21621895430

Committed 03 Feb 2026 07:57AM UTC coverage: 81.72% (-0.04%) from 81.763%

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Pull #3413

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Merge 81c4418d8 into b41e22f68

Pull Request Pull Request #3413: More interpolation types in Tabular.

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94.84

/src/math_functions.cpp

1	#include "openmc/math_functions.h"
2
3	#include "openmc/external/Faddeeva.hh"
4
5	#include "openmc/constants.h"
6	#include "openmc/random_lcg.h"
7
8	namespace openmc {
9
10	//==============================================================================
11	// Mathematical methods
12	//==============================================================================
13
14	double normal_percentile(double p)	168✔
15	{
16	constexpr double p_low = 0.02425;	168✔
17	constexpr double a[6] = {-3.969683028665376e1, 2.209460984245205e2,
18	-2.759285104469687e2, 1.383577518672690e2, -3.066479806614716e1,
19	2.506628277459239e0};
20	constexpr double b[5] = {-5.447609879822406e1, 1.615858368580409e2,
21	-1.556989798598866e2, 6.680131188771972e1, -1.328068155288572e1};
22	constexpr double c[6] = {-7.784894002430293e-3, -3.223964580411365e-1,
23	-2.400758277161838, -2.549732539343734, 4.374664141464968,
24	2.938163982698783};
25	constexpr double d[4] = {7.784695709041462e-3, 3.224671290700398e-1,
26	2.445134137142996, 3.754408661907416};
27
28	// The rational approximation used here is from an unpublished work at
29	// http://home.online.no/~pjacklam/notes/invnorm/
30
31	double z;
32	double q;
33
34	if (p < p_low) {	168✔
35	// Rational approximation for lower region.
36
37	q = std::sqrt(-2.0 * std::log(p));	11✔
38	z = (((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]) /	11✔
39	((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1.0);	11✔
40
41	} else if (p <= 1.0 - p_low) {	157✔
42	// Rational approximation for central region
43	q = p - 0.5;	146✔
44	double r = q * q;	146✔
45	z = (((((a[0] * r + a[1]) * r + a[2]) * r + a[3]) * r + a[4]) * r + a[5]) *	146✔
46	q /
47	(((((b[0] * r + b[1]) * r + b[2]) * r + b[3]) * r + b[4]) * r + 1.0);	146✔
48
49	} else {
50	// Rational approximation for upper region
51
52	q = std::sqrt(-2.0 * std::log(1.0 - p));	11✔
53	z = -(((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]) /	11✔
54	((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1.0);	11✔
55	}
56
57	// Refinement based on Newton's method
58
59	z = z - (0.5 * std::erfc(-z / std::sqrt(2.0)) - p) * std::sqrt(2.0 * PI) *	168✔
60	std::exp(0.5 * z * z);	168✔
61
62	return z;	168✔
63	}
64
65	double t_percentile(double p, int df)	310✔
66	{
67	double t;
68
69	if (df == 1) {	310✔
70	// For one degree of freedom, the t-distribution becomes a Cauchy
71	// distribution whose cdf we can invert directly
72
73	t = std::tan(PI * (p - 0.5));	71✔
74	} else if (df == 2) {	239✔
75	// For two degrees of freedom, the cdf is given by 1/2 + x/(2*sqrt(x^2 +
76	// 2)). This can be directly inverted to yield the solution below
77
78	t = 2.0 * std::sqrt(2.0) * (p - 0.5) /	71✔
79	std::sqrt(1. - 4. * std::pow(p - 0.5, 2.));	71✔
80	} else {
81	// This approximation is from E. Olusegun George and Meenakshi Sivaram, "A
82	// modification of the Fisher-Cornish approximation for the student t
83	// percentiles," Communication in Statistics - Simulation and Computation,
84	// 16 (4), pp. 1123-1132 (1987).
85	double n = df;	168✔
86	double k = 1. / (n - 2.);	168✔
87	double z = normal_percentile(p);	168✔
88	double z2 = z * z;	168✔
89	t = std::sqrt(n * k) *	168✔
90	(z + (z2 - 3.) * z * k / 4. +	168✔
91	((5. * z2 - 56.) * z2 + 75.) * z * k * k / 96. +	168✔
92	(((z2 - 27.) * 3. * z2 + 417.) * z2 - 315.) * z * k * k * k / 384.);	168✔
93	}
94
95	return t;	310✔
96	}
97
98	void calc_pn_c(int n, double x, double pnx[])	367,667,679✔
99	{
100	pnx[0] = 1.;	367,667,679✔
101	if (n >= 1) {	367,667,679✔
102	pnx[1] = x;	114,455,363✔
103	}
104
105	// Use recursion relation to build the higher orders
106	for (int l = 1; l < n; l++) {	373,241,016✔
107	pnx[l + 1] = ((2 * l + 1) * x * pnx[l] - l * pnx[l - 1]) / (l + 1);	5,573,337✔
108	}
109	}	367,667,679✔
110
111	double evaluate_legendre(int n, const double data[], double x)	97,874,021✔
112	{
113	double* pnx = new double[n + 1];	97,874,021!
114	double val = 0.0;	97,874,021✔
115	calc_pn_c(n, x, pnx);	97,874,021✔
116	for (int l = 0; l <= n; l++) {	244,981,809✔
117	val += (l + 0.5) * data[l] * pnx[l];	147,107,788✔
118	}
119	delete[] pnx;	97,874,021!
120	return val;	97,874,021✔
121	}
122
123	void calc_rn_c(int n, const double uvw[3], double rn[])	11✔
124	{
125	Direction u {uvw};	11✔
126	calc_rn(n, u, rn);	11✔
127	}	11✔
128
129	void calc_rn(int n, Direction u, double rn[])	4,568,344✔
130	{
131	// rn[] is assumed to have already been allocated to the correct size
132
133	// Store the cosine of the polar angle and the azimuthal angle
134	double w = u.z;	4,568,344✔
135	double phi;
136	if (u.x == 0.) {	4,568,344!
137	phi = 0.;	×
138	} else {
139	phi = std::atan2(u.y, u.x);	4,568,344✔
140	}
141
142	// Store the shorthand of 1-w * w
143	double w2m1 = 1. - w * w;	4,568,344✔
144
145	// Now evaluate the spherical harmonics function
146	rn[0] = 1.;	4,568,344✔
147	int i = 0;	4,568,344✔
148	for (int l = 1; l <= n; l++) {	22,844,778✔
149	// Set the index to the start of this order
150	i += 2 * (l - 1) + 1;	18,276,434✔
151
152	// Now evaluate each
153	switch (l) {	18,276,434!
