34176864186

Committed 08 Sep 2026 01:30AM UTC coverage: 81.371% (+0.01%) from 81.36%

Build # 34176864186

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Pull #4113

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Merge 239d78ca0 into d7d3284a1

Pull Request Pull Request #4113: Extract the combined k-effective estimator and fix its two-estimate branch

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95.73

/src/math_functions.cpp

1	#include "openmc/math_functions.h"
2
3	#include <cmath> // for abs, sqrt
4	#include <limits> // for numeric_limits
5
6	#include "openmc/external/Faddeeva.hh"
7
8	#include "openmc/array.h"
9	#include "openmc/constants.h"
10	#include "openmc/random_lcg.h"
11
12	namespace openmc {
13
14	//==============================================================================
15	// Mathematical methods
16	//==============================================================================
17
18	double normal_percentile(double p)	163 ✔
19	{
20	constexpr double p_low = 0.02425;	163 ✔
21	constexpr double a[6] = {-3.969683028665376e1, 2.209460984245205e2,	163 ✔
22	-2.759285104469687e2, 1.383577518672690e2, -3.066479806614716e1,
23	2.506628277459239e0};
24	constexpr double b[5] = {-5.447609879822406e1, 1.615858368580409e2,	163 ✔
25	-1.556989798598866e2, 6.680131188771972e1, -1.328068155288572e1};
26	constexpr double c[6] = {-7.784894002430293e-3, -3.223964580411365e-1,	163 ✔
27	-2.400758277161838, -2.549732539343734, 4.374664141464968,
28	2.938163982698783};
29	constexpr double d[4] = {7.784695709041462e-3, 3.224671290700398e-1,	163 ✔
30	2.445134137142996, 3.754408661907416};
31
32	// The rational approximation used here is from an unpublished work at
33	// http://home.online.no/~pjacklam/notes/invnorm/
34
35	double z;	163 ✔
36	double q;	163 ✔
37
38	if (p < p_low) {	163 ✔
39	// Rational approximation for lower region.
40
41	q = std::sqrt(-2.0 * std::log(p));	11 ✔
42	z = (((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]) /	11 ✔
43	((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1.0);	11 ✔
44
45	} else if (p <= 1.0 - p_low) {	152 ✔
46	// Rational approximation for central region
47	q = p - 0.5;	141 ✔
48	double r = q * q;	141 ✔
49	z = (((((a[0] * r + a[1]) * r + a[2]) * r + a[3]) * r + a[4]) * r + a[5]) *	141 ✔
50	q /
51	(((((b[0] * r + b[1]) * r + b[2]) * r + b[3]) * r + b[4]) * r + 1.0);	141 ✔
52
53	} else {
54	// Rational approximation for upper region
55
56	q = std::sqrt(-2.0 * std::log(1.0 - p));	11 ✔
57	z = -(((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]) /	11 ✔
58	((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1.0);	11 ✔
59	}
60
61	// Refinement based on Newton's method
62
63	z = z - (0.5 * std::erfc(-z / std::sqrt(2.0)) - p) * std::sqrt(2.0 * PI) *	163 ✔
64	std::exp(0.5 * z * z);	163 ✔
65
66	return z;	163 ✔
67	}
68
69	double t_percentile(double p, int df)	303 ✔
70	{
71	double t;	303 ✔
72
73	if (df == 1) {	303 ✔
74	// For one degree of freedom, the t-distribution becomes a Cauchy
75	// distribution whose cdf we can invert directly
76
77	t = std::tan(PI * (p - 0.5));	70 ✔
78	} else if (df == 2) {	233 ✔
79	// For two degrees of freedom, the cdf is given by 1/2 + x/(2*sqrt(x^2 +
80	// 2)). This can be directly inverted to yield the solution below
81
82	t = 2.0 * std::sqrt(2.0) * (p - 0.5) /	140 ✔
83	std::sqrt(1. - 4. * std::pow(p - 0.5, 2.));	70 ✔
84	} else {
85	// This approximation is from E. Olusegun George and Meenakshi Sivaram, "A
86	// modification of the Fisher-Cornish approximation for the student t
87	// percentiles," Communication in Statistics - Simulation and Computation,
88	// 16 (4), pp. 1123-1132 (1987).
89	double n = df;	163 ✔
90	double k = 1. / (n - 2.);	163 ✔
91	double z = normal_percentile(p);	163 ✔
92	double z2 = z * z;	163 ✔
93	t = std::sqrt(n * k) *	163 ✔
94	(z + (z2 - 3.) * z * k / 4. +	163 ✔
95	((5. * z2 - 56.) * z2 + 75.) * z * k * k / 96. +	163 ✔
96	(((z2 - 27.) * 3. * z2 + 417.) * z2 - 315.) * z * k * k * k / 384.);	163 ✔
97	}
98
99	return t;	303 ✔
100	}
101
102	double standard_normal_cdf(double z)	110 ✔
103	{
104	// Use the complementary error function to compute the standard normal CDF
105	// Phi(z) = 0.5 * (1 + erf(z / sqrt(2))) = 0.5 * erfc(-z / sqrt(2))
106	return 0.5 * std::erfc(-z / std::sqrt(2.0));	110 ✔
107	}
108
109	void calc_pn_c(int n, double x, double pnx[])	634,591,185 ✔
110	{
111	pnx[0] = 1.;	634,591,185 ✔
112	if (n >= 1) {	634,591,185 ✔
113	pnx[1] = x;	144,845,151 ✔
114	}
115
116	// Use recursion relation to build the higher orders
117	for (int l = 1; l < n; l++) {	640,733,156 ✔
118	pnx[l + 1] = ((2 * l + 1) * x * pnx[l] - l * pnx[l - 1]) / (l + 1);	6,141,971 ✔
119	}
120	}	634,591,185 ✔
121
122	double evaluate_legendre(int n, const double data[], double x)	127,558,640 ✔
123	{
124	double* pnx = new double[n + 1];	127,558,640 !
125	double val = 0.0;	127,558,640 ✔
126	calc_pn_c(n, x, pnx);	127,558,640 ✔
127	for (int l = 0; l <= n; l++) {	334,801,819 ✔
128	val += (l + 0.5) * data[l] * pnx[l];	207,243,179 ✔
129	}
130	delete[] pnx;	127,558,640 ✔
131	return val;	127,558,640 ✔
132	}
133
134	void calc_rn_c(int n, const double uvw[3], double rn[])	11 ✔
135	{
136	Direction u {uvw};	11 ✔
137	calc_rn(n, u, rn);	11 ✔
138	}	11 ✔
139
140	void calc_rn(int n, Direction u, double rn[])	4,568,344 ✔
141	{
142	// rn[] is assumed to have already been allocated to the correct size
143
144	// Store the cosine of the polar angle and the azimuthal angle
145	double w = u.z;	4,568,344 ✔
146	double phi;	4,568,344 ✔
147	if (u.x == 0.) {	4,568,344 !
148	phi = 0.;
149	} else {
150	phi = std::atan2(u.y, u.x);	4,568,344 ✔
151	}
152
153	// Store the shorthand of 1-w * w
154	double w2m1 = 1. - w * w;	4,568,344 ✔
155
156	// Now evaluate the spherical harmonics function
157	rn[0] = 1.;	4,568,344 ✔
158	int i = 0;	4,568,344 ✔
159	for (int l = 1; l <= n; l++) {	22,844,778 ✔
160	// Set the index to the start of this order
161	i += 2 * (l - 1) + 1;	18,276,434 ✔
162
163	// Now evaluate each
164	switch (l) {	18,276,434 !
