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55918ebabf2d18d43813dd5bfa81fc1a34aefb05
lammps/src/compute_orientorder_atom.cpp
Axel Kohlmeyer 92c118b29f replace for (;;) with while (true)
2023-06-25 05:56:51 -04:00

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/* ----------------------------------------------------------------------
LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
https://www.lammps.org/, Sandia National Laboratories
LAMMPS development team: developers@lammps.org
Copyright (2003) Sandia Corporation. Under the terms of Contract
DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
certain rights in this software. This software is distributed under
the GNU General Public License.
See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */
/* ----------------------------------------------------------------------
Contributing authors: Aidan Thompson (SNL), Axel Kohlmeyer (Temple U)
Tomas Oppelstrup (LLNL): Optimization which reduces the number
of iterations in the L,m1,m2 loops (by a factor of up to 10), and
avoids evaluation of Ylm functions of negative m
------------------------------------------------------------------------- */
#include "compute_orientorder_atom.h"
#include "atom.h"
#include "comm.h"
#include "error.h"
#include "force.h"
#include "math_const.h"
#include "math_special.h"
#include "memory.h"
#include "modify.h"
#include "neigh_list.h"
#include "neighbor.h"
#include "pair.h"
#include "update.h"
#include <cmath>
#include <cstring>
using namespace LAMMPS_NS;
using namespace MathConst;
using MathSpecial::factorial;
#ifdef DBL_EPSILON
static constexpr double MY_EPSILON = (10.0 * DBL_EPSILON);
#else
static constexpr double MY_EPSILON = (10.0 * 2.220446049250313e-16);
#endif
static constexpr double QEPSILON = 1.0e-6;
/* ---------------------------------------------------------------------- */
ComputeOrientOrderAtom::ComputeOrientOrderAtom(LAMMPS *lmp, int narg, char **arg) :
Compute(lmp, narg, arg), qlist(nullptr), qnormfac(nullptr), qnormfac2(nullptr), distsq(nullptr),
nearest(nullptr), rlist(nullptr), qnarray(nullptr), qnm_r(nullptr), qnm_i(nullptr),
w3jlist(nullptr)
{
if (narg < 3) error->all(FLERR, "Illegal compute orientorder/atom command");
// set default values for optional args
nnn = 12;
cutsq = 0.0;
wlflag = 0;
wlhatflag = 0;
qlcompflag = 0;
chunksize = 16384;
// specify which orders to request
nqlist = 5;
memory->create(qlist, nqlist, "orientorder/atom:qlist");
qlist[0] = 4;
qlist[1] = 6;
qlist[2] = 8;
qlist[3] = 10;
qlist[4] = 12;
qmax = 12;
// process optional args
int iarg = 3;
while (iarg < narg) {
if (strcmp(arg[iarg], "nnn") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
if (strcmp(arg[iarg + 1], "NULL") == 0) {
nnn = 0;
} else {
nnn = utils::numeric(FLERR, arg[iarg + 1], false, lmp);
if (nnn <= 0) error->all(FLERR, "Illegal compute orientorder/atom command");
}
iarg += 2;
} else if (strcmp(arg[iarg], "degrees") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
nqlist = utils::numeric(FLERR, arg[iarg + 1], false, lmp);
if (nqlist <= 0) error->all(FLERR, "Illegal compute orientorder/atom command");
memory->destroy(qlist);
memory->create(qlist, nqlist, "orientorder/atom:qlist");
iarg += 2;
if (iarg + nqlist > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
qmax = 0;
for (int il = 0; il < nqlist; il++) {
qlist[il] = utils::numeric(FLERR, arg[iarg + il], false, lmp);
if (qlist[il] < 0) error->all(FLERR, "Illegal compute orientorder/atom command");
if (qlist[il] > qmax) qmax = qlist[il];
}
iarg += nqlist;
} else if (strcmp(arg[iarg], "wl") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
