1 // Copyright Matthew Pulver 2018 - 2019.
2 // Distributed under the Boost Software License, Version 1.0.
3 // (See accompanying file LICENSE_1_0.txt or copy at
4 // https://www.boost.org/LICENSE_1_0.txt)
5
6 #ifndef BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP
7 #define BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP
8
9 #include <boost/cstdfloat.hpp>
10 #include <boost/math/constants/constants.hpp>
11 #include <boost/math/special_functions/trunc.hpp>
12 #include <boost/math/special_functions/round.hpp>
13 #include <boost/math/special_functions/acosh.hpp>
14 #include <boost/math/special_functions/asinh.hpp>
15 #include <boost/math/special_functions/atanh.hpp>
16 #include <boost/math/special_functions/digamma.hpp>
17 #include <boost/math/special_functions/polygamma.hpp>
18 #include <boost/math/special_functions/erf.hpp>
19 #include <boost/math/special_functions/lambert_w.hpp>
20 #include <boost/math/tools/config.hpp>
21 #include <boost/math/tools/promotion.hpp>
22
23 #include <algorithm>
24 #include <array>
25 #include <cmath>
26 #include <functional>
27 #include <limits>
28 #include <numeric>
29 #include <ostream>
30 #include <tuple>
31 #include <type_traits>
32
33 namespace boost {
34 namespace math {
35 namespace differentiation {
36 // Automatic Differentiation v1
37 inline namespace autodiff_v1 {
38 namespace detail {
39
40 template <typename RealType, typename... RealTypes>
41 struct promote_args_n {
42 using type = typename tools::promote_args_2<RealType, typename promote_args_n<RealTypes...>::type>::type;
43 };
44
45 template <typename RealType>
46 struct promote_args_n<RealType> {
47 using type = typename tools::promote_arg<RealType>::type;
48 };
49
50 } // namespace detail
51
52 template <typename RealType, typename... RealTypes>
53 using promote = typename detail::promote_args_n<RealType, RealTypes...>::type;
54
55 namespace detail {
56
57 template <typename RealType, size_t Order>
58 class fvar;
59
60 template <typename T>
61 struct is_fvar_impl : std::false_type {};
62
63 template <typename RealType, size_t Order>
64 struct is_fvar_impl<fvar<RealType, Order>> : std::true_type {};
65
66 template <typename T>
67 using is_fvar = is_fvar_impl<typename std::decay<T>::type>;
68
69 template <typename RealType, size_t Order, size_t... Orders>
70 struct nest_fvar {
71 using type = fvar<typename nest_fvar<RealType, Orders...>::type, Order>;
72 };
73
74 template <typename RealType, size_t Order>
75 struct nest_fvar<RealType, Order> {
76 using type = fvar<RealType, Order>;
77 };
78
79 template <typename>
80 struct get_depth_impl : std::integral_constant<size_t, 0> {};
81
82 template <typename RealType, size_t Order>
83 struct get_depth_impl<fvar<RealType, Order>>
84 : std::integral_constant<size_t, get_depth_impl<RealType>::value + 1> {};
85
86 template <typename T>
87 using get_depth = get_depth_impl<typename std::decay<T>::type>;
88
89 template <typename>
90 struct get_order_sum_t : std::integral_constant<size_t, 0> {};
91
92 template <typename RealType, size_t Order>
93 struct get_order_sum_t<fvar<RealType, Order>>
94 : std::integral_constant<size_t, get_order_sum_t<RealType>::value + Order> {};
95
96 template <typename T>
97 using get_order_sum = get_order_sum_t<typename std::decay<T>::type>;
98
99 template <typename RealType>
100 struct get_root_type {
101 using type = RealType;
102 };
103
104 template <typename RealType, size_t Order>
105 struct get_root_type<fvar<RealType, Order>> {
106 using type = typename get_root_type<RealType>::type;
107 };
108
109 template <typename RealType, size_t Depth>
110 struct type_at {
111 using type = RealType;
112 };
113
114 template <typename RealType, size_t Order, size_t Depth>
115 struct type_at<fvar<RealType, Order>, Depth> {
116 using type = typename conditional<Depth == 0,
117 fvar<RealType, Order>,
118 typename type_at<RealType, Depth - 1>::type>::type;
119 };
120
121 template <typename RealType, size_t Depth>
122 using get_type_at = typename type_at<RealType, Depth>::type;
123
124 // Satisfies Boost's Conceptual Requirements for Real Number Types.
125 // https://www.boost.org/libs/math/doc/html/math_toolkit/real_concepts.html
126 template <typename RealType, size_t Order>
127 class fvar {
128 std::array<RealType, Order + 1> v;
129
130 public:
131 using root_type = typename get_root_type<RealType>::type; // RealType in the root fvar<RealType,Order>.
132
133 fvar() = default;
134
135 // Initialize a variable or constant.
136 fvar(root_type const&, bool const is_variable);
137
138 // RealType(cr) | RealType | RealType is copy constructible.
139 fvar(fvar const&) = default;
140
141 // Be aware of implicit casting from one fvar<> type to another by this copy constructor.
142 template <typename RealType2, size_t Order2>
143 fvar(fvar<RealType2, Order2> const&);
144
145 // RealType(ca) | RealType | RealType is copy constructible from the arithmetic types.
146 explicit fvar(root_type const&); // Initialize a constant. (No epsilon terms.)
147
148 template <typename RealType2>
149 fvar(RealType2 const& ca); // Supports any RealType2 for which static_cast<root_type>(ca) compiles.
150
151 // r = cr | RealType& | Assignment operator.
152 fvar& operator=(fvar const&) = default;
153
154 // r = ca | RealType& | Assignment operator from the arithmetic types.
155 // Handled by constructor that takes a single parameter of generic type.
156 // fvar& operator=(root_type const&); // Set a constant.
157
158 // r += cr | RealType& | Adds cr to r.
159 template <typename RealType2, size_t Order2>
160 fvar& operator+=(fvar<RealType2, Order2> const&);
161
162 // r += ca | RealType& | Adds ar to r.
163 fvar& operator+=(root_type const&);
164
165 // r -= cr | RealType& | Subtracts cr from r.
166 template <typename RealType2, size_t Order2>
167 fvar& operator-=(fvar<RealType2, Order2> const&);
168
169 // r -= ca | RealType& | Subtracts ca from r.
170 fvar& operator-=(root_type const&);
171
172 // r *= cr | RealType& | Multiplies r by cr.
173 template <typename RealType2, size_t Order2>
174 fvar& operator*=(fvar<RealType2, Order2> const&);
175
176 // r *= ca | RealType& | Multiplies r by ca.
177 fvar& operator*=(root_type const&);
178
179 // r /= cr | RealType& | Divides r by cr.
180 template <typename RealType2, size_t Order2>
181 fvar& operator/=(fvar<RealType2, Order2> const&);
182
183 // r /= ca | RealType& | Divides r by ca.
184 fvar& operator/=(root_type const&);
185
186 // -r | RealType | Unary Negation.
187 fvar operator-() const;
188
189 // +r | RealType& | Identity Operation.
190 fvar const& operator+() const;
191
192 // cr + cr2 | RealType | Binary Addition
193 template <typename RealType2, size_t Order2>
194 promote<fvar, fvar<RealType2, Order2>> operator+(fvar<RealType2, Order2> const&) const;
195
196 // cr + ca | RealType | Binary Addition
197 fvar operator+(root_type const&) const;
198
199 // ca + cr | RealType | Binary Addition
200 template <typename RealType2, size_t Order2>
201 friend fvar<RealType2, Order2> operator+(typename fvar<RealType2, Order2>::root_type const&,
202 fvar<RealType2, Order2> const&);
203
204 // cr - cr2 | RealType | Binary Subtraction
205 template <typename RealType2, size_t Order2>
206 promote<fvar, fvar<RealType2, Order2>> operator-(fvar<RealType2, Order2> const&) const;
207
208 // cr - ca | RealType | Binary Subtraction
209 fvar operator-(root_type const&) const;
210
211 // ca - cr | RealType | Binary Subtraction
212 template <typename RealType2, size_t Order2>
213 friend fvar<RealType2, Order2> operator-(typename fvar<RealType2, Order2>::root_type const&,
214 fvar<RealType2, Order2> const&);
215
216 // cr * cr2 | RealType | Binary Multiplication
217 template <typename RealType2, size_t Order2>
218 promote<fvar, fvar<RealType2, Order2>> operator*(fvar<RealType2, Order2> const&)const;
219
220 // cr * ca | RealType | Binary Multiplication
221 fvar operator*(root_type const&)const;
222
223 // ca * cr | RealType | Binary Multiplication
224 template <typename RealType2, size_t Order2>
225 friend fvar<RealType2, Order2> operator*(typename fvar<RealType2, Order2>::root_type const&,
226 fvar<RealType2, Order2> const&);
227
228 // cr / cr2 | RealType | Binary Subtraction
229 template <typename RealType2, size_t Order2>
230 promote<fvar, fvar<RealType2, Order2>> operator/(fvar<RealType2, Order2> const&) const;
231
232 // cr / ca | RealType | Binary Subtraction
233 fvar operator/(root_type const&) const;
234
235 // ca / cr | RealType | Binary Subtraction
236 template <typename RealType2, size_t Order2>
237 friend fvar<RealType2, Order2> operator/(typename fvar<RealType2, Order2>::root_type const&,
238 fvar<RealType2, Order2> const&);
239
240 // For all comparison overloads, only the root term is compared.
241
242 // cr == cr2 | bool | Equality Comparison
243 template <typename RealType2, size_t Order2>
244 bool operator==(fvar<RealType2, Order2> const&) const;
245
246 // cr == ca | bool | Equality Comparison
247 bool operator==(root_type const&) const;
248
249 // ca == cr | bool | Equality Comparison
250 template <typename RealType2, size_t Order2>
251 friend bool operator==(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
252
253 // cr != cr2 | bool | Inequality Comparison
254 template <typename RealType2, size_t Order2>
255 bool operator!=(fvar<RealType2, Order2> const&) const;
256
257 // cr != ca | bool | Inequality Comparison
258 bool operator!=(root_type const&) const;
259
260 // ca != cr | bool | Inequality Comparison
261 template <typename RealType2, size_t Order2>
262 friend bool operator!=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
263
264 // cr <= cr2 | bool | Less than equal to.
265 template <typename RealType2, size_t Order2>
266 bool operator<=(fvar<RealType2, Order2> const&) const;
267
268 // cr <= ca | bool | Less than equal to.
269 bool operator<=(root_type const&) const;
270
271 // ca <= cr | bool | Less than equal to.
272 template <typename RealType2, size_t Order2>
273 friend bool operator<=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
274
275 // cr >= cr2 | bool | Greater than equal to.
276 template <typename RealType2, size_t Order2>
277 bool operator>=(fvar<RealType2, Order2> const&) const;
278
279 // cr >= ca | bool | Greater than equal to.
280 bool operator>=(root_type const&) const;
281
282 // ca >= cr | bool | Greater than equal to.
283 template <typename RealType2, size_t Order2>
284 friend bool operator>=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
285
286 // cr < cr2 | bool | Less than comparison.
287 template <typename RealType2, size_t Order2>
288 bool operator<(fvar<RealType2, Order2> const&) const;
289
290 // cr < ca | bool | Less than comparison.
291 bool operator<(root_type const&) const;
292
293 // ca < cr | bool | Less than comparison.
294 template <typename RealType2, size_t Order2>
295 friend bool operator<(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
296
297 // cr > cr2 | bool | Greater than comparison.
298 template <typename RealType2, size_t Order2>
299 bool operator>(fvar<RealType2, Order2> const&) const;
300
301 // cr > ca | bool | Greater than comparison.