165	case 1:	4,568,344 ✔
166	// l = 1, m = -1
167	rn[i] = -(std::sqrt(w2m1) * std::sin(phi));	4,568,344 ✔
168	// l = 1, m = 0
169	rn[i + 1] = w;	4,568,344 ✔
170	// l = 1, m = 1
171	rn[i + 2] = -(std::sqrt(w2m1) * std::cos(phi));	4,568,344 ✔
172	break;	4,568,344 ✔
173	case 2:	4,568,344 ✔
174	// l = 2, m = -2
175	rn[i] = 0.288675134594813 * (-3. * w * w + 3.) * std::sin(2. * phi);	4,568,344 ✔
176	// l = 2, m = -1
177	rn[i + 1] = -(1.73205080756888 * w * std::sqrt(w2m1) * std::sin(phi));	4,568,344 ✔
178	// l = 2, m = 0
179	rn[i + 2] = 1.5 * w * w - 0.5;	4,568,344 ✔
180	// l = 2, m = 1
181	rn[i + 3] = -(1.73205080756888 * w * std::sqrt(w2m1) * std::cos(phi));	4,568,344 ✔
182	// l = 2, m = 2
183	rn[i + 4] = 0.288675134594813 * (-3. * w * w + 3.) * std::cos(2. * phi);	4,568,344 ✔
184	break;	4,568,344 ✔
185	case 3:	4,568,344 ✔
186	// l = 3, m = -3
187	rn[i] = -(0.790569415042095 * std::pow(w2m1, 1.5) * std::sin(3. * phi));	4,568,344 ✔
188	// l = 3, m = -2
189	rn[i + 1] = 1.93649167310371 * w * (w2m1)std::sin(2. phi);	4,568,344 ✔
190	// l = 3, m = -1
191	rn[i + 2] = -(0.408248290463863 * std::sqrt(w2m1) *	4,568,344 ✔
192	((7.5) * w * w - 3. / 2.) * std::sin(phi));	4,568,344 ✔
193	// l = 3, m = 0
194	rn[i + 3] = 2.5 * std::pow(w, 3) - 1.5 * w;	4,568,344 ✔
195	// l = 3, m = 1
196	rn[i + 4] = -(0.408248290463863 * std::sqrt(w2m1) *	4,568,344 ✔
197	((7.5) * w * w - 3. / 2.) * std::cos(phi));	4,568,344 ✔
198	// l = 3, m = 2
199	rn[i + 5] = 1.93649167310371 * w * (w2m1)std::cos(2. phi);	4,568,344 ✔
200	// l = 3, m = 3
201	rn[i + 6] =	9,136,688 ✔
202	-(0.790569415042095 * std::pow(w2m1, 1.5) * std::cos(3. * phi));	4,568,344 ✔
203	break;	4,568,344 ✔
204	case 4:	4,568,344 ✔
205	// l = 4, m = -4
206	rn[i] = 0.739509972887452 * (w2m1 * w2m1) * std::sin(4.0 * phi);	4,568,344 ✔
207	// l = 4, m = -3
208	rn[i + 1] =	9,136,688 ✔
209	-(2.09165006633519 * w * std::pow(w2m1, 1.5) * std::sin(3. * phi));	4,568,344 ✔
210	// l = 4, m = -2
211	rn[i + 2] =	9,136,688 ✔
212	0.074535599249993 * (w2m1) * (52.5 * w * w - 7.5) * std::sin(2. * phi);	4,568,344 ✔
213	// l = 4, m = -1
214	rn[i + 3] = -(0.316227766016838 * std::sqrt(w2m1) *	4,568,344 ✔
215	(17.5 * std::pow(w, 3) - 7.5 * w) * std::sin(phi));	4,568,344 ✔
216	// l = 4, m = 0
217	rn[i + 4] = 4.375 * std::pow(w, 4) - 3.75 * w * w + 0.375;	4,568,344 ✔
218	// l = 4, m = 1
219	rn[i + 5] = -(0.316227766016838 * std::sqrt(w2m1) *	4,568,344 ✔
220	(17.5 * std::pow(w, 3) - 7.5 * w) * std::cos(phi));	4,568,344 ✔
221	// l = 4, m = 2
222	rn[i + 6] =	9,136,688 ✔
223	0.074535599249993 * (w2m1) * (52.5 * w * w - 7.5) * std::cos(2. * phi);	4,568,344 ✔
224	// l = 4, m = 3
225	rn[i + 7] =	9,136,688 ✔
226	-(2.09165006633519 * w * std::pow(w2m1, 1.5) * std::cos(3. * phi));	4,568,344 ✔
227	// l = 4, m = 4
228	rn[i + 8] = 0.739509972887452 * w2m1 * w2m1 * std::cos(4.0 * phi);	4,568,344 ✔
229	break;	4,568,344 ✔
230	case 5:	3,003 ✔
231	// l = 5, m = -5
232	rn[i] = -(0.701560760020114 * std::pow(w2m1, 2.5) * std::sin(5.0 * phi));	3,003 ✔
233	// l = 5, m = -4
234	rn[i + 1] = 2.21852991866236 * w * w2m1 * w2m1 * std::sin(4.0 * phi);	3,003 ✔
235	// l = 5, m = -3
236	rn[i + 2] = -(0.00996023841111995 * std::pow(w2m1, 1.5) *	3,003 ✔
237	((945.0 / 2.) * w * w - 52.5) * std::sin(3. * phi));	3,003 ✔
238	// l = 5, m = -2
239	rn[i + 3] = 0.0487950036474267 * (w2m1) *	3,003 ✔
240	((315.0 / 2.) * std::pow(w, 3) - 52.5 * w) *	6,006 ✔
241	std::sin(2. * phi);	3,003 ✔
242	// l = 5, m = -1
243	rn[i + 4] =	6,006 ✔
244	-(0.258198889747161 * std::sqrt(w2m1) *	3,003 ✔
245	(39.375 * std::pow(w, 4) - 105.0 / 4.0 * w * w + 15.0 / 8.0) *	6,006 ✔
246	std::sin(phi));	3,003 ✔
247	// l = 5, m = 0
248	rn[i + 5] = 7.875 * std::pow(w, 5) - 8.75 * std::pow(w, 3) + 1.875 * w;	3,003 ✔
249	// l = 5, m = 1
250	rn[i + 6] =	6,006 ✔
251	-(0.258198889747161 * std::sqrt(w2m1) *	3,003 ✔
252	(39.375 * std::pow(w, 4) - 105.0 / 4.0 * w * w + 15.0 / 8.0) *	6,006 ✔
253	std::cos(phi));	3,003 ✔
254	// l = 5, m = 2
255	rn[i + 7] = 0.0487950036474267 * (w2m1) *	3,003 ✔
256	((315.0 / 2.) * std::pow(w, 3) - 52.5 * w) *	6,006 ✔
257	std::cos(2. * phi);	3,003 ✔
258	// l = 5, m = 3
259	rn[i + 8] = -(0.00996023841111995 * std::pow(w2m1, 1.5) *	3,003 ✔
260	((945.0 / 2.) * w * w - 52.5) * std::cos(3. * phi));	3,003 ✔
261	// l = 5, m = 4
262	rn[i + 9] = 2.21852991866236 * w * w2m1 * w2m1 * std::cos(4.0 * phi);	3,003 ✔
263	// l = 5, m = 5
264	rn[i + 10] =	6,006 ✔
265	-(0.701560760020114 * std::pow(w2m1, 2.5) * std::cos(5.0 * phi));	3,003 ✔
266	break;	3,003 ✔
267	case 6:	11 ✔
268	// l = 6, m = -6
269	rn[i] = 0.671693289381396 * std::pow(w2m1, 3) * std::sin(6.0 * phi);	11 ✔
270	// l = 6, m = -5
271	rn[i + 1] =	22 ✔
272	-(2.32681380862329 * w * std::pow(w2m1, 2.5) * std::sin(5.0 * phi));	11 ✔
273	// l = 6, m = -4
274	rn[i + 2] = 0.00104990131391452 * w2m1 * w2m1 *	22 ✔
275	((10395.0 / 2.) * w * w - 945.0 / 2.) * std::sin(4.0 * phi);	11 ✔
276	// l = 6, m = -3
277	rn[i + 3] = -(0.00575054632785295 * std::pow(w2m1, 1.5) *	11 ✔
278	((3465.0 / 2.) * std::pow(w, 3) - 945.0 / 2. * w) *	22 ✔
279	std::sin(3. * phi));	11 ✔
280	// l = 6, m = -2
281	rn[i + 4] =	22 ✔
282	0.0345032779671177 * (w2m1) *	11 ✔
283	((3465.0 / 8.0) * std::pow(w, 4) - 945.0 / 4.0 * w * w + 105.0 / 8.0) *	22 ✔
284	std::sin(2. * phi);	11 ✔
285	// l = 6, m = -1
286	rn[i + 5] = -(0.218217890235992 * std::sqrt(w2m1) *	11 ✔
287	((693.0 / 8.0) * std::pow(w, 5) -	11 ✔
288	315.0 / 4.0 * std::pow(w, 3) + (105.0 / 8.0) * w) *	22 ✔
289	std::sin(phi));	11 ✔
290	// l = 6, m = 0
291	rn[i + 6] = 14.4375 * std::pow(w, 6) - 19.6875 * std::pow(w, 4) +	11 ✔
292	6.5625 * w * w - 0.3125;	11 ✔
293	// l = 6, m = 1
294	rn[i + 7] = -(0.218217890235992 * std::sqrt(w2m1) *	11 ✔
295	((693.0 / 8.0) * std::pow(w, 5) -	11 ✔
296	315.0 / 4.0 * std::pow(w, 3) + (105.0 / 8.0) * w) *	22 ✔
297	std::cos(phi));	11 ✔
298	// l = 6, m = 2
299	rn[i + 8] =	22 ✔
300	0.0345032779671177 * w2m1 *	11 ✔
301	((3465.0 / 8.0) * std::pow(w, 4) - 945.0 / 4.0 * w * w + 105.0 / 8.0) *	22 ✔
302	std::cos(2. * phi);	11 ✔
303	// l = 6, m = 3
304	rn[i + 9] = -(0.00575054632785295 * std::pow(w2m1, 1.5) *	11 ✔
305	((3465.0 / 2.) * std::pow(w, 3) - 945.0 / 2. * w) *	22 ✔
306	std::cos(3. * phi));	11 ✔
307	// l = 6, m = 4
308	rn[i + 10] = 0.00104990131391452 * w2m1 * w2m1 *	22 ✔
309	((10395.0 / 2.) * w * w - 945.0 / 2.) * std::cos(4.0 * phi);	11 ✔
310	// l = 6, m = 5
311	rn[i + 11] =	22 ✔
312	-(2.32681380862329 * w * std::pow(w2m1, 2.5) * std::cos(5.0 * phi));	11 ✔
313	// l = 6, m = 6
314	rn[i + 12] = 0.671693289381396 * std::pow(w2m1, 3) * std::cos(6.0 * phi);	11 ✔
315	break;	11 ✔
316	case 7:	11 ✔
317	// l = 7, m = -7
318	rn[i] = -(0.647259849287749 * std::pow(w2m1, 3.5) * std::sin(7.0 * phi));	11 ✔
319	// l = 7, m = -6
320	rn[i + 1] =	22 ✔
321	2.42182459624969 * w * std::pow(w2m1, 3) * std::sin(6.0 * phi);	11 ✔
322	// l = 7, m = -5
323	rn[i + 2] =	22 ✔
324	-(9.13821798555235e-5 * std::pow(w2m1, 2.5) *	11 ✔
325	((135135.0 / 2.) * w * w - 10395.0 / 2.) * std::sin(5.0 * phi));	11 ✔
326	// l = 7, m = -4
327	rn[i + 3] = 0.000548293079133141 * w2m1 * w2m1 *	11 ✔
328	((45045.0 / 2.) * std::pow(w, 3) - 10395.0 / 2. * w) *	22 ✔
329	std::sin(4.0 * phi);	11 ✔
330	// l = 7, m = -3
331	rn[i + 4] = -(0.00363696483726654 * std::pow(w2m1, 1.5) *	11 ✔
332	((45045.0 / 8.0) * std::pow(w, 4) - 10395.0 / 4.0 * w * w +	11 ✔
333	945.0 / 8.0) *	11 ✔
334	std::sin(3. * phi));	11 ✔
335	// l = 7, m = -2
336	rn[i + 5] = 0.025717224993682 * (w2m1) *	11 ✔
337	((9009.0 / 8.0) * std::pow(w, 5) -	11 ✔
338	3465.0 / 4.0 * std::pow(w, 3) + (945.0 / 8.0) * w) *	22 ✔
339	std::sin(2. * phi);	11 ✔
340	// l = 7, m = -1
341	rn[i + 6] =	22 ✔
342	-(0.188982236504614 * std::sqrt(w2m1) *	11 ✔
343	((3003.0 / 16.0) * std::pow(w, 6) - 3465.0 / 16.0 * std::pow(w, 4) +	11 ✔
344	(945.0 / 16.0) * w * w - 35.0 / 16.0) *	22 ✔
345	std::sin(phi));	11 ✔
346	// l = 7, m = 0
347	rn[i + 7] = 26.8125 * std::pow(w, 7) - 43.3125 * std::pow(w, 5) +	11 ✔
348	19.6875 * std::pow(w, 3) - 2.1875 * w;	11 ✔
349	// l = 7, m = 1
350	rn[i + 8] =	22 ✔
351	-(0.188982236504614 * std::sqrt(w2m1) *	11 ✔
352	((3003.0 / 16.0) * std::pow(w, 6) - 3465.0 / 16.0 * std::pow(w, 4) +	11 ✔
353	(945.0 / 16.0) * w * w - 35.0 / 16.0) *	22 ✔
354	std::cos(phi));	11 ✔
355	// l = 7, m = 2
356	rn[i + 9] = 0.025717224993682 * (w2m1) *	11 ✔
357	((9009.0 / 8.0) * std::pow(w, 5) -	11 ✔
358	3465.0 / 4.0 * std::pow(w, 3) + (945.0 / 8.0) * w) *	22 ✔
359	std::cos(2. * phi);	11 ✔
360	// l = 7, m = 3
361	rn[i + 10] = -(0.00363696483726654 * std::pow(w2m1, 1.5) *	11 ✔
362	((45045.0 / 8.0) * std::pow(w, 4) - 10395.0 / 4.0 * w * w +	11 ✔
363	945.0 / 8.0) *	11 ✔
364	std::cos(3. * phi));	11 ✔
365	// l = 7, m = 4
366	rn[i + 11] = 0.000548293079133141 * w2m1 * w2m1 *	11 ✔
367	((45045.0 / 2.) * std::pow(w, 3) - 10395.0 / 2. * w) *	22 ✔
368	std::cos(4.0 * phi);	11 ✔
369	// l = 7, m = 5
370	rn[i + 12] =	22 ✔
371	-(9.13821798555235e-5 * std::pow(w2m1, 2.5) *	11 ✔
372	((135135.0 / 2.) * w * w - 10395.0 / 2.) * std::cos(5.0 * phi));	11 ✔
373	// l = 7, m = 6
374	rn[i + 13] =	22 ✔
375	2.42182459624969 * w * std::pow(w2m1, 3) * std::cos(6.0 * phi);	11 ✔
376	// l = 7, m = 7
377	rn[i + 14] =	22 ✔
378	-(0.647259849287749 * std::pow(w2m1, 3.5) * std::cos(7.0 * phi));	11 ✔
379	break;	11 ✔
380	case 8:	11 ✔
381	// l = 8, m = -8
382	rn[i] = 0.626706654240044 * std::pow(w2m1, 4) * std::sin(8.0 * phi);	11 ✔
383	// l = 8, m = -7
384	rn[i + 1] =	22 ✔
385	-(2.50682661696018 * w * std::pow(w2m1, 3.5) * std::sin(7.0 * phi));	11 ✔
386	// l = 8, m = -6
387	rn[i + 2] = 6.77369783729086e-6 * std::pow(w2m1, 3) *	11 ✔
388	((2027025.0 / 2.) * w * w - 135135.0 / 2.) *	22 ✔
389	std::sin(6.0 * phi);	11 ✔
390	// l = 8, m = -5
391	rn[i + 3] = -(4.38985792528482e-5 * std::pow(w2m1, 2.5) *	11 ✔
392	((675675.0 / 2.) * std::pow(w, 3) - 135135.0 / 2. * w) *	22 ✔
393	std::sin(5.0 * phi));	11 ✔
394	// l = 8, m = -4
395	rn[i + 4] = 0.000316557156832328 * w2m1 * w2m1 *	11 ✔
396	((675675.0 / 8.0) * std::pow(w, 4) - 135135.0 / 4.0 * w * w +	11 ✔
397	10395.0 / 8.0) *	11 ✔
398	std::sin(4.0 * phi);	11 ✔
399	// l = 8, m = -3
400	rn[i + 5] = -(0.00245204119306875 * std::pow(w2m1, 1.5) *	11 ✔
401	((135135.0 / 8.0) * std::pow(w, 5) -	11 ✔
402	45045.0 / 4.0 * std::pow(w, 3) + (10395.0 / 8.0) * w) *	22 ✔
403	std::sin(3. * phi));	11 ✔
404	// l = 8, m = -2
405	rn[i + 6] =	22 ✔
406	0.0199204768222399 * (w2m1) *	11 ✔
407	((45045.0 / 16.0) * std::pow(w, 6) - 45045.0 / 16.0 * std::pow(w, 4) +	11 ✔
408	(10395.0 / 16.0) * w * w - 315.0 / 16.0) *	22 ✔
409	std::sin(2. * phi);	11 ✔
410	// l = 8, m = -1
411	rn[i + 7] =	22 ✔
412	-(0.166666666666667 * std::sqrt(w2m1) *	11 ✔
413	((6435.0 / 16.0) * std::pow(w, 7) - 9009.0 / 16.0 * std::pow(w, 5) +	11 ✔
414	(3465.0 / 16.0) * std::pow(w, 3) - 315.0 / 16.0 * w) *	22 ✔
415	std::sin(phi));	11 ✔
416	// l = 8, m = 0