wlflag = utils::logical(FLERR, arg[iarg + 1], false, lmp);
iarg += 2;
} else if (strcmp(arg[iarg], "wl/hat") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
wlhatflag = utils::logical(FLERR, arg[iarg + 1], false, lmp);
iarg += 2;
} else if (strcmp(arg[iarg], "components") == 0) {
qlcompflag = 1;
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
qlcomp = utils::numeric(FLERR, arg[iarg + 1], false, lmp);
iqlcomp = -1;
for (int il = 0; il < nqlist; il++)
if (qlcomp == qlist[il]) {
iqlcomp = il;
break;
}
if (iqlcomp == -1) error->all(FLERR, "Illegal compute orientorder/atom command");
iarg += 2;
} else if (strcmp(arg[iarg], "cutoff") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
double cutoff = utils::numeric(FLERR, arg[iarg + 1], false, lmp);
if (cutoff <= 0.0) error->all(FLERR, "Illegal compute orientorder/atom command");
cutsq = cutoff * cutoff;
iarg += 2;
} else if (strcmp(arg[iarg], "chunksize") == 0) {
if (iarg + 2 > narg) error->all(FLERR, "Illegal compute orientorder/atom command");
chunksize = utils::numeric(FLERR, arg[iarg + 1], false, lmp);
if (chunksize <= 0) error->all(FLERR, "Illegal compute orientorder/atom command");
iarg += 2;
} else
error->all(FLERR, "Illegal compute orientorder/atom command");
}
ncol = nqlist;
if (wlflag) ncol += nqlist;
if (wlhatflag) ncol += nqlist;
if (qlcompflag) ncol += 2 * (2 * qlcomp + 1);
peratom_flag = 1;
size_peratom_cols = ncol;
nmax = 0;
maxneigh = 0;
memory->create(qnormfac, nqlist, "orientorder/atom:qnormfac");
memory->create(qnormfac2, nqlist, "orientorder/atom:qnormfac2");
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
qnormfac[il] = sqrt(MY_4PI / (2.0 * l + 1.0));
qnormfac2[il] = sqrt(2.0 * l + 1.0);
}
}
/* --------------------------------------------------------------------- */
ComputeOrientOrderAtom::~ComputeOrientOrderAtom()
{
if (copymode) return;
memory->destroy(qnarray);
memory->destroy(distsq);
memory->destroy(rlist);
memory->destroy(nearest);
memory->destroy(qlist);
memory->destroy(qnormfac);
memory->destroy(qnormfac2);
memory->destroy(qnm_r);
memory->destroy(qnm_i);
memory->destroy(w3jlist);
}
/* ---------------------------------------------------------------------- */
void ComputeOrientOrderAtom::init()
{
if (force->pair == nullptr)
error->all(FLERR, "Compute orientorder/atom requires a pair style be defined");
if (cutsq == 0.0)
cutsq = force->pair->cutforce * force->pair->cutforce;
else if (sqrt(cutsq) > force->pair->cutforce)
error->all(FLERR, "Compute orientorder/atom cutoff is longer than pairwise cutoff");
memory->destroy(qnm_r);
memory->destroy(qnm_i);
memory->create(qnm_r, nqlist, qmax + 1, "orientorder/atom:qnm_r");
memory->create(qnm_i, nqlist, qmax + 1, "orientorder/atom:qnm_i");
// need an occasional full neighbor list
neighbor->add_request(this, NeighConst::REQ_FULL | NeighConst::REQ_OCCASIONAL);
if ((modify->get_compute_by_style("orientorder/atom").size() > 1) && (comm->me == 0))
error->warning(FLERR, "More than one instance of compute orientorder/atom");
if (wlflag || wlhatflag) init_wigner3j();
}
/* ---------------------------------------------------------------------- */
void ComputeOrientOrderAtom::init_list(int /*id*/, NeighList *ptr)
{
list = ptr;
}
/* ---------------------------------------------------------------------- */
void ComputeOrientOrderAtom::compute_peratom()
{
int i, j, ii, jj, inum, jnum;
double xtmp, ytmp, ztmp, delx, dely, delz, rsq;
int *ilist, *jlist, *numneigh, **firstneigh;
invoked_peratom = update->ntimestep;
// grow order parameter array if necessary
if (atom->nmax > nmax) {
