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Reference Guide
RooMath.cxx
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1/*****************************************************************************
2 * Project: RooFit *
3 * Package: RooFitCore *
4 * @(#)root/roofitcore:$Id$
5 * Authors: *
6 * WV, Wouter Verkerke, UC Santa Barbara, verkerke@slac.stanford.edu *
7 * DK, David Kirkby, UC Irvine, dkirkby@uci.edu *
8 * *
9 * Copyright (c) 2000-2005, Regents of the University of California *
10 * and Stanford University. All rights reserved. *
11 * *
12 * Redistribution and use in source and binary forms, *
13 * with or without modification, are permitted according to the terms *
14 * listed in LICENSE (http://roofit.sourceforge.net/license.txt) *
15 *****************************************************************************/
16
17// -- CLASS DESCRIPTION [MISC] --
18// RooMath is a singleton class implementing various mathematical
19// functions not found in TMath, mostly involving complex algebra
20//
21//
22
23#include <complex>
24#include <cmath>
25#include <algorithm>
26
27#include "RooFit.h"
28
29#include "RooMath.h"
30#include "Riostream.h"
31#include "RooMsgService.h"
32
33using namespace std;
34
35namespace faddeeva_impl {
36 static inline void cexp(double& re, double& im)
37 {
38 // with gcc on unix machines and on x86_64, we can gain by hand-coding
39 // exp(z) for the x87 coprocessor; other platforms have the default
40 // routines as fallback implementation, and compilers other than gcc on
41 // x86_64 generate better code with the default routines; also avoid
42 // the inline assembly code when the copiler is not optimising code, or
43 // is optimising for code size
44 // (we insist on __unix__ here, since the assemblers on other OSs
45 // typically do not speak AT&T syntax as gas does...)
46#if !(defined(__GNUC__) || defined(__clang__)) || \
47 !defined(__unix__) || !defined(__x86_64__) || \
48 !defined(__OPTIMIZE__) || defined(__OPTIMIZE_SIZE__) || \
49 defined(__INTEL_COMPILER) || \
50 defined(__OPEN64__) || defined(__PATHSCALE__)
51 const double e = std::exp(re);
52 re = e * std::cos(im);
53 im = e * std::sin(im);
54#else
55 __asm__ (
56 "fxam\n\t" // examine st(0): NaN? Inf?
57 "fstsw %%ax\n\t"
58 "movb $0x45,%%dh\n\t"
59 "andb %%ah,%%dh\n\t"
60 "cmpb $0x05,%%dh\n\t"
61 "jz 1f\n\t" // have NaN or infinity, handle below
62 "fldl2e\n\t" // load log2(e)
63 "fmulp\n\t" // re * log2(e)
64 "fld %%st(0)\n\t" // duplicate re * log2(e)
65 "frndint\n\t" // int(re * log2(e))
66 "fsubr %%st,%%st(1)\n\t" // st(1) = x = frac(re * log2(e))
67 "fxch\n\t" // swap st(0), st(1)
68 "f2xm1\n\t" // 2^x - 1
69 "fld1\n\t" // st(0) = 1
70 "faddp\n\t" // st(0) = 2^x
71 "fscale\n\t" // 2 ^ (int(re * log2(e)) + x)
72 "fstp %%st(1)\n\t" // pop st(1)
73 "jmp 2f\n\t"
74 "1:\n\t" // handle NaN, Inf...
75 "testl $0x200, %%eax\n\t"// -infinity?
76 "jz 2f\n\t"
77 "fstp %%st\n\t" // -Inf, so pop st(0)
78 "fldz\n\t" // st(0) = 0
79 "2:\n\t" // here. we have st(0) == exp(re)
80 "fxch\n\t" // st(0) = im, st(1) = exp(re)
81 "fsincos\n\t" // st(0) = cos(im), st(1) = sin(im)
82 "fnstsw %%ax\n\t"
83 "testl $0x400,%%eax\n\t"
84 "jz 4f\n\t" // |im| too large for fsincos?
85 "fldpi\n\t" // st(0) = pi
86 "fadd %%st(0)\n\t" // st(0) *= 2;
87 "fxch %%st(1)\n\t" // st(0) = im, st(1) = 2 * pi
88 "3:\n\t"
89 "fprem1\n\t" // st(0) = fmod(im, 2 * pi)
90 "fnstsw %%ax\n\t"
91 "testl $0x400,%%eax\n\t"
92 "jnz 3b\n\t" // fmod done?