154	case 1:	4,568,344✔
155	// l = 1, m = -1
156	rn[i] = -(std::sqrt(w2m1) * std::sin(phi));	4,568,344✔
157	// l = 1, m = 0
158	rn[i + 1] = w;	4,568,344✔
159	// l = 1, m = 1
160	rn[i + 2] = -(std::sqrt(w2m1) * std::cos(phi));	4,568,344✔
161	break;	4,568,344✔
162	case 2:	4,568,344✔
163	// l = 2, m = -2
164	rn[i] = 0.288675134594813 * (-3. * w * w + 3.) * std::sin(2. * phi);	4,568,344✔
165	// l = 2, m = -1
166	rn[i + 1] = -(1.73205080756888 * w * std::sqrt(w2m1) * std::sin(phi));	4,568,344✔
167	// l = 2, m = 0
168	rn[i + 2] = 1.5 * w * w - 0.5;	4,568,344✔
169	// l = 2, m = 1
170	rn[i + 3] = -(1.73205080756888 * w * std::sqrt(w2m1) * std::cos(phi));	4,568,344✔
171	// l = 2, m = 2
172	rn[i + 4] = 0.288675134594813 * (-3. * w * w + 3.) * std::cos(2. * phi);	4,568,344✔
173	break;	4,568,344✔
174	case 3:	4,568,344✔
175	// l = 3, m = -3
176	rn[i] = -(0.790569415042095 * std::pow(w2m1, 1.5) * std::sin(3. * phi));	4,568,344✔
177	// l = 3, m = -2
178	rn[i + 1] = 1.93649167310371 * w * (w2m1)std::sin(2. phi);	4,568,344✔
179	// l = 3, m = -1
180	rn[i + 2] = -(0.408248290463863 * std::sqrt(w2m1) *	4,568,344✔
181	((7.5) * w * w - 3. / 2.) * std::sin(phi));	4,568,344✔
182	// l = 3, m = 0
183	rn[i + 3] = 2.5 * std::pow(w, 3) - 1.5 * w;	4,568,344✔
184	// l = 3, m = 1
185	rn[i + 4] = -(0.408248290463863 * std::sqrt(w2m1) *	4,568,344✔
186	((7.5) * w * w - 3. / 2.) * std::cos(phi));	4,568,344✔
187	// l = 3, m = 2
188	rn[i + 5] = 1.93649167310371 * w * (w2m1)std::cos(2. phi);	4,568,344✔
189	// l = 3, m = 3
190	rn[i + 6] =	4,568,344✔
191	-(0.790569415042095 * std::pow(w2m1, 1.5) * std::cos(3. * phi));	4,568,344✔
192	break;	4,568,344✔
193	case 4:	4,568,344✔
194	// l = 4, m = -4
195	rn[i] = 0.739509972887452 * (w2m1 * w2m1) * std::sin(4.0 * phi);	4,568,344✔
196	// l = 4, m = -3
197	rn[i + 1] =	4,568,344✔
198	-(2.09165006633519 * w * std::pow(w2m1, 1.5) * std::sin(3. * phi));	4,568,344✔
199	// l = 4, m = -2
200	rn[i + 2] =	4,568,344✔
201	0.074535599249993 * (w2m1) * (52.5 * w * w - 7.5) * std::sin(2. * phi);	4,568,344✔
202	// l = 4, m = -1
203	rn[i + 3] = -(0.316227766016838 * std::sqrt(w2m1) *	9,136,688✔
204	(17.5 * std::pow(w, 3) - 7.5 * w) * std::sin(phi));	4,568,344✔
205	// l = 4, m = 0
206	rn[i + 4] = 4.375 * std::pow(w, 4) - 3.75 * w * w + 0.375;	4,568,344✔
207	// l = 4, m = 1
208	rn[i + 5] = -(0.316227766016838 * std::sqrt(w2m1) *	9,136,688✔
209	(17.5 * std::pow(w, 3) - 7.5 * w) * std::cos(phi));	4,568,344✔
210	// l = 4, m = 2
211	rn[i + 6] =	4,568,344✔
212	0.074535599249993 * (w2m1) * (52.5 * w * w - 7.5) * std::cos(2. * phi);	4,568,344✔
213	// l = 4, m = 3
214	rn[i + 7] =	4,568,344✔
215	-(2.09165006633519 * w * std::pow(w2m1, 1.5) * std::cos(3. * phi));	4,568,344✔
216	// l = 4, m = 4
217	rn[i + 8] = 0.739509972887452 * w2m1 * w2m1 * std::cos(4.0 * phi);	4,568,344✔
218	break;	4,568,344✔
219	case 5:	3,003✔
220	// l = 5, m = -5
221	rn[i] = -(0.701560760020114 * std::pow(w2m1, 2.5) * std::sin(5.0 * phi));	3,003✔
222	// l = 5, m = -4
223	rn[i + 1] = 2.21852991866236 * w * w2m1 * w2m1 * std::sin(4.0 * phi);	3,003✔
224	// l = 5, m = -3
225	rn[i + 2] = -(0.00996023841111995 * std::pow(w2m1, 1.5) *	3,003✔
226	((945.0 / 2.) * w * w - 52.5) * std::sin(3. * phi));	3,003✔
227	// l = 5, m = -2
228	rn[i + 3] = 0.0487950036474267 * (w2m1) *	6,006✔
229	((315.0 / 2.) * std::pow(w, 3) - 52.5 * w) *	3,003✔
230	std::sin(2. * phi);	3,003✔
231	// l = 5, m = -1
232	rn[i + 4] =	3,003✔
233	-(0.258198889747161 * std::sqrt(w2m1) *	6,006✔
234	(39.375 * std::pow(w, 4) - 105.0 / 4.0 * w * w + 15.0 / 8.0) *	3,003✔
235	std::sin(phi));	3,003✔
236	// l = 5, m = 0
237	rn[i + 5] = 7.875 * std::pow(w, 5) - 8.75 * std::pow(w, 3) + 1.875 * w;	3,003✔
238	// l = 5, m = 1
239	rn[i + 6] =	3,003✔
240	-(0.258198889747161 * std::sqrt(w2m1) *	6,006✔
241	(39.375 * std::pow(w, 4) - 105.0 / 4.0 * w * w + 15.0 / 8.0) *	3,003✔
242	std::cos(phi));	3,003✔
243	// l = 5, m = 2
244	rn[i + 7] = 0.0487950036474267 * (w2m1) *	6,006✔
245	((315.0 / 2.) * std::pow(w, 3) - 52.5 * w) *	3,003✔
246	std::cos(2. * phi);	3,003✔
247	// l = 5, m = 3
248	rn[i + 8] = -(0.00996023841111995 * std::pow(w2m1, 1.5) *	3,003✔
249	((945.0 / 2.) * w * w - 52.5) * std::cos(3. * phi));	3,003✔
250	// l = 5, m = 4
251	rn[i + 9] = 2.21852991866236 * w * w2m1 * w2m1 * std::cos(4.0 * phi);	3,003✔
252	// l = 5, m = 5
253	rn[i + 10] =	3,003✔
254	-(0.701560760020114 * std::pow(w2m1, 2.5) * std::cos(5.0 * phi));	3,003✔
255	break;	3,003✔
256	case 6:	11✔
257	// l = 6, m = -6
258	rn[i] = 0.671693289381396 * std::pow(w2m1, 3) * std::sin(6.0 * phi);	11✔