memory->destroy(qnarray);
nmax = atom->nmax;
memory->create(qnarray, nmax, ncol, "orientorder/atom:qnarray");
array_atom = qnarray;
}
// invoke full neighbor list (will copy or build if necessary)
neighbor->build_one(list);
inum = list->inum;
ilist = list->ilist;
numneigh = list->numneigh;
firstneigh = list->firstneigh;
// compute order parameter for each atom in group
// use full neighbor list to count atoms less than cutoff
double **x = atom->x;
int *mask = atom->mask;
memset(&qnarray[0][0], 0, sizeof(double) * nmax * ncol);
for (ii = 0; ii < inum; ii++) {
i = ilist[ii];
double *qn = qnarray[i];
if (mask[i] & groupbit) {
xtmp = x[i][0];
ytmp = x[i][1];
ztmp = x[i][2];
jlist = firstneigh[i];
jnum = numneigh[i];
// ensure distsq and nearest arrays are long enough
if (jnum > maxneigh) {
memory->destroy(distsq);
memory->destroy(rlist);
memory->destroy(nearest);
maxneigh = jnum;
memory->create(distsq, maxneigh, "orientorder/atom:distsq");
memory->create(rlist, maxneigh, 3, "orientorder/atom:rlist");
memory->create(nearest, maxneigh, "orientorder/atom:nearest");
}
// loop over list of all neighbors within force cutoff
// distsq[] = distance sq to each
// rlist[] = distance vector to each
// nearest[] = atom indices of neighbors
int ncount = 0;
for (jj = 0; jj < jnum; jj++) {
j = jlist[jj];
j &= NEIGHMASK;
delx = xtmp - x[j][0];
dely = ytmp - x[j][1];
delz = ztmp - x[j][2];
rsq = delx * delx + dely * dely + delz * delz;
if (rsq < cutsq) {
distsq[ncount] = rsq;
rlist[ncount][0] = delx;
rlist[ncount][1] = dely;
rlist[ncount][2] = delz;
nearest[ncount++] = j;
}
}
// if not nnn neighbors, order parameter = 0;
if ((ncount == 0) || (ncount < nnn)) {
for (jj = 0; jj < ncol; jj++) qn[jj] = 0.0;
continue;
}
// if nnn > 0, use only nearest nnn neighbors
if (nnn > 0) {
select3(nnn, ncount, distsq, nearest, rlist);
ncount = nnn;
}
calc_boop(rlist, ncount, qn, qlist, nqlist);
}
}
}
/* ----------------------------------------------------------------------
memory usage of local atom-based array
------------------------------------------------------------------------- */
double ComputeOrientOrderAtom::memory_usage()
{
double bytes = (double) ncol * nmax * sizeof(double);
bytes += (double) (qmax * (2 * qmax + 1) + maxneigh * 4) * sizeof(double);
bytes += (double) (nqlist + maxneigh) * sizeof(int);
return bytes;
}
/* ----------------------------------------------------------------------
select3 routine from Numerical Recipes (slightly modified)
find k smallest values in array of length n
sort auxiliary arrays at same time
------------------------------------------------------------------------- */
// Use no-op do while to create single statement
#define SWAP(a, b) \
do { \
tmp = a; \
(a) = b; \
(b) = tmp; \
} while (0)
#define ISWAP(a, b) \
do { \
itmp = a; \
(a) = b; \
(b) = itmp; \
} while (0)
#define SWAP3(a, b) \
do { \
tmp = (a)[0]; \
(a)[0] = (b)[0]; \
(b)[0] = tmp; \
tmp = (a)[1]; \
(a)[1] = (b)[1]; \
(b)[1] = tmp; \
tmp = (a)[2]; \
(a)[2] = (b)[2]; \
(b)[2] = tmp; \
} while (0)
/* ---------------------------------------------------------------------- */
void ComputeOrientOrderAtom::select3(int k, int n, double *arr, int *iarr, double **arr3)
{
int i, ir, j, l, mid, ia, itmp;
double a, tmp, a3[3];
arr--;
iarr--;
arr3--;
l = 1;
ir = n;
while (true) {
if (ir <= l + 1) {
if (ir == l + 1 && arr[ir] < arr[l]) {
SWAP(arr[l], arr[ir]);
ISWAP(iarr[l], iarr[ir]);
SWAP3(arr3[l], arr3[ir]);
}
return;
} else {
mid = (l + ir) >> 1;
SWAP(arr[mid], arr[l + 1]);
ISWAP(iarr[mid], iarr[l + 1]);
SWAP3(arr3[mid], arr3[l + 1]);
if (arr[l] > arr[ir]) {