93 "fstp %%st(1)\n\t" // yes, pop st(1) == 2 * pi
94 "fsincos\n\t" // st(0) = cos(im), st(1) = sin(im)
95 "4:\n\t" // all fine, fsincos succeeded
96 "fmul %%st(2)\n\t" // st(0) *= st(2)
97 "fxch %%st(2)\n\t" // st(2)=exp(re)*cos(im),st(0)=exp(im)
98 "fmulp %%st(1)\n\t" // st(1)=exp(re)*sin(im), pop st(0)
99 : "=t" (im), "=u" (re): "0" (re), "1" (im) :
100 "eax", "dh", "cc"
101#ifndef __clang__
102 // normal compilers (like gcc) want to be told that we
103 // clobber x87 registers, even if we pop them afterwards
104 // (so they can make sure they don't save anything there)
105 , "st(5)", "st(6)", "st(7)"
106#else // __clang__
107 // clang produces an error message with the clobber list
108 // from above - not sure why; it seems harmless to leave
109 // the popped x87 registers out of the clobber list for
110 // clang, and that is in fact what seems to be recommended
111 // here:
112 // http://lists.cs.uiuc.edu/pipermail/cfe-dev/2012-May/021715.html
113#endif // __clang__
114 );
115#endif
116 }
117
118 template <class T, unsigned N, unsigned NTAYLOR, unsigned NCF>
119 static inline std::complex<T> faddeeva_smabmq_impl(
120 T zre, T zim, const T tm,
121 const T (&a)[N], const T (&npi)[N],
122 const T (&taylorarr)[N * NTAYLOR * 2])
123 {
124 // catch singularities in the Fourier representation At
125 // z = n pi / tm, and provide a Taylor series expansion in those
126 // points, and only use it when we're close enough to the real axis
127 // that there is a chance we need it
128 const T zim2 = zim * zim;
129 const T maxnorm = T(9) / T(1000000);
130 if (zim2 < maxnorm) {
131 // we're close enough to the real axis that we need to worry about
132 // singularities
133 const T dnsing = tm * zre / npi[1];
134 const T dnsingmax2 = (T(N) - T(1) / T(2)) * (T(N) - T(1) / T(2));
135 if (dnsing * dnsing < dnsingmax2) {
136 // we're in the interesting range of the real axis as well...
137 // deal with Re(z) < 0 so we only need N different Taylor
138 // expansions; use w(-x+iy) = conj(w(x+iy))
139 const bool negrez = zre < T(0);
140 // figure out closest singularity
141 const int nsing = int(std::abs(dnsing) + T(1) / T(2));
142 // and calculate just how far we are from it
143 const T zmnpire = std::abs(zre) - npi[nsing];
144 const T zmnpinorm = zmnpire * zmnpire + zim2;
145 // close enough to one of the singularities?
146 if (zmnpinorm < maxnorm) {
147 const T* coeffs = &taylorarr[nsing * NTAYLOR * 2];
148 // calculate value of taylor expansion...
149 // (note: there's no chance to vectorize this one, since
150 // the value of the next iteration depend on the ones from
151 // the previous iteration)
152 T sumre = coeffs[0], sumim = coeffs[1];
153 for (unsigned i = 1; i < NTAYLOR; ++i) {
154 const T re = sumre * zmnpire - sumim * zim;
155 const T im = sumim * zmnpire + sumre * zim;
156 sumre = re + coeffs[2 * i + 0];
157 sumim = im + coeffs[2 * i + 1];
158 }
159 // undo the flip in real part of z if needed
160 if (negrez) return std::complex<T>(sumre, -sumim);
161 else return std::complex<T>(sumre, sumim);
162 }
163 }
164 }
165 // negative Im(z) is treated by calculating for -z, and using the
166 // symmetry properties of erfc(z)
167 const bool negimz = zim < T(0);
168 if (negimz) {
169 zre = -zre;
170 zim = -zim;
171 }
172 const T znorm = zre * zre + zim2;
173 if (znorm > tm * tm) {
174 // use continued fraction approximation for |z| large
175 const T isqrtpi = 5.64189583547756287e-01;
176 const T z2re = (zre + zim) * (zre - zim);
