1
2 // normal_misc_examples.cpp
3
4 // Copyright Paul A. Bristow 2007, 2010, 2014, 2016.
5
6 // Use, modification and distribution are subject to the
7 // Boost Software License, Version 1.0.
8 // (See accompanying file LICENSE_1_0.txt
9 // or copy at http://www.boost.org/LICENSE_1_0.txt)
10
11 // Example of using normal distribution.
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
16 /*`
17 First we need some includes to access the normal distribution
18 (and some std output of course).
19 */
20
21 #include <boost/cstdfloat.hpp> // MUST be first include!!!
22 // See Implementation of Float128 type, Overloading template functions with float128_t.
23
24 #include <boost/math/distributions/normal.hpp> // for normal_distribution.
25 using boost::math::normal; // typedef provides default type of double.
26
27 #include <iostream>
28 //using std::cout; using std::endl;
29 //using std::left; using std::showpoint; using std::noshowpoint;
30 #include <iomanip>
31 //using std::setw; using std::setprecision;
32 #include <limits>
33 //using std::numeric_limits;
34
35 /*!
36 Function max_digits10
37 Returns maximum number of possibly significant decimal digits for a floating-point type FPT,
38 even for older compilers/standard libraries that
39 lack support for std::std::numeric_limits<FPT>::max_digits10,
40 when the Kahan formula 2 + binary_digits * 0.3010 is used instead.
41 Also provides the correct result for Visual Studio 2010 where the max_digits10 provided for float is wrong.
42 */
43 namespace boost
44 {
45 namespace math
46 {
47 template <typename FPT>
max_digits10()48 int max_digits10()
49 {
50 // Since max_digits10 is not defined (or wrong) on older systems, define a local max_digits10.
51
52 // Usage: int m = max_digits10<boost::float64_t>();
53 const int m =
54 #if (defined BOOST_NO_CXX11_NUMERIC_LIMITS) || (_MSC_VER == 1600) // is wrongly 8 not 9 for VS2010.
55 2 + std::numeric_limits<FPT>::digits * 3010/10000;
56 #else
57 std::numeric_limits<FPT>::max_digits10;
58 #endif
59 return m;
60 }
61 } // namespace math
62 } // namespace boost
63
64 template <typename FPT>
normal_table()65 void normal_table()
66 {
67 using namespace boost::math;
68
69 FPT step = static_cast<FPT>(1.); // step in z.
70 FPT range = static_cast<FPT>(10.); // min and max z = -range to +range.
71
72 // Traditional tables are only computed to much lower precision.
73 // but @c std::std::numeric_limits<double>::max_digits10;
74 // on new Standard Libraries gives 17,
75 // the maximum number of digits from 64-bit double that can possibly be significant.
76 // @c std::std::numeric_limits<double>::digits10; == 15
77 // is number of @b guaranteed digits, the other two digits being 'noisy'.
78 // Here we use a custom version of max_digits10 which deals with those platforms
79 // where @c std::numeric_limits is not specialized,
80 // or @c std::numeric_limits<>::max_digits10 not implemented, or wrong.
81 int precision = boost::math::max_digits10<FPT>();
82
83 // std::cout << typeid(FPT).name() << std::endl;
84 // demo_normal.cpp:85: undefined reference to `typeinfo for __float128'
85 // [@http://gcc.gnu.org/bugzilla/show_bug.cgi?id=43622 GCC 43622]
86 // typeinfo for __float128 was missing GCC 4.9 Mar 2014, but OK for GCC 6.1.1.
87
88 // Construct a standard normal distribution s, with
89 // (default mean = zero, and standard deviation = unity)
90 normal s;
91 std::cout << "\nStandard normal distribution, mean = "<< s.mean()
92 << ", standard deviation = " << s.standard_deviation() << std::endl;
93
94 std::cout << "maxdigits_10 is " << precision
95 << ", digits10 is " << std::numeric_limits<FPT>::digits10 << std::endl;
96
97 std::cout << "Probability distribution function values" << std::endl;
98
99 std::cout << " z " " PDF " << std::endl;
100 for (FPT z = -range; z < range + step; z += step)
101 {
102 std::cout << std::left << std::setprecision(3) << std::setw(6) << z << " "
103 << std::setprecision(precision) << std::setw(12) << pdf(s, z) << std::endl;
104 }
105 std::cout.precision(6); // Restore to default precision.
106
107 /*`And the area under the normal curve from -[infin] up to z,
108 the cumulative distribution function (CDF).