417	rn[i + 8] = 50.2734375 * std::pow(w, 8) - 93.84375 * std::pow(w, 6) +	11 ✔
418	54.140625 * std::pow(w, 4) - 9.84375 * w * w + 0.2734375;	11 ✔
419	// l = 8, m = 1
420	rn[i + 9] =	22 ✔
421	-(0.166666666666667 * std::sqrt(w2m1) *	11 ✔
422	((6435.0 / 16.0) * std::pow(w, 7) - 9009.0 / 16.0 * std::pow(w, 5) +	11 ✔
423	(3465.0 / 16.0) * std::pow(w, 3) - 315.0 / 16.0 * w) *	22 ✔
424	std::cos(phi));	11 ✔
425	// l = 8, m = 2
426	rn[i + 10] =	22 ✔
427	0.0199204768222399 * (w2m1) *	11 ✔
428	((45045.0 / 16.0) * std::pow(w, 6) - 45045.0 / 16.0 * std::pow(w, 4) +	11 ✔
429	(10395.0 / 16.0) * w * w - 315.0 / 16.0) *	22 ✔
430	std::cos(2. * phi);	11 ✔
431	// l = 8, m = 3
432	rn[i + 11] = -(0.00245204119306875 * std::pow(w2m1, 1.5) *	11 ✔
433	((135135.0 / 8.0) * std::pow(w, 5) -	11 ✔
434	45045.0 / 4.0 * std::pow(w, 3) + (10395.0 / 8.0) * w) *	22 ✔
435	std::cos(3. * phi));	11 ✔
436	// l = 8, m = 4
437	rn[i + 12] = 0.000316557156832328 * w2m1 * w2m1 *	11 ✔
438	((675675.0 / 8.0) * std::pow(w, 4) - 135135.0 / 4.0 * w * w +	11 ✔
439	10395.0 / 8.0) *	11 ✔
440	std::cos(4.0 * phi);	11 ✔
441	// l = 8, m = 5
442	rn[i + 13] = -(4.38985792528482e-5 * std::pow(w2m1, 2.5) *	11 ✔
443	((675675.0 / 2.) * std::pow(w, 3) - 135135.0 / 2. * w) *	22 ✔
444	std::cos(5.0 * phi));	11 ✔
445	// l = 8, m = 6
446	rn[i + 14] = 6.77369783729086e-6 * std::pow(w2m1, 3) *	11 ✔
447	((2027025.0 / 2.) * w * w - 135135.0 / 2.) *	11 ✔
448	std::cos(6.0 * phi);	11 ✔
449	// l = 8, m = 7
450	rn[i + 15] =	22 ✔
451	-(2.50682661696018 * w * std::pow(w2m1, 3.5) * std::cos(7.0 * phi));	11 ✔
452	// l = 8, m = 8
453	rn[i + 16] = 0.626706654240044 * std::pow(w2m1, 4) * std::cos(8.0 * phi);	11 ✔
454	break;	11 ✔
455	case 9:	11 ✔
456	// l = 9, m = -9
457	rn[i] = -(0.609049392175524 * std::pow(w2m1, 4.5) * std::sin(9.0 * phi));	11 ✔
458	// l = 9, m = -8
459	rn[i + 1] =	22 ✔
460	2.58397773170915 * w * std::pow(w2m1, 4) * std::sin(8.0 * phi);	11 ✔
461	// l = 9, m = -7
462	rn[i + 2] =	22 ✔
463	-(4.37240315267812e-7 * std::pow(w2m1, 3.5) *	11 ✔
464	((34459425.0 / 2.) * w * w - 2027025.0 / 2.) * std::sin(7.0 * phi));	11 ✔
465	// l = 9, m = -6
466	rn[i + 3] = 3.02928976464514e-6 * std::pow(w2m1, 3) *	11 ✔
467	((11486475.0 / 2.) * std::pow(w, 3) - 2027025.0 / 2. * w) *	22 ✔
468	std::sin(6.0 * phi);	11 ✔
469	// l = 9, m = -5
470	rn[i + 4] = -(2.34647776186144e-5 * std::pow(w2m1, 2.5) *	11 ✔
471	((11486475.0 / 8.0) * std::pow(w, 4) -	11 ✔
472	2027025.0 / 4.0 * w * w + 135135.0 / 8.0) *	22 ✔
473	std::sin(5.0 * phi));	11 ✔
474	// l = 9, m = -4
475	rn[i + 5] = 0.000196320414650061 * w2m1 * w2m1 *	11 ✔
476	((2297295.0 / 8.0) * std::pow(w, 5) -	11 ✔
477	675675.0 / 4.0 * std::pow(w, 3) + (135135.0 / 8.0) * w) *	22 ✔
478	std::sin(4.0 * phi);	11 ✔
479	// l = 9, m = -3
480	rn[i + 6] = -(	22 ✔
481	0.00173385495536766 * std::pow(w2m1, 1.5) *	11 ✔
482	((765765.0 / 16.0) * std::pow(w, 6) - 675675.0 / 16.0 * std::pow(w, 4) +	11 ✔
483	(135135.0 / 16.0) * w * w - 3465.0 / 16.0) *	22 ✔
484	std::sin(3. * phi));	11 ✔
485	// l = 9, m = -2
486	rn[i + 7] =	22 ✔
487	0.0158910431540932 * (w2m1) *	11 ✔
488	((109395.0 / 16.0) * std::pow(w, 7) - 135135.0 / 16.0 * std::pow(w, 5) +	11 ✔
489	(45045.0 / 16.0) * std::pow(w, 3) - 3465.0 / 16.0 * w) *	22 ✔
490	std::sin(2. * phi);	11 ✔
491	// l = 9, m = -1
492	rn[i + 8] = -(	22 ✔
493	0.149071198499986 * std::sqrt(w2m1) *	11 ✔
494	((109395.0 / 128.0) * std::pow(w, 8) - 45045.0 / 32.0 * std::pow(w, 6) +	11 ✔
495	(45045.0 / 64.0) * std::pow(w, 4) - 3465.0 / 32.0 * w * w +	11 ✔
496	315.0 / 128.0) *	11 ✔
497	std::sin(phi));	11 ✔
498	// l = 9, m = 0
499	rn[i + 9] = 94.9609375 * std::pow(w, 9) - 201.09375 * std::pow(w, 7) +	11 ✔
500	140.765625 * std::pow(w, 5) - 36.09375 * std::pow(w, 3) +	11 ✔
501	2.4609375 * w;	11 ✔
502	// l = 9, m = 1
503	rn[i + 10] = -(	22 ✔
504	0.149071198499986 * std::sqrt(w2m1) *	11 ✔
505	((109395.0 / 128.0) * std::pow(w, 8) - 45045.0 / 32.0 * std::pow(w, 6) +	11 ✔
506	(45045.0 / 64.0) * std::pow(w, 4) - 3465.0 / 32.0 * w * w +	11 ✔
507	315.0 / 128.0) *	11 ✔
508	std::cos(phi));	11 ✔
509	// l = 9, m = 2
510	rn[i + 11] =	22 ✔
511	0.0158910431540932 * (w2m1) *	11 ✔
512	((109395.0 / 16.0) * std::pow(w, 7) - 135135.0 / 16.0 * std::pow(w, 5) +	11 ✔
513	(45045.0 / 16.0) * std::pow(w, 3) - 3465.0 / 16.0 * w) *	22 ✔
514	std::cos(2. * phi);	11 ✔
515	// l = 9, m = 3
516	rn[i + 12] = -(	22 ✔
517	0.00173385495536766 * std::pow(w2m1, 1.5) *	11 ✔
518	((765765.0 / 16.0) * std::pow(w, 6) - 675675.0 / 16.0 * std::pow(w, 4) +	11 ✔
519	(135135.0 / 16.0) * w * w - 3465.0 / 16.0) *	22 ✔
520	std::cos(3. * phi));	11 ✔
521	// l = 9, m = 4
522	rn[i + 13] = 0.000196320414650061 * w2m1 * w2m1 *	11 ✔
523	((2297295.0 / 8.0) * std::pow(w, 5) -	11 ✔
524	675675.0 / 4.0 * std::pow(w, 3) + (135135.0 / 8.0) * w) *	22 ✔
525	std::cos(4.0 * phi);	11 ✔
526	// l = 9, m = 5
527	rn[i + 14] = -(2.34647776186144e-5 * std::pow(w2m1, 2.5) *	11 ✔
528	((11486475.0 / 8.0) * std::pow(w, 4) -	11 ✔
529	2027025.0 / 4.0 * w * w + 135135.0 / 8.0) *	22 ✔
530	std::cos(5.0 * phi));	11 ✔
531	// l = 9, m = 6
532	rn[i + 15] = 3.02928976464514e-6 * std::pow(w2m1, 3) *	11 ✔
533	((11486475.0 / 2.) * std::pow(w, 3) - 2027025.0 / 2. * w) *	22 ✔
534	std::cos(6.0 * phi);	11 ✔
535	// l = 9, m = 7
536	rn[i + 16] =	22 ✔
537	-(4.37240315267812e-7 * std::pow(w2m1, 3.5) *	11 ✔
538	((34459425.0 / 2.) * w * w - 2027025.0 / 2.) * std::cos(7.0 * phi));	11 ✔
539	// l = 9, m = 8
540	rn[i + 17] =	22 ✔
