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1 | // Copyright Paul A. Bristow 2015 |
2 | ||
3 | // Use, modification and distribution are subject to the | |
4 | // Boost Software License, Version 1.0. | |
5 | // (See accompanying file LICENSE_1_0.txt | |
6 | // or copy at http://www.boost.org/LICENSE_1_0.txt) | |
7 | ||
8 | // Comparison of finding roots using TOMS748, Newton-Raphson, Halley & Schroder algorithms. | |
9 | // root_n_finding_algorithms.cpp Generalised for nth root version. | |
10 | ||
11 | // http://en.wikipedia.org/wiki/Cube_root | |
12 | ||
13 | // Note that this file contains Quickbook mark-up as well as code | |
14 | // and comments, don't change any of the special comment mark-ups! | |
15 | // This program also writes files in Quickbook tables mark-up format. | |
16 | ||
17 | #include <boost/cstdlib.hpp> | |
18 | #include <boost/config.hpp> | |
19 | #include <boost/array.hpp> | |
20 | #include <boost/type_traits/is_floating_point.hpp> | |
21 | #include <boost/math/tools/roots.hpp> | |
22 | #include <boost/math/special_functions/ellint_1.hpp> | |
23 | #include <boost/math/special_functions/ellint_2.hpp> | |
24 | ||
25 | //using boost::math::policies::policy; | |
26 | //using boost::math::tools::eps_tolerance; // Binary functor for specified number of bits. | |
27 | //using boost::math::tools::bracket_and_solve_root; | |
28 | //using boost::math::tools::toms748_solve; | |
29 | //using boost::math::tools::halley_iterate; | |
30 | //using boost::math::tools::newton_raphson_iterate; | |
31 | //using boost::math::tools::schroder_iterate; | |
32 | ||
33 | #include <boost/math/special_functions/next.hpp> // For float_distance. | |
34 | ||
35 | #include <boost/multiprecision/cpp_bin_float.hpp> // is binary. | |
36 | using boost::multiprecision::cpp_bin_float_100; | |
37 | using boost::multiprecision::cpp_bin_float_50; | |
38 | ||
39 | #include <boost/timer/timer.hpp> | |
40 | #include <boost/system/error_code.hpp> | |
41 | #include <boost/preprocessor/stringize.hpp> | |
42 | ||
43 | // STL | |
44 | #include <iostream> | |
45 | #include <iomanip> | |
46 | #include <string> | |
47 | #include <vector> | |
48 | #include <limits> | |
49 | #include <fstream> // std::ofstream | |
50 | #include <cmath> | |
51 | #include <typeinfo> // for type name using typid(thingy).name(); | |
52 | ||
53 | #ifdef __FILE__ | |
54 | std::string sourcefilename = __FILE__; | |
55 | #else | |
56 | std::string sourcefilename(""); | |
57 | #endif | |
58 | ||
59 | std::string chop_last(std::string s) | |
60 | { | |
61 | std::string::size_type pos = s.find_last_of("\\/"); | |
62 | if(pos != std::string::npos) | |
63 | s.erase(pos); | |
64 | else if(s.empty()) | |
65 | abort(); | |
66 | else | |
67 | s.erase(); | |
68 | return s; | |
69 | } | |
70 | ||
71 | std::string make_root() | |
72 | { | |
73 | std::string result; | |
74 | if(sourcefilename.find_first_of(":") != std::string::npos) | |
75 | { | |
76 | result = chop_last(sourcefilename); // lose filename part | |
77 | result = chop_last(result); // lose /example/ | |
78 | result = chop_last(result); // lose /math/ | |
79 | result = chop_last(result); // lose /libs/ | |
80 | } | |
81 | else | |
82 | { | |
83 | result = chop_last(sourcefilename); // lose filename part | |
84 | if(result.empty()) | |
85 | result = "."; | |
86 | result += "/../../.."; | |
87 | } | |
88 | return result; | |
89 | } | |
90 | ||
91 | std::string short_file_name(std::string s) | |
92 | { | |
93 | std::string::size_type pos = s.find_last_of("\\/"); | |
94 | if(pos != std::string::npos) | |
95 | s.erase(0, pos + 1); | |
96 | return s; | |
97 | } | |
98 | ||
99 | std::string boost_root = make_root(); | |
100 | ||
101 | ||
102 | std::string fp_hardware; // Any hardware features like SEE or AVX | |
103 | ||
104 | const std::string roots_name = "libs/math/doc/roots/"; | |
105 | ||
106 | const std::string full_roots_name(boost_root + "/libs/math/doc/roots/"); | |
107 | ||
108 | const std::size_t nooftypes = 4; | |
109 | const std::size_t noofalgos = 4; | |
110 | const std::size_t noofroots = 3; | |
111 | ||
112 | double digits_accuracy = 1.0; // 1 == maximum possible accuracy. | |
113 | ||
114 | std::stringstream ss; | |
115 | ||
116 | std::ofstream fout; | |
117 | ||
118 | std::vector<std::string> algo_names = | |
119 | { | |
120 | "TOMS748", "Newton", "Halley", "Schr'''ö'''der" | |
121 | }; | |
122 | ||
123 | std::vector<std::string> names = | |
124 | { | |
125 | "float", "double", "long double", "cpp_bin_float50" | |
126 | }; | |
127 | ||
128 | uintmax_t iters; // Global as value of iterations is not returned. | |
129 | ||
130 | struct root_info | |