302 bool operator>(root_type const&) const;
303
304 // ca > cr | bool | Greater than comparison.
305 template <typename RealType2, size_t Order2>
306 friend bool operator>(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&);
307
308 // Will throw std::out_of_range if Order < order.
309 template <typename... Orders>
310 get_type_at<RealType, sizeof...(Orders)> at(size_t order, Orders... orders) const;
311
312 template <typename... Orders>
313 get_type_at<fvar, sizeof...(Orders)> derivative(Orders... orders) const;
314
315 const RealType& operator[](size_t) const;
316
317 fvar inverse() const; // Multiplicative inverse.
318
319 fvar& negate(); // Negate and return reference to *this.
320
321 static constexpr size_t depth = get_depth<fvar>::value; // Number of nested std::array<RealType,Order>.
322
323 static constexpr size_t order_sum = get_order_sum<fvar>::value;
324
325 explicit operator root_type() const; // Must be explicit, otherwise overloaded operators are ambiguous.
326
327 template <typename T, typename = typename std::enable_if<std::is_arithmetic<typename std::decay<T>::type>::value>>
328 explicit operator T() const; // Must be explicit; multiprecision has trouble without the std::enable_if
329
330 fvar& set_root(root_type const&);
331
332 // Apply coefficients using horner method.
333 template <typename Func, typename Fvar, typename... Fvars>
334 promote<fvar<RealType, Order>, Fvar, Fvars...> apply_coefficients(size_t const order,
335 Func const& f,
336 Fvar const& cr,
337 Fvars&&... fvars) const;
338
339 template <typename Func>
340 fvar apply_coefficients(size_t const order, Func const& f) const;
341
342 // Use when function returns derivative(i)/factorial(i) and may have some infinite derivatives.
343 template <typename Func, typename Fvar, typename... Fvars>
344 promote<fvar<RealType, Order>, Fvar, Fvars...> apply_coefficients_nonhorner(size_t const order,
345 Func const& f,
346 Fvar const& cr,
347 Fvars&&... fvars) const;
348
349 template <typename Func>
350 fvar apply_coefficients_nonhorner(size_t const order, Func const& f) const;
351
352 // Apply derivatives using horner method.
353 template <typename Func, typename Fvar, typename... Fvars>
354 promote<fvar<RealType, Order>, Fvar, Fvars...> apply_derivatives(size_t const order,
355 Func const& f,
356 Fvar const& cr,
357 Fvars&&... fvars) const;
358
359 template <typename Func>
360 fvar apply_derivatives(size_t const order, Func const& f) const;
361
362 // Use when function returns derivative(i) and may have some infinite derivatives.
363 template <typename Func, typename Fvar, typename... Fvars>
364 promote<fvar<RealType, Order>, Fvar, Fvars...> apply_derivatives_nonhorner(size_t const order,
365 Func const& f,
366 Fvar const& cr,
367 Fvars&&... fvars) const;
368
369 template <typename Func>
370 fvar apply_derivatives_nonhorner(size_t const order, Func const& f) const;
371
372 private:
373 RealType epsilon_inner_product(size_t z0,
374 size_t isum0,
375 size_t m0,
376 fvar const& cr,
377 size_t z1,
378 size_t isum1,
379 size_t m1,
380 size_t j) const;
381
382 fvar epsilon_multiply(size_t z0, size_t isum0, fvar const& cr, size_t z1, size_t isum1) const;
383
384 fvar epsilon_multiply(size_t z0, size_t isum0, root_type const& ca) const;
385
386 fvar inverse_apply() const;
387
388 fvar& multiply_assign_by_root_type(bool is_root, root_type const&);
389
390 template <typename RealType2, size_t Orders2>
391 friend class fvar;
392
393 template <typename RealType2, size_t Order2>
394 friend std::ostream& operator<<(std::ostream&, fvar<RealType2, Order2> const&);
395
396 // C++11 Compatibility
397 #ifdef BOOST_NO_CXX17_IF_CONSTEXPR
398 template <typename RootType>
399 void fvar_cpp11(std::true_type, RootType const& ca, bool const is_variable);
400
401 template <typename RootType>
402 void fvar_cpp11(std::false_type, RootType const& ca, bool const is_variable);
403
404 template <typename... Orders>
405 get_type_at<RealType, sizeof...(Orders)> at_cpp11(std::true_type, size_t order, Orders... orders) const;
406
407 template <typename... Orders>
408 get_type_at<RealType, sizeof...(Orders)> at_cpp11(std::false_type, size_t order, Orders... orders) const;
409
410 template <typename SizeType>
411 fvar epsilon_multiply_cpp11(std::true_type,
412 SizeType z0,
413 size_t isum0,
414 fvar const& cr,
415 size_t z1,
416 size_t isum1) const;
417
418 template <typename SizeType>
419 fvar epsilon_multiply_cpp11(std::false_type,
420 SizeType z0,
421 size_t isum0,
422 fvar const& cr,
423 size_t z1,
424 size_t isum1) const;
425
426 template <typename SizeType>
427 fvar epsilon_multiply_cpp11(std::true_type, SizeType z0, size_t isum0, root_type const& ca) const;
428
429 template <typename SizeType>
430 fvar epsilon_multiply_cpp11(std::false_type, SizeType z0, size_t isum0, root_type const& ca) const;
431
432 template <typename RootType>
433 fvar& multiply_assign_by_root_type_cpp11(std::true_type, bool is_root, RootType const& ca);
434
435 template <typename RootType>
436 fvar& multiply_assign_by_root_type_cpp11(std::false_type, bool is_root, RootType const& ca);
437
438 template <typename RootType>
439 fvar& negate_cpp11(std::true_type, RootType const&);
440
441 template <typename RootType>
442 fvar& negate_cpp11(std::false_type, RootType const&);
443
444 template <typename RootType>
445 fvar& set_root_cpp11(std::true_type, RootType const& root);
446
447 template <typename RootType>
448 fvar& set_root_cpp11(std::false_type, RootType const& root);
449 #endif
450 };
451
452 // Standard Library Support Requirements
453
454 // fabs(cr1) | RealType
455 template <typename RealType, size_t Order>
456 fvar<RealType, Order> fabs(fvar<RealType, Order> const&);
457
458 // abs(cr1) | RealType
459 template <typename RealType, size_t Order>
460 fvar<RealType, Order> abs(fvar<RealType, Order> const&);
461
462 // ceil(cr1) | RealType
463 template <typename RealType, size_t Order>
464 fvar<RealType, Order> ceil(fvar<RealType, Order> const&);
465
466 // floor(cr1) | RealType
467 template <typename RealType, size_t Order>
468 fvar<RealType, Order> floor(fvar<RealType, Order> const&);
469
470 // exp(cr1) | RealType
471 template <typename RealType, size_t Order>
472 fvar<RealType, Order> exp(fvar<RealType, Order> const&);
473
474 // pow(cr, ca) | RealType
475 template <typename RealType, size_t Order>
476 fvar<RealType, Order> pow(fvar<RealType, Order> const&, typename fvar<RealType, Order>::root_type const&);
477
478 // pow(ca, cr) | RealType
479 template <typename RealType, size_t Order>
480 fvar<RealType, Order> pow(typename fvar<RealType, Order>::root_type const&, fvar<RealType, Order> const&);
481
482 // pow(cr1, cr2) | RealType
483 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
484 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> pow(fvar<RealType1, Order1> const&,
485 fvar<RealType2, Order2> const&);
486
487 // sqrt(cr1) | RealType
488 template <typename RealType, size_t Order>
489 fvar<RealType, Order> sqrt(fvar<RealType, Order> const&);
490
491 // log(cr1) | RealType
492 template <typename RealType, size_t Order>
493 fvar<RealType, Order> log(fvar<RealType, Order> const&);
494
495 // frexp(cr1, &i) | RealType
496 template <typename RealType, size_t Order>
497 fvar<RealType, Order> frexp(fvar<RealType, Order> const&, int*);
498
499 // ldexp(cr1, i) | RealType
500 template <typename RealType, size_t Order>
501 fvar<RealType, Order> ldexp(fvar<RealType, Order> const&, int);
502
503 // cos(cr1) | RealType
504 template <typename RealType, size_t Order>
505 fvar<RealType, Order> cos(fvar<RealType, Order> const&);
506
507 // sin(cr1) | RealType
508 template <typename RealType, size_t Order>
509 fvar<RealType, Order> sin(fvar<RealType, Order> const&);
510
511 // asin(cr1) | RealType
512 template <typename RealType, size_t Order>
513 fvar<RealType, Order> asin(fvar<RealType, Order> const&);
514
515 // tan(cr1) | RealType
516 template <typename RealType, size_t Order>
517 fvar<RealType, Order> tan(fvar<RealType, Order> const&);
518
519 // atan(cr1) | RealType
520 template <typename RealType, size_t Order>
521 fvar<RealType, Order> atan(fvar<RealType, Order> const&);
522
523 // atan2(cr, ca) | RealType
524 template <typename RealType, size_t Order>
525 fvar<RealType, Order> atan2(fvar<RealType, Order> const&, typename fvar<RealType, Order>::root_type const&);
526
527 // atan2(ca, cr) | RealType
528 template <typename RealType, size_t Order>
529 fvar<RealType, Order> atan2(typename fvar<RealType, Order>::root_type const&, fvar<RealType, Order> const&);
530
531 // atan2(cr1, cr2) | RealType
532 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
533 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> atan2(fvar<RealType1, Order1> const&,
534 fvar<RealType2, Order2> const&);
535
536 // fmod(cr1,cr2) | RealType
537 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
538 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> fmod(fvar<RealType1, Order1> const&,
539 fvar<RealType2, Order2> const&);
540
541 // round(cr1) | RealType
542 template <typename RealType, size_t Order>
543 fvar<RealType, Order> round(fvar<RealType, Order> const&);
544
545 // iround(cr1) | int
546 template <typename RealType, size_t Order>
547 int iround(fvar<RealType, Order> const&);
548
549 template <typename RealType, size_t Order>
550 long lround(fvar<RealType, Order> const&);
551
552 template <typename RealType, size_t Order>
553 long long llround(fvar<RealType, Order> const&);
554
555 // trunc(cr1) | RealType
556 template <typename RealType, size_t Order>
557 fvar<RealType, Order> trunc(fvar<RealType, Order> const&);
558
559 template <typename RealType, size_t Order>
560 long double truncl(fvar<RealType, Order> const&);
561
562 // itrunc(cr1) | int
563 template <typename RealType, size_t Order>
564 int itrunc(fvar<RealType, Order> const&);
565
566 template <typename RealType, size_t Order>
567 long long lltrunc(fvar<RealType, Order> const&);
568
569 // Additional functions
570 template <typename RealType, size_t Order>
571 fvar<RealType, Order> acos(fvar<RealType, Order> const&);
572
573 template <typename RealType, size_t Order>
574 fvar<RealType, Order> acosh(fvar<RealType, Order> const&);
575
576 template <typename RealType, size_t Order>
577 fvar<RealType, Order> asinh(fvar<RealType, Order> const&);
578
579 template <typename RealType, size_t Order>
580 fvar<RealType, Order> atanh(fvar<RealType, Order> const&);
581
582 template <typename RealType, size_t Order>
583 fvar<RealType, Order> cosh(fvar<RealType, Order> const&);
584
585 template <typename RealType, size_t Order>
586 fvar<RealType, Order> digamma(fvar<RealType, Order> const&);
587
588 template <typename RealType, size_t Order>