177 const T z2im = T(2) * zre * zim;
178 T cfre = T(1), cfim = T(0), cfnorm = T(1);
179 for (unsigned k = NCF; k; --k) {
180 cfre = +(T(k) / T(2)) * cfre / cfnorm;
181 cfim = -(T(k) / T(2)) * cfim / cfnorm;
182 if (k & 1) cfre -= z2re, cfim -= z2im;
183 else cfre += T(1);
184 cfnorm = cfre * cfre + cfim * cfim;
185 }
186 T sumre = (zim * cfre - zre * cfim) * isqrtpi / cfnorm;
187 T sumim = -(zre * cfre + zim * cfim) * isqrtpi / cfnorm;
188 if (negimz) {
189 // use erfc(-z) = 2 - erfc(z) to get good accuracy for
190 // Im(z) < 0: 2 / exp(z^2) - w(z)
191 T ez2re = -z2re, ez2im = -z2im;
192 faddeeva_impl::cexp(ez2re, ez2im);
193 return std::complex<T>(T(2) * ez2re - sumre,
194 T(2) * ez2im - sumim);
195 } else {
196 return std::complex<T>(sumre, sumim);
197 }
198 }
199 const T twosqrtpi = 3.54490770181103205e+00;
200 const T tmzre = tm * zre, tmzim = tm * zim;
201 // calculate exp(i tm z)
202 T eitmzre = -tmzim, eitmzim = tmzre;
203 faddeeva_impl::cexp(eitmzre, eitmzim);
204 // form 1 +/- exp (i tm z)
205 const T numerarr[4] = {
206 T(1) - eitmzre, -eitmzim, T(1) + eitmzre, +eitmzim
207 };
208 // form tm z * (1 +/- exp(i tm z))
209 const T numertmz[4] = {
210 tmzre * numerarr[0] - tmzim * numerarr[1],
211 tmzre * numerarr[1] + tmzim * numerarr[0],
212 tmzre * numerarr[2] - tmzim * numerarr[3],
213 tmzre * numerarr[3] + tmzim * numerarr[2]
214 };
215 // common subexpressions for use inside the loop
216 const T reimtmzm2 = T(-2) * tmzre * tmzim;
217 const T imtmz2 = tmzim * tmzim;
218 const T reimtmzm22 = reimtmzm2 * reimtmzm2;
219 // on non-x86_64 architectures, when the compiler is producing
220 // unoptimised code and when optimising for code size, we use the
221 // straightforward implementation, but for x86_64, we use the
222 // brainf*cked code below that the gcc vectorizer likes to gain a few
223 // clock cycles; non-gcc compilers also get the normal code, since they
224 // usually do a better job with the default code (and yes, it's a pain
225 // that they're all pretending to be gcc)
226#if (!defined(__x86_64__)) || !defined(__OPTIMIZE__) || \
227 defined(__OPTIMIZE_SIZE__) || defined(__INTEL_COMPILER) || \
228 defined(__clang__) || defined(__OPEN64__) || \
229 defined(__PATHSCALE__) || !defined(__GNUC__)
230 T sumre = (-a[0] / znorm) * (numerarr[0] * zre + numerarr[1] * zim);
231 T sumim = (-a[0] / znorm) * (numerarr[1] * zre - numerarr[0] * zim);
232 for (unsigned i = 0; i < N; ++i) {
233 const unsigned j = (i << 1) & 2;
234 // denominator
235 const T wk = imtmz2 + (npi[i] + tmzre) * (npi[i] - tmzre);
236 // norm of denominator
237 const T norm = wk * wk + reimtmzm22;
238 const T f = T(2) * tm * a[i] / norm;
239 // sum += a[i] * numer / wk
240 sumre -= f * (numertmz[j] * wk + numertmz[j + 1] * reimtmzm2);
241 sumim -= f * (numertmz[j + 1] * wk - numertmz[j] * reimtmzm2);
242 }
243#else
244 // BEGIN fully vectorisable code - enjoy reading... ;)
245 T tmp[2 * N];
246 for (unsigned i = 0; i < N; ++i) {
247 const T wk = imtmz2 + (npi[i] + tmzre) * (npi[i] - tmzre);
248 tmp[2 * i + 0] = wk;
249 tmp[2 * i + 1] = T(2) * tm * a[i] / (wk * wk + reimtmzm22);
250 }
251 for (unsigned i = 0; i < N / 2; ++i) {
252 T wk = tmp[4 * i + 0], f = tmp[4 * i + 1];
253 tmp[4 * i + 0] = -f * (numertmz[0] * wk + numertmz[1] * reimtmzm2);
254 tmp[4 * i + 1] = -f * (numertmz[1] * wk - numertmz[0] * reimtmzm2);
255 wk = tmp[4 * i + 2], f = tmp[4 * i + 3];
256 tmp[4 * i + 2] = -f * (numertmz[2] * