return result;
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}
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//] [/root_finding_nth_function_2deriv]
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template <int N, typename T = double>
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T show_nth_root(T value)
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{ // Demonstrate by printing the nth root using all possibly significant digits.
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//std::cout.precision(std::numeric_limits<T>::max_digits10);
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// or use cout.precision(max_digits10 = 2 + std::numeric_limits<double>::digits * 3010/10000);
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// Or guaranteed significant digits:
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std::cout.precision(std::numeric_limits<T>::digits10);
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T r = nth_2deriv<N>(value);
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std::cout << "Type " << typeid(T).name() << " value = " << value << ", " << N << "th root = " << r << std::endl;
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return r;
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} // print_nth_root
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int main()
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{
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std::cout << "nth Root finding Example." << std::endl;
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using boost::multiprecision::cpp_dec_float_50; // decimal.
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using boost::multiprecision::cpp_bin_float_50; // binary.
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#ifndef _MSC_VER // Not supported by Microsoft compiler.
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using boost::multiprecision::float128; // Requires libquadmath
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#endif
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try
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{ // Always use try'n'catch blocks with Boost.Math to get any error messages.
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//[root_finding_n_example_1
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double r1 = nth_2deriv<5, double>(2); // Integral value converted to double.
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// double r2 = nth_2deriv<5>(2); // Only floating-point type types can be used!
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//] [/root_finding_n_example_1
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//show_nth_root<5, float>(2); // Integral value converted to float.
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//show_nth_root<5, float>(2.F); // 'initializing' : conversion from 'double' to 'float', possible loss of data
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//[root_finding_n_example_2
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show_nth_root<5, double>(2.);
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show_nth_root<5, long double>(2.);
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#ifndef _MSC_VER // float128 is not supported by Microsoft compiler 2013.
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show_nth_root<5, float128>(2);
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#endif
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show_nth_root<5, cpp_dec_float_50>(2); // dec
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show_nth_root<5, cpp_bin_float_50>(2); // bin
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//] [/root_finding_n_example_2
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// show_nth_root<1000000>(2.); // Type double value = 2, 555th root = 1.00124969405651
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// Type double value = 2, 1000th root = 1.00069338746258
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// Type double value = 2, 1000000th root = 1.00000069314783
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}
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catch (const std::exception& e)
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{ // Always useful to include try & catch blocks because default policies
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// are to throw exceptions on arguments that cause errors like underflow, overflow.
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// Lacking try & catch blocks, the program will abort without a message below,
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// which may give some helpful clues as to the cause of the exception.
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std::cout <<
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"\n""Message from thrown exception was:\n " << e.what() << std::endl;
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}
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return 0;
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} // int main()
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/*
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//[root_finding_example_output_1
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Using MSVC 2013
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nth Root finding Example.
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Type double value = 2, 5th root = 1.14869835499704
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Type long double value = 2, 5th root = 1.14869835499704
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Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_dec_float<50,int,void>,1> value = 2,
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5th root = 1.1486983549970350067986269467779275894438508890978
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Type class boost::multiprecision::number<class boost::multiprecision::backends::cpp_bin_float<50,10,void,int,0,0>,0> value = 2,
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5th root = 1.1486983549970350067986269467779275894438508890978
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//] [/root_finding_example_output_1]
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//[root_finding_example_output_2
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Using GCC 4.91 (includes float_128 type)
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nth Root finding Example.
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Type d value = 2, 5th root = 1.14869835499704
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Type e value = 2, 5th root = 1.14869835499703501
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Type N5boost14multiprecision6numberINS0_8backends16float128_backendELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.148698354997035006798626946777928
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Type N5boost14multiprecision6numberINS0_8backends13cpp_dec_floatILj50EivEELNS0_26expression_template_optionE1EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978
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Type N5boost14multiprecision6numberINS0_8backends13cpp_bin_floatILj50ELNS2_15digit_base_typeE10EviLi0ELi0EEELNS0_26expression_template_optionE0EEE value = 2, 5th root = 1.1486983549970350067986269467779275894438508890978
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RUN SUCCESSFUL (total time: 63ms)
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//] [/root_finding_example_output_2]
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*/
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/*
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Throw out of range using GCC release mode :-(
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*/
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