109 */
110 // For a standard normal distribution:
111 std::cout << "Standard normal mean = "<< s.mean()
112 << ", standard deviation = " << s.standard_deviation() << std::endl;
113 std::cout << "Integral (area under the curve) from - infinity up to z." << std::endl;
114 std::cout << " z " " CDF " << std::endl;
115 for (FPT z = -range; z < range + step; z += step)
116 {
117 std::cout << std::left << std::setprecision(3) << std::setw(6) << z << " "
118 << std::setprecision(precision) << std::setw(12) << cdf(s, z) << std::endl;
119 }
120 std::cout.precision(6); // Reset to default precision.
121 } // template <typename FPT> void normal_table()
122
main()123 int main()
124 {
125 std::cout << "\nExample: Normal distribution tables." << std::endl;
126
127 using namespace boost::math;
128
129 try
130 {// Tip - always use try'n'catch blocks to ensure that messages from thrown exceptions are shown.
131
132 //[normal_table_1
133 #ifdef BOOST_FLOAT32_C
134 normal_table<boost::float32_t>(); // Usually type float
135 #endif
136 normal_table<boost::float64_t>(); // Usually type double. Assume that float64_t is always available.
137 #ifdef BOOST_FLOAT80_C
138 normal_table<boost::float80_t>(); // Type long double on some X86 platforms.
139 #endif
140 #ifdef BOOST_FLOAT128_C
141 normal_table<boost::float128_t>(); // Type _Quad on some Intel and __float128 on some GCC platforms.
142 #endif
143 normal_table<boost::floatmax_t>();
144 //] [/normal_table_1 ]
145 }
146 catch(std::exception ex)
147 {
148 std::cout << "exception thrown " << ex.what() << std::endl;
149 }
150
151 return 0;
152 } // int main()