131 | { // for a floating-point type, float, double ... | |
132 | std::size_t max_digits10; // for type. | |
133 | std::string full_typename; // for type from type_id.name(). | |
134 | std::string short_typename; // for type "float", "double", "cpp_bin_float_50" .... | |
135 | std::size_t bin_digits; // binary in floating-point type numeric_limits<T>::digits; | |
136 | int get_digits; // fraction of maximum possible accuracy required. | |
137 | // = digits * digits_accuracy | |
138 | // Vector of values (4) for each algorithm, TOMS748, Newton, Halley & Schroder. | |
139 | //std::vector< boost::int_least64_t> times; converted to int. | |
140 | std::vector<int> times; // arbirary units (ticks). | |
141 | //boost::int_least64_t min_time = std::numeric_limits<boost::int_least64_t>::max(); // Used to normalize times (as int). | |
142 | std::vector<double> normed_times; | |
143 | int min_time = (std::numeric_limits<int>::max)(); // Used to normalize times. | |
144 | std::vector<uintmax_t> iterations; | |
145 | std::vector<long int> distances; | |
146 | std::vector<cpp_bin_float_100> full_results; | |
147 | }; // struct root_info | |
148 | ||
149 | std::vector<root_info> root_infos; // One element for each floating-point type used. | |
150 | ||
151 | inline std::string build_test_name(const char* type_name, const char* test_name) | |
152 | { | |
153 | std::string result(BOOST_COMPILER); | |
154 | result += "|"; | |
155 | result += BOOST_STDLIB; | |
156 | result += "|"; | |
157 | result += BOOST_PLATFORM; | |
158 | result += "|"; | |
159 | result += type_name; | |
160 | result += "|"; | |
161 | result += test_name; | |
162 | #if defined(_DEBUG) || !defined(NDEBUG) | |
163 | result += "|"; | |
164 | result += " debug"; | |
165 | #else | |
166 | result += "|"; | |
167 | result += " release"; | |
168 | #endif | |
169 | result += "|"; | |
170 | return result; | |
171 | } // std::string build_test_name | |
172 | ||
173 | // Algorithms ////////////////////////////////////////////// | |
174 | ||
175 | // No derivatives - using TOMS748 internally. | |
176 | //[elliptic_noderv_func | |
177 | template <typename T = double> | |
178 | struct elliptic_root_functor_noderiv | |
179 | { // Nth root of x using only function - no derivatives. | |
180 | elliptic_root_functor_noderiv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius) | |
181 | { // Constructor just stores value a to find root of. | |
182 | } | |
183 | T operator()(T const& x) | |
184 | { | |
185 | using std::sqrt; | |
186 | // return the difference between required arc-length, and the calculated arc-length for an | |
187 | // ellipse with radii m_radius and x: | |
188 | T a = (std::max)(m_radius, x); | |
189 | T b = (std::min)(m_radius, x); | |
190 | T k = sqrt(1 - b * b / (a * a)); | |
191 | return 4 * a * boost::math::ellint_2(k) - m_arc; | |
192 | } | |
193 | private: | |
194 | T m_arc; // length of arc. | |
195 | T m_radius; // one of the two radii of the ellipse | |
196 | }; // template <class T> struct elliptic_root_functor_noderiv | |
197 | //] | |
198 | //[elliptic_root_noderiv | |
199 | template <class T = double> | |
200 | T elliptic_root_noderiv(T radius, T arc) | |
201 | { // return the other radius of an ellipse, given one radii and the arc-length | |
202 | using namespace std; // Help ADL of std functions. | |
203 | using namespace boost::math::tools; // For bracket_and_solve_root. | |
204 | ||
205 | T guess = sqrt(arc * arc / 16 - radius * radius); | |
206 | T factor = 1.2; // How big steps to take when searching. | |
207 | ||
208 | const boost::uintmax_t maxit = 50; // Limit to maximum iterations. | |
209 | boost::uintmax_t it = maxit; // Initally our chosen max iterations, but updated with actual. | |
210 | bool is_rising = true; // arc-length increases if one radii increases, so function is rising | |
211 | // Define a termination condition, stop when nearly all digits are correct, but allow for | |
212 | // the fact that we are returning a range, and must have some inaccuracy in the elliptic integral: | |
213 | eps_tolerance<T> tol(std::numeric_limits<T>::digits - 2); | |
214 | // Call bracket_and_solve_root to find the solution, note that this is a rising function: | |
215 | std::pair<T, T> r = bracket_and_solve_root(elliptic_root_functor_noderiv<T>(arc, radius), guess, factor, is_rising, tol, it); | |
216 | //<- | |
217 | iters = it; | |
218 | //-> | |
219 | // Result is midway between the endpoints of the range: | |
220 | return r.first + (r.second - r.first) / 2; | |
221 | } // template <class T> T elliptic_root_noderiv(T x) | |
222 | //] | |
223 | // Using 1st derivative only Newton-Raphson | |
224 | //[elliptic_1deriv_func | |