259	// l = 6, m = -5
260	rn[i + 1] =	11✔
261	-(2.32681380862329 * w * std::pow(w2m1, 2.5) * std::sin(5.0 * phi));	11✔
262	// l = 6, m = -4
263	rn[i + 2] = 0.00104990131391452 * w2m1 * w2m1 *	11✔
264	((10395.0 / 2.) * w * w - 945.0 / 2.) * std::sin(4.0 * phi);	11✔
265	// l = 6, m = -3
266	rn[i + 3] = -(0.00575054632785295 * std::pow(w2m1, 1.5) *	22✔
267	((3465.0 / 2.) * std::pow(w, 3) - 945.0 / 2. * w) *	11✔
268	std::sin(3. * phi));	11✔
269	// l = 6, m = -2
270	rn[i + 4] =	11✔
271	0.0345032779671177 * (w2m1) *	22✔
272	((3465.0 / 8.0) * std::pow(w, 4) - 945.0 / 4.0 * w * w + 105.0 / 8.0) *	11✔
273	std::sin(2. * phi);	11✔
274	// l = 6, m = -1
275	rn[i + 5] = -(0.218217890235992 * std::sqrt(w2m1) *	22✔
276	((693.0 / 8.0) * std::pow(w, 5) -	11✔
277	315.0 / 4.0 * std::pow(w, 3) + (105.0 / 8.0) * w) *	11✔
278	std::sin(phi));	11✔
279	// l = 6, m = 0
280	rn[i + 6] = 14.4375 * std::pow(w, 6) - 19.6875 * std::pow(w, 4) +	11✔
281	6.5625 * w * w - 0.3125;	11✔
282	// l = 6, m = 1
283	rn[i + 7] = -(0.218217890235992 * std::sqrt(w2m1) *	22✔
284	((693.0 / 8.0) * std::pow(w, 5) -	11✔
285	315.0 / 4.0 * std::pow(w, 3) + (105.0 / 8.0) * w) *	11✔
286	std::cos(phi));	11✔
287	// l = 6, m = 2
288	rn[i + 8] =	11✔
289	0.0345032779671177 * w2m1 *	22✔
290	((3465.0 / 8.0) * std::pow(w, 4) - 945.0 / 4.0 * w * w + 105.0 / 8.0) *	11✔
291	std::cos(2. * phi);	11✔
292	// l = 6, m = 3
293	rn[i + 9] = -(0.00575054632785295 * std::pow(w2m1, 1.5) *	22✔
294	((3465.0 / 2.) * std::pow(w, 3) - 945.0 / 2. * w) *	11✔
295	std::cos(3. * phi));	11✔
296	// l = 6, m = 4
297	rn[i + 10] = 0.00104990131391452 * w2m1 * w2m1 *	11✔
298	((10395.0 / 2.) * w * w - 945.0 / 2.) * std::cos(4.0 * phi);	11✔
299	// l = 6, m = 5
300	rn[i + 11] =	11✔
301	-(2.32681380862329 * w * std::pow(w2m1, 2.5) * std::cos(5.0 * phi));	11✔
302	// l = 6, m = 6
303	rn[i + 12] = 0.671693289381396 * std::pow(w2m1, 3) * std::cos(6.0 * phi);	11✔
304	break;	11✔
305	case 7:	11✔
306	// l = 7, m = -7
307	rn[i] = -(0.647259849287749 * std::pow(w2m1, 3.5) * std::sin(7.0 * phi));	11✔
308	// l = 7, m = -6
309	rn[i + 1] =	11✔
310	2.42182459624969 * w * std::pow(w2m1, 3) * std::sin(6.0 * phi);	11✔
311	// l = 7, m = -5
312	rn[i + 2] =	11✔
313	-(9.13821798555235e-5 * std::pow(w2m1, 2.5) *	11✔
314	((135135.0 / 2.) * w * w - 10395.0 / 2.) * std::sin(5.0 * phi));	11✔
315	// l = 7, m = -4
316	rn[i + 3] = 0.000548293079133141 * w2m1 * w2m1 *	22✔
317	((45045.0 / 2.) * std::pow(w, 3) - 10395.0 / 2. * w) *	11✔
318	std::sin(4.0 * phi);	11✔
319	// l = 7, m = -3
320	rn[i + 4] = -(0.00363696483726654 * std::pow(w2m1, 1.5) *	22✔
321	((45045.0 / 8.0) * std::pow(w, 4) - 10395.0 / 4.0 * w * w +	11✔
322	945.0 / 8.0) *	11✔
323	std::sin(3. * phi));	11✔
324	// l = 7, m = -2
325	rn[i + 5] = 0.025717224993682 * (w2m1) *	22✔
326	((9009.0 / 8.0) * std::pow(w, 5) -	11✔
327	3465.0 / 4.0 * std::pow(w, 3) + (945.0 / 8.0) * w) *	11✔
328	std::sin(2. * phi);	11✔
329	// l = 7, m = -1
330	rn[i + 6] =	11✔
331	-(0.188982236504614 * std::sqrt(w2m1) *	22✔
332	((3003.0 / 16.0) * std::pow(w, 6) - 3465.0 / 16.0 * std::pow(w, 4) +	11✔
333	(945.0 / 16.0) * w * w - 35.0 / 16.0) *	11✔
334	std::sin(phi));	11✔
335	// l = 7, m = 0
336	rn[i + 7] = 26.8125 * std::pow(w, 7) - 43.3125 * std::pow(w, 5) +	11✔
337	19.6875 * std::pow(w, 3) - 2.1875 * w;	11✔
338	// l = 7, m = 1
339	rn[i + 8] =	11✔
340	-(0.188982236504614 * std::sqrt(w2m1) *	22✔
341	((3003.0 / 16.0) * std::pow(w, 6) - 3465.0 / 16.0 * std::pow(w, 4) +	11✔
342	(945.0 / 16.0) * w * w - 35.0 / 16.0) *	11✔
343	std::cos(phi));	11✔
344	// l = 7, m = 2
345	rn[i + 9] = 0.025717224993682 * (w2m1) *	22✔
346	((9009.0 / 8.0) * std::pow(w, 5) -	11✔
347	3465.0 / 4.0 * std::pow(w, 3) + (945.0 / 8.0) * w) *	11✔
348	std::cos(2. * phi);	11✔
349	// l = 7, m = 3
350	rn[i + 10] = -(0.00363696483726654 * std::pow(w2m1, 1.5) *	22✔
351	((45045.0 / 8.0) * std::pow(w, 4) - 10395.0 / 4.0 * w * w +	11✔
352	945.0 / 8.0) *	11✔
353	std::cos(3. * phi));	11✔
354	// l = 7, m = 4
355	rn[i + 11] = 0.000548293079133141 * w2m1 * w2m1 *	22✔
356	((45045.0 / 2.) * std::pow(w, 3) - 10395.0 / 2. * w) *	11✔
357	std::cos(4.0 * phi);	11✔
358	// l = 7, m = 5
359	rn[i + 12] =	11✔
360	-(9.13821798555235e-5 * std::pow(w2m1, 2.5) *	11✔
361	((135135.0 / 2.) * w * w - 10395.0 / 2.) * std::cos(5.0 * phi));	11✔
362	// l = 7, m = 6
363	rn[i + 13] =	11✔
364	2.42182459624969 * w * std::pow(w2m1, 3) * std::cos(6.0 * phi);	11✔
365	// l = 7, m = 7
366	rn[i + 14] =	11✔
367	-(0.647259849287749 * std::pow(w2m1, 3.5) * std::cos(7.0 * phi));	11✔
368	break;	11✔