153
154
155 /*
156
157 GCC 4.8.1 with quadmath
158
159 Example: Normal distribution tables.
160
161 Standard normal distribution, mean = 0, standard deviation = 1
162 maxdigits_10 is 9, digits10 is 6
163 Probability distribution function values
164 z PDF
165 -10 7.69459863e-023
166 -9 1.02797736e-018
167 -8 5.05227108e-015
168 -7 9.13472041e-012
169 -6 6.07588285e-009
170 -5 1.48671951e-006
171 -4 0.000133830226
172 -3 0.00443184841
173 -2 0.0539909665
174 -1 0.241970725
175 0 0.39894228
176 1 0.241970725
177 2 0.0539909665
178 3 0.00443184841
179 4 0.000133830226
180 5 1.48671951e-006
181 6 6.07588285e-009
182 7 9.13472041e-012
183 8 5.05227108e-015
184 9 1.02797736e-018
185 10 7.69459863e-023
186 Standard normal mean = 0, standard deviation = 1
187 Integral (area under the curve) from - infinity up to z.
188 z CDF
189 -10 7.61985302e-024
190 -9 1.12858841e-019
191 -8 6.22096057e-016
192 -7 1.27981254e-012
193 -6 9.86587645e-010
194 -5 2.86651572e-007
195 -4 3.16712418e-005
196 -3 0.00134989803
197 -2 0.0227501319
198 -1 0.158655254
199 0 0.5
200 1 0.841344746
201 2 0.977249868
202 3 0.998650102
203 4 0.999968329
204 5 0.999999713
205 6 0.999999999
206 7 1
207 8 1
208 9 1
209 10 1
210
211 Standard normal distribution, mean = 0, standard deviation = 1
212 maxdigits_10 is 17, digits10 is 15
213 Probability distribution function values
214 z PDF
215 -10 7.6945986267064199e-023
216 -9 1.0279773571668917e-018
217 -8 5.0522710835368927e-015
218 -7 9.1347204083645953e-012
219 -6 6.0758828498232861e-009
220 -5 1.4867195147342979e-006
221 -4 0.00013383022576488537
222 -3 0.0044318484119380075
223 -2 0.053990966513188063
224 -1 0.24197072451914337
225 0 0.3989422804014327
226 1 0.24197072451914337
227 2 0.053990966513188063
228 3 0.0044318484119380075
229 4 0.00013383022576488537
230 5 1.4867195147342979e-006
231 6 6.0758828498232861e-009
232 7 9.1347204083645953e-012
233 8 5.0522710835368927e-015
234 9 1.0279773571668917e-018
235 10 7.6945986267064199e-023
236 Standard normal mean = 0, standard deviation = 1
237 Integral (area under the curve) from - infinity up to z.
238 z CDF
239 -10 7.6198530241605945e-024
240 -9 1.1285884059538422e-019
241 -8 6.2209605742718204e-016
242 -7 1.279812543885835e-012
243 -6 9.865876450377014e-010
244 -5 2.8665157187919455e-007
245 -4 3.1671241833119972e-005
246 -3 0.0013498980316300957
247 -2 0.022750131948179216
248 -1 0.15865525393145705
249 0 0.5
250 1 0.84134474606854293
251 2 0.97724986805182079
252 3 0.9986501019683699
253 4 0.99996832875816688
254 5 0.99999971334842808
255 6 0.9999999990134123
256 7 0.99999999999872013
257 8 0.99999999999999933
258 9 1
259 10 1
260
261 Standard normal distribution, mean = 0, standard deviation = 1
262 maxdigits_10 is 21, digits10 is 18
263 Probability distribution function values
264 z PDF
265 -10 7.69459862670641993759e-023
266 -9 1.0279773571668916523e-018
267 -8 5.05227108353689273243e-015
268 -7 9.13472040836459525705e-012
269 -6 6.07588284982328608733e-009
270 -5 1.48671951473429788965e-006
271 -4 0.00013383022576488536764
272 -3 0.00443184841193800752729
273 -2 0.0539909665131880628364
274 -1 0.241970724519143365328
275 0 0.398942280401432702863
276 1 0.241970724519143365328
277 2 0.0539909665131880628364
278 3 0.00443184841193800752729
279 4 0.00013383022576488536764
280 5 1.48671951473429788965e-006
281 6 6.07588284982328608733e-009
282 7 9.13472040836459525705e-012
283 8 5.05227108353689273243e-015
284 9 1.0279773571668916523e-018
285 10 7.69459862670641993759e-023
286 Standard normal mean = 0, standard deviation = 1
287 Integral (area under the curve) from - infinity up to z.
288 z CDF
289 -10 7.61985302416059451083e-024
290 -9 1.12858840595384222719e-019
291 -8 6.22096057427182035917e-016
292 -7 1.279812543885834962e-012
293 -6 9.86587645037701399241e-010
294 -5 2.86651571879194547129e-007
295 -4 3.16712418331199717608e-005
296 -3 0.00134989803163009566139
297 -2 0.0227501319481792155242
298 -1 0.158655253931457046468
299 0 0.5
300 1 0.841344746068542925777
301 2 0.977249868051820791415
302 3 0.998650101968369896532
303 4 0.999968328758166880021
304 5 0.999999713348428076465
305 6 0.999999999013412299576
306 7 0.999999999998720134897
307 8 0.999999999999999333866
308 9 1
309 10 1
310
311 Standard normal distribution, mean = 0, standard deviation = 1
312 maxdigits_10 is 36, digits10 is 34
313 Probability distribution function values
314 z PDF
315 -10 7.69459862670641993759264402330435296e-023
316 -9 1.02797735716689165230378750485667109e-018
317 -8 5.0522710835368927324337437844893081e-015
318 -7 9.13472040836459525705208369548147081e-012
319 -6 6.07588284982328608733411870229841611e-009
320 -5 1.48671951473429788965346931561839483e-006
321 -4 0.00013383022576488536764006964663309418