SWAP(arr[l], arr[ir]);
ISWAP(iarr[l], iarr[ir]);
SWAP3(arr3[l], arr3[ir]);
}
if (arr[l + 1] > arr[ir]) {
SWAP(arr[l + 1], arr[ir]);
ISWAP(iarr[l + 1], iarr[ir]);
SWAP3(arr3[l + 1], arr3[ir]);
}
if (arr[l] > arr[l + 1]) {
SWAP(arr[l], arr[l + 1]);
ISWAP(iarr[l], iarr[l + 1]);
SWAP3(arr3[l], arr3[l + 1]);
}
i = l + 1;
j = ir;
a = arr[l + 1];
ia = iarr[l + 1];
a3[0] = arr3[l + 1][0];
a3[1] = arr3[l + 1][1];
a3[2] = arr3[l + 1][2];
while (true) {
do i++;
while (arr[i] < a);
do j--;
while (arr[j] > a);
if (j < i) break;
SWAP(arr[i], arr[j]);
ISWAP(iarr[i], iarr[j]);
SWAP3(arr3[i], arr3[j]);
}
arr[l + 1] = arr[j];
arr[j] = a;
iarr[l + 1] = iarr[j];
iarr[j] = ia;
arr3[l + 1][0] = arr3[j][0];
arr3[l + 1][1] = arr3[j][1];
arr3[l + 1][2] = arr3[j][2];
arr3[j][0] = a3[0];
arr3[j][1] = a3[1];
arr3[j][2] = a3[2];
if (j >= k) ir = j - 1;
if (j <= k) l = i;
}
}
}
/* ----------------------------------------------------------------------
calculate the bond orientational order parameters
------------------------------------------------------------------------- */
void ComputeOrientOrderAtom::calc_boop(double **rlist, int ncount, double qn[], int qlist[],
int nqlist)
{
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
for (int m = 0; m < l + 1; m++) {
qnm_r[il][m] = 0.0;
qnm_i[il][m] = 0.0;
}
}
for (int ineigh = 0; ineigh < ncount; ineigh++) {
const double *const r = rlist[ineigh];
double rmag = sqrt(r[0] * r[0] + r[1] * r[1] + r[2] * r[2]);
if (rmag <= MY_EPSILON) { return; }
double costheta = r[2] / rmag;
double expphi_r = r[0];
double expphi_i = r[1];
double rxymag = sqrt(expphi_r * expphi_r + expphi_i * expphi_i);
if (rxymag <= MY_EPSILON) {
expphi_r = 1.0;
expphi_i = 0.0;
} else {
double rxymaginv = 1.0 / rxymag;
expphi_r *= rxymaginv;
expphi_i *= rxymaginv;
}
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
// calculate spherical harmonics
// Ylm, -l <= m <= l
// sign convention: sign(Yll(0,0)) = (-1)^l
qnm_r[il][0] += polar_prefactor(l, 0, costheta);
double expphim_r = expphi_r;
double expphim_i = expphi_i;
for (int m = 1; m <= +l; m++) {
double prefactor = polar_prefactor(l, m, costheta);
double ylm_r = prefactor * expphim_r;
double ylm_i = prefactor * expphim_i;
qnm_r[il][m] += ylm_r;
qnm_i[il][m] += ylm_i;
// Skip calculation of qnm for m<0 due to symmetry
double tmp_r = expphim_r * expphi_r - expphim_i * expphi_i;
double tmp_i = expphim_r * expphi_i + expphim_i * expphi_r;
expphim_r = tmp_r;
expphim_i = tmp_i;
}
}
}
// convert sums to averages
double facn = 1.0 / ncount;
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
for (int m = 0; m < l + 1; m++) {
qnm_r[il][m] *= facn;
qnm_i[il][m] *= facn;
}
}
// calculate Q_l
// NOTE: optional W_l_hat and components of Q_qlcomp use these stored Q_l values
int jj = 0;
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
double qm_sum = qnm_r[il][0] * qnm_r[il][0];
for (int m = 1; m < l + 1; m++)
qm_sum += 2.0 * (qnm_r[il][m] * qnm_r[il][m] + qnm_i[il][m] * qnm_i[il][m]);
qn[jj++] = qnormfac[il] * sqrt(qm_sum);
}
// calculate W_l
int nterms = 0;
int widx_count = 0;
if (wlflag || wlhatflag) {
for (int il = 0; il < nqlist; il++) {
int l = qlist[il];
double wlsum = 0.0;
for (int m1 = -l; m1 <= 0; m1++) {
const int sgn = 1 - 2 * (m1 & 1); // sgn = (-1)^m1
for (int m2 = 0; m2 <= ((-m1) >> 1); m2++) {
const int m3 = -(m1 + m2);
// Loop enforces -L <= m1 <= 0 <= m2 <= m3 <= L, and m1 + m2 + m3 = 0
// For even L, W3j is invariant under permutation of
// (m1, m2, m3) and (m1, m2, m3) -> (-m1, -m2, -m3). The loop
// structure enforces visiting only one member of each
// such symmetry (invariance) group.