225 | template <class T = double> | |
226 | struct elliptic_root_functor_1deriv | |
227 | { // Functor also returning 1st derviative. | |
228 | BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); | |
229 | ||
230 | elliptic_root_functor_1deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius) | |
231 | { // Constructor just stores value a to find root of. | |
232 | } | |
233 | std::pair<T, T> operator()(T const& x) | |
234 | { | |
235 | using std::sqrt; | |
236 | // Return the difference between required arc-length, and the calculated arc-length for an | |
237 | // ellipse with radii m_radius and x, plus it's derivative. | |
238 | // See http://www.wolframalpha.com/input/?i=d%2Fda+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29] | |
239 | // We require two elliptic integral calls, but from these we can calculate both | |
240 | // the function and it's derivative: | |
241 | T a = (std::max)(m_radius, x); | |
242 | T b = (std::min)(m_radius, x); | |
243 | T a2 = a * a; | |
244 | T b2 = b * b; | |
245 | T k = sqrt(1 - b2 / a2); | |
246 | T Ek = boost::math::ellint_2(k); | |
247 | T Kk = boost::math::ellint_1(k); | |
248 | T fx = 4 * a * Ek - m_arc; | |
249 | T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2); | |
250 | return std::make_pair(fx, dfx); | |
251 | } | |
252 | private: | |
253 | T m_arc; // length of arc. | |
254 | T m_radius; // one of the two radii of the ellipse | |
255 | }; // struct elliptic_root__functor_1deriv | |
256 | //] | |
257 | //[elliptic_1deriv | |
258 | template <class T = double> | |
259 | T elliptic_root_1deriv(T radius, T arc) | |
260 | { | |
261 | using namespace std; // Help ADL of std functions. | |
262 | using namespace boost::math::tools; // For newton_raphson_iterate. | |
263 | ||
264 | BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); | |
265 | ||
266 | T guess = sqrt(arc * arc / 16 - radius * radius); | |
267 | T min = 0; // Minimum possible value is zero. | |
268 | T max = arc; // Maximum possible value is the arc length. | |
269 | ||
270 | // Accuracy doubles at each step, so stop when just over half of the digits are | |
271 | // correct, and rely on that step to polish off the remainder: | |
272 | int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.6); | |
273 | const boost::uintmax_t maxit = 20; | |
274 | boost::uintmax_t it = maxit; | |
275 | T result = newton_raphson_iterate(elliptic_root_functor_1deriv<T>(arc, radius), guess, min, max, get_digits, it); | |
276 | //<- | |
277 | iters = it; | |
278 | //-> | |
279 | return result; | |
280 | } // T elliptic_root_1_deriv Newton-Raphson | |
281 | //] | |
282 | ||
283 | // Using 1st and 2nd derivatives with Halley algorithm. | |
284 | //[elliptic_2deriv_func | |
285 | template <class T = double> | |
286 | struct elliptic_root_functor_2deriv | |
287 | { // Functor returning both 1st and 2nd derivatives. | |
288 | BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); | |
289 | ||
290 | elliptic_root_functor_2deriv(T const& arc, T const& radius) : m_arc(arc), m_radius(radius) {} | |
291 | std::tuple<T, T, T> operator()(T const& x) | |
292 | { | |
293 | using std::sqrt; | |
294 | // Return the difference between required arc-length, and the calculated arc-length for an | |
295 | // ellipse with radii m_radius and x, plus it's derivative. | |
296 | // See http://www.wolframalpha.com/input/?i=d^2%2Fda^2+[4+*+a+*+EllipticE%281+-+b^2%2Fa^2%29] | |
297 | // for the second derivative. | |
298 | T a = (std::max)(m_radius, x); | |
299 | T b = (std::min)(m_radius, x); | |
300 | T a2 = a * a; | |
301 | T b2 = b * b; | |
302 | T k = sqrt(1 - b2 / a2); | |
303 | T Ek = boost::math::ellint_2(k); | |
304 | T Kk = boost::math::ellint_1(k); | |
305 | T fx = 4 * a * Ek - m_arc; | |
306 | T dfx = 4 * (a2 * Ek - b2 * Kk) / (a2 - b2); | |
307 | T dfx2 = 4 * b2 * ((a2 + b2) * Kk - 2 * a2 * Ek) / (a * (a2 - b2) * (a2 - b2)); | |
308 | return std::make_tuple(fx, dfx, dfx2); | |
309 | } | |
310 | private: | |
311 | T m_arc; // length of arc. | |
312 | T m_radius; // one of the two radii of the ellipse | |
313 | }; | |
314 | //] | |
315 | //[elliptic_2deriv | |
316 | template <class T = double> | |
317 | T elliptic_root_2deriv(T radius, T arc) | |
318 | { | |
319 | using namespace std; // Help ADL of std functions. | |
320 | using namespace boost::math::tools; // For halley_iterate. | |
321 | ||
322 | BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); | |
323 | ||
324 | T guess = sqrt(arc * arc / 16 - radius * radius); | |
325 | T min = 0; // Minimum possible value is zero. | |
326 | T max = arc; // radius can't be larger than the arc length. | |
327 | ||
328 | // Accuracy triples at each step, so stop when just over one-third of the digits | |
329 | // are correct, and the last iteration will polish off the remaining digits: | |
330 | int get_digits = static_cast<int>(std::numeric_limits<T>::digits * 0.4); | |
331 | const boost::uintmax_t maxit = 20; | |
332 | boost::uintmax_t it = maxit; | |
333 | T result = halley_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it); | |
334 | //<- | |
335 | iters = it; | |
336 | //-> | |
337 | return result; | |
338 | } // nth_2deriv Halley | |
339 | //] | |
340 | // Using 1st and 2nd derivatives using Schroder algorithm. | |
341 | ||
342 | template <class T = double> | |
343 | T elliptic_root_2deriv_s(T arc, T radius) | |
344 | { // return nth root of x using 1st and 2nd derivatives and Schroder. | |
345 | ||
346 | using namespace std; // Help ADL of std functions. | |
347 | using namespace boost::math::tools; // For schroder_iterate. | |
348 | ||
349 | BOOST_STATIC_ASSERT_MSG(boost::is_integral<T>::value == false, "Only floating-point type types can be used!"); | |
350 | ||
351 | T guess = sqrt(arc * arc / 16 - radius * radius); | |
352 | T min = 0; // Minimum possible value is zero. | |
353 | T max = arc; // radius can't be larger than the arc length. | |
354 | ||
355 | int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T. | |
356 | int get_digits = static_cast<int>(digits * digits_accuracy); | |
357 | const boost::uintmax_t maxit = 20; | |
358 | boost::uintmax_t it = maxit; | |
359 | T result = schroder_iterate(elliptic_root_functor_2deriv<T>(arc, radius), guess, min, max, get_digits, it); | |
360 | iters = it; | |
361 | ||
362 | return result; | |
363 | } // T elliptic_root_2deriv_s Schroder | |
364 | ||
365 | //////////////////////////////////////////////////////// end of algorithms - perhaps in a separate .hpp? | |
366 | ||
367 | //! Print 4 floating-point types info: max_digits10, digits and required accuracy digits as a Quickbook table. | |
368 | int table_type_info(double digits_accuracy) | |
369 | { | |
370 | std::string qbk_name = full_roots_name; // Prefix by boost_root file. | |
371 | ||
372 | qbk_name += "type_info_table"; | |
373 | std::stringstream ss; | |
374 | ss.precision(3); | |
375 | ss << "_" << digits_accuracy * 100; | |
376 | qbk_name += ss.str(); | |
377 | ||
378 | #ifdef _MSC_VER | |
379 | qbk_name += "_msvc.qbk"; | |
380 | #else // assume GCC | |
381 | qbk_name += "_gcc.qbk"; | |
382 | #endif | |
383 | ||
384 | // Example: type_info_table_100_msvc.qbk | |
385 | fout.open(qbk_name, std::ios_base::out); | |
386 | ||
387 | if (fout.is_open()) | |
388 | { | |
389 | std::cout << "Output type table to " << qbk_name << std::endl; | |
390 | } | |
391 | else | |
392 | { // Failed to open. | |
393 | std::cout << " Open file " << qbk_name << " for output failed!" << std::endl; | |
394 | std::cout << "errno " << errno << std::endl; | |
395 | return errno; | |
396 | } | |
397 | ||
398 | fout << | |
399 | "[/" | |
400 | << qbk_name | |
401 | << "\n" | |
402 | "Copyright 2015 Paul A. Bristow.""\n" | |
403 | "Copyright 2015 John Maddock.""\n" | |
404 | "Distributed under the Boost Software License, Version 1.0.""\n" | |
405 | "(See accompanying file LICENSE_1_0.txt or copy at""\n" | |
406 | "http://www.boost.org/LICENSE_1_0.txt).""\n" | |
407 | "]""\n" | |
408 | << std::endl; | |
409 | ||
410 | fout << "[h6 Fraction of maximum possible bits of accuracy required is " << digits_accuracy << ".]\n" << std::endl; | |
411 | ||
412 | std::string table_id("type_info"); | |
413 | table_id += ss.str(); // Fraction digits accuracy. | |
414 | ||
415 | #ifdef _MSC_VER | |
416 | table_id += "_msvc"; | |
417 | #else // assume GCC | |
418 | table_id += "_gcc"; | |
419 | #endif | |
420 | ||
421 | fout << "[table:" << table_id << " Digits for float, double, long double and cpp_bin_float_50\n" | |
422 | << "[[type name] [max_digits10] [binary digits] [required digits]]\n";// header. | |
423 | ||
424 | // For all fout types: | |
425 | ||
426 | fout << "[[" << "float" << "]" | |
427 | << "[" << std::numeric_limits<float>::max_digits10 << "]" // max_digits10 | |
428 | << "[" << std::numeric_limits<float>::digits << "]"// < "Binary digits | |
429 | << "[" << static_cast<int>(std::numeric_limits<float>::digits * digits_accuracy) << "]]\n"; // Accuracy digits. | |
430 | ||
431 | fout << "[[" << "float" << "]" | |
432 | << "[" << std::numeric_limits<double>::max_digits10 << "]" // max_digits10 | |
433 | << "[" << std::numeric_limits<double>::digits << "]"// < "Binary digits | |
434 | << "[" << static_cast<int>(std::numeric_limits<double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits. | |
435 | ||