369	case 8:	11✔
370	// l = 8, m = -8
371	rn[i] = 0.626706654240044 * std::pow(w2m1, 4) * std::sin(8.0 * phi);	11✔
372	// l = 8, m = -7
373	rn[i + 1] =	11✔
374	-(2.50682661696018 * w * std::pow(w2m1, 3.5) * std::sin(7.0 * phi));	11✔
375	// l = 8, m = -6
376	rn[i + 2] = 6.77369783729086e-6 * std::pow(w2m1, 3) *	11✔
377	((2027025.0 / 2.) * w * w - 135135.0 / 2.) *	11✔
378	std::sin(6.0 * phi);	11✔
379	// l = 8, m = -5
380	rn[i + 3] = -(4.38985792528482e-5 * std::pow(w2m1, 2.5) *	22✔
381	((675675.0 / 2.) * std::pow(w, 3) - 135135.0 / 2. * w) *	11✔
382	std::sin(5.0 * phi));	11✔
383	// l = 8, m = -4
384	rn[i + 4] = 0.000316557156832328 * w2m1 * w2m1 *	22✔
385	((675675.0 / 8.0) * std::pow(w, 4) - 135135.0 / 4.0 * w * w +	11✔
386	10395.0 / 8.0) *	11✔
387	std::sin(4.0 * phi);	11✔
388	// l = 8, m = -3
389	rn[i + 5] = -(0.00245204119306875 * std::pow(w2m1, 1.5) *	22✔
390	((135135.0 / 8.0) * std::pow(w, 5) -	11✔
391	45045.0 / 4.0 * std::pow(w, 3) + (10395.0 / 8.0) * w) *	11✔
392	std::sin(3. * phi));	11✔
393	// l = 8, m = -2
394	rn[i + 6] =	11✔
395	0.0199204768222399 * (w2m1) *	22✔
396	((45045.0 / 16.0) * std::pow(w, 6) - 45045.0 / 16.0 * std::pow(w, 4) +	11✔
397	(10395.0 / 16.0) * w * w - 315.0 / 16.0) *	11✔
398	std::sin(2. * phi);	11✔
399	// l = 8, m = -1
400	rn[i + 7] =	11✔
401	-(0.166666666666667 * std::sqrt(w2m1) *	22✔
402	((6435.0 / 16.0) * std::pow(w, 7) - 9009.0 / 16.0 * std::pow(w, 5) +	11✔
403	(3465.0 / 16.0) * std::pow(w, 3) - 315.0 / 16.0 * w) *	11✔
404	std::sin(phi));	11✔
405	// l = 8, m = 0
406	rn[i + 8] = 50.2734375 * std::pow(w, 8) - 93.84375 * std::pow(w, 6) +	11✔
407	54.140625 * std::pow(w, 4) - 9.84375 * w * w + 0.2734375;	11✔
408	// l = 8, m = 1
409	rn[i + 9] =	11✔
410	-(0.166666666666667 * std::sqrt(w2m1) *	22✔
411	((6435.0 / 16.0) * std::pow(w, 7) - 9009.0 / 16.0 * std::pow(w, 5) +	11✔
412	(3465.0 / 16.0) * std::pow(w, 3) - 315.0 / 16.0 * w) *	11✔
413	std::cos(phi));	11✔
414	// l = 8, m = 2
415	rn[i + 10] =	11✔
416	0.0199204768222399 * (w2m1) *	22✔
417	((45045.0 / 16.0) * std::pow(w, 6) - 45045.0 / 16.0 * std::pow(w, 4) +	11✔
418	(10395.0 / 16.0) * w * w - 315.0 / 16.0) *	11✔
419	std::cos(2. * phi);	11✔
420	// l = 8, m = 3
421	rn[i + 11] = -(0.00245204119306875 * std::pow(w2m1, 1.5) *	22✔
422	((135135.0 / 8.0) * std::pow(w, 5) -	11✔
423	45045.0 / 4.0 * std::pow(w, 3) + (10395.0 / 8.0) * w) *	11✔
424	std::cos(3. * phi));	11✔
425	// l = 8, m = 4
426	rn[i + 12] = 0.000316557156832328 * w2m1 * w2m1 *	22✔
427	((675675.0 / 8.0) * std::pow(w, 4) - 135135.0 / 4.0 * w * w +	11✔
428	10395.0 / 8.0) *	11✔
429	std::cos(4.0 * phi);	11✔
430	// l = 8, m = 5
431	rn[i + 13] = -(4.38985792528482e-5 * std::pow(w2m1, 2.5) *	22✔
432	((675675.0 / 2.) * std::pow(w, 3) - 135135.0 / 2. * w) *	11✔
433	std::cos(5.0 * phi));	11✔
434	// l = 8, m = 6
435	rn[i + 14] = 6.77369783729086e-6 * std::pow(w2m1, 3) *	11✔
436	((2027025.0 / 2.) * w * w - 135135.0 / 2.) *	11✔
437	std::cos(6.0 * phi);	11✔
438	// l = 8, m = 7
439	rn[i + 15] =	11✔
440	-(2.50682661696018 * w * std::pow(w2m1, 3.5) * std::cos(7.0 * phi));	11✔
441	// l = 8, m = 8
442	rn[i + 16] = 0.626706654240044 * std::pow(w2m1, 4) * std::cos(8.0 * phi);	11✔
443	break;	11✔
444	case 9:	11✔
445	// l = 9, m = -9
446	rn[i] = -(0.609049392175524 * std::pow(w2m1, 4.5) * std::sin(9.0 * phi));	11✔
447	// l = 9, m = -8
448	rn[i + 1] =	11✔
449	2.58397773170915 * w * std::pow(w2m1, 4) * std::sin(8.0 * phi);	11✔
450	// l = 9, m = -7
451	rn[i + 2] =	11✔
452	-(4.37240315267812e-7 * std::pow(w2m1, 3.5) *	11✔
453	((34459425.0 / 2.) * w * w - 2027025.0 / 2.) * std::sin(7.0 * phi));	11✔
454	// l = 9, m = -6
455	rn[i + 3] = 3.02928976464514e-6 * std::pow(w2m1, 3) *	11✔
456	((11486475.0 / 2.) * std::pow(w, 3) - 2027025.0 / 2. * w) *	11✔
457	std::sin(6.0 * phi);	11✔
458	// l = 9, m = -5
459	rn[i + 4] = -(2.34647776186144e-5 * std::pow(w2m1, 2.5) *	22✔
460	((11486475.0 / 8.0) * std::pow(w, 4) -	11✔
461	2027025.0 / 4.0 * w * w + 135135.0 / 8.0) *	11✔
462	std::sin(5.0 * phi));	11✔
463	// l = 9, m = -4
464	rn[i + 5] = 0.000196320414650061 * w2m1 * w2m1 *	22✔
465	((2297295.0 / 8.0) * std::pow(w, 5) -	11✔
466	675675.0 / 4.0 * std::pow(w, 3) + (135135.0 / 8.0) * w) *	11✔
467	std::sin(4.0 * phi);	11✔
468	// l = 9, m = -3
469	rn[i + 6] = -(	11✔
470	0.00173385495536766 * std::pow(w2m1, 1.5) *	22✔
471	((765765.0 / 16.0) * std::pow(w, 6) - 675675.0 / 16.0 * std::pow(w, 4) +	11✔
472	(135135.0 / 16.0) * w * w - 3465.0 / 16.0) *	11✔
473	std::sin(3. * phi));	11✔
474	// l = 9, m = -2
475	rn[i + 7] =	11✔