// m1 <= 0, and Qlm[-m] = (-1)^m * conjg(Qlm[m])
const double Q1Q2_r =
(qnm_r[il][-m1] * qnm_r[il][m2] + qnm_i[il][-m1] * qnm_i[il][m2]) * sgn;
const double Q1Q2_i =
(qnm_r[il][-m1] * qnm_i[il][m2] - qnm_i[il][-m1] * qnm_r[il][m2]) * sgn;
const double Q1Q2Q3 = Q1Q2_r * qnm_r[il][m3] - Q1Q2_i * qnm_i[il][m3];
const double c = w3jlist[widx_count++];
wlsum += Q1Q2Q3 * c;
}
}
qn[jj++] = wlsum / qnormfac2[il];
nterms++;
}
}
// calculate W_l_hat
if (wlhatflag) {
const int jptr = jj - nterms;
if (!wlflag) jj = jptr;
for (int il = 0; il < nqlist; il++) {
if (qn[il] < QEPSILON)
qn[jj++] = 0.0;
else {
double qnfac = qnormfac[il] / qn[il];
qn[jj++] = qn[jptr + il] * (qnfac * qnfac * qnfac) * qnormfac2[il];
}
}
}
// Calculate components of Q_l/|Q_l|, for l=qlcomp
if (qlcompflag) {
int il = iqlcomp;
int l = qlcomp;
if (qn[il] < QEPSILON)
for (int m = 0; m < 2 * l + 1; m++) {
qn[jj++] = 0.0;
qn[jj++] = 0.0;
}
else {
double qnfac = qnormfac[il] / qn[il];
for (int m = -l; m < 0; m++) {
// Computed only qnm for m>=0.
// qnm[-m] = (-1)^m * conjg(qnm[m])
const int sgn = 1 - 2 * (m & 1); // sgn = (-1)^m
qn[jj++] = qnm_r[il][-m] * qnfac * sgn;
qn[jj++] = -qnm_i[il][-m] * qnfac * sgn;
}
for (int m = 0; m < l + 1; m++) {
qn[jj++] = qnm_r[il][m] * qnfac;
qn[jj++] = qnm_i[il][m] * qnfac;
}
}
}
}
/* ----------------------------------------------------------------------
polar prefactor for spherical harmonic Y_l^m, where
Y_l^m (theta, phi) = prefactor(l, m, cos(theta)) * exp(i*m*phi)
------------------------------------------------------------------------- */
double ComputeOrientOrderAtom::polar_prefactor(int l, int m, double costheta)
{
const int mabs = abs(m);
double prefactor = 1.0;
for (int i = l - mabs + 1; i < l + mabs + 1; ++i) prefactor *= static_cast<double>(i);
prefactor = sqrt(static_cast<double>(2 * l + 1) / (MY_4PI * prefactor)) *
associated_legendre(l, mabs, costheta);
if ((m < 0) && (m % 2)) prefactor = -prefactor;
return prefactor;
}
/* ----------------------------------------------------------------------
associated legendre polynomial
sign convention: P(l,l) = (2l-1)!!(-sqrt(1-x^2))^l
------------------------------------------------------------------------- */
double ComputeOrientOrderAtom::associated_legendre(int l, int m, double x)
{
if (l < m) return 0.0;
double p(1.0), pm1(0.0), pm2(0.0);
if (m != 0) {
const double msqx = -sqrt(1.0 - x * x);
for (int i = 1; i < m + 1; ++i) p *= static_cast<double>(2 * i - 1) * msqx;
}
for (int i = m + 1; i < l + 1; ++i) {
pm2 = pm1;
pm1 = p;
p = (static_cast<double>(2 * i - 1) * x * pm1 - static_cast<double>(i + m - 1) * pm2) /
static_cast<double>(i - m);
}
return p;
}
/* ----------------------------------------------------------------------
Initialize table of Wigner 3j symbols