436 | fout << "[[" << "long double" << "]" | |
437 | << "[" << std::numeric_limits<long double>::max_digits10 << "]" // max_digits10 | |
438 | << "[" << std::numeric_limits<long double>::digits << "]"// < "Binary digits | |
439 | << "[" << static_cast<int>(std::numeric_limits<long double>::digits * digits_accuracy) << "]]\n"; // Accuracy digits. | |
440 | ||
441 | fout << "[[" << "cpp_bin_float_50" << "]" | |
442 | << "[" << std::numeric_limits<cpp_bin_float_50>::max_digits10 << "]" // max_digits10 | |
443 | << "[" << std::numeric_limits<cpp_bin_float_50>::digits << "]"// < "Binary digits | |
444 | << "[" << static_cast<int>(std::numeric_limits<cpp_bin_float_50>::digits * digits_accuracy) << "]]\n"; // Accuracy digits. | |
445 | ||
446 | fout << "] [/table table_id_msvc] \n" << std::endl; // End of table. | |
447 | ||
448 | fout.close(); | |
449 | return 0; | |
450 | } // type_table | |
451 | ||
452 | //! Evaluate root N timing for each algorithm, and for one floating-point type T. | |
453 | template <typename T> | |
454 | int test_root(cpp_bin_float_100 big_radius, cpp_bin_float_100 big_arc, cpp_bin_float_100 answer, const char* type_name, std::size_t type_no) | |
455 | { | |
456 | std::size_t max_digits = 2 + std::numeric_limits<T>::digits * 3010 / 10000; | |
457 | // For new versions use max_digits10 | |
458 | // std::cout.precision(std::numeric_limits<T>::max_digits10); | |
459 | std::cout.precision(max_digits); | |
460 | std::cout << std::showpoint << std::endl; // Show trailing zeros too. | |
461 | ||
462 | root_infos.push_back(root_info()); | |
463 | ||
464 | root_infos[type_no].max_digits10 = max_digits; | |
465 | root_infos[type_no].full_typename = typeid(T).name(); // Full typename. | |
466 | root_infos[type_no].short_typename = type_name; // Short typename. | |
467 | root_infos[type_no].bin_digits = std::numeric_limits<T>::digits; | |
468 | root_infos[type_no].get_digits = static_cast<int>(std::numeric_limits<T>::digits * digits_accuracy); | |
469 | ||
470 | T radius = static_cast<T>(big_radius); | |
471 | T arc = static_cast<T>(big_arc); | |
472 | ||
473 | T result; // root | |
474 | T sum = 0; | |
475 | T ans = static_cast<T>(answer); | |
476 | ||
477 | using boost::timer::nanosecond_type; | |
478 | using boost::timer::cpu_times; | |
479 | using boost::timer::cpu_timer; | |
480 | ||
481 | long eval_count = boost::is_floating_point<T>::value ? 1000000 : 10000; // To give a sufficiently stable timing for the fast built-in types, | |
482 | // This takes an inconveniently long time for multiprecision cpp_bin_float_50 etc types. | |
483 | ||
484 | cpu_times now; // Holds wall, user and system times. | |
485 | ||
486 | { // Evaluate times etc for each algorithm. | |
487 | //algorithm_names.push_back("TOMS748"); // | |
488 | cpu_timer ti; // Can start, pause, resume and stop, and read elapsed. | |
489 | ti.start(); | |
490 | for(long i = eval_count; i >= 0; --i) | |
491 | { | |
492 | result = elliptic_root_noderiv(radius, arc); // | |
493 | sum += result; | |
494 | } | |
495 | now = ti.elapsed(); | |
496 | int time = static_cast<int>(now.user / eval_count); | |
497 | root_infos[type_no].times.push_back(time); // CPU time taken. | |
498 | if (time < root_infos[type_no].min_time) | |
499 | { | |
500 | root_infos[type_no].min_time = time; | |
501 | } | |
502 | ti.stop(); | |
503 | long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans)); | |
504 | root_infos[type_no].distances.push_back(distance); | |
505 | root_infos[type_no].iterations.push_back(iters); // | |
506 | root_infos[type_no].full_results.push_back(result); | |
507 | } | |
508 | { | |
509 | // algorithm_names.push_back("Newton"); // algorithm | |
510 | cpu_timer ti; // Can start, pause, resume and stop, and read elapsed. | |
511 | ti.start(); | |
512 | for(long i = eval_count; i >= 0; --i) | |
513 | { | |
514 | result = elliptic_root_1deriv(radius, arc); // | |
515 | sum += result; | |
516 | } | |
517 | now = ti.elapsed(); | |
518 | int time = static_cast<int>(now.user / eval_count); | |
519 | root_infos[type_no].times.push_back(time); // CPU time taken. | |
520 | if (time < root_infos[type_no].min_time) | |
521 | { | |
522 | root_infos[type_no].min_time = time; | |
523 | } | |
524 | ||
525 | ti.stop(); | |
526 | long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans)); | |
527 | root_infos[type_no].distances.push_back(distance); | |
528 | root_infos[type_no].iterations.push_back(iters); // | |
529 | root_infos[type_no].full_results.push_back(result); | |
530 | } | |
531 | { | |
532 | //algorithm_names.push_back("Halley"); // algorithm | |
533 | cpu_timer ti; // Can start, pause, resume and stop, and read elapsed. | |