476	0.0158910431540932 * (w2m1) *	22✔
477	((109395.0 / 16.0) * std::pow(w, 7) - 135135.0 / 16.0 * std::pow(w, 5) +	11✔
478	(45045.0 / 16.0) * std::pow(w, 3) - 3465.0 / 16.0 * w) *	11✔
479	std::sin(2. * phi);	11✔
480	// l = 9, m = -1
481	rn[i + 8] = -(	11✔
482	0.149071198499986 * std::sqrt(w2m1) *	22✔
483	((109395.0 / 128.0) * std::pow(w, 8) - 45045.0 / 32.0 * std::pow(w, 6) +	11✔
484	(45045.0 / 64.0) * std::pow(w, 4) - 3465.0 / 32.0 * w * w +	11✔
485	315.0 / 128.0) *	11✔
486	std::sin(phi));	11✔
487	// l = 9, m = 0
488	rn[i + 9] = 94.9609375 * std::pow(w, 9) - 201.09375 * std::pow(w, 7) +	11✔
489	140.765625 * std::pow(w, 5) - 36.09375 * std::pow(w, 3) +	11✔
490	2.4609375 * w;	11✔
491	// l = 9, m = 1
492	rn[i + 10] = -(	11✔
493	0.149071198499986 * std::sqrt(w2m1) *	22✔
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::cos(phi));	11✔
498	// l = 9, m = 2
499	rn[i + 11] =	11✔
500	0.0158910431540932 * (w2m1) *	22✔
501	((109395.0 / 16.0) * std::pow(w, 7) - 135135.0 / 16.0 * std::pow(w, 5) +	11✔
502	(45045.0 / 16.0) * std::pow(w, 3) - 3465.0 / 16.0 * w) *	11✔
503	std::cos(2. * phi);	11✔
504	// l = 9, m = 3
505	rn[i + 12] = -(	11✔
506	0.00173385495536766 * std::pow(w2m1, 1.5) *	22✔
507	((765765.0 / 16.0) * std::pow(w, 6) - 675675.0 / 16.0 * std::pow(w, 4) +	11✔
508	(135135.0 / 16.0) * w * w - 3465.0 / 16.0) *	11✔
509	std::cos(3. * phi));	11✔
510	// l = 9, m = 4
511	rn[i + 13] = 0.000196320414650061 * w2m1 * w2m1 *	22✔
512	((2297295.0 / 8.0) * std::pow(w, 5) -	11✔
513	675675.0 / 4.0 * std::pow(w, 3) + (135135.0 / 8.0) * w) *	11✔
514	std::cos(4.0 * phi);	11✔
515	// l = 9, m = 5
516	rn[i + 14] = -(2.34647776186144e-5 * std::pow(w2m1, 2.5) *	22✔
517	((11486475.0 / 8.0) * std::pow(w, 4) -	11✔
518	2027025.0 / 4.0 * w * w + 135135.0 / 8.0) *	11✔
519	std::cos(5.0 * phi));	11✔
520	// l = 9, m = 6
521	rn[i + 15] = 3.02928976464514e-6 * std::pow(w2m1, 3) *	11✔
522	((11486475.0 / 2.) * std::pow(w, 3) - 2027025.0 / 2. * w) *	11✔
523	std::cos(6.0 * phi);	11✔
524	// l = 9, m = 7
525	rn[i + 16] =	11✔
526	-(4.37240315267812e-7 * std::pow(w2m1, 3.5) *	11✔
527	((34459425.0 / 2.) * w * w - 2027025.0 / 2.) * std::cos(7.0 * phi));	11✔
528	// l = 9, m = 8
529	rn[i + 17] =	11✔
530	2.58397773170915 * w * std::pow(w2m1, 4) * std::cos(8.0 * phi);	11✔
531	// l = 9, m = 9
532	rn[i + 18] =	11✔
533	-(0.609049392175524 * std::pow(w2m1, 4.5) * std::cos(9.0 * phi));	11✔
534	break;	11✔
535	case 10:	11✔
536	// l = 10, m = -10
537	rn[i] = 0.593627917136573 * std::pow(w2m1, 5) * std::sin(10.0 * phi);	11✔
538	// l = 10, m = -9
539	rn[i + 1] =	11✔
540	-(2.65478475211798 * w * std::pow(w2m1, 4.5) * std::sin(9.0 * phi));	11✔
541	// l = 10, m = -8
542	rn[i + 2] = 2.49953651452314e-8 * std::pow(w2m1, 4) *	11✔
543	((654729075.0 / 2.) * w * w - 34459425.0 / 2.) *	11✔
544	std::sin(8.0 * phi);	11✔
545	// l = 10, m = -7
546	rn[i + 3] =	11✔
547	-(1.83677671621093e-7 * std::pow(w2m1, 3.5) *	22✔
548	((218243025.0 / 2.) * std::pow(w, 3) - 34459425.0 / 2. * w) *	11✔
549	std::sin(7.0 * phi));	11✔
550	// l = 10, m = -6
551	rn[i + 4] = 1.51464488232257e-6 * std::pow(w2m1, 3) *	11✔
552	((218243025.0 / 8.0) * std::pow(w, 4) -	11✔
553	34459425.0 / 4.0 * w * w + 2027025.0 / 8.0) *	11✔
554	std::sin(6.0 * phi);	11✔
555	// l = 10, m = -5
556	rn[i + 5] =	11✔
557	-(1.35473956745817e-5 * std::pow(w2m1, 2.5) *	22✔
558	((43648605.0 / 8.0) * std::pow(w, 5) -	11✔
559	11486475.0 / 4.0 * std::pow(w, 3) + (2027025.0 / 8.0) * w) *	11✔
560	std::sin(5.0 * phi));	11✔
561	// l = 10, m = -4
562	rn[i + 6] = 0.000128521880085575 * w2m1 * w2m1 *	22✔
563	((14549535.0 / 16.0) * std::pow(w, 6) -	11✔
564	11486475.0 / 16.0 * std::pow(w, 4) +	11✔
565	(2027025.0 / 16.0) * w * w - 45045.0 / 16.0) *	11✔
566	std::sin(4.0 * phi);	11✔
567	// l = 10, m = -3
568	rn[i + 7] = -(0.00127230170115096 * std::pow(w2m1, 1.5) *	22✔
569	((2078505.0 / 16.0) * std::pow(w, 7) -	11✔
570	2297295.0 / 16.0 * std::pow(w, 5) +	11✔
571	(675675.0 / 16.0) * std::pow(w, 3) - 45045.0 / 16.0 * w) *	11✔
572	std::sin(3. * phi));	11✔
573	// l = 10, m = -2
574	rn[i + 8] = 0.012974982402692 * (w2m1) *	22✔
575	((2078505.0 / 128.0) * std::pow(w, 8) -	11✔
576	765765.0 / 32.0 * std::pow(w, 6) +	11✔
577	(675675.0 / 64.0) * std::pow(w, 4) -	11✔
578	45045.0 / 32.0 * w * w + 3465.0 / 128.0) *	11✔
579	std::sin(2. * phi);	11✔
580	// l = 10, m = -1
581	rn[i + 9] = -(0.134839972492648 * std::sqrt(w2m1) *	22✔
582	((230945.0 / 128.0) * std::pow(w, 9) -	11✔
583	109395.0 / 32.0 * std::pow(w, 7) +	11✔
584	(135135.0 / 64.0) * std::pow(w, 5) -	11✔
585	15015.0 / 32.0 * std::pow(w, 3) + (3465.0 / 128.0) * w) *	11✔