541	2.58397773170915 * w * std::pow(w2m1, 4) * std::cos(8.0 * phi);	11 ✔
542	// l = 9, m = 9
543	rn[i + 18] =	22 ✔
544	-(0.609049392175524 * std::pow(w2m1, 4.5) * std::cos(9.0 * phi));	11 ✔
545	break;	11 ✔
546	case 10:	11 ✔
547	// l = 10, m = -10
548	rn[i] = 0.593627917136573 * std::pow(w2m1, 5) * std::sin(10.0 * phi);	11 ✔
549	// l = 10, m = -9
550	rn[i + 1] =	22 ✔
551	-(2.65478475211798 * w * std::pow(w2m1, 4.5) * std::sin(9.0 * phi));	11 ✔
552	// l = 10, m = -8
553	rn[i + 2] = 2.49953651452314e-8 * std::pow(w2m1, 4) *	11 ✔
554	((654729075.0 / 2.) * w * w - 34459425.0 / 2.) *	22 ✔
555	std::sin(8.0 * phi);	11 ✔
556	// l = 10, m = -7
557	rn[i + 3] =	22 ✔
558	-(1.83677671621093e-7 * std::pow(w2m1, 3.5) *	11 ✔
559	((218243025.0 / 2.) * std::pow(w, 3) - 34459425.0 / 2. * w) *	22 ✔
560	std::sin(7.0 * phi));	11 ✔
561	// l = 10, m = -6
562	rn[i + 4] = 1.51464488232257e-6 * std::pow(w2m1, 3) *	11 ✔
563	((218243025.0 / 8.0) * std::pow(w, 4) -	11 ✔
564	34459425.0 / 4.0 * w * w + 2027025.0 / 8.0) *	22 ✔
565	std::sin(6.0 * phi);	11 ✔
566	// l = 10, m = -5
567	rn[i + 5] =	22 ✔
568	-(1.35473956745817e-5 * std::pow(w2m1, 2.5) *	11 ✔
569	((43648605.0 / 8.0) * std::pow(w, 5) -	11 ✔
570	11486475.0 / 4.0 * std::pow(w, 3) + (2027025.0 / 8.0) * w) *	22 ✔
571	std::sin(5.0 * phi));	11 ✔
572	// l = 10, m = -4
573	rn[i + 6] = 0.000128521880085575 * w2m1 * w2m1 *	11 ✔
574	((14549535.0 / 16.0) * std::pow(w, 6) -	11 ✔
575	11486475.0 / 16.0 * std::pow(w, 4) +	11 ✔
576	(2027025.0 / 16.0) * w * w - 45045.0 / 16.0) *	22 ✔
577	std::sin(4.0 * phi);	11 ✔
578	// l = 10, m = -3
579	rn[i + 7] = -(0.00127230170115096 * std::pow(w2m1, 1.5) *	11 ✔
580	((2078505.0 / 16.0) * std::pow(w, 7) -	11 ✔
581	2297295.0 / 16.0 * std::pow(w, 5) +	11 ✔
582	(675675.0 / 16.0) * std::pow(w, 3) - 45045.0 / 16.0 * w) *	22 ✔
583	std::sin(3. * phi));	11 ✔
584	// l = 10, m = -2
585	rn[i + 8] = 0.012974982402692 * (w2m1) *	11 ✔
586	((2078505.0 / 128.0) * std::pow(w, 8) -	11 ✔
587	765765.0 / 32.0 * std::pow(w, 6) +	11 ✔
588	(675675.0 / 64.0) * std::pow(w, 4) -	11 ✔
589	45045.0 / 32.0 * w * w + 3465.0 / 128.0) *	22 ✔
590	std::sin(2. * phi);	11 ✔
591	// l = 10, m = -1
592	rn[i + 9] = -(0.134839972492648 * std::sqrt(w2m1) *	11 ✔
593	((230945.0 / 128.0) * std::pow(w, 9) -	11 ✔
594	109395.0 / 32.0 * std::pow(w, 7) +	11 ✔
595	(135135.0 / 64.0) * std::pow(w, 5) -	11 ✔
596	15015.0 / 32.0 * std::pow(w, 3) + (3465.0 / 128.0) * w) *	22 ✔
597	std::sin(phi));	11 ✔
598	// l = 10, m = 0
599	rn[i + 10] = 180.42578125 * std::pow(w, 10) -	11 ✔
600	427.32421875 * std::pow(w, 8) +	11 ✔
601	351.9140625 * std::pow(w, 6) - 117.3046875 * std::pow(w, 4) +	11 ✔
602	13.53515625 * w * w - 0.24609375;	11 ✔
603	// l = 10, m = 1
604	rn[i + 11] = -(0.134839972492648 * std::sqrt(w2m1) *	11 ✔
605	((230945.0 / 128.0) * std::pow(w, 9) -	11 ✔
606	109395.0 / 32.0 * std::pow(w, 7) +	11 ✔
607	(135135.0 / 64.0) * std::pow(w, 5) -	11 ✔
608	15015.0 / 32.0 * std::pow(w, 3) + (3465.0 / 128.0) * w) *	22 ✔
609	std::cos(phi));	11 ✔
610	// l = 10, m = 2
611	rn[i + 12] = 0.012974982402692 * (w2m1) *	11 ✔
612	((2078505.0 / 128.0) * std::pow(w, 8) -	11 ✔
613	765765.0 / 32.0 * std::pow(w, 6) +	11 ✔
614	(675675.0 / 64.0) * std::pow(w, 4) -	11 ✔
615	45045.0 / 32.0 * w * w + 3465.0 / 128.0) *	22 ✔
616	std::cos(2. * phi);	11 ✔
617	// l = 10, m = 3
618	rn[i + 13] =	22 ✔
619	-(0.00127230170115096 * std::pow(w2m1, 1.5) *	11 ✔
620	((2078505.0 / 16.0) * std::pow(w, 7) -	11 ✔
621	2297295.0 / 16.0 * std::pow(w, 5) +	11 ✔
622	(675675.0 / 16.0) * std::pow(w, 3) - 45045.0 / 16.0 * w) *	22 ✔
623	std::cos(3. * phi));	11 ✔
624	// l = 10, m = 4
625	rn[i + 14] = 0.000128521880085575 * w2m1 * w2m1 *	11 ✔
626	((14549535.0 / 16.0) * std::pow(w, 6) -	11 ✔
627	11486475.0 / 16.0 * std::pow(w, 4) +	11 ✔
628	(2027025.0 / 16.0) * w * w - 45045.0 / 16.0) *	22 ✔
629	std::cos(4.0 * phi);	11 ✔
630	// l = 10, m = 5
631	rn[i + 15] =	22 ✔
632	-(1.35473956745817e-5 * std::pow(w2m1, 2.5) *	11 ✔
633	((43648605.0 / 8.0) * std::pow(w, 5) -	11 ✔
634	11486475.0 / 4.0 * std::pow(w, 3) + (2027025.0 / 8.0) * w) *	22 ✔
635	std::cos(5.0 * phi));	11 ✔
636	// l = 10, m = 6
637	rn[i + 16] = 1.51464488232257e-6 * std::pow(w2m1, 3) *	11 ✔
638	((218243025.0 / 8.0) * std::pow(w, 4) -	11 ✔
639	34459425.0 / 4.0 * w * w + 2027025.0 / 8.0) *	22 ✔
640	std::cos(6.0 * phi);	11 ✔
641	// l = 10, m = 7
642	rn[i + 17] =	22 ✔
643	-(1.83677671621093e-7 * std::pow(w2m1, 3.5) *	11 ✔
644	((218243025.0 / 2.) * std::pow(w, 3) - 34459425.0 / 2. * w) *	22 ✔
645	std::cos(7.0 * phi));	11 ✔
646	// l = 10, m = 8
647	rn[i + 18] = 2.49953651452314e-8 * std::pow(w2m1, 4) *	11 ✔
648	((654729075.0 / 2.) * w * w - 34459425.0 / 2.) *	11 ✔
649	std::cos(8.0 * phi);	11 ✔
650	// l = 10, m = 9
651	rn[i + 19] =	22 ✔
652	-(2.65478475211798 * w * std::pow(w2m1, 4.5) * std::cos(9.0 * phi));	11 ✔
653	// l = 10, m = 10
654	rn[i + 20] = 0.593627917136573 * std::pow(w2m1, 5) * std::cos(10.0 * phi);	11 ✔
655	}
656	}
657	}	4,568,344 ✔
658
659	void calc_zn(int n, double rho, double phi, double zn[])	2,704,570 ✔
660	{
661	// ===========================================================================
662	// Determine vector of sin(nphi) and cos(nphi). This takes advantage of the
663	// following recurrence relations so that only a single sin/cos have to be
664	// evaluated (https://mathworld.wolfram.com/Multiple-AngleFormulas.html)
665	//
666	// sin(nx) = 2 cos(x) sin((n-1)x) - sin((n-2)x)