322 -3 0.00443184841193800752728870762098267733
323 -2 0.0539909665131880628363703067407186609
324 -1 0.241970724519143365327522587904240936
325 0 0.398942280401432702863218082711682655
326 1 0.241970724519143365327522587904240936
327 2 0.0539909665131880628363703067407186609
328 3 0.00443184841193800752728870762098267733
329 4 0.00013383022576488536764006964663309418
330 5 1.48671951473429788965346931561839483e-006
331 6 6.07588284982328608733411870229841611e-009
332 7 9.13472040836459525705208369548147081e-012
333 8 5.0522710835368927324337437844893081e-015
334 9 1.02797735716689165230378750485667109e-018
335 10 7.69459862670641993759264402330435296e-023
336 Standard normal mean = 0, standard deviation = 1
337 Integral (area under the curve) from - infinity up to z.
338 z CDF
339 -10 7.61985302416059451083278826816793623e-024
340 -9 1.1285884059538422271881384555435713e-019
341 -8 6.22096057427182035917417257601387863e-016
342 -7 1.27981254388583496200054074948511201e-012
343 -6 9.86587645037701399241244820583623953e-010
344 -5 2.86651571879194547128505464808623238e-007
345 -4 3.16712418331199717608064048146587766e-005
346 -3 0.001349898031630095661392854111682027
347 -2 0.0227501319481792155241528519127314212
348 -1 0.158655253931457046467912164189328905
349 0 0.5
350 1 0.841344746068542925776512220181757584
351 2 0.977249868051820791414741051994496956
352 3 0.998650101968369896532351503992686048
353 4 0.999968328758166880021462930017150939
354 5 0.999999713348428076464813329948810861
355 6 0.999999999013412299575520592043176293
356 7 0.999999999998720134897212119540199637
357 8 0.999999999999999333866185224906075746
358 9 1
359 10 1
360
361 Standard normal distribution, mean = 0, standard deviation = 1
362 maxdigits_10 is 36, digits10 is 34
363 Probability distribution function values
364 z PDF
365 -10 7.69459862670641993759264402330435296e-023
366 -9 1.02797735716689165230378750485667109e-018
367 -8 5.0522710835368927324337437844893081e-015
368 -7 9.13472040836459525705208369548147081e-012
369 -6 6.07588284982328608733411870229841611e-009
370 -5 1.48671951473429788965346931561839483e-006
371 -4 0.00013383022576488536764006964663309418
372 -3 0.00443184841193800752728870762098267733
373 -2 0.0539909665131880628363703067407186609
374 -1 0.241970724519143365327522587904240936
375 0 0.398942280401432702863218082711682655
376 1 0.241970724519143365327522587904240936
377 2 0.0539909665131880628363703067407186609
378 3 0.00443184841193800752728870762098267733
379 4 0.00013383022576488536764006964663309418
380 5 1.48671951473429788965346931561839483e-006
381 6 6.07588284982328608733411870229841611e-009
382 7 9.13472040836459525705208369548147081e-012
383 8 5.0522710835368927324337437844893081e-015
384 9 1.02797735716689165230378750485667109e-018
385 10 7.69459862670641993759264402330435296e-023
386 Standard normal mean = 0, standard deviation = 1
387 Integral (area under the curve) from - infinity up to z.
388 z CDF
389 -10 7.61985302416059451083278826816793623e-024
390 -9 1.1285884059538422271881384555435713e-019
391 -8 6.22096057427182035917417257601387863e-016
392 -7 1.27981254388583496200054074948511201e-012
393 -6 9.86587645037701399241244820583623953e-010
394 -5 2.86651571879194547128505464808623238e-007
395 -4 3.16712418331199717608064048146587766e-005
396 -3 0.001349898031630095661392854111682027
397 -2 0.0227501319481792155241528519127314212
398 -1 0.158655253931457046467912164189328905
399 0 0.5
400 1 0.841344746068542925776512220181757584
401 2 0.977249868051820791414741051994496956
402 3 0.998650101968369896532351503992686048
403 4 0.999968328758166880021462930017150939
404 5 0.999999713348428076464813329948810861
405 6 0.999999999013412299575520592043176293
406 7 0.999999999998720134897212119540199637
407 8 0.999999999999999333866185224906075746
408 9 1
409 10 1
410
411 MSVC 2013 64-bit
412 1>
413 1> Example: Normal distribution tables.
414 1>
415 1> Standard normal distribution, mean = 0, standard deviation = 1
416 1> maxdigits_10 is 9, digits10 is 6
417 1> Probability distribution function values
418 1> z PDF
419 1> -10 7.69459863e-023
420 1> -9 1.02797736e-018
421 1> -8 5.05227108e-015
422 1> -7 9.13472041e-012
423 1> -6 6.07588285e-009
424 1> -5 1.48671951e-006
425 1> -4 0.000133830226
426 1> -3 0.00443184841
427 1> -2 0.0539909665
428 1> -1 0.241970725
429 1> 0 0.39894228
430 1> 1 0.241970725
431 1> 2 0.0539909665
432 1> 3 0.00443184841
433 1> 4 0.000133830226
434 1> 5 1.48671951e-006
435 1> 6 6.07588285e-009
436 1> 7 9.13472041e-012
437 1> 8 5.05227108e-015
438 1> 9 1.02797736e-018
439 1> 10 7.69459863e-023
440 1> Standard normal mean = 0, standard deviation = 1
441 1> Integral (area under the curve) from - infinity up to z.