586	std::sin(phi));	11✔
587	// l = 10, m = 0
588	rn[i + 10] = 180.42578125 * std::pow(w, 10) -	11✔
589	427.32421875 * std::pow(w, 8) +	11✔
590	351.9140625 * std::pow(w, 6) - 117.3046875 * std::pow(w, 4) +	11✔
591	13.53515625 * w * w - 0.24609375;	11✔
592	// l = 10, m = 1
593	rn[i + 11] = -(0.134839972492648 * std::sqrt(w2m1) *	22✔
594	((230945.0 / 128.0) * std::pow(w, 9) -	11✔
595	109395.0 / 32.0 * std::pow(w, 7) +	11✔
596	(135135.0 / 64.0) * std::pow(w, 5) -	11✔
597	15015.0 / 32.0 * std::pow(w, 3) + (3465.0 / 128.0) * w) *	11✔
598	std::cos(phi));	11✔
599	// l = 10, m = 2
600	rn[i + 12] = 0.012974982402692 * (w2m1) *	22✔
601	((2078505.0 / 128.0) * std::pow(w, 8) -	11✔
602	765765.0 / 32.0 * std::pow(w, 6) +	11✔
603	(675675.0 / 64.0) * std::pow(w, 4) -	11✔
604	45045.0 / 32.0 * w * w + 3465.0 / 128.0) *	11✔
605	std::cos(2. * phi);	11✔
606	// l = 10, m = 3
607	rn[i + 13] =	11✔
608	-(0.00127230170115096 * std::pow(w2m1, 1.5) *	22✔
609	((2078505.0 / 16.0) * std::pow(w, 7) -	11✔
610	2297295.0 / 16.0 * std::pow(w, 5) +	11✔
611	(675675.0 / 16.0) * std::pow(w, 3) - 45045.0 / 16.0 * w) *	11✔
612	std::cos(3. * phi));	11✔
613	// l = 10, m = 4
614	rn[i + 14] = 0.000128521880085575 * w2m1 * w2m1 *	22✔
615	((14549535.0 / 16.0) * std::pow(w, 6) -	11✔
616	11486475.0 / 16.0 * std::pow(w, 4) +	11✔
617	(2027025.0 / 16.0) * w * w - 45045.0 / 16.0) *	11✔
618	std::cos(4.0 * phi);	11✔
619	// l = 10, m = 5
620	rn[i + 15] =	11✔
621	-(1.35473956745817e-5 * std::pow(w2m1, 2.5) *	22✔
622	((43648605.0 / 8.0) * std::pow(w, 5) -	11✔
623	11486475.0 / 4.0 * std::pow(w, 3) + (2027025.0 / 8.0) * w) *	11✔
624	std::cos(5.0 * phi));	11✔
625	// l = 10, m = 6
626	rn[i + 16] = 1.51464488232257e-6 * std::pow(w2m1, 3) *	11✔
627	((218243025.0 / 8.0) * std::pow(w, 4) -	11✔
628	34459425.0 / 4.0 * w * w + 2027025.0 / 8.0) *	11✔
629	std::cos(6.0 * phi);	11✔
630	// l = 10, m = 7
631	rn[i + 17] =	11✔
632	-(1.83677671621093e-7 * std::pow(w2m1, 3.5) *	22✔
633	((218243025.0 / 2.) * std::pow(w, 3) - 34459425.0 / 2. * w) *	11✔
634	std::cos(7.0 * phi));	11✔
635	// l = 10, m = 8
636	rn[i + 18] = 2.49953651452314e-8 * std::pow(w2m1, 4) *	11✔
637	((654729075.0 / 2.) * w * w - 34459425.0 / 2.) *	11✔
638	std::cos(8.0 * phi);	11✔
639	// l = 10, m = 9
640	rn[i + 19] =	11✔
641	-(2.65478475211798 * w * std::pow(w2m1, 4.5) * std::cos(9.0 * phi));	11✔
642	// l = 10, m = 10
643	rn[i + 20] = 0.593627917136573 * std::pow(w2m1, 5) * std::cos(10.0 * phi);	11✔
644	}
645	}
646	}	4,568,344✔
647
648	void calc_zn(int n, double rho, double phi, double zn[])	2,704,570✔
649	{
650	// ===========================================================================
651	// Determine vector of sin(nphi) and cos(nphi). This takes advantage of the
652	// following recurrence relations so that only a single sin/cos have to be
653	// evaluated (https://mathworld.wolfram.com/Multiple-AngleFormulas.html)
654	//
655	// sin(nx) = 2 cos(x) sin((n-1)x) - sin((n-2)x)
656	// cos(nx) = 2 cos(x) cos((n-1)x) - cos((n-2)x)
657
658	double sin_phi = std::sin(phi);	2,704,570✔
659	double cos_phi = std::cos(phi);	2,704,570✔
660
661	vector<double> sin_phi_vec(n + 1); // Sin[n * phi]	2,704,570✔
662	vector<double> cos_phi_vec(n + 1); // Cos[n * phi]	2,704,570✔
663	sin_phi_vec[0] = 1.0;	2,704,570✔
664	cos_phi_vec[0] = 1.0;	2,704,570✔
665	sin_phi_vec[1] = 2.0 * cos_phi;	2,704,570✔
666	cos_phi_vec[1] = cos_phi;	2,704,570✔
667
668	for (int i = 2; i <= n; i++) {	13,519,715✔
669	sin_phi_vec[i] = 2. * cos_phi * sin_phi_vec[i - 1] - sin_phi_vec[i - 2];	10,815,145✔
670	cos_phi_vec[i] = 2. * cos_phi * cos_phi_vec[i - 1] - cos_phi_vec[i - 2];	10,815,145✔
671	}
672
673	for (int i = 0; i <= n; i++) {	18,928,855✔
674	sin_phi_vec[i] *= sin_phi;	16,224,285✔
675	}
676
677	// ===========================================================================
678	// Calculate R_pq(rho)
679	// Matrix forms of the coefficients which are easier to work with
680	vector<vector<double>> zn_mat(n + 1, vector<double>(n + 1));	5,409,140✔
681
682	// Fill the main diagonal first (Eq 3.9 in Chong)
683	for (int p = 0; p <= n; p++) {	18,928,855✔
684	zn_mat[p][p] = std::pow(rho, p);	16,224,285✔
685	}
686
687	// Fill the 2nd diagonal (Eq 3.10 in Chong)
688	for (int q = 0; q <= n - 2; q++) {	13,519,715✔
689	zn_mat[q][q + 2] = (q + 2) * zn_mat[q + 2][q + 2] - (q + 1) * zn_mat[q][q];	10,815,145✔
690	}
691
692	// Fill in the rest of the values using the original results (Eq. 3.8 in
693	// Chong)
694	for (int p = 4; p <= n; p++) {	8,110,575✔