------------------------------------------------------------------------- */
void ComputeOrientOrderAtom::init_wigner3j()
{
int widx_count = 0;
for (int il = 0; il < nqlist; il++) {
const int l = qlist[il];
for (int m1 = -l; m1 <= 0; m1++) {
for (int m2 = 0; m2 <= ((-m1) >> 1); m2++) { widx_count++; }
}
}
widx_max = widx_count;
memory->destroy(w3jlist);
memory->create(w3jlist, widx_max, "computeorientorderatom:w3jlist");
widx_count = 0;
for (int il = 0; il < nqlist; il++) {
const int l = qlist[il];
for (int m1 = -l; m1 <= 0; m1++) {
for (int m2 = 0; m2 <= ((-m1) >> 1); m2++) {
const int m3 = -(m1 + m2);
// Loop enforces -L<=m1<=0<=m2<=m3<=L, and m1+m2+m3=0
// For even L, W3j is invariant under permutation of
// (m1,m2,m3) and (m1,m2,m3)->(-m1,-m2,-m3). The loop
// structure enforces visiting only one member of each
// such symmetry (invariance) group.
// Determine number of elements in symmetry group of (m1,m2,m3)
// Concise determination exploiting (m1,m2,m3) loop structure.
int pfac;
if (m1 == 0)
pfac = 1; // m1 = m2 = m3 = 0
else if (m2 == 0 || m2 == m3) {
// reduced group when only 3 permutations, or sign inversion
// is equivalent to permutation
pfac = 6;
} else
pfac = 12; // 6 permutations * 2 signs
w3jlist[widx_count] = w3j(l, m1, m2, m3) * pfac;
widx_count++;
}
}
}
}
/* ---------------------------------------------------------------------- */
double ComputeOrientOrderAtom::triangle_coeff(const int a, const int b, const int c)
{
return factorial(a + b - c) * factorial(a - b + c) * factorial(-a + b + c) /
factorial(a + b + c + 1);
}
/* ---------------------------------------------------------------------- */
double ComputeOrientOrderAtom::w3j(const int lmax, const int j1, const int j2, const int j3)
{
const int a = lmax, b = lmax, c = lmax;
const int alpha = j1, beta = j2, gamma = j3;
struct {
double operator()(const int a, const int b, const int c, const int alpha, const int beta,
const int t)
{
return factorial(t) * factorial(c - b + t + alpha) * factorial(c - a + t - beta) *
factorial(a + b - c - t) * factorial(a - t - alpha) * factorial(b - t + beta);
}
} x;
const double sgn = 1 - 2 * ((a - b - gamma) & 1);
const double g = sqrt(triangle_coeff(lmax, lmax, lmax)) *
sqrt(factorial(a + alpha) * factorial(a - alpha) * factorial(b + beta) * factorial(b - beta) *
factorial(c + gamma) * factorial(c - gamma));
double s = 0;
int t = 0;
while (c - b + t + alpha < 0 || c - a + t - beta < 0) t++;
// ^^ t>=-j1 ^^ t>=j2
while (true) {
if (a + b - c - t < 0) break; // t<=lmax
if (a - t - alpha < 0) break; // t<=lmax-j1
if (b - t + beta < 0) break; // t<=lmax+j2
const int m1t = 1 - 2 * (t & 1);
s += m1t / x(lmax, lmax, lmax, alpha, beta, t);
t++;
}
return sgn * g * s;
}
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