667	// cos(nx) = 2 cos(x) cos((n-1)x) - cos((n-2)x)
668
669	double sin_phi = std::sin(phi);	2,704,570 ✔
670	double cos_phi = std::cos(phi);	2,704,570 ✔
671
672	vector<double> sin_phi_vec(n + 1); // Sin[n * phi]	2,704,570 ✔
673	vector<double> cos_phi_vec(n + 1); // Cos[n * phi]	2,704,570 ✔
674	sin_phi_vec[0] = 1.0;	2,704,570 ✔
675	cos_phi_vec[0] = 1.0;	2,704,570 ✔
676	sin_phi_vec[1] = 2.0 * cos_phi;	2,704,570 ✔
677	cos_phi_vec[1] = cos_phi;	2,704,570 ✔
678
679	for (int i = 2; i <= n; i++) {	13,519,715 ✔
680	sin_phi_vec[i] = 2. * cos_phi * sin_phi_vec[i - 1] - sin_phi_vec[i - 2];	10,815,145 ✔
681	cos_phi_vec[i] = 2. * cos_phi * cos_phi_vec[i - 1] - cos_phi_vec[i - 2];	10,815,145 ✔
682	}
683
684	for (int i = 0; i <= n; i++) {	18,928,855 ✔
685	sin_phi_vec[i] *= sin_phi;	16,224,285 ✔
686	}
687
688	// ===========================================================================
689	// Calculate R_pq(rho)
690	// Matrix forms of the coefficients which are easier to work with
691	vector<vector<double>> zn_mat(n + 1, vector<double>(n + 1));	4,425,660 ✔
692
693	// Fill the main diagonal first (Eq 3.9 in Chong)
694	for (int p = 0; p <= n; p++) {	18,928,855 ✔
695	zn_mat[p][p] = std::pow(rho, p);	16,224,285 ✔
696	}
697
698	// Fill the 2nd diagonal (Eq 3.10 in Chong)
699	for (int q = 0; q <= n - 2; q++) {	13,519,715 ✔
700	zn_mat[q][q + 2] = (q + 2) * zn_mat[q + 2][q + 2] - (q + 1) * zn_mat[q][q];	10,815,145 ✔
701	}
702
703	// Fill in the rest of the values using the original results (Eq. 3.8 in
704	// Chong)
705	for (int p = 4; p <= n; p++) {	8,110,575 ✔
706	double k2 = 2 * p * (p - 1) * (p - 2);	5,406,005 ✔
707	for (int q = p - 4; q >= 0; q -= 2) {	10,812,109 ✔
708	double k1 = ((p + q) * (p - q) * (p - 2)) / 2.;	5,406,104 ✔
709	double k3 = -q * q * (p - 1) - p * (p - 1) * (p - 2);	5,406,104 ✔
710	double k4 = (-p * (p + q - 2) * (p - q - 2)) / 2.;	5,406,104 ✔
711	zn_mat[q][p] =	5,406,104 ✔
712	((k2 * rho * rho + k3) * zn_mat[q][p - 2] + k4 * zn_mat[q][p - 4]) / k1;	5,406,104 ✔
713	}
714	}
715
716	// Roll into a single vector for easier computation later
717	// The vector is ordered (0,0), (1,-1), (1,1), (2,-2), (2,0),
718	// (2, 2), .... in (n,m) indices
719	// Note that the cos and sin vectors are offset by one
720	// sin_phi_vec = [sin(x), sin(2x), sin(3x) ...]
721	// cos_phi_vec = [1.0, cos(x), cos(2x)... ]
722	int i = 0;
723	for (int p = 0; p <= n; p++) {	18,928,855 ✔
724	for (int q = -p; q <= p; q += 2) {	73,003,106 ✔
725	if (q < 0) {	56,778,821 ✔
726	zn[i] = zn_mat[std::abs(q)][p] * sin_phi_vec[std::abs(q) - 1];	24,333,287 ✔
727	} else if (q == 0) {	32,445,534 ✔
728	zn[i] = zn_mat[q][p];	8,112,247 ✔
729	} else {
730	zn[i] = zn_mat[q][p] * cos_phi_vec[q];	24,333,287 ✔
731	}
732	i++;	56,778,821 ✔
733	}
734	}
735	}	2,704,570 ✔
736
737	void calc_zn_rad(int n, double rho, double zn_rad[])	44 ✔
738	{
739	// Calculate R_p0(rho) as Zn_p0(rho)
740	// Set up the array of the coefficients
741
742	double q = 0;	44 ✔
743
744	// R_00 is always 1
745	zn_rad[0] = 1;	44 ✔
746
747	// Fill in the rest of the array (Eq 3.8 and Eq 3.10 in Chong)
748	for (int p = 2; p <= n; p += 2) {	264 ✔
749	int index = int(p / 2);	220 ✔
750	if (p == 2) {	220 ✔
751	// Setting up R_22 to calculate R_20 (Eq 3.10 in Chong)
752	double R_22 = rho * rho;	44 ✔
753	zn_rad[index] = 2 * R_22 - zn_rad[0];	44 ✔
754	} else {
755	double k1 = ((p + q) * (p - q) * (p - 2)) / 2.;	176 ✔
756	double k2 = 2 * p * (p - 1) * (p - 2);	176 ✔
757	double k3 = -q * q * (p - 1) - p * (p - 1) * (p - 2);	176 ✔
758	double k4 = (-p * (p + q - 2) * (p - q - 2)) / 2.;	176 ✔
759	zn_rad[index] =	176 ✔
760	((k2 * rho * rho + k3) * zn_rad[index - 1] + k4 * zn_rad[index - 2]) /	176 ✔
761	k1;
762	}
763	}
764	}	44 ✔
765
766	void rotate_angle_c(double uvw[3], double mu, const double* phi, uint64_t* seed)	33 ✔
767	{
768	Direction u = rotate_angle({uvw}, mu, phi, seed);	33 ✔
769	uvw[0] = u.x;	33 ✔
770	uvw[1] = u.y;	33 ✔
771	uvw[2] = u.z;	33 ✔
772	}	33 ✔
773
774	Direction rotate_angle(	2,147,483,647 ✔
775	Direction u, double mu, const double* phi, uint64_t* seed)
776	{
777	// Sample azimuthal angle in [0,2pi) if none provided
778	double phi_;	2,147,483,647 ✔
779	if (phi != nullptr) {	2,147,483,647 ✔
780	phi_ = (*phi);	49,164,912 ✔
781	} else {
782	phi_ = 2.0 * PI * prn(seed);	2,147,483,647 ✔
783	}
784
785	// Precompute factors to save flops
786	double sinphi = std::sin(phi_);	2,147,483,647 ✔
787	double cosphi = std::cos(phi_);	2,147,483,647 ✔
788	double a = std::sqrt(std::fmax(0., 1. - mu * mu));	2,147,483,647 ✔
789	double b = std::sqrt(std::fmax(0., 1. - u.z * u.z));	2,147,483,647 ✔
790
791	// Need to treat special case where sqrt(1 - w**2) is close to zero by
792	// expanding about the v component rather than the w component
793	if (b > 1e-10) {	2,147,483,647 ✔
794	return {mu * u.x + a * (u.x * u.z * cosphi - u.y * sinphi) / b,	2,147,483,647 ✔
795	mu * u.y + a * (u.y * u.z * cosphi + u.x * sinphi) / b,	2,147,483,647 ✔
796	mu * u.z - a * b * cosphi};	2,147,483,647 ✔
797	} else {
798	b = std::sqrt(1. - u.y * u.y);	391,457 ✔
799	return {mu * u.x + a * (-u.x * u.y * sinphi + u.z * cosphi) / b,	391,457 ✔
800	mu * u.y + a * b * sinphi,	391,457 ✔
801	mu * u.z - a * (u.y * u.z * sinphi + u.x * cosphi) / b};	391,457 ✔
802	}
803	}
804
805	void spline(int n, const double x[], const double y[], double z[])	196,560 ✔
806	{
807	vector<double> c_new(n - 1);	196,560 ✔
808