695	double k2 = 2 * p * (p - 1) * (p - 2);	5,406,005✔
696	for (int q = p - 4; q >= 0; q -= 2) {	10,812,109✔
697	double k1 = ((p + q) * (p - q) * (p - 2)) / 2.;	5,406,104✔
698	double k3 = -q * q * (p - 1) - p * (p - 1) * (p - 2);	5,406,104✔
699	double k4 = (-p * (p + q - 2) * (p - q - 2)) / 2.;	5,406,104✔
700	zn_mat[q][p] =	5,406,104✔
701	((k2 * rho * rho + k3) * zn_mat[q][p - 2] + k4 * zn_mat[q][p - 4]) / k1;	5,406,104✔
702	}
703	}
704
705	// Roll into a single vector for easier computation later
706	// The vector is ordered (0,0), (1,-1), (1,1), (2,-2), (2,0),
707	// (2, 2), .... in (n,m) indices
708	// Note that the cos and sin vectors are offset by one
709	// sin_phi_vec = [sin(x), sin(2x), sin(3x) ...]
710	// cos_phi_vec = [1.0, cos(x), cos(2x)... ]
711	int i = 0;	2,704,570✔
712	for (int p = 0; p <= n; p++) {	18,928,855✔
713	for (int q = -p; q <= p; q += 2) {	73,003,106✔
714	if (q < 0) {	56,778,821✔
715	zn[i] = zn_mat[std::abs(q)][p] * sin_phi_vec[std::abs(q) - 1];	24,333,287✔
716	} else if (q == 0) {	32,445,534✔
717	zn[i] = zn_mat[q][p];	8,112,247✔
718	} else {
719	zn[i] = zn_mat[q][p] * cos_phi_vec[q];	24,333,287✔
720	}
721	i++;	56,778,821✔
722	}
723	}
724	}	2,704,570✔
725
726	void calc_zn_rad(int n, double rho, double zn_rad[])	44✔
727	{
728	// Calculate R_p0(rho) as Zn_p0(rho)
729	// Set up the array of the coefficients
730
731	double q = 0;	44✔
732
733	// R_00 is always 1
734	zn_rad[0] = 1;	44✔
735
736	// Fill in the rest of the array (Eq 3.8 and Eq 3.10 in Chong)
737	for (int p = 2; p <= n; p += 2) {	264✔
738	int index = int(p / 2);	220✔
739	if (p == 2) {	220✔
740	// Setting up R_22 to calculate R_20 (Eq 3.10 in Chong)
741	double R_22 = rho * rho;	44✔
742	zn_rad[index] = 2 * R_22 - zn_rad[0];	44✔
743	} else {
744	double k1 = ((p + q) * (p - q) * (p - 2)) / 2.;	176✔
745	double k2 = 2 * p * (p - 1) * (p - 2);	176✔
746	double k3 = -q * q * (p - 1) - p * (p - 1) * (p - 2);	176✔
747	double k4 = (-p * (p + q - 2) * (p - q - 2)) / 2.;	176✔
748	zn_rad[index] =	176✔
749	((k2 * rho * rho + k3) * zn_rad[index - 1] + k4 * zn_rad[index - 2]) /	176✔
750	k1;
751	}
752	}
753	}	44✔
754
755	void rotate_angle_c(double uvw[3], double mu, const double* phi, uint64_t* seed)	33✔
756	{
757	Direction u = rotate_angle({uvw}, mu, phi, seed);	33✔
758	uvw[0] = u.x;	33✔
759	uvw[1] = u.y;	33✔
760	uvw[2] = u.z;	33✔
761	}	33✔
762
763	Direction rotate_angle(	2,418,878,303✔
764	Direction u, double mu, const double* phi, uint64_t* seed)
765	{
766	// Sample azimuthal angle in [0,2pi) if none provided
767	double phi_;
768	if (phi != nullptr) {	2,418,878,303✔
769	phi_ = (*phi);	22,156,525✔
770	} else {
771	phi_ = 2.0 * PI * prn(seed);	2,416,864,664✔
772	}
773
774	// Precompute factors to save flops
775	double sinphi = std::sin(phi_);	2,418,878,303✔
776	double cosphi = std::cos(phi_);	2,418,878,303✔
777	double a = std::sqrt(std::fmax(0., 1. - mu * mu));	2,418,878,303✔
778	double b = std::sqrt(std::fmax(0., 1. - u.z * u.z));	2,418,878,303✔
779
780	// Need to treat special case where sqrt(1 - w**2) is close to zero by
781	// expanding about the v component rather than the w component
782	if (b > 1e-10) {	2,418,878,303✔
783	return {mu * u.x + a * (u.x * u.z * cosphi - u.y * sinphi) / b,	2,418,842,768✔
784	mu * u.y + a * (u.y * u.z * cosphi + u.x * sinphi) / b,	2,418,842,768✔
785	mu * u.z - a * b * cosphi};	2,418,842,768✔
786	} else {
787	b = std::sqrt(1. - u.y * u.y);	390,885✔
788	return {mu * u.x + a * (-u.x * u.y * sinphi + u.z * cosphi) / b,	390,885✔
789	mu * u.y + a * b * sinphi,	390,885✔
790	mu * u.z - a * (u.y * u.z * sinphi + u.x * cosphi) / b};	390,885✔
791	}
792	}
793
794	void spline(int n, const double x[], const double y[], double z[])	108,688✔
795	{
796	vector<double> c_new(n - 1);	108,688✔
797
798	// Set natural boundary conditions
799	c_new[0] = 0.0;	108,688✔
800	z[0] = 0.0;	108,688✔
801	z[n - 1] = 0.0;	108,688✔
802
803	// Solve using tridiagonal matrix algorithm; first do forward sweep
804	for (int i = 1; i < n - 1; i++) {	10,903,612✔
805	double a = x[i] - x[i - 1];	10,794,924✔
806	double c = x[i + 1] - x[i];	10,794,924✔
807	double b = 2.0 * (a + c);	10,794,924✔
808	double d = 6.0 * ((y[i + 1] - y[i]) / c - (y[i] - y[i - 1]) / a);	10,794,924✔
809
810	c_new[i] = c / (b - a * c_new[i - 1]);	10,794,924✔
811	z[i] = (d - a * z[i - 1]) / (b - a * c_new[i - 1]);	10,794,924✔
812	}
813
814	// Back substitution
815	for (int i = n - 2; i >= 0; i--) {	11,012,300✔
816	z[i] = z[i] - c_new[i] * z[i + 1];	10,903,612✔
817	}
818	}	108,688✔
819
820	double spline_interpolate(	×