589 fvar<RealType, Order> erf(fvar<RealType, Order> const&);
590
591 template <typename RealType, size_t Order>
592 fvar<RealType, Order> erfc(fvar<RealType, Order> const&);
593
594 template <typename RealType, size_t Order>
595 fvar<RealType, Order> lambert_w0(fvar<RealType, Order> const&);
596
597 template <typename RealType, size_t Order>
598 fvar<RealType, Order> lgamma(fvar<RealType, Order> const&);
599
600 template <typename RealType, size_t Order>
601 fvar<RealType, Order> sinc(fvar<RealType, Order> const&);
602
603 template <typename RealType, size_t Order>
604 fvar<RealType, Order> sinh(fvar<RealType, Order> const&);
605
606 template <typename RealType, size_t Order>
607 fvar<RealType, Order> tanh(fvar<RealType, Order> const&);
608
609 template <typename RealType, size_t Order>
610 fvar<RealType, Order> tgamma(fvar<RealType, Order> const&);
611
612 template <size_t>
613 struct zero : std::integral_constant<size_t, 0> {};
614
615 } // namespace detail
616
617 template <typename RealType, size_t Order, size_t... Orders>
618 using autodiff_fvar = typename detail::nest_fvar<RealType, Order, Orders...>::type;
619
620 template <typename RealType, size_t Order, size_t... Orders>
make_fvar(RealType const & ca)621 autodiff_fvar<RealType, Order, Orders...> make_fvar(RealType const& ca) {
622 return autodiff_fvar<RealType, Order, Orders...>(ca, true);
623 }
624
625 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
626 namespace detail {
627
628 template <typename RealType, size_t Order, size_t... Is>
make_fvar_for_tuple(std::index_sequence<Is...>,RealType const & ca)629 auto make_fvar_for_tuple(std::index_sequence<Is...>, RealType const& ca) {
630 return make_fvar<RealType, zero<Is>::value..., Order>(ca);
631 }
632
633 template <typename RealType, size_t... Orders, size_t... Is, typename... RealTypes>
make_ftuple_impl(std::index_sequence<Is...>,RealTypes const &...ca)634 auto make_ftuple_impl(std::index_sequence<Is...>, RealTypes const&... ca) {
635 return std::make_tuple(make_fvar_for_tuple<RealType, Orders>(std::make_index_sequence<Is>{}, ca)...);
636 }
637
638 } // namespace detail
639
640 template <typename RealType, size_t... Orders, typename... RealTypes>
make_ftuple(RealTypes const &...ca)641 auto make_ftuple(RealTypes const&... ca) {
642 static_assert(sizeof...(Orders) == sizeof...(RealTypes),
643 "Number of Orders must match number of function parameters.");
644 return detail::make_ftuple_impl<RealType, Orders...>(std::index_sequence_for<RealTypes...>{}, ca...);
645 }
646 #endif
647
648 namespace detail {
649
650 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
651 template <typename RealType, size_t Order>
fvar(root_type const & ca,bool const is_variable)652 fvar<RealType, Order>::fvar(root_type const& ca, bool const is_variable) {
653 if constexpr (is_fvar<RealType>::value) {
654 v.front() = RealType(ca, is_variable);
655 if constexpr (0 < Order)
656 std::fill(v.begin() + 1, v.end(), static_cast<RealType>(0));
657 } else {
658 v.front() = ca;
659 if constexpr (0 < Order)
660 v[1] = static_cast<root_type>(static_cast<int>(is_variable));
661 if constexpr (1 < Order)
662 std::fill(v.begin() + 2, v.end(), static_cast<RealType>(0));
663 }
664 }
665 #endif
666
667 template <typename RealType, size_t Order>
668 template <typename RealType2, size_t Order2>
fvar(fvar<RealType2,Order2> const & cr)669 fvar<RealType, Order>::fvar(fvar<RealType2, Order2> const& cr) {
670 for (size_t i = 0; i <= (std::min)(Order, Order2); ++i)
671 v[i] = static_cast<RealType>(cr.v[i]);
672 BOOST_IF_CONSTEXPR (Order2 < Order)
673 std::fill(v.begin() + (Order2 + 1), v.end(), static_cast<RealType>(0));
674 }
675
676 template <typename RealType, size_t Order>
fvar(root_type const & ca)677 fvar<RealType, Order>::fvar(root_type const& ca) : v{{static_cast<RealType>(ca)}} {}
678
679 // Can cause compiler error if RealType2 cannot be cast to root_type.
680 template <typename RealType, size_t Order>
681 template <typename RealType2>
fvar(RealType2 const & ca)682 fvar<RealType, Order>::fvar(RealType2 const& ca) : v{{static_cast<RealType>(ca)}} {}
683
684 /*
685 template<typename RealType, size_t Order>
686 fvar<RealType,Order>& fvar<RealType,Order>::operator=(root_type const& ca)
687 {
688 v.front() = static_cast<RealType>(ca);
689 if constexpr (0 < Order)
690 std::fill(v.begin()+1, v.end(), static_cast<RealType>(0));
691 return *this;
692 }
693 */
694
695 template <typename RealType, size_t Order>
696 template <typename RealType2, size_t Order2>
operator +=(fvar<RealType2,Order2> const & cr)697 fvar<RealType, Order>& fvar<RealType, Order>::operator+=(fvar<RealType2, Order2> const& cr) {
698 for (size_t i = 0; i <= (std::min)(Order, Order2); ++i)
699 v[i] += cr.v[i];
700 return *this;
701 }
702
703 template <typename RealType, size_t Order>
operator +=(root_type const & ca)704 fvar<RealType, Order>& fvar<RealType, Order>::operator+=(root_type const& ca) {
705 v.front() += ca;
706 return *this;
707 }
708
709 template <typename RealType, size_t Order>
710 template <typename RealType2, size_t Order2>
operator -=(fvar<RealType2,Order2> const & cr)711 fvar<RealType, Order>& fvar<RealType, Order>::operator-=(fvar<RealType2, Order2> const& cr) {
712 for (size_t i = 0; i <= Order; ++i)
713 v[i] -= cr.v[i];
714 return *this;
715 }
716
717 template <typename RealType, size_t Order>
operator -=(root_type const & ca)718 fvar<RealType, Order>& fvar<RealType, Order>::operator-=(root_type const& ca) {
719 v.front() -= ca;
720 return *this;
721 }
722
723 template <typename RealType, size_t Order>
724 template <typename RealType2, size_t Order2>
operator *=(fvar<RealType2,Order2> const & cr)725 fvar<RealType, Order>& fvar<RealType, Order>::operator*=(fvar<RealType2, Order2> const& cr) {
726 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
727 promote<RealType, RealType2> const zero(0);
728 BOOST_IF_CONSTEXPR (Order <= Order2)
729 for (size_t i = 0, j = Order; i <= Order; ++i, --j)
730 v[j] = std::inner_product(v.cbegin(), v.cend() - diff_t(i), cr.v.crbegin() + diff_t(i), zero);
731 else {
732 for (size_t i = 0, j = Order; i <= Order - Order2; ++i, --j)
733 v[j] = std::inner_product(cr.v.cbegin(), cr.v.cend(), v.crbegin() + diff_t(i), zero);
734 for (size_t i = Order - Order2 + 1, j = Order2 - 1; i <= Order; ++i, --j)
735 v[j] = std::inner_product(cr.v.cbegin(), cr.v.cbegin() + diff_t(j + 1), v.crbegin() + diff_t(i), zero);
736 }
737 return *this;
738 }
739
740 template <typename RealType, size_t Order>
operator *=(root_type const & ca)741 fvar<RealType, Order>& fvar<RealType, Order>::operator*=(root_type const& ca) {
742 return multiply_assign_by_root_type(true, ca);
743 }
744
745 template <typename RealType, size_t Order>
746 template <typename RealType2, size_t Order2>
operator /=(fvar<RealType2,Order2> const & cr)747 fvar<RealType, Order>& fvar<RealType, Order>::operator/=(fvar<RealType2, Order2> const& cr) {
748 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
749 RealType const zero(0);
750 v.front() /= cr.v.front();
751 BOOST_IF_CONSTEXPR (Order < Order2)
752 for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, --j, --k)
753 (v[i] -= std::inner_product(
754 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero)) /= cr.v.front();
755 else BOOST_IF_CONSTEXPR (0 < Order2)
756 for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, j && --j, --k)
757 (v[i] -= std::inner_product(
758 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero)) /= cr.v.front();
759 else
760 for (size_t i = 1; i <= Order; ++i)
761 v[i] /= cr.v.front();
762 return *this;
763 }
764
765 template <typename RealType, size_t Order>
operator /=(root_type const & ca)766 fvar<RealType, Order>& fvar<RealType, Order>::operator/=(root_type const& ca) {
767 std::for_each(v.begin(), v.end(), [&ca](RealType& x) { x /= ca; });
768 return *this;
769 }
770
771 template <typename RealType, size_t Order>
operator -() const772 fvar<RealType, Order> fvar<RealType, Order>::operator-() const {
773 fvar<RealType, Order> retval(*this);
774 retval.negate();
775 return retval;
776 }
777
778 template <typename RealType, size_t Order>
operator +() const779 fvar<RealType, Order> const& fvar<RealType, Order>::operator+() const {
780 return *this;
781 }
782
783 template <typename RealType, size_t Order>
784 template <typename RealType2, size_t Order2>
operator +(fvar<RealType2,Order2> const & cr) const785 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator+(
786 fvar<RealType2, Order2> const& cr) const {
787 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval;
788 for (size_t i = 0; i <= (std::min)(Order, Order2); ++i)
789 retval.v[i] = v[i] + cr.v[i];
790 BOOST_IF_CONSTEXPR (Order < Order2)
791 for (size_t i = Order + 1; i <= Order2; ++i)
792 retval.v[i] = cr.v[i];
793 else BOOST_IF_CONSTEXPR (Order2 < Order)
794 for (size_t i = Order2 + 1; i <= Order; ++i)
795 retval.v[i] = v[i];
796 return retval;
797 }
798
799 template <typename RealType, size_t Order>
operator +(root_type const & ca) const800 fvar<RealType, Order> fvar<RealType, Order>::operator+(root_type const& ca) const {
801 fvar<RealType, Order> retval(*this);
802 retval.v.front() += ca;
803 return retval;
804 }
805
806 template <typename RealType, size_t Order>
operator +(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)807 fvar<RealType, Order> operator+(typename fvar<RealType, Order>::root_type const& ca,
808 fvar<RealType, Order> const& cr) {
809 return cr + ca;
810 }
811
812 template <typename RealType, size_t Order>
813 template <typename RealType2, size_t Order2>
operator -(fvar<RealType2,Order2> const & cr) const814 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator-(
815 fvar<RealType2, Order2> const& cr) const {
816 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval;
817 for (size_t i = 0; i <= (std::min)(Order, Order2); ++i)
818 retval.v[i] = v[i] - cr.v[i];
819 BOOST_IF_CONSTEXPR (Order < Order2)
820 for (auto i = Order + 1; i <= Order2; ++i)
821 retval.v[i] = -cr.v[i];
822 else BOOST_IF_CONSTEXPR (Order2 < Order)
823 for (auto i = Order2 + 1; i <= Order; ++i)
824 retval.v[i] = v[i];
825 return retval;
826 }
827
828 template <typename RealType, size_t Order>
operator -(root_type const & ca) const829 fvar<RealType, Order> fvar<RealType, Order>::operator-(root_type const& ca) const {
830 fvar<RealType, Order> retval(*this);
831 retval.v.front() -= ca;
832 return retval;
833 }
834
835 template <typename RealType, size_t Order>
operator -(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)836 fvar<RealType, Order> operator-(typename fvar<RealType, Order>::root_type const& ca,
837 fvar<RealType, Order> const& cr) {
838 fvar<RealType, Order> mcr = -cr; // Has same address as retval in operator-() due to NRVO.