442 1> z CDF
443 1> -10 7.61985302e-024
444 1> -9 1.12858841e-019
445 1> -8 6.22096057e-016
446 1> -7 1.27981254e-012
447 1> -6 9.86587645e-010
448 1> -5 2.86651572e-007
449 1> -4 3.16712418e-005
450 1> -3 0.00134989803
451 1> -2 0.0227501319
452 1> -1 0.158655254
453 1> 0 0.5
454 1> 1 0.841344746
455 1> 2 0.977249868
456 1> 3 0.998650102
457 1> 4 0.999968329
458 1> 5 0.999999713
459 1> 6 0.999999999
460 1> 7 1
461 1> 8 1
462 1> 9 1
463 1> 10 1
464 1>
465 1> Standard normal distribution, mean = 0, standard deviation = 1
466 1> maxdigits_10 is 17, digits10 is 15
467 1> Probability distribution function values
468 1> z PDF
469 1> -10 7.6945986267064199e-023
470 1> -9 1.0279773571668917e-018
471 1> -8 5.0522710835368927e-015
472 1> -7 9.1347204083645953e-012
473 1> -6 6.0758828498232861e-009
474 1> -5 1.4867195147342979e-006
475 1> -4 0.00013383022576488537
476 1> -3 0.0044318484119380075
477 1> -2 0.053990966513188063
478 1> -1 0.24197072451914337
479 1> 0 0.3989422804014327
480 1> 1 0.24197072451914337
481 1> 2 0.053990966513188063
482 1> 3 0.0044318484119380075
483 1> 4 0.00013383022576488537
484 1> 5 1.4867195147342979e-006
485 1> 6 6.0758828498232861e-009
486 1> 7 9.1347204083645953e-012
487 1> 8 5.0522710835368927e-015
488 1> 9 1.0279773571668917e-018
489 1> 10 7.6945986267064199e-023
490 1> Standard normal mean = 0, standard deviation = 1
491 1> Integral (area under the curve) from - infinity up to z.
492 1> z CDF
493 1> -10 7.6198530241605813e-024
494 1> -9 1.1285884059538408e-019
495 1> -8 6.2209605742718292e-016
496 1> -7 1.2798125438858352e-012
497 1> -6 9.8658764503770161e-010
498 1> -5 2.8665157187919439e-007
499 1> -4 3.1671241833119979e-005
500 1> -3 0.0013498980316300957
501 1> -2 0.022750131948179219
502 1> -1 0.15865525393145707
503 1> 0 0.5
504 1> 1 0.84134474606854293
505 1> 2 0.97724986805182079
506 1> 3 0.9986501019683699
507 1> 4 0.99996832875816688
508 1> 5 0.99999971334842808
509 1> 6 0.9999999990134123
510 1> 7 0.99999999999872013
511 1> 8 0.99999999999999933
512 1> 9 1
513 1> 10 1
514 1>
515 1> Standard normal distribution, mean = 0, standard deviation = 1
516 1> maxdigits_10 is 17, digits10 is 15
517 1> Probability distribution function values
518 1> z PDF
519 1> -10 7.6945986267064199e-023
520 1> -9 1.0279773571668917e-018
521 1> -8 5.0522710835368927e-015
522 1> -7 9.1347204083645953e-012
523 1> -6 6.0758828498232861e-009
524 1> -5 1.4867195147342979e-006
525 1> -4 0.00013383022576488537
526 1> -3 0.0044318484119380075
527 1> -2 0.053990966513188063
528 1> -1 0.24197072451914337
529 1> 0 0.3989422804014327
530 1> 1 0.24197072451914337
531 1> 2 0.053990966513188063
532 1> 3 0.0044318484119380075
533 1> 4 0.00013383022576488537
534 1> 5 1.4867195147342979e-006
535 1> 6 6.0758828498232861e-009
536 1> 7 9.1347204083645953e-012
537 1> 8 5.0522710835368927e-015
538 1> 9 1.0279773571668917e-018
539 1> 10 7.6945986267064199e-023
540 1> Standard normal mean = 0, standard deviation = 1
541 1> Integral (area under the curve) from - infinity up to z.
542 1> z CDF
543 1> -10 7.6198530241605813e-024
544 1> -9 1.1285884059538408e-019
545 1> -8 6.2209605742718292e-016
546 1> -7 1.2798125438858352e-012
547 1> -6 9.8658764503770161e-010
548 1> -5 2.8665157187919439e-007
549 1> -4 3.1671241833119979e-005
550 1> -3 0.0013498980316300957
551 1> -2 0.022750131948179219
552 1> -1 0.15865525393145707
553 1> 0 0.5
554 1> 1 0.84134474606854293
555 1> 2 0.97724986805182079
556 1> 3 0.9986501019683699
557 1> 4 0.99996832875816688
558 1> 5 0.99999971334842808
559 1> 6 0.9999999990134123
560 1> 7 0.99999999999872013
561 1> 8 0.99999999999999933
562 1> 9 1
563 1> 10 1
564
565
566 */
567