821	int n, const double x[], const double y[], const double z[], double xint)
822	{
823	// Find the lower bounding index in x of xint
824	int i = n - 1;	×
825	while (--i) {	×
826	if (xint >= x[i])	×
827	break;	×
828	}
829
830	double h = x[i + 1] - x[i];	×
831	double r = xint - x[i];	×
832
833	// Compute the coefficients
834	double b = (y[i + 1] - y[i]) / h - (h / 6.0) * (z[i + 1] + 2.0 * z[i]);	×
835	double c = z[i] / 2.0;	×
836	double d = (z[i + 1] - z[i]) / (h * 6.0);	×
837
838	return y[i] + b * r + c * r * r + d * r * r * r;	×
839	}
840
841	double spline_integrate(int n, const double x[], const double y[],	10,903,612✔
842	const double z[], double xa, double xb)
843	{
844	// Find the lower bounding index in x of the lower limit of integration.
845	int ia = n - 1;	10,903,612✔
846	while (--ia) {	729,358,176✔
847	if (xa >= x[ia])	729,249,488✔
848	break;	10,794,924✔
849	}
850
851	// Find the lower bounding index in x of the upper limit of integration.
852	int ib = n - 1;	10,903,612✔
853	while (--ib) {	718,563,252!
854	if (xb >= x[ib])	718,563,252✔
855	break;	10,903,612✔
856	}
857
858	// Evaluate the integral
859	double s = 0.0;	10,903,612✔
860	for (int i = ia; i <= ib; i++) {	32,602,148✔
861	double h = x[i + 1] - x[i];	21,698,536✔
862
863	// Compute the coefficients
864	double b = (y[i + 1] - y[i]) / h - (h / 6.0) * (z[i + 1] + 2.0 * z[i]);	21,698,536✔
865	double c = z[i] / 2.0;	21,698,536✔
866	double d = (z[i + 1] - z[i]) / (h * 6.0);	21,698,536✔
867
868	// Subtract the integral from x[ia] to xa
869	if (i == ia) {	21,698,536✔
870	double r = xa - x[ia];	10,903,612✔
871	s = s - (y[i] * r + b / 2.0 * r * r + c / 3.0 * r * r * r +	10,903,612✔
872	d / 4.0 * r * r * r * r);	10,903,612✔
873	}
874
875	// Integrate from x[ib] to xb in final interval
876	if (i == ib) {	21,698,536✔
877	h = xb - x[ib];	10,903,612✔
878	}
879
880	// Accumulate the integral
881	s = s + y[i] * h + b / 2.0 * h * h + c / 3.0 * h * h * h +	21,698,536✔
882	d / 4.0 * h * h * h * h;	21,698,536✔
883	}
884
885	return s;	10,903,612✔
886	}
887
888	std::complex<double> faddeeva(std::complex<double> z)	3,505,777✔
889	{
890	// Technically, the value we want is given by the equation:
891	// w(z) = I/pi * Integrate[Exp[-t^2]/(z-t), {t, -Infinity, Infinity}]
892	// as shown in Equation 63 from Hwang, R. N. "A rigorous pole
893	// representation of multilevel cross sections and its practical
894	// applications." Nucl. Sci. Eng. 96.3 (1987): 192-209.
895	//
896	// The MIT Faddeeva function evaluates w(z) = exp(-z^2)erfc(-iz). These
897	// two forms of the Faddeeva function are related by a transformation.
898	//
899	// If we call the integral form w_int, and the function form w_fun:
900	// For imag(z) > 0, w_int(z) = w_fun(z)
901	// For imag(z) < 0, w_int(z) = -conjg(w_fun(conjg(z)))
902
903	// Note that Faddeeva::w will interpret zero as machine epsilon
904	return z.imag() > 0.0 ? Faddeeva::w(z)	3,505,777!
905	: -std::conj(Faddeeva::w(std::conj(z)));	7,011,554!
906	}
907
908	std::complex<double> w_derivative(std::complex<double> z, int order)	4,232,547✔
909	{
910	using namespace std::complex_literals;
911	switch (order) {	4,232,547✔
912	case 0:	1,410,849✔
913	return faddeeva(z);	1,410,849✔
914	case 1:	1,410,849✔
915	return -2.0 * z * faddeeva(z) + 2.0i / SQRT_PI;	2,821,698✔
916	default:	1,410,849✔
917	return -2.0 * z * w_derivative(z, order - 1) -	1,410,849✔
918	2.0 * (order - 1) * w_derivative(z, order - 2);	4,232,547✔
919	}
920	}
921
NEW 922	double exprel(double x)	×
923	{
NEW 924	if (std::abs(x) < 1e-16)	×
NEW 925	return 1.0;	×
926	else {
NEW 927	return std::expm1(x) / x;	×
928	}
929	}
930
NEW 931	double log1prel(double x)	×
932	{
NEW 933	if (std::abs(x) < 1e-16)	×
NEW 934	return 1.0;	×
935	else {
NEW 936	return std::log1p(x) / x;	×
937	}
938	}
939
940	// Helper function to get index and interpolation function on an incident energy
941	// grid
942	void get_energy_index(	854,035,615✔
943	const vector<double>& energies, double E, int& i, double& f)
944	{
945	// Get index and interpolation factor for linear-linear energy grid
946	i = 0;	854,035,615✔
947	f = 0.0;	854,035,615✔
948	if (E >= energies.front()) {	854,035,615✔
949	i = lower_bound_index(energies.begin(), energies.end(), E);	854,035,234✔
950	if (i + 1 < energies.size())	854,035,234✔
951	f = (E - energies[i]) / (energies[i + 1] - energies[i]);	854,033,892✔
952	}
953	}	854,035,615✔
954
955	} // namespace openmc

openmc-dev / openmc / 21621895430

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