839 mcr += ca;
840 return mcr; // <-- This allows for NRVO. The following does not. --> return mcr += ca;
841 }
842
843 template <typename RealType, size_t Order>
844 template <typename RealType2, size_t Order2>
operator *(fvar<RealType2,Order2> const & cr) const845 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator*(
846 fvar<RealType2, Order2> const& cr) const {
847 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
848 promote<RealType, RealType2> const zero(0);
849 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval;
850 BOOST_IF_CONSTEXPR (Order < Order2)
851 for (size_t i = 0, j = Order, k = Order2; i <= Order2; ++i, j && --j, --k)
852 retval.v[i] = std::inner_product(v.cbegin(), v.cend() - diff_t(j), cr.v.crbegin() + diff_t(k), zero);
853 else
854 for (size_t i = 0, j = Order2, k = Order; i <= Order; ++i, j && --j, --k)
855 retval.v[i] = std::inner_product(cr.v.cbegin(), cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero);
856 return retval;
857 }
858
859 template <typename RealType, size_t Order>
operator *(root_type const & ca) const860 fvar<RealType, Order> fvar<RealType, Order>::operator*(root_type const& ca) const {
861 fvar<RealType, Order> retval(*this);
862 retval *= ca;
863 return retval;
864 }
865
866 template <typename RealType, size_t Order>
operator *(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)867 fvar<RealType, Order> operator*(typename fvar<RealType, Order>::root_type const& ca,
868 fvar<RealType, Order> const& cr) {
869 return cr * ca;
870 }
871
872 template <typename RealType, size_t Order>
873 template <typename RealType2, size_t Order2>
operator /(fvar<RealType2,Order2> const & cr) const874 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator/(
875 fvar<RealType2, Order2> const& cr) const {
876 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
877 promote<RealType, RealType2> const zero(0);
878 promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval;
879 retval.v.front() = v.front() / cr.v.front();
880 BOOST_IF_CONSTEXPR (Order < Order2) {
881 for (size_t i = 1, j = Order2 - 1; i <= Order; ++i, --j)
882 retval.v[i] =
883 (v[i] - std::inner_product(
884 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero)) /
885 cr.v.front();
886 for (size_t i = Order + 1, j = Order2 - Order - 1; i <= Order2; ++i, --j)
887 retval.v[i] =
888 -std::inner_product(
889 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero) /
890 cr.v.front();
891 } else BOOST_IF_CONSTEXPR (0 < Order2)
892 for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, j && --j, --k)
893 retval.v[i] =
894 (v[i] - std::inner_product(
895 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(k), zero)) /
896 cr.v.front();
897 else
898 for (size_t i = 1; i <= Order; ++i)
899 retval.v[i] = v[i] / cr.v.front();
900 return retval;
901 }
902
903 template <typename RealType, size_t Order>
operator /(root_type const & ca) const904 fvar<RealType, Order> fvar<RealType, Order>::operator/(root_type const& ca) const {
905 fvar<RealType, Order> retval(*this);
906 retval /= ca;
907 return retval;
908 }
909
910 template <typename RealType, size_t Order>
operator /(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)911 fvar<RealType, Order> operator/(typename fvar<RealType, Order>::root_type const& ca,
912 fvar<RealType, Order> const& cr) {
913 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
914 fvar<RealType, Order> retval;
915 retval.v.front() = ca / cr.v.front();
916 BOOST_IF_CONSTEXPR (0 < Order) {
917 RealType const zero(0);
918 for (size_t i = 1, j = Order - 1; i <= Order; ++i, --j)
919 retval.v[i] =
920 -std::inner_product(
921 cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero) /
922 cr.v.front();
923 }
924 return retval;
925 }
926
927 template <typename RealType, size_t Order>
928 template <typename RealType2, size_t Order2>
operator ==(fvar<RealType2,Order2> const & cr) const929 bool fvar<RealType, Order>::operator==(fvar<RealType2, Order2> const& cr) const {
930 return v.front() == cr.v.front();
931 }
932
933 template <typename RealType, size_t Order>
operator ==(root_type const & ca) const934 bool fvar<RealType, Order>::operator==(root_type const& ca) const {
935 return v.front() == ca;
936 }
937
938 template <typename RealType, size_t Order>
operator ==(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)939 bool operator==(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
940 return ca == cr.v.front();
941 }
942
943 template <typename RealType, size_t Order>
944 template <typename RealType2, size_t Order2>
operator !=(fvar<RealType2,Order2> const & cr) const945 bool fvar<RealType, Order>::operator!=(fvar<RealType2, Order2> const& cr) const {
946 return v.front() != cr.v.front();
947 }
948
949 template <typename RealType, size_t Order>
operator !=(root_type const & ca) const950 bool fvar<RealType, Order>::operator!=(root_type const& ca) const {
951 return v.front() != ca;
952 }
953
954 template <typename RealType, size_t Order>
operator !=(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)955 bool operator!=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
956 return ca != cr.v.front();
957 }
958
959 template <typename RealType, size_t Order>
960 template <typename RealType2, size_t Order2>
operator <=(fvar<RealType2,Order2> const & cr) const961 bool fvar<RealType, Order>::operator<=(fvar<RealType2, Order2> const& cr) const {
962 return v.front() <= cr.v.front();
963 }
964
965 template <typename RealType, size_t Order>
operator <=(root_type const & ca) const966 bool fvar<RealType, Order>::operator<=(root_type const& ca) const {
967 return v.front() <= ca;
968 }
969
970 template <typename RealType, size_t Order>
operator <=(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)971 bool operator<=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
972 return ca <= cr.v.front();
973 }
974
975 template <typename RealType, size_t Order>
976 template <typename RealType2, size_t Order2>
operator >=(fvar<RealType2,Order2> const & cr) const977 bool fvar<RealType, Order>::operator>=(fvar<RealType2, Order2> const& cr) const {
978 return v.front() >= cr.v.front();
979 }
980
981 template <typename RealType, size_t Order>
operator >=(root_type const & ca) const982 bool fvar<RealType, Order>::operator>=(root_type const& ca) const {
983 return v.front() >= ca;
984 }
985
986 template <typename RealType, size_t Order>
operator >=(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)987 bool operator>=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
988 return ca >= cr.v.front();
989 }
990
991 template <typename RealType, size_t Order>
992 template <typename RealType2, size_t Order2>
operator <(fvar<RealType2,Order2> const & cr) const993 bool fvar<RealType, Order>::operator<(fvar<RealType2, Order2> const& cr) const {
994 return v.front() < cr.v.front();
995 }
996
997 template <typename RealType, size_t Order>
operator <(root_type const & ca) const998 bool fvar<RealType, Order>::operator<(root_type const& ca) const {
999 return v.front() < ca;
1000 }
1001
1002 template <typename RealType, size_t Order>
operator <(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)1003 bool operator<(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
1004 return ca < cr.v.front();
1005 }
1006
1007 template <typename RealType, size_t Order>
1008 template <typename RealType2, size_t Order2>
operator >(fvar<RealType2,Order2> const & cr) const1009 bool fvar<RealType, Order>::operator>(fvar<RealType2, Order2> const& cr) const {
1010 return v.front() > cr.v.front();
1011 }
1012
1013 template <typename RealType, size_t Order>
operator >(root_type const & ca) const1014 bool fvar<RealType, Order>::operator>(root_type const& ca) const {
1015 return v.front() > ca;
1016 }
1017
1018 template <typename RealType, size_t Order>
operator >(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)1019 bool operator>(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) {
1020 return ca > cr.v.front();
1021 }
1022
1023 /*** Other methods and functions ***/
1024
1025 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1026 // f : order -> derivative(order)/factorial(order)
1027 // Use this when you have the polynomial coefficients, rather than just the derivatives. E.g. See atan2().
1028 template <typename RealType, size_t Order>
1029 template <typename Func, typename Fvar, typename... Fvars>
apply_coefficients(size_t const order,Func const & f,Fvar const & cr,Fvars &&...fvars) const1030 promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_coefficients(
1031 size_t const order,
1032 Func const& f,
1033 Fvar const& cr,
1034 Fvars&&... fvars) const {
1035 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1036 size_t i = (std::min)(order, order_sum);
1037 promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_coefficients(
1038 order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...);
1039 while (i--)
1040 (accumulator *= epsilon) += cr.apply_coefficients(
1041 order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...);
1042 return accumulator;
1043 }
1044 #endif
1045
1046 // f : order -> derivative(order)/factorial(order)
1047 // Use this when you have the polynomial coefficients, rather than just the derivatives. E.g. See atan().