534 | ti.start(); | |
535 | for(long i = eval_count; i >= 0; --i) | |
536 | { | |
537 | result = elliptic_root_2deriv(radius, arc); // | |
538 | sum += result; | |
539 | } | |
540 | now = ti.elapsed(); | |
541 | int time = static_cast<int>(now.user / eval_count); | |
542 | root_infos[type_no].times.push_back(time); // CPU time taken. | |
543 | ti.stop(); | |
544 | if (time < root_infos[type_no].min_time) | |
545 | { | |
546 | root_infos[type_no].min_time = time; | |
547 | } | |
548 | long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans)); | |
549 | root_infos[type_no].distances.push_back(distance); | |
550 | root_infos[type_no].iterations.push_back(iters); // | |
551 | root_infos[type_no].full_results.push_back(result); | |
552 | } | |
553 | { | |
554 | // algorithm_names.push_back("Schr'''ö'''der"); // algorithm | |
555 | cpu_timer ti; // Can start, pause, resume and stop, and read elapsed. | |
556 | ti.start(); | |
557 | for(long i = eval_count; i >= 0; --i) | |
558 | { | |
559 | result = elliptic_root_2deriv_s(arc, radius); // | |
560 | sum += result; | |
561 | } | |
562 | now = ti.elapsed(); | |
563 | int time = static_cast<int>(now.user / eval_count); | |
564 | root_infos[type_no].times.push_back(time); // CPU time taken. | |
565 | if (time < root_infos[type_no].min_time) | |
566 | { | |
567 | root_infos[type_no].min_time = time; | |
568 | } | |
569 | ti.stop(); | |
570 | long int distance = static_cast<int>(boost::math::float_distance<T>(result, ans)); | |
571 | root_infos[type_no].distances.push_back(distance); | |
572 | root_infos[type_no].iterations.push_back(iters); // | |
573 | root_infos[type_no].full_results.push_back(result); | |
574 | } | |
575 | for (size_t i = 0; i != root_infos[type_no].times.size(); i++) // For each time. | |
576 | { // Normalize times. | |
577 | root_infos[type_no].normed_times.push_back(static_cast<double>(root_infos[type_no].times[i]) / root_infos[type_no].min_time); | |
578 | } | |
579 | ||
580 | std::cout << "Accumulated result was: " << sum << std::endl; | |
581 | ||
582 | return 4; // eval_count of how many algorithms used. | |
583 | } // test_root | |
584 | ||
585 | /*! Fill array of times, interations, etc for Nth root for all 4 types, | |
586 | and write a table of results in Quickbook format. | |
587 | */ | |
588 | void table_root_info(cpp_bin_float_100 radius, cpp_bin_float_100 arc) | |
589 | { | |
590 | using std::abs; | |
591 | ||
592 | std::cout << nooftypes << " floating-point types tested:" << std::endl; | |
593 | #if defined(_DEBUG) || !defined(NDEBUG) | |
594 | std::cout << "Compiled in debug mode." << std::endl; | |
595 | #else | |
596 | std::cout << "Compiled in optimise mode." << std::endl; | |
597 | #endif | |
598 | std::cout << "FP hardware " << fp_hardware << std::endl; | |
599 | // Compute the 'right' answer for root N at 100 decimal digits. | |
600 | cpp_bin_float_100 full_answer = elliptic_root_noderiv(radius, arc); | |
601 | ||
602 | int type_count = 0; | |
603 | root_infos.clear(); // Erase any previous data. | |
604 | // Fill the elements of the array for each floating-point type. | |
605 | ||
606 | type_count = test_root<float>(radius, arc, full_answer, "float", 0); | |
607 | type_count = test_root<double>(radius, arc, full_answer, "double", 1); | |
608 | type_count = test_root<long double>(radius, arc, full_answer, "long double", 2); | |
609 | type_count = test_root<cpp_bin_float_50>(radius, arc, full_answer, "cpp_bin_float_50", 3); | |
610 | ||
611 | // Use info from 4 floating point types to | |
612 | ||
613 | // Prepare Quickbook table for a single root | |
614 | // with columns of times, iterations, distances repeated for various floating-point types, | |
615 | // and 4 rows for each algorithm. | |
616 | ||
617 | std::stringstream table_info; | |
618 | table_info.precision(3); | |
619 | table_info << "[table:elliptic root with radius " << radius << " and arc length " << arc << ") for float, double, long double and cpp_bin_float_50 types"; | |
620 | if (fp_hardware != "") | |
621 | { | |
622 | table_info << ", using " << fp_hardware; | |
623 | } | |
624 | table_info << std::endl; | |
625 | ||
626 | fout << table_info.str() | |
627 | << "[[][float][][][] [][double][][][] [][long d][][][] [][cpp50][][]]\n" | |
628 | << "[[Algo ]"; | |
629 | for (size_t tp = 0; tp != nooftypes; tp++) | |
630 | { // For all types: | |
631 | fout << "[Its]" << "[Times]" << "[Norm]" << "[Dis]" << "[ ]"; | |
632 | } | |
633 | fout << "]" << std::endl; | |
634 | ||
635 | // Row for all algorithms. | |
636 | for (std::size_t algo = 0; algo != noofalgos; algo++) | |
637 | { | |
638 | fout << "[[" << std::left << std::setw(9) << algo_names[algo] << "]"; | |