809	// Set natural boundary conditions
810	c_new[0] = 0.0;	196,560 ✔
811	z[0] = 0.0;	196,560 ✔
812	z[n - 1] = 0.0;	196,560 ✔
813
814	// Solve using tridiagonal matrix algorithm; first do forward sweep
815	for (int i = 1; i < n - 1; i++) {	19,731,104 ✔
816	double a = x[i] - x[i - 1];	19,534,544 ✔
817	double c = x[i + 1] - x[i];	19,534,544 ✔
818	double b = 2.0 * (a + c);	19,534,544 ✔
819	double d = 6.0 * ((y[i + 1] - y[i]) / c - (y[i] - y[i - 1]) / a);	19,534,544 ✔
820
821	c_new[i] = c / (b - a * c_new[i - 1]);	19,534,544 ✔
822	z[i] = (d - a * z[i - 1]) / (b - a * c_new[i - 1]);	19,534,544 ✔
823	}
824
825	// Back substitution
826	for (int i = n - 2; i >= 0; i--) {	19,927,664 ✔
827	z[i] = z[i] - c_new[i] * z[i + 1];	19,731,104 ✔
828	}
829	}	196,560 ✔
830
UNCOV 831	double spline_interpolate(	×
832	int n, const double x[], const double y[], const double z[], double xint)
833	{
834	// Find the lower bounding index in x of xint
UNCOV 835	int i = n - 1;	×
UNCOV 836	while (--i) {	×
837	if (xint >= x[i])	×
838	break;
839	}
840
UNCOV 841	double h = x[i + 1] - x[i];	×
UNCOV 842	double r = xint - x[i];	×
843
844	// Compute the coefficients
UNCOV 845	double b = (y[i + 1] - y[i]) / h - (h / 6.0) * (z[i + 1] + 2.0 * z[i]);	×
UNCOV 846	double c = z[i] / 2.0;	×
847	double d = (z[i + 1] - z[i]) / (h * 6.0);	×
848
849	return y[i] + b * r + c * r * r + d * r * r * r;	×
850	}
851
852	double spline_integrate(int n, const double x[], const double y[],	19,731,104 ✔
853	const double z[], double xa, double xb)
854	{
855	// Find the lower bounding index in x of the lower limit of integration.
856	int ia = n - 1;	19,731,104 ✔
857	while (--ia) {	1,320,604,202 ✔
858	if (xa >= x[ia])	1,320,407,642 ✔
859	break;
860	}
861
862	// Find the lower bounding index in x of the upper limit of integration.
863	int ib = n - 1;
864	while (--ib) {	1,301,069,658 !
865	if (xb >= x[ib])	1,301,069,658 ✔
866	break;
867	}
868
869	// Evaluate the integral
870	double s = 0.0;
871	for (int i = ia; i <= ib; i++) {	58,996,752 ✔
872	double h = x[i + 1] - x[i];	39,265,648 ✔
873
874	// Compute the coefficients
875	double b = (y[i + 1] - y[i]) / h - (h / 6.0) * (z[i + 1] + 2.0 * z[i]);	39,265,648 ✔
876	double c = z[i] / 2.0;	39,265,648 ✔
877	double d = (z[i + 1] - z[i]) / (h * 6.0);	39,265,648 ✔
878
879	// Subtract the integral from x[ia] to xa
880	if (i == ia) {	39,265,648 ✔
881	double r = xa - x[ia];	19,731,104 ✔
882	s = s - (y[i] * r + b / 2.0 * r * r + c / 3.0 * r * r * r +	19,731,104 ✔
883	d / 4.0 * r * r * r * r);	19,731,104 ✔
884	}
885
886	// Integrate from x[ib] to xb in final interval
887	if (i == ib) {	39,265,648 ✔
888	h = xb - x[ib];	19,731,104 ✔
889	}
890
891	// Accumulate the integral
892	s = s + y[i] * h + b / 2.0 * h * h + c / 3.0 * h * h * h +	39,265,648 ✔
893	d / 4.0 * h * h * h * h;	39,265,648 ✔
894	}
895
896	return s;	19,731,104 ✔
897	}
898
899	std::complex<double> faddeeva(std::complex<double> z)	254,391,841 ✔
900	{
901	// Technically, the value we want is given by the equation:
902	// w(z) = I/pi * Integrate[Exp[-t^2]/(z-t), {t, -Infinity, Infinity}]
903	// as shown in Equation 63 from Hwang, R. N. "A rigorous pole
904	// representation of multilevel cross sections and its practical
905	// applications." Nucl. Sci. Eng. 96.3 (1987): 192-209.
906	//
907	// The MIT Faddeeva function evaluates w(z) = exp(-z^2)erfc(-iz). These
908	// two forms of the Faddeeva function are related by a transformation.
909	//
910	// If we call the integral form w_int, and the function form w_fun:
911	// For imag(z) > 0, w_int(z) = w_fun(z)
912	// For imag(z) < 0, w_int(z) = -conjg(w_fun(conjg(z)))
913
914	// Note that Faddeeva::w will interpret zero as machine epsilon
915	return z.imag() > 0.0 ? Faddeeva::w(z)	254,391,841 !
916	: -std::conj(Faddeeva::w(std::conj(z)));	254,391,841 !
917	}
918
919	std::complex<double> w_derivative(std::complex<double> z, int order)	4,232,547 ✔
920	{
921	using namespace std::complex_literals;	4,232,547 ✔
922	switch (order) {	4,232,547 ✔
923	case 0:	1,410,849 ✔
924	return faddeeva(z);	1,410,849 ✔
925	case 1:	1,410,849 ✔
926	return -2.0 * z * faddeeva(z) + 2.0i / SQRT_PI;	1,410,849 ✔
927	default:	1,410,849 ✔
928	return -2.0 * z * w_derivative(z, order - 1) -	1,410,849 ✔
929	2.0 * (order - 1) * w_derivative(z, order - 2);	1,410,849 ✔
930	}
931	}
932
UNCOV 933	double exprel(double x)	×
934	{
935	if (std::abs(x) < 1e-16)	×
936	return 1.0;
937	else {
UNCOV 938	return std::expm1(x) / x;	×
939	}
940	}
941
UNCOV 942	double log1prel(double x)	×
943	{
944	if (std::abs(x) < 1e-16)	×
945	return 1.0;
946	else {
UNCOV 947	return std::log1p(x) / x;	×
948	}
949	}
950
951	double cyl_bessel_j(int n, double x)	264 ✔
952	{
953	// Handle negative arguments via the parity relation
954	// J_n(-x) = (-1)^n J_n(x); std::cyl_bessel_j has a domain error for x < 0.
955	double sign = 1.0;	264 ✔
956	if (x < 0.0) {	264 ✔
957	x = -x;	88 ✔
958	if (n % 2 == 1)	88 ✔
959	sign = -1.0;	44 ✔
960	}
961
962	#if defined(__cpp_lib_math_special_functions) && \
963	__cpp_lib_math_special_functions >= 201603L
964	return sign * std::cyl_bessel_j(static_cast<double>(n), x);	264 ✔
965	#else
966	// Ascending power series (e.g., Abramowitz & Stegun eq. 9.1.10):
967	// J_n(x) = sum_{m=0}^inf (-1)^m / (m! (m+n)!) * (x/2)^(2m+n)
968	// The term ratio is -(x/2)^2 / (m*(m+n)), so for \|x\| <= 2 the series
969	// converges to machine precision within ~20 terms.
970	double half_x = 0.5 * x;
971
972	// First term: (x/2)^n / n!
973	double term = 1.0;
974	for (int k = 1; k <= n; ++k) {
975	term *= half_x / k;
976	}
977
978	double sum = term;
979	double neg_half_x_sq = -half_x * half_x;
980	for (int m = 1; m <= 50; ++m) {
981	term = neg_half_x_sq / (m (m + n));
982	sum += term;
983	if (std::abs(term) <=
984	std::numeric_limits<double>::epsilon() * std::abs(sum))
985	break;
986	}
987	return sign * sum;
988	#endif
989	}
990
991	// Helper function to get index and interpolation function on an incident energy
992	// grid
993	void get_energy_index(	1,652,593,314 ✔
994	const vector<double>& energies, double E, int& i, double& f)
995	{
996	// Get index and interpolation factor for linear-linear energy grid
997	i = 0;	1,652,593,314 ✔
998	f = 0.0;	1,652,593,314 ✔
999	if (E >= energies.front()) {	1,652,593,314 ✔
1000	i = lower_bound_index(energies.begin(), energies.end(), E);	1,652,591,860 ✔
1001	if (i + 1 < energies.size())	1,652,591,860 ✔
1002	f = (E - energies[i]) / (energies[i + 1] - energies[i]);	1,652,590,518 ✔
1003	}
1004	}	1,652,593,314 ✔
1005
1006	// Return true if two floating-point values are approximately equal within a
1007	// combined relative and absolute tolerance.
1008	bool isclose(double a, double b, double rel_tol, double abs_tol)	2,147,483,647 ✔
1009	{
1010	return std::abs(a - b) <=	2,147,483,647 ✔
1011	std::max(rel_tol * std::max(std::abs(a), std::abs(b)), abs_tol);	2,147,483,647 ✔
1012	}
1013
1014	bool combine_estimates(const array<double, 3>& estimates,	11,166 ✔
1015	const tensor::StaticTensor2D<double, 3, 3>& cov, int64_t n,
1016	array<double, 2>& combined)
1017	{
1018	combined[0] = 0.0;	11,166 ✔
1019	combined[1] = 0.0;	11,166 ✔
1020
1021	// The three-estimate expression has an n-3 term in a denominator, and the
1022	// two-estimate expression an n-2 term
1023	if (n <= 3)	11,166 ✔
1024	return false;
1025
1026	// Check to see if two estimates are the same. If they are, the three
1027	// estimate expressions are singular and will produce floating-point
1028	// exceptions, so an expression specifically derived for the combination of
1029	// two estimates (vice three) is used instead.
1030
1031	// First we will identify if there are any matching estimates
1032	int i, j;	10,352 ✔
1033	bool use_three = false;	10,352 ✔
1034	if ((std::abs(estimates[0] - estimates[1]) / estimates[0] <	10,352 ✔
1035	FP_REL_PRECISION) &&	10,352 ✔
1036	(std::abs(cov(0, 0) - cov(1, 1)) / cov(0, 0) < FP_REL_PRECISION)) {	66 ✔
1037	// 0 and 1 match, so only use 0 and 2 in our comparisons
1038	i = 0;
1039	j = 2;
1040
1041	} else if ((std::abs(estimates[0] - estimates[2]) / estimates[0] <	10,297 ✔
1042	FP_REL_PRECISION) &&	10,297 !
1043	(std::abs(cov(0, 0) - cov(2, 2)) / cov(0, 0) < FP_REL_PRECISION)) {	11 !
1044	// 0 and 2 match, so only use 0 and 1 in our comparisons
1045	i = 0;
1046	j = 1;
1047
1048	} else if ((std::abs(estimates[1] - estimates[2]) / estimates[1] <	10,297 ✔
1049	FP_REL_PRECISION) &&	10,297 !
1050	(std::abs(cov(1, 1) - cov(2, 2)) / cov(1, 1) < FP_REL_PRECISION)) {	11 !
1051	// 1 and 2 match, so only use 0 and 1 in our comparisons
1052	i = 0;
1053	j = 1;
1054
1055	} else {
1056	// No two estimates match, so set boolean to use all three estimates.
1057	use_three = true;	10,297 ✔
1058	}
1059
1060	if (use_three) {	10,297 ✔
1061	// Use three estimates as derived in the paper by Urbatsch
1062
1063	// Initialize variables
1064	double g = 0.0;	10,297 ✔
1065	array<double, 3> S {};	10,297 ✔
1066
1067	for (int l = 0; l < 3; ++l) {	41,188 ✔
1068	// Permutations of the three estimates
1069	int k;	30,891 ✔
1070	switch (l) {	30,891 ✔
1071	case 0:
1072	i = 0;
1073	j = 1;
1074	k = 2;
1075	break;
1076	case 1:	10,297 ✔
1077	i = 1;	10,297 ✔
1078	j = 2;	10,297 ✔
1079	k = 0;	10,297 ✔
1080	break;	10,297 ✔
1081	case 2:	10,297 ✔
1082	i = 2;	10,297 ✔
1083	j = 0;	10,297 ✔
1084	k = 1;	10,297 ✔
1085	break;	10,297 ✔
1086	}
1087
1088	// Calculate weighting
1089	double f = cov(j, j) * (cov(k, k) - cov(i, k)) - cov(k, k) * cov(i, j) +	30,891 ✔
1090	cov(j, k) * (cov(i, j) + cov(i, k) - cov(j, k));	30,891 ✔
1091
1092	// Add to S sums for variance of combined estimate
1093	S[0] += f * cov(0, l);	30,891 ✔
1094	S[1] +=	30,891 ✔
1095	(cov(j, j) + cov(k, k) - 2.0 * cov(j, k)) * estimates[l] * estimates[l];	30,891 ✔
1096	S[2] += (cov(k, k) + cov(i, j) - cov(j, k) - cov(i, k)) * estimates[l] *	30,891 ✔
1097	estimates[j];	30,891 ✔
1098
1099	// Add to sum for the combination
1100	combined[0] += f * estimates[l];	30,891 ✔
1101	g += f;	30,891 ✔
1102	}
1103
1104	// Complete calculations of S sums
1105	for (auto& S_i : S) {	41,188 ✔
1106	S_i *= (n - 1);	30,891 ✔
1107	}
1108	S[0] = (n - 1) (n - 1);	10,297 ✔
1109
1110	// Calculate the combination
1111	combined[0] /= g;	10,297 ✔
1112
1113	// Calculate standard deviation of the combination
1114	g = (n - 1) (n - 1);	10,297 ✔
1115	combined[1] =	10,297 ✔
1116	std::sqrt(S[0] / (g * n * (n - 3)) * (1 + n * ((S[1] - 2 * S[2]) / g)));	10,297 ✔
1117
1118	} else {
1119	// Use only two estimates
1120	// These equations are derived analogously to that done in the paper by
1121	// Urbatsch, but are simpler than for the three estimate case since the
1122	// block matrices of the three estimate equations reduces to scalars here
1123
1124	// Store the commonly used term
1125	double f = estimates[i] - estimates[j];	55 ✔
1126	double g = cov(i, i) + cov(j, j) - 2.0 * cov(i, j);	55 ✔
1127
1128	// Calculate the combination
1129	combined[0] = estimates[i] - (cov(i, i) - cov(i, j)) / g * f;	55 ✔
1130
1131	// Calculate standard deviation of the combination. Urbatsch's Eq. 40 is
1132	// written in terms of the matrix S rather than the sample covariance
1133	// Sigma = S / (n - 1). The factor cancels in the combination itself but
1134	// not here, and omitting it understates the standard deviation by up to
1135	// sqrt(n - 1).
1136	combined[1] = (cov(i, i) * cov(j, j) - cov(i, j) * cov(i, j)) *	55 ✔
1137	((n - 1) * g + n * f * f) / (n * (n - 2) * g * g);	55 ✔
1138	combined[1] = std::sqrt(combined[1]);	55 ✔
1139	}
1140
1141	return true;
1142	}
1143
1144	} // namespace openmc

openmc-dev / openmc / 34176864186

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