1048 template <typename RealType, size_t Order>
1049 template <typename Func>
apply_coefficients(size_t const order,Func const & f) const1050 fvar<RealType, Order> fvar<RealType, Order>::apply_coefficients(size_t const order, Func const& f) const {
1051 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1052 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1053 size_t i = (std::min)(order, order_sum);
1054 #else // ODR-use of static constexpr
1055 size_t i = order < order_sum ? order : order_sum;
1056 #endif
1057 fvar<RealType, Order> accumulator = f(i);
1058 while (i--)
1059 (accumulator *= epsilon) += f(i);
1060 return accumulator;
1061 }
1062
1063 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1064 // f : order -> derivative(order)
1065 template <typename RealType, size_t Order>
1066 template <typename Func, typename Fvar, typename... Fvars>
apply_coefficients_nonhorner(size_t const order,Func const & f,Fvar const & cr,Fvars &&...fvars) const1067 promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_coefficients_nonhorner(
1068 size_t const order,
1069 Func const& f,
1070 Fvar const& cr,
1071 Fvars&&... fvars) const {
1072 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1073 fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i
1074 promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_coefficients_nonhorner(
1075 order,
1076 [&f](auto... indices) { return f(0, static_cast<std::size_t>(indices)...); },
1077 std::forward<Fvars>(fvars)...);
1078 size_t const i_max = (std::min)(order, order_sum);
1079 for (size_t i = 1; i <= i_max; ++i) {
1080 epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0);
1081 accumulator += epsilon_i.epsilon_multiply(
1082 i,
1083 0,
1084 cr.apply_coefficients_nonhorner(
1085 order - i,
1086 [&f, i](auto... indices) { return f(i, static_cast<std::size_t>(indices)...); },
1087 std::forward<Fvars>(fvars)...),
1088 0,
1089 0);
1090 }
1091 return accumulator;
1092 }
1093 #endif
1094
1095 // f : order -> coefficient(order)
1096 template <typename RealType, size_t Order>
1097 template <typename Func>
apply_coefficients_nonhorner(size_t const order,Func const & f) const1098 fvar<RealType, Order> fvar<RealType, Order>::apply_coefficients_nonhorner(size_t const order,
1099 Func const& f) const {
1100 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1101 fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i
1102 fvar<RealType, Order> accumulator = fvar<RealType, Order>(f(0u));
1103 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1104 size_t const i_max = (std::min)(order, order_sum);
1105 #else // ODR-use of static constexpr
1106 size_t const i_max = order < order_sum ? order : order_sum;
1107 #endif
1108 for (size_t i = 1; i <= i_max; ++i) {
1109 epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0);
1110 accumulator += epsilon_i.epsilon_multiply(i, 0, f(i));
1111 }
1112 return accumulator;
1113 }
1114
1115 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1116 // f : order -> derivative(order)
1117 template <typename RealType, size_t Order>
1118 template <typename Func, typename Fvar, typename... Fvars>
apply_derivatives(size_t const order,Func const & f,Fvar const & cr,Fvars &&...fvars) const1119 promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_derivatives(
1120 size_t const order,
1121 Func const& f,
1122 Fvar const& cr,
1123 Fvars&&... fvars) const {
1124 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1125 size_t i = (std::min)(order, order_sum);
1126 promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator =
1127 cr.apply_derivatives(
1128 order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...) /
1129 factorial<root_type>(static_cast<unsigned>(i));
1130 while (i--)
1131 (accumulator *= epsilon) +=
1132 cr.apply_derivatives(
1133 order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...) /
1134 factorial<root_type>(static_cast<unsigned>(i));
1135 return accumulator;
1136 }
1137 #endif
1138
1139 // f : order -> derivative(order)
1140 template <typename RealType, size_t Order>
1141 template <typename Func>
apply_derivatives(size_t const order,Func const & f) const1142 fvar<RealType, Order> fvar<RealType, Order>::apply_derivatives(size_t const order, Func const& f) const {
1143 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1144 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1145 size_t i = (std::min)(order, order_sum);
1146 #else // ODR-use of static constexpr
1147 size_t i = order < order_sum ? order : order_sum;
1148 #endif
1149 fvar<RealType, Order> accumulator = f(i) / factorial<root_type>(static_cast<unsigned>(i));
1150 while (i--)
1151 (accumulator *= epsilon) += f(i) / factorial<root_type>(static_cast<unsigned>(i));
1152 return accumulator;
1153 }
1154
1155 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1156 // f : order -> derivative(order)
1157 template <typename RealType, size_t Order>
1158 template <typename Func, typename Fvar, typename... Fvars>
apply_derivatives_nonhorner(size_t const order,Func const & f,Fvar const & cr,Fvars &&...fvars) const1159 promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_derivatives_nonhorner(
1160 size_t const order,
1161 Func const& f,
1162 Fvar const& cr,
1163 Fvars&&... fvars) const {
1164 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1165 fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i
1166 promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_derivatives_nonhorner(
1167 order,
1168 [&f](auto... indices) { return f(0, static_cast<std::size_t>(indices)...); },
1169 std::forward<Fvars>(fvars)...);
1170 size_t const i_max = (std::min)(order, order_sum);
1171 for (size_t i = 1; i <= i_max; ++i) {
1172 epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0);
1173 accumulator += epsilon_i.epsilon_multiply(
1174 i,
1175 0,
1176 cr.apply_derivatives_nonhorner(
1177 order - i,
1178 [&f, i](auto... indices) { return f(i, static_cast<std::size_t>(indices)...); },
1179 std::forward<Fvars>(fvars)...) /
1180 factorial<root_type>(static_cast<unsigned>(i)),
1181 0,
1182 0);
1183 }
1184 return accumulator;
1185 }
1186 #endif
1187
1188 // f : order -> derivative(order)
1189 template <typename RealType, size_t Order>
1190 template <typename Func>
apply_derivatives_nonhorner(size_t const order,Func const & f) const1191 fvar<RealType, Order> fvar<RealType, Order>::apply_derivatives_nonhorner(size_t const order,
1192 Func const& f) const {
1193 fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0);
1194 fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i
1195 fvar<RealType, Order> accumulator = fvar<RealType, Order>(f(0u));
1196 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1197 size_t const i_max = (std::min)(order, order_sum);
1198 #else // ODR-use of static constexpr
1199 size_t const i_max = order < order_sum ? order : order_sum;
1200 #endif
1201 for (size_t i = 1; i <= i_max; ++i) {
1202 epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0);
1203 accumulator += epsilon_i.epsilon_multiply(i, 0, f(i) / factorial<root_type>(static_cast<unsigned>(i)));
1204 }
1205 return accumulator;
1206 }
1207
1208 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1209 // Can throw "std::out_of_range: array::at: __n (which is 7) >= _Nm (which is 7)"
1210 template <typename RealType, size_t Order>
1211 template <typename... Orders>
at(size_t order,Orders...orders) const1212 get_type_at<RealType, sizeof...(Orders)> fvar<RealType, Order>::at(size_t order, Orders... orders) const {
1213 if constexpr (0 < sizeof...(Orders))
1214 return v.at(order).at(static_cast<std::size_t>(orders)...);
1215 else
1216 return v.at(order);
1217 }
1218 #endif
1219
1220 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1221 // Can throw "std::out_of_range: array::at: __n (which is 7) >= _Nm (which is 7)"
1222 template <typename RealType, size_t Order>
1223 template <typename... Orders>
derivative(Orders...orders) const1224 get_type_at<fvar<RealType, Order>, sizeof...(Orders)> fvar<RealType, Order>::derivative(
1225 Orders... orders) const {
1226 static_assert(sizeof...(Orders) <= depth,
1227 "Number of parameters to derivative(...) cannot exceed fvar::depth.");
1228 return at(static_cast<std::size_t>(orders)...) *
1229 (... * factorial<root_type>(static_cast<unsigned>(orders)));
1230 }
1231 #endif
1232
1233 template <typename RealType, size_t Order>
operator [](size_t i) const1234 const RealType& fvar<RealType, Order>::operator[](size_t i) const {
1235 return v[i];
1236 }
1237
1238 template <typename RealType, size_t Order>
1239 RealType fvar<RealType, Order>::epsilon_inner_product(size_t z0,
1240 size_t const isum0,
1241 size_t const m0,
1242 fvar<RealType, Order> const& cr,
1243 size_t z1,
1244 size_t const isum1,
1245 size_t const m1,
1246 size_t const j) const {
1247 static_assert(is_fvar<RealType>::value, "epsilon_inner_product() must have 1 < depth.");
1248 RealType accumulator = RealType();
1249 auto const i0_max = m1 < j ? j - m1 : 0;
1250 for (auto i0 = m0, i1 = j - m0; i0 <= i0_max; ++i0, --i1)
1251 accumulator += v[i0].epsilon_multiply(z0, isum0 + i0, cr.v[i1], z1, isum1 + i1);
1252 return accumulator;
1253 }
1254
1255 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1256 template <typename RealType, size_t Order>
epsilon_multiply(size_t z0,size_t isum0,fvar<RealType,Order> const & cr,size_t z1,size_t isum1) const1257 fvar<RealType, Order> fvar<RealType, Order>::epsilon_multiply(size_t z0,
1258 size_t isum0,
1259 fvar<RealType, Order> const& cr,
1260 size_t z1,
1261 size_t isum1) const {
1262 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
1263 RealType const zero(0);
1264 size_t const m0 = order_sum + isum0 < Order + z0 ? Order + z0 - (order_sum + isum0) : 0;
1265 size_t const m1 = order_sum + isum1 < Order + z1 ? Order + z1 - (order_sum + isum1) : 0;
1266 size_t const i_max = m0 + m1 < Order ? Order - (m0 + m1) : 0;
1267 fvar<RealType, Order> retval = fvar<RealType, Order>();
1268 if constexpr (is_fvar<RealType>::value)
1269 for (size_t i = 0, j = Order; i <= i_max; ++i, --j)
1270 retval.v[j] = epsilon_inner_product(z0, isum0, m0, cr, z1, isum1, m1, j);
1271 else
1272 for (size_t i = 0, j = Order; i <= i_max; ++i, --j)
1273 retval.v[j] = std::inner_product(
1274 v.cbegin() + diff_t(m0), v.cend() - diff_t(i + m1), cr.v.crbegin() + diff_t(i + m0), zero);
1275 return retval;
1276 }
1277 #endif
1278
1279 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1280 // When called from outside this method, z0 should be non-zero. Otherwise if z0=0 then it will give an
1281 // incorrect result of 0 when the root value is 0 and ca=inf, when instead the correct product is nan.
1282 // If z0=0 then use the regular multiply operator*() instead.
1283 template <typename RealType, size_t Order>
epsilon_multiply(size_t z0,size_t isum0,root_type const & ca) const1284 fvar<RealType, Order> fvar<RealType, Order>::epsilon_multiply(size_t z0,
1285 size_t isum0,
1286 root_type const& ca) const {
1287 fvar<RealType, Order> retval(*this);
1288 size_t const m0 = order_sum + isum0 < Order + z0 ? Order + z0 - (order_sum + isum0) : 0;
1289 if constexpr (is_fvar<RealType>::value)
1290 for (size_t i = m0; i <= Order; ++i)
1291 retval.v[i] = retval.v[i].epsilon_multiply(z0, isum0 + i, ca);
1292 else
1293 for (size_t i = m0; i <= Order; ++i)
1294 if (retval.v[i] != static_cast<RealType>(0))
1295 retval.v[i] *= ca;
1296 return retval;
1297 }
1298 #endif
1299
1300 template <typename RealType, size_t Order>
inverse() const1301 fvar<RealType, Order> fvar<RealType, Order>::inverse() const {
1302 return static_cast<root_type>(*this) == 0 ? inverse_apply() : 1 / *this;
1303 }
1304
1305 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1306 template <typename RealType, size_t Order>
negate()1307 fvar<RealType, Order>& fvar<RealType, Order>::negate() {
1308 if constexpr (is_fvar<RealType>::value)
1309 std::for_each(v.begin(), v.end(), [](RealType& r) { r.negate(); });
1310 else
1311 std::for_each(v.begin(), v.end(), [](RealType& a) { a = -a; });
1312 return *this;
1313 }
1314 #endif
1315
1316 // This gives log(0.0) = depth(1)(-inf,inf,-inf,inf,-inf,inf)
1317 // 1 / *this: log(0.0) = depth(1)(-inf,inf,-inf,-nan,-nan,-nan)
1318 template <typename RealType, size_t Order>
inverse_apply() const1319 fvar<RealType, Order> fvar<RealType, Order>::inverse_apply() const {
1320 root_type derivatives[order_sum + 1]; // LCOV_EXCL_LINE This causes a false negative on lcov coverage test.
1321 root_type const x0 = static_cast<root_type>(*this);
1322 *derivatives = 1 / x0;
1323 for (size_t i = 1; i <= order_sum; ++i)
1324 derivatives[i] = -derivatives[i - 1] * i / x0;
1325 return apply_derivatives_nonhorner(order_sum, [&derivatives](size_t j) { return derivatives[j]; });
1326 }
1327
1328 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1329 template <typename RealType, size_t Order>
multiply_assign_by_root_type(bool is_root,root_type const & ca)1330 fvar<RealType, Order>& fvar<RealType, Order>::multiply_assign_by_root_type(bool is_root,
1331 root_type const& ca) {
1332 auto itr = v.begin();
1333 if constexpr (is_fvar<RealType>::value) {
1334 itr->multiply_assign_by_root_type(is_root, ca);
1335 for (++itr; itr != v.end(); ++itr)
1336 itr->multiply_assign_by_root_type(false, ca);
1337 } else {
1338 if (is_root || *itr != 0)
1339 *itr *= ca; // Skip multiplication of 0 by ca=inf to avoid nan, except when is_root.
1340 for (++itr; itr != v.end(); ++itr)
1341 if (*itr != 0)
1342 *itr *= ca;
1343 }
1344 return *this;
1345 }
1346 #endif
1347
1348 template <typename RealType, size_t Order>
operator root_type() const1349 fvar<RealType, Order>::operator root_type() const {
1350 return static_cast<root_type>(v.front());
1351 }
1352
1353 template <typename RealType, size_t Order>
1354 template <typename T, typename>
operator T() const1355 fvar<RealType, Order>::operator T() const {
1356 return static_cast<T>(static_cast<root_type>(v.front()));
1357 }
1358
1359 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1360 template <typename RealType, size_t Order>
set_root(root_type const & root)1361 fvar<RealType, Order>& fvar<RealType, Order>::set_root(root_type const& root) {
1362 if constexpr (is_fvar<RealType>::value)
1363 v.front().set_root(root);
1364 else
1365 v.front() = root;
1366 return *this;
1367 }
1368 #endif
1369
1370 // Standard Library Support Requirements
1371
1372 template <typename RealType, size_t Order>
fabs(fvar<RealType,Order> const & cr)1373 fvar<RealType, Order> fabs(fvar<RealType, Order> const& cr) {
1374 typename fvar<RealType, Order>::root_type const zero(0);
1375 return cr < zero ? -cr
1376 : cr == zero ? fvar<RealType, Order>() // Canonical fabs'(0) = 0.
1377 : cr; // Propagate NaN.
1378 }
1379
1380 template <typename RealType, size_t Order>
abs(fvar<RealType,Order> const & cr)1381 fvar<RealType, Order> abs(fvar<RealType, Order> const& cr) {
1382 return fabs(cr);
1383 }
1384
1385 template <typename RealType, size_t Order>
ceil(fvar<RealType,Order> const & cr)1386 fvar<RealType, Order> ceil(fvar<RealType, Order> const& cr) {
1387 using std::ceil;
1388 return fvar<RealType, Order>(ceil(static_cast<typename fvar<RealType, Order>::root_type>(cr)));
1389 }
1390
1391 template <typename RealType, size_t Order>
floor(fvar<RealType,Order> const & cr)1392 fvar<RealType, Order> floor(fvar<RealType, Order> const& cr) {
1393 using std::floor;
1394 return fvar<RealType, Order>(floor(static_cast<typename fvar<RealType, Order>::root_type>(cr)));
1395 }
1396
1397 template <typename RealType, size_t Order>
exp(fvar<RealType,Order> const & cr)1398 fvar<RealType, Order> exp(fvar<RealType, Order> const& cr) {
1399 using std::exp;
1400 constexpr size_t order = fvar<RealType, Order>::order_sum;
1401 using root_type = typename fvar<RealType, Order>::root_type;
1402 root_type const d0 = exp(static_cast<root_type>(cr));
1403 return cr.apply_derivatives(order, [&d0](size_t) { return d0; });
1404 }
1405
1406 template <typename RealType, size_t Order>
pow(fvar<RealType,Order> const & x,typename fvar<RealType,Order>::root_type const & y)1407 fvar<RealType, Order> pow(fvar<RealType, Order> const& x,
1408 typename fvar<RealType, Order>::root_type const& y) {
1409 BOOST_MATH_STD_USING
1410 using root_type = typename fvar<RealType, Order>::root_type;
1411 constexpr size_t order = fvar<RealType, Order>::order_sum;
1412 root_type const x0 = static_cast<root_type>(x);
1413 root_type derivatives[order + 1]{pow(x0, y)};
1414 if (fabs(x0) < std::numeric_limits<root_type>::epsilon()) {
1415 root_type coef = 1;
1416 for (size_t i = 0; i < order && y - i != 0; ++i) {
1417 coef *= y - i;
1418 derivatives[i + 1] = coef * pow(x0, y - (i + 1));
1419 }
1420 return x.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; });
1421 } else {
1422 for (size_t i = 0; i < order && y - i != 0; ++i)
1423 derivatives[i + 1] = (y - i) * derivatives[i] / x0;
1424 return x.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i]; });
1425 }
1426 }
1427
1428 template <typename RealType, size_t Order>
pow(typename fvar<RealType,Order>::root_type const & x,fvar<RealType,Order> const & y)1429 fvar<RealType, Order> pow(typename fvar<RealType, Order>::root_type const& x,
1430 fvar<RealType, Order> const& y) {
1431 BOOST_MATH_STD_USING
1432 using root_type = typename fvar<RealType, Order>::root_type;
1433 constexpr size_t order = fvar<RealType, Order>::order_sum;
1434 root_type const y0 = static_cast<root_type>(y);
1435 root_type derivatives[order + 1];
1436 *derivatives = pow(x, y0);
1437 root_type const logx = log(x);
1438 for (size_t i = 0; i < order; ++i)
1439 derivatives[i + 1] = derivatives[i] * logx;
1440 return y.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i]; });
1441 }
1442
1443 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
pow(fvar<RealType1,Order1> const & x,fvar<RealType2,Order2> const & y)1444 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> pow(fvar<RealType1, Order1> const& x,
1445 fvar<RealType2, Order2> const& y) {
1446 BOOST_MATH_STD_USING
1447 using return_type = promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>>;
1448 using root_type = typename return_type::root_type;
1449 constexpr size_t order = return_type::order_sum;
1450 root_type const x0 = static_cast<root_type>(x);
1451 root_type const y0 = static_cast<root_type>(y);
1452 root_type dxydx[order + 1]{pow(x0, y0)};
1453 BOOST_IF_CONSTEXPR (order == 0)
1454 return return_type(*dxydx);
1455 else {
1456 for (size_t i = 0; i < order && y0 - i != 0; ++i)
1457 dxydx[i + 1] = (y0 - i) * dxydx[i] / x0;
1458 std::array<fvar<root_type, order>, order + 1> lognx;
1459 lognx.front() = fvar<root_type, order>(1);
1460 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1461 lognx[1] = log(make_fvar<root_type, order>(x0));
1462 #else // for compilers that compile this branch when order == 0.
1463 lognx[(std::min)(size_t(1), order)] = log(make_fvar<root_type, order>(x0));
1464 #endif
1465 for (size_t i = 1; i < order; ++i)
1466 lognx[i + 1] = lognx[i] * lognx[1];
1467 auto const f = [&dxydx, &lognx](size_t i, size_t j) {
1468 size_t binomial = 1;
1469 root_type sum = dxydx[i] * static_cast<root_type>(lognx[j]);
1470 for (size_t k = 1; k <= i; ++k) {
1471 (binomial *= (i - k + 1)) /= k; // binomial_coefficient(i,k)
1472 sum += binomial * dxydx[i - k] * lognx[j].derivative(k);
1473 }
1474 return sum;
1475 };
1476 if (fabs(x0) < std::numeric_limits<root_type>::epsilon())
1477 return x.apply_derivatives_nonhorner(order, f, y);
1478 return x.apply_derivatives(order, f, y);
1479 }
1480 }
1481
1482 template <typename RealType, size_t Order>
sqrt(fvar<RealType,Order> const & cr)1483 fvar<RealType, Order> sqrt(fvar<RealType, Order> const& cr) {
1484 using std::sqrt;
1485 using root_type = typename fvar<RealType, Order>::root_type;
1486 constexpr size_t order = fvar<RealType, Order>::order_sum;
1487 root_type derivatives[order + 1];
1488 root_type const x = static_cast<root_type>(cr);
1489 *derivatives = sqrt(x);
1490 BOOST_IF_CONSTEXPR (order == 0)
1491 return fvar<RealType, Order>(*derivatives);
1492 else {
1493 root_type numerator = 0.5;
1494 root_type powers = 1;
1495 #ifndef BOOST_NO_CXX17_IF_CONSTEXPR
1496 derivatives[1] = numerator / *derivatives;
1497 #else // for compilers that compile this branch when order == 0.
1498 derivatives[(std::min)(size_t(1), order)] = numerator / *derivatives;
1499 #endif
1500 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
1501 for (size_t i = 2; i <= order; ++i) {
1502 numerator *= static_cast<root_type>(-0.5) * ((static_cast<diff_t>(i) << 1) - 3);
1503 powers *= x;
1504 derivatives[i] = numerator / (powers * *derivatives);
1505 }
1506 auto const f = [&derivatives](size_t i) { return derivatives[i]; };
1507 if (cr < std::numeric_limits<root_type>::epsilon())
1508 return cr.apply_derivatives_nonhorner(order, f);
1509 return cr.apply_derivatives(order, f);
1510 }
1511 }
1512
1513 // Natural logarithm. If cr==0 then derivative(i) may have nans due to nans from inverse().
1514 template <typename RealType, size_t Order>
log(fvar<RealType,Order> const & cr)1515 fvar<RealType, Order> log(fvar<RealType, Order> const& cr) {
1516 using std::log;
1517 using root_type = typename fvar<RealType, Order>::root_type;
1518 constexpr size_t order = fvar<RealType, Order>::order_sum;
1519 root_type const d0 = log(static_cast<root_type>(cr));
1520 BOOST_IF_CONSTEXPR (order == 0)
1521 return fvar<RealType, Order>(d0);
1522 else {
1523 auto const d1 = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)).inverse(); // log'(x) = 1 / x
1524 return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1525 }
1526 }
1527
1528 template <typename RealType, size_t Order>
frexp(fvar<RealType,Order> const & cr,int * exp)1529 fvar<RealType, Order> frexp(fvar<RealType, Order> const& cr, int* exp) {
1530 using std::exp2;
1531 using std::frexp;
1532 using root_type = typename fvar<RealType, Order>::root_type;
1533 frexp(static_cast<root_type>(cr), exp);
1534 return cr * static_cast<root_type>(exp2(-*exp));
1535 }
1536
1537 template <typename RealType, size_t Order>
ldexp(fvar<RealType,Order> const & cr,int exp)1538 fvar<RealType, Order> ldexp(fvar<RealType, Order> const& cr, int exp) {
1539 // argument to std::exp2 must be casted to root_type, otherwise std::exp2 returns double (always)
1540 using std::exp2;
1541 return cr * exp2(static_cast<typename fvar<RealType, Order>::root_type>(exp));
1542 }
1543
1544 template <typename RealType, size_t Order>
cos(fvar<RealType,Order> const & cr)1545 fvar<RealType, Order> cos(fvar<RealType, Order> const& cr) {
1546 BOOST_MATH_STD_USING
1547 using root_type = typename fvar<RealType, Order>::root_type;
1548 constexpr size_t order = fvar<RealType, Order>::order_sum;
1549 root_type const d0 = cos(static_cast<root_type>(cr));
1550 BOOST_IF_CONSTEXPR (order == 0)
1551 return fvar<RealType, Order>(d0);
1552 else {
1553 root_type const d1 = -sin(static_cast<root_type>(cr));
1554 root_type const derivatives[4]{d0, d1, -d0, -d1};
1555 return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 3]; });
1556 }
1557 }
1558
1559 template <typename RealType, size_t Order>
sin(fvar<RealType,Order> const & cr)1560 fvar<RealType, Order> sin(fvar<RealType, Order> const& cr) {
1561 BOOST_MATH_STD_USING
1562 using root_type = typename fvar<RealType, Order>::root_type;
1563 constexpr size_t order = fvar<RealType, Order>::order_sum;
1564 root_type const d0 = sin(static_cast<root_type>(cr));
1565 BOOST_IF_CONSTEXPR (order == 0)
1566 return fvar<RealType, Order>(d0);
1567 else {
1568 root_type const d1 = cos(static_cast<root_type>(cr));
1569 root_type const derivatives[4]{d0, d1, -d0, -d1};
1570 return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 3]; });
1571 }
1572 }
1573
1574 template <typename RealType, size_t Order>
asin(fvar<RealType,Order> const & cr)1575 fvar<RealType, Order> asin(fvar<RealType, Order> const& cr) {
1576 using std::asin;
1577 using root_type = typename fvar<RealType, Order>::root_type;
1578 constexpr size_t order = fvar<RealType, Order>::order_sum;
1579 root_type const d0 = asin(static_cast<root_type>(cr));
1580 BOOST_IF_CONSTEXPR (order == 0)
1581 return fvar<RealType, Order>(d0);
1582 else {
1583 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1584 auto const d1 = sqrt((x *= x).negate() += 1).inverse(); // asin'(x) = 1 / sqrt(1-x*x).
1585 return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1586 }
1587 }
1588
1589 template <typename RealType, size_t Order>
tan(fvar<RealType,Order> const & cr)1590 fvar<RealType, Order> tan(fvar<RealType, Order> const& cr) {
1591 using std::tan;
1592 using root_type = typename fvar<RealType, Order>::root_type;
1593 constexpr size_t order = fvar<RealType, Order>::order_sum;
1594 root_type const d0 = tan(static_cast<root_type>(cr));
1595 BOOST_IF_CONSTEXPR (order == 0)
1596 return fvar<RealType, Order>(d0);
1597 else {
1598 auto c = cos(make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)));
1599 auto const d1 = (c *= c).inverse(); // tan'(x) = 1 / cos(x)^2
1600 return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1601 }
1602 }
1603
1604 template <typename RealType, size_t Order>
atan(fvar<RealType,Order> const & cr)1605 fvar<RealType, Order> atan(fvar<RealType, Order> const& cr) {
1606 using std::atan;
1607 using root_type = typename fvar<RealType, Order>::root_type;
1608 constexpr size_t order = fvar<RealType, Order>::order_sum;
1609 root_type const d0 = atan(static_cast<root_type>(cr));
1610 BOOST_IF_CONSTEXPR (order == 0)
1611 return fvar<RealType, Order>(d0);
1612 else {
1613 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1614 auto const d1 = ((x *= x) += 1).inverse(); // atan'(x) = 1 / (x*x+1).
1615 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1616 }
1617 }
1618
1619 template <typename RealType, size_t Order>
atan2(fvar<RealType,Order> const & cr,typename fvar<RealType,Order>::root_type const & ca)1620 fvar<RealType, Order> atan2(fvar<RealType, Order> const& cr,
1621 typename fvar<RealType, Order>::root_type const& ca) {
1622 using std::atan2;
1623 using root_type = typename fvar<RealType, Order>::root_type;
1624 constexpr size_t order = fvar<RealType, Order>::order_sum;
1625 root_type const d0 = atan2(static_cast<root_type>(cr), ca);
1626 BOOST_IF_CONSTEXPR (order == 0)
1627 return fvar<RealType, Order>(d0);
1628 else {
1629 auto y = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1630 auto const d1 = ca / ((y *= y) += (ca * ca)); // (d/dy)atan2(y,x) = x / (y*y+x*x)
1631 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1632 }
1633 }
1634
1635 template <typename RealType, size_t Order>
atan2(typename fvar<RealType,Order>::root_type const & ca,fvar<RealType,Order> const & cr)1636 fvar<RealType, Order> atan2(typename fvar<RealType, Order>::root_type const& ca,
1637 fvar<RealType, Order> const& cr) {
1638 using std::atan2;
1639 using root_type = typename fvar<RealType, Order>::root_type;
1640 constexpr size_t order = fvar<RealType, Order>::order_sum;
1641 root_type const d0 = atan2(ca, static_cast<root_type>(cr));
1642 BOOST_IF_CONSTEXPR (order == 0)
1643 return fvar<RealType, Order>(d0);
1644 else {
1645 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1646 auto const d1 = -ca / ((x *= x) += (ca * ca)); // (d/dx)atan2(y,x) = -y / (x*x+y*y)
1647 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1648 }
1649 }
1650
1651 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
atan2(fvar<RealType1,Order1> const & cr1,fvar<RealType2,Order2> const & cr2)1652 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> atan2(fvar<RealType1, Order1> const& cr1,
1653 fvar<RealType2, Order2> const& cr2) {
1654 using std::atan2;
1655 using return_type = promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>>;
1656 using root_type = typename return_type::root_type;
1657 constexpr size_t order = return_type::order_sum;
1658 root_type const y = static_cast<root_type>(cr1);
1659 root_type const x = static_cast<root_type>(cr2);
1660 root_type const d00 = atan2(y, x);
1661 BOOST_IF_CONSTEXPR (order == 0)
1662 return return_type(d00);
1663 else {
1664 constexpr size_t order1 = fvar<RealType1, Order1>::order_sum;
1665 constexpr size_t order2 = fvar<RealType2, Order2>::order_sum;
1666 auto x01 = make_fvar<typename fvar<RealType2, Order2>::root_type, order2 - 1>(x);
1667 auto const d01 = -y / ((x01 *= x01) += (y * y));
1668 auto y10 = make_fvar<typename fvar<RealType1, Order1>::root_type, order1 - 1>(y);
1669 auto x10 = make_fvar<typename fvar<RealType2, Order2>::root_type, 0, order2>(x);
1670 auto const d10 = x10 / ((x10 * x10) + (y10 *= y10));
1671 auto const f = [&d00, &d01, &d10](size_t i, size_t j) {
1672 return i ? d10[i - 1][j] / i : j ? d01[j - 1] / j : d00;
1673 };
1674 return cr1.apply_coefficients(order, f, cr2);
1675 }
1676 }
1677
1678 template <typename RealType1, size_t Order1, typename RealType2, size_t Order2>
fmod(fvar<RealType1,Order1> const & cr1,fvar<RealType2,Order2> const & cr2)1679 promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> fmod(fvar<RealType1, Order1> const& cr1,
1680 fvar<RealType2, Order2> const& cr2) {
1681 using boost::math::trunc;
1682 auto const numer = static_cast<typename fvar<RealType1, Order1>::root_type>(cr1);
1683 auto const denom = static_cast<typename fvar<RealType2, Order2>::root_type>(cr2);
1684 return cr1 - cr2 * trunc(numer / denom);
1685 }
1686
1687 template <typename RealType, size_t Order>
round(fvar<RealType,Order> const & cr)1688 fvar<RealType, Order> round(fvar<RealType, Order> const& cr) {
1689 using boost::math::round;
1690 return fvar<RealType, Order>(round(static_cast<typename fvar<RealType, Order>::root_type>(cr)));
1691 }
1692
1693 template <typename RealType, size_t Order>
iround(fvar<RealType,Order> const & cr)1694 int iround(fvar<RealType, Order> const& cr) {
1695 using boost::math::iround;
1696 return iround(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1697 }
1698
1699 template <typename RealType, size_t Order>
lround(fvar<RealType,Order> const & cr)1700 long lround(fvar<RealType, Order> const& cr) {
1701 using boost::math::lround;
1702 return lround(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1703 }
1704
1705 template <typename RealType, size_t Order>
llround(fvar<RealType,Order> const & cr)1706 long long llround(fvar<RealType, Order> const& cr) {
1707 using boost::math::llround;
1708 return llround(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1709 }
1710
1711 template <typename RealType, size_t Order>
trunc(fvar<RealType,Order> const & cr)1712 fvar<RealType, Order> trunc(fvar<RealType, Order> const& cr) {
1713 using boost::math::trunc;
1714 return fvar<RealType, Order>(trunc(static_cast<typename fvar<RealType, Order>::root_type>(cr)));
1715 }
1716
1717 template <typename RealType, size_t Order>
truncl(fvar<RealType,Order> const & cr)1718 long double truncl(fvar<RealType, Order> const& cr) {
1719 using std::truncl;
1720 return truncl(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1721 }
1722
1723 template <typename RealType, size_t Order>
itrunc(fvar<RealType,Order> const & cr)1724 int itrunc(fvar<RealType, Order> const& cr) {
1725 using boost::math::itrunc;
1726 return itrunc(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1727 }
1728
1729 template <typename RealType, size_t Order>
lltrunc(fvar<RealType,Order> const & cr)1730 long long lltrunc(fvar<RealType, Order> const& cr) {
1731 using boost::math::lltrunc;
1732 return lltrunc(static_cast<typename fvar<RealType, Order>::root_type>(cr));
1733 }
1734
1735 template <typename RealType, size_t Order>
operator <<(std::ostream & out,fvar<RealType,Order> const & cr)1736 std::ostream& operator<<(std::ostream& out, fvar<RealType, Order> const& cr) {
1737 out << "depth(" << cr.depth << ")(" << cr.v.front();
1738 for (size_t i = 1; i <= Order; ++i)
1739 out << ',' << cr.v[i];
1740 return out << ')';
1741 }
1742
1743 // Additional functions
1744
1745 template <typename RealType, size_t Order>
acos(fvar<RealType,Order> const & cr)1746 fvar<RealType, Order> acos(fvar<RealType, Order> const& cr) {
1747 using std::acos;
1748 using root_type = typename fvar<RealType, Order>::root_type;
1749 constexpr size_t order = fvar<RealType, Order>::order_sum;
1750 root_type const d0 = acos(static_cast<root_type>(cr));
1751 BOOST_IF_CONSTEXPR (order == 0)
1752 return fvar<RealType, Order>(d0);
1753 else {
1754 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1755 auto const d1 = sqrt((x *= x).negate() += 1).inverse().negate(); // acos'(x) = -1 / sqrt(1-x*x).
1756 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1757 }
1758 }
1759
1760 template <typename RealType, size_t Order>
acosh(fvar<RealType,Order> const & cr)1761 fvar<RealType, Order> acosh(fvar<RealType, Order> const& cr) {
1762 using boost::math::acosh;
1763 using root_type = typename fvar<RealType, Order>::root_type;
1764 constexpr size_t order = fvar<RealType, Order>::order_sum;
1765 root_type const d0 = acosh(static_cast<root_type>(cr));
1766 BOOST_IF_CONSTEXPR (order == 0)
1767 return fvar<RealType, Order>(d0);
1768 else {
1769 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1770 auto const d1 = sqrt((x *= x) -= 1).inverse(); // acosh'(x) = 1 / sqrt(x*x-1).
1771 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1772 }
1773 }
1774
1775 template <typename RealType, size_t Order>
asinh(fvar<RealType,Order> const & cr)1776 fvar<RealType, Order> asinh(fvar<RealType, Order> const& cr) {
1777 using boost::math::asinh;
1778 using root_type = typename fvar<RealType, Order>::root_type;
1779 constexpr size_t order = fvar<RealType, Order>::order_sum;
1780 root_type const d0 = asinh(static_cast<root_type>(cr));
1781 BOOST_IF_CONSTEXPR (order == 0)
1782 return fvar<RealType, Order>(d0);
1783 else {
1784 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1785 auto const d1 = sqrt((x *= x) += 1).inverse(); // asinh'(x) = 1 / sqrt(x*x+1).
1786 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1787 }
1788 }
1789
1790 template <typename RealType, size_t Order>
atanh(fvar<RealType,Order> const & cr)1791 fvar<RealType, Order> atanh(fvar<RealType, Order> const& cr) {
1792 using boost::math::atanh;
1793 using root_type = typename fvar<RealType, Order>::root_type;
1794 constexpr size_t order = fvar<RealType, Order>::order_sum;
1795 root_type const d0 = atanh(static_cast<root_type>(cr));
1796 BOOST_IF_CONSTEXPR (order == 0)
1797 return fvar<RealType, Order>(d0);
1798 else {
1799 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr));
1800 auto const d1 = ((x *= x).negate() += 1).inverse(); // atanh'(x) = 1 / (1-x*x)
1801 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1802 }
1803 }
1804
1805 template <typename RealType, size_t Order>
cosh(fvar<RealType,Order> const & cr)1806 fvar<RealType, Order> cosh(fvar<RealType, Order> const& cr) {
1807 BOOST_MATH_STD_USING
1808 using root_type = typename fvar<RealType, Order>::root_type;
1809 constexpr size_t order = fvar<RealType, Order>::order_sum;
1810 root_type const d0 = cosh(static_cast<root_type>(cr));
1811 BOOST_IF_CONSTEXPR (order == 0)
1812 return fvar<RealType, Order>(d0);
1813 else {
1814 root_type const derivatives[2]{d0, sinh(static_cast<root_type>(cr))};
1815 return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 1]; });
1816 }
1817 }
1818
1819 template <typename RealType, size_t Order>
digamma(fvar<RealType,Order> const & cr)1820 fvar<RealType, Order> digamma(fvar<RealType, Order> const& cr) {
1821 using boost::math::digamma;
1822 using root_type = typename fvar<RealType, Order>::root_type;
1823 constexpr size_t order = fvar<RealType, Order>::order_sum;
1824 root_type const x = static_cast<root_type>(cr);
1825 root_type const d0 = digamma(x);
1826 BOOST_IF_CONSTEXPR (order == 0)
1827 return fvar<RealType, Order>(d0);
1828 else {
1829 static_assert(order <= static_cast<size_t>((std::numeric_limits<int>::max)()),
1830 "order exceeds maximum derivative for boost::math::polygamma().");
1831 return cr.apply_derivatives(
1832 order, [&x, &d0](size_t i) { return i ? boost::math::polygamma(static_cast<int>(i), x) : d0; });
1833 }
1834 }
1835
1836 template <typename RealType, size_t Order>
erf(fvar<RealType,Order> const & cr)1837 fvar<RealType, Order> erf(fvar<RealType, Order> const& cr) {
1838 using boost::math::erf;
1839 using root_type = typename fvar<RealType, Order>::root_type;
1840 constexpr size_t order = fvar<RealType, Order>::order_sum;
1841 root_type const d0 = erf(static_cast<root_type>(cr));
1842 BOOST_IF_CONSTEXPR (order == 0)
1843 return fvar<RealType, Order>(d0);
1844 else {
1845 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); // d1 = 2/sqrt(pi)*exp(-x*x)
1846 auto const d1 = 2 * constants::one_div_root_pi<root_type>() * exp((x *= x).negate());
1847 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1848 }
1849 }
1850
1851 template <typename RealType, size_t Order>
erfc(fvar<RealType,Order> const & cr)1852 fvar<RealType, Order> erfc(fvar<RealType, Order> const& cr) {
1853 using boost::math::erfc;
1854 using root_type = typename fvar<RealType, Order>::root_type;
1855 constexpr size_t order = fvar<RealType, Order>::order_sum;
1856 root_type const d0 = erfc(static_cast<root_type>(cr));
1857 BOOST_IF_CONSTEXPR (order == 0)
1858 return fvar<RealType, Order>(d0);
1859 else {
1860 auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); // erfc'(x) = -erf'(x)
1861 auto const d1 = -2 * constants::one_div_root_pi<root_type>() * exp((x *= x).negate());
1862 return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; });
1863 }
1864 }
1865
1866 template <typename RealType, size_t Order>
lambert_w0(fvar<RealType,Order> const & cr)1867 fvar<RealType, Order> lambert_w0(fvar<RealType, Order> const& cr) {
1868 using std::exp;
1869 using boost::math::lambert_w0;
1870 using root_type = typename fvar<RealType, Order>::root_type;
1871 constexpr size_t order = fvar<RealType, Order>::order_sum;
1872 root_type derivatives[order + 1];
1873 *derivatives = lambert_w0(static_cast<root_type>(cr));
1874 BOOST_IF_CONSTEXPR (order == 0)
1875 return fvar<RealType, Order>(*derivatives);
1876 else {
1877 root_type const expw = exp(*derivatives);
1878 derivatives[1] = 1 / (static_cast<root_type>(cr) + expw);
1879 BOOST_IF_CONSTEXPR (order == 1)
1880 return cr.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; });
1881 else {
1882 using diff_t = typename std::array<RealType, Order + 1>::difference_type;
1883 root_type d1powers = derivatives[1] * derivatives[1];
1884 root_type const x = derivatives[1] * expw;
1885 derivatives[2] = d1powers * (-1 - x);
1886 std::array<root_type, order> coef{{-1, -1}}; // as in derivatives[2].
1887 for (size_t n = 3; n <= order; ++n) {
1888 coef[n - 1] = coef[n - 2] * -static_cast<root_type>(2 * n - 3);
1889 for (size_t j = n - 2; j != 0; --j)
1890 (coef[j] *= -static_cast<root_type>(n - 1)) -= (n + j - 2) * coef[j - 1];
1891 coef[0] *= -static_cast<root_type>(n - 1);
1892 d1powers *= derivatives[1];
1893 derivatives[n] =
1894 d1powers * std::accumulate(coef.crend() - diff_t(n - 1),
1895 coef.crend(),
1896 coef[n - 1],
1897 [&x](root_type const& a, root_type const& b) { return a * x + b; });
1898 }
1899 return cr.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; });
1900 }
1901 }
1902 }
1903
1904 template <typename RealType, size_t Order>
lgamma(fvar<RealType,Order> const & cr)1905 fvar<RealType, Order> lgamma(fvar<RealType, Order> const& cr) {
1906 using std::lgamma;
1907 using root_type = typename fvar<RealType, Order>::root_type;
1908 constexpr size_t order = fvar<RealType, Order>::order_sum;
1909 root_type const x = static_cast<root_type>(cr);
1910 root_type const d0 = lgamma(x);
1911 BOOST_IF_CONSTEXPR (order == 0)
1912 return fvar<RealType, Order>(d0);
1913 else {
1914 static_assert(order <= static_cast<size_t>((std::numeric_limits<int>::max)()) + 1,
1915 "order exceeds maximum derivative for boost::math::polygamma().");
1916 return cr.apply_derivatives(
1917 order, [&x, &d0](size_t i) { return i ? boost::math::polygamma(static_cast<int>(i - 1), x) : d0; });
1918 }
1919 }
1920
1921 template <typename RealType, size_t Order>
sinc(fvar<RealType,Order> const & cr)1922 fvar<RealType, Order> sinc(fvar<RealType, Order> const& cr) {
1923 if (cr != 0)
1924 return sin(cr) / cr;
1925 using root_type = typename fvar<RealType, Order>::root_type;
1926 constexpr size_t order = fvar<RealType, Order>::order_sum;
1927 root_type taylor[order + 1]{1}; // sinc(0) = 1
1928 BOOST_IF_CONSTEXPR (order == 0)
1929 return fvar<RealType, Order>(*taylor);
1930 else {
1931 for (size_t n = 2; n <= order; n += 2)
1932 taylor[n] = (1 - static_cast<int>(n & 2)) / factorial<root_type>(static_cast<unsigned>(n + 1));
1933 return cr.apply_coefficients_nonhorner(order, [&taylor](size_t i) { return taylor[i]; });
1934 }
1935 }
1936
1937 template <typename RealType, size_t Order>
sinh(fvar<RealType,Order> const & cr)1938 fvar<RealType, Order> sinh(fvar<RealType, Order> const& cr) {
1939 BOOST_MATH_STD_USING
1940 using root_type = typename fvar<RealType, Order>::root_type;
1941 constexpr size_t order = fvar<RealType, Order>::order_sum;
1942 root_type const d0 = sinh(static_cast<root_type>(cr));
1943 BOOST_IF_CONSTEXPR (fvar<RealType, Order>::order_sum == 0)
1944 return fvar<RealType, Order>(d0);
1945 else {
1946 root_type const derivatives[2]{d0, cosh(static_cast<root_type>(cr))};
1947 return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 1]; });
1948 }
1949 }
1950
1951 template <typename RealType, size_t Order>
tanh(fvar<RealType,Order> const & cr)1952 fvar<RealType, Order> tanh(fvar<RealType, Order> const& cr) {
1953 fvar<RealType, Order> retval = exp(cr * 2);
1954 fvar<RealType, Order> const denom = retval + 1;
1955 (retval -= 1) /= denom;
1956 return retval;
1957 }
1958
1959 template <typename RealType, size_t Order>
tgamma(fvar<RealType,Order> const & cr)1960 fvar<RealType, Order> tgamma(fvar<RealType, Order> const& cr) {
1961 using std::tgamma;
1962 using root_type = typename fvar<RealType, Order>::root_type;
1963 constexpr size_t order = fvar<RealType, Order>::order_sum;
1964 BOOST_IF_CONSTEXPR (order == 0)
1965 return fvar<RealType, Order>(tgamma(static_cast<root_type>(cr)));
1966 else {
1967 if (cr < 0)
1968 return constants::pi<root_type>() / (sin(constants::pi<root_type>() * cr) * tgamma(1 - cr));
1969 return exp(lgamma(cr)).set_root(tgamma(static_cast<root_type>(cr)));
1970 }
1971 }
1972
1973 } // namespace detail
1974 } // namespace autodiff_v1
1975 } // namespace differentiation
1976 } // namespace math
1977 } // namespace boost
1978
1979 namespace std {
1980
1981 // boost::math::tools::digits<RealType>() is handled by this std::numeric_limits<> specialization,
1982 // and similarly for max_value, min_value, log_max_value, log_min_value, and epsilon.
1983 template <typename RealType, size_t Order>
1984 class numeric_limits<boost::math::differentiation::detail::fvar<RealType, Order>>
1985 : public numeric_limits<typename boost::math::differentiation::detail::fvar<RealType, Order>::root_type> {
1986 };
1987
1988 } // namespace std
1989
1990 namespace boost {
1991 namespace math {
1992 namespace tools {
1993 namespace detail {
1994
1995 template <typename RealType, std::size_t Order>
1996 using autodiff_fvar_type = differentiation::detail::fvar<RealType, Order>;
1997
1998 template <typename RealType, std::size_t Order>
1999 using autodiff_root_type = typename autodiff_fvar_type<RealType, Order>::root_type;
2000 } // namespace detail
2001
2002 // See boost/math/tools/promotion.hpp
2003 template <typename RealType0, size_t Order0, typename RealType1, size_t Order1>
2004 struct promote_args_2<detail::autodiff_fvar_type<RealType0, Order0>,
2005 detail::autodiff_fvar_type<RealType1, Order1>> {
2006 using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type,
2007 #ifndef BOOST_NO_CXX14_CONSTEXPR
2008 (std::max)(Order0, Order1)>;
2009 #else
2010 Order0<Order1 ? Order1 : Order0>;
2011 #endif
2012 };
2013
2014 template <typename RealType, size_t Order>
2015 struct promote_args<detail::autodiff_fvar_type<RealType, Order>> {
2016 using type = detail::autodiff_fvar_type<typename promote_args<RealType>::type, Order>;
2017 };
2018
2019 template <typename RealType0, size_t Order0, typename RealType1>
2020 struct promote_args_2<detail::autodiff_fvar_type<RealType0, Order0>, RealType1> {
2021 using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type, Order0>;
2022 };
2023
2024 template <typename RealType0, typename RealType1, size_t Order1>
2025 struct promote_args_2<RealType0, detail::autodiff_fvar_type<RealType1, Order1>> {
2026 using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type, Order1>;
2027 };
2028
2029 template <typename destination_t, typename RealType, std::size_t Order>
real_cast(detail::autodiff_fvar_type<RealType,Order> const & from_v)2030 inline BOOST_MATH_CONSTEXPR destination_t real_cast(detail::autodiff_fvar_type<RealType, Order> const& from_v)
2031 BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(destination_t) && BOOST_MATH_IS_FLOAT(RealType)) {
2032 return real_cast<destination_t>(static_cast<detail::autodiff_root_type<RealType, Order>>(from_v));
2033 }
2034
2035 } // namespace tools
2036
2037 namespace policies {
2038
2039 template <class Policy, std::size_t Order>
2040 using fvar_t = differentiation::detail::fvar<Policy, Order>;
2041 template <class Policy, std::size_t Order>
2042 struct evaluation<fvar_t<float, Order>, Policy> {
2043 using type = fvar_t<typename conditional<Policy::promote_float_type::value, double, float>::type, Order>;
2044 };
2045
2046 template <class Policy, std::size_t Order>
2047 struct evaluation<fvar_t<double, Order>, Policy> {
2048 using type =
2049 fvar_t<typename conditional<Policy::promote_double_type::value, long double, double>::type, Order>;
2050 };
2051
2052 } // namespace policies
2053 } // namespace math
2054 } // namespace boost
2055
2056 #ifdef BOOST_NO_CXX17_IF_CONSTEXPR
2057 #include "autodiff_cpp11.hpp"
2058 #endif
2059
2060 #endif // BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP
2061