639 | for (size_t tp = 0; tp != nooftypes; tp++) | |
640 | { // For all types: | |
641 | fout | |
642 | << "[" << std::right << std::showpoint | |
643 | << std::setw(3) << std::setprecision(2) << root_infos[tp].iterations[algo] << "][" | |
644 | << std::setw(5) << std::setprecision(5) << root_infos[tp].times[algo] << "]["; | |
645 | fout << std::setw(3) << std::setprecision(3); | |
646 | double normed_time = root_infos[tp].normed_times[algo]; | |
647 | if (abs(normed_time - 1.00) <= 0.05) | |
648 | { // At or near the best time, so show as blue. | |
649 | fout << "[role blue " << normed_time << "]"; | |
650 | } | |
651 | else if (abs(normed_time) > 4.) | |
652 | { // markedly poor so show as red. | |
653 | fout << "[role red " << normed_time << "]"; | |
654 | } | |
655 | else | |
656 | { // Not the best, so normal black. | |
657 | fout << normed_time; | |
658 | } | |
659 | fout << "][" | |
660 | << std::setw(3) << std::setprecision(2) << root_infos[tp].distances[algo] << "][ ]"; | |
661 | } // tp | |
662 | fout << "]" << std::endl; | |
663 | } // for algo | |
664 | fout << "] [/end of table root]\n"; | |
665 | } // void table_root_info | |
666 | ||
667 | /*! Output program header, table of type info, and tables for 4 algorithms and 4 floating-point types, | |
668 | for Nth root required digits_accuracy. | |
669 | */ | |
670 | ||
671 | int roots_tables(cpp_bin_float_100 radius, cpp_bin_float_100 arc, double digits_accuracy) | |
672 | { | |
673 | ::digits_accuracy = digits_accuracy; | |
674 | // Save globally so that it is available to root-finding algorithms. Ugly :-( | |
675 | ||
676 | #if defined(_DEBUG) || !defined(NDEBUG) | |
677 | std::string debug_or_optimize("Compiled in debug mode."); | |
678 | #else | |
679 | std::string debug_or_optimize("Compiled in optimise mode."); | |
680 | #endif | |
681 | ||
682 | // Create filename for roots_table | |
683 | std::string qbk_name = full_roots_name; | |
684 | qbk_name += "elliptic_table"; | |
685 | ||
686 | std::stringstream ss; | |
687 | ss.precision(3); | |
688 | // ss << "_" << N // now put all the tables in one .qbk file? | |
689 | ss << "_" << digits_accuracy * 100 | |
690 | << std::flush; | |
691 | // Assume only save optimize mode runs, so don't add any _DEBUG info. | |
692 | qbk_name += ss.str(); | |
693 | ||
694 | #ifdef _MSC_VER | |
695 | qbk_name += "_msvc"; | |
696 | #else // assume GCC | |
697 | qbk_name += "_gcc"; | |
698 | #endif | |
699 | if (fp_hardware != "") | |
700 | { | |
701 | qbk_name += fp_hardware; | |
702 | } | |
703 | qbk_name += ".qbk"; | |
704 | ||
705 | fout.open(qbk_name, std::ios_base::out); | |
706 | ||
707 | if (fout.is_open()) | |
708 | { | |
709 | std::cout << "Output root table to " << qbk_name << std::endl; | |
710 | } | |
711 | else | |
712 | { // Failed to open. | |
713 | std::cout << " Open file " << qbk_name << " for output failed!" << std::endl; | |
714 | std::cout << "errno " << errno << std::endl; | |
715 | return errno; | |
716 | } | |
717 | ||
718 | fout << | |
719 | "[/" | |
720 | << qbk_name | |
721 | << "\n" | |
722 | "Copyright 2015 Paul A. Bristow.""\n" | |
723 | "Copyright 2015 John Maddock.""\n" | |
724 | "Distributed under the Boost Software License, Version 1.0.""\n" | |
725 | "(See accompanying file LICENSE_1_0.txt or copy at""\n" | |
726 | "http://www.boost.org/LICENSE_1_0.txt).""\n" | |
727 | "]""\n" | |
728 | << std::endl; | |
729 | ||
730 | // Print out the program/compiler/stdlib/platform names as a Quickbook comment: | |
731 | fout << "\n[h6 Program [@../../example/" << short_file_name(sourcefilename) << " " << short_file_name(sourcefilename) << "],\n " | |
732 | << BOOST_COMPILER << ", " | |
733 | << BOOST_STDLIB << ", " | |
734 | << BOOST_PLATFORM << "\n" | |
735 | << debug_or_optimize | |
736 | << ((fp_hardware != "") ? ", " + fp_hardware : "") | |
737 | << "]" // [h6 close]. | |
738 | << std::endl; | |
739 | ||
740 | //fout << "Fraction of full accuracy " << digits_accuracy << std::endl; | |
741 | ||
742 | table_root_info(radius, arc); | |
743 | ||
744 | fout.close(); | |
745 | ||
746 | // table_type_info(digits_accuracy); | |
747 | ||
748 | return 0; | |
749 | } // roots_tables | |
750 | ||
751 | ||
752 | int main() | |
753 | { | |
754 | using namespace boost::multiprecision; | |
755 | using namespace boost::math; | |
756 | ||
757 | ||
758 | try | |
759 | { | |
760 | std::cout << "Tests run with " << BOOST_COMPILER << ", " | |
761 | << BOOST_STDLIB << ", " << BOOST_PLATFORM << ", "; | |
762 | ||
763 | // How to: Configure Visual C++ Projects to Target 64-Bit Platforms | |
764 | // https://msdn.microsoft.com/en-us/library/9yb4317s.aspx | |
765 | ||
766 | #ifdef _M_X64 // Defined for compilations that target x64 processors. | |
767 | std::cout << "X64 " << std::endl; | |
768 | fp_hardware += "_X64"; | |
769 | #else | |
770 | # ifdef _M_IX86 | |
771 | std::cout << "X32 " << std::endl; | |
772 | fp_hardware += "_X86"; | |
773 | # endif | |
774 | #endif | |
775 | ||
776 | #ifdef _M_AMD64 | |
777 | std::cout << "AMD64 " << std::endl; | |
778 | // fp_hardware += "_AMD64"; | |
779 | #endif | |
780 | ||
781 | // https://msdn.microsoft.com/en-us/library/7t5yh4fd.aspx | |
782 | // /arch (x86) options /arch:[IA32|SSE|SSE2|AVX|AVX2] | |
783 | // default is to use SSE and SSE2 instructions by default. | |
784 | // https://msdn.microsoft.com/en-us/library/jj620901.aspx | |
785 | // /arch (x64) options /arch:AVX and /arch:AVX2 | |
786 | ||
787 | // MSVC doesn't bother to set these SSE macros! | |
788 | // http://stackoverflow.com/questions/18563978/sse-sse2-is-enabled-control-in-visual-studio | |
789 | // https://msdn.microsoft.com/en-us/library/b0084kay.aspx predefined macros. | |
790 | ||
791 | // But some of these macros are *not* defined by MSVC, | |
792 | // unlike AVX (but *are* defined by GCC and Clang). | |
793 | // So the macro code above does define them. | |
794 | #if (defined(_M_AMD64) || defined (_M_X64)) | |
795 | # define _M_X64 | |
796 | # define __SSE2__ | |
797 | #else | |
798 | # ifdef _M_IX86_FP // Expands to an integer literal value indicating which /arch compiler option was used: | |
799 | std::cout << "Floating-point _M_IX86_FP = " << _M_IX86_FP << std::endl; | |
800 | # if (_M_IX86_FP == 2) // 2 if /arch:SSE2, /arch:AVX or /arch:AVX2 | |
801 | # define __SSE2__ // x32 | |
802 | # elif (_M_IX86_FP == 1) // 1 if /arch:SSE was used. | |
803 | # define __SSE__ // x32 | |
804 | # elif (_M_IX86_FP == 0) // 0 if /arch:IA32 was used. | |
805 | # define _X32 // No special FP instructions. | |
806 | # endif | |
807 | # endif | |
808 | #endif | |
809 | // Set the fp_hardware that is used in the .qbk filename. | |
810 | #ifdef __AVX2__ | |
811 | std::cout << "Floating-point AVX2 " << std::endl; | |
812 | fp_hardware += "_AVX2"; | |
813 | # else | |
814 | # ifdef __AVX__ | |
815 | std::cout << "Floating-point AVX " << std::endl; | |
816 | fp_hardware += "_AVX"; | |
817 | # else | |
818 | # ifdef __SSE2__ | |
819 | std::cout << "Floating-point SSE2 " << std::endl; | |
820 | fp_hardware += "_SSE2"; | |
821 | # else | |
822 | # ifdef __SSE__ | |
823 | std::cout << "Floating-point SSE " << std::endl; | |
824 | fp_hardware += "_SSE"; | |
825 | # endif | |
826 | # endif | |
827 | # endif | |
828 | # endif | |
829 | ||
830 | #ifdef _M_IX86 | |
831 | std::cout << "Floating-point X86 _M_IX86 = " << _M_IX86 << std::endl; | |
832 | // https://msdn.microsoft.com/en-us/library/aa273918%28v=vs.60%29.aspx#_predir_table_1..3 | |
833 | // 600 = Pentium Pro | |
834 | #endif | |
835 | ||
836 | #ifdef _MSC_FULL_VER | |
837 | std::cout << "Floating-point _MSC_FULL_VER " << _MSC_FULL_VER << std::endl; | |
838 | #endif | |
839 | ||
840 | #ifdef __MSVC_RUNTIME_CHECKS | |
841 | std::cout << "Runtime __MSVC_RUNTIME_CHECKS " << std::endl; | |
842 | #endif | |
843 | ||
844 | BOOST_MATH_CONTROL_FP; | |
845 | ||
846 | cpp_bin_float_100 radius("28."); | |
847 | cpp_bin_float_100 arc("300."); | |
848 | // Compute full answer to more than precision of tests. | |
849 | //T value = 28.; // integer (exactly representable as floating-point) | |
850 | // whose cube root is *not* exactly representable. | |
851 | // Wolfram Alpha command N[28 ^ (1 / 3), 100] computes cube root to 100 decimal digits. | |
852 | // 3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895 | |
853 | ||
854 | std::cout.precision(100); | |
855 | std::cout << "radius 1" << radius << std::endl; | |
856 | std::cout << "arc length" << arc << std::endl; | |
857 | // std::cout << ",\n""answer = " << full_answer << std::endl; | |
858 | std::cout.precision(6); | |
859 | // cbrt cpp_bin_float_100 full_answer("3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895"); | |
860 | ||
861 | // Output the table of types, maxdigits10 and digits and required digits for some accuracies. | |
862 | ||
863 | // Output tables for some roots at full accuracy. | |
864 | roots_tables(radius, arc, 1.); | |
865 | ||
866 | // Output tables for some roots at less accuracy. | |
867 | //roots_tables(full_value, 0.75); | |
868 | ||
869 | return boost::exit_success; | |
870 | } | |
871 | catch (std::exception ex) | |
872 | { | |
873 | std::cout << "exception thrown: " << ex.what() << std::endl; | |
874 | return boost::exit_failure; | |
875 | } | |
876 | } // int main() | |
877 | ||
878 | /* | |
879 | ||
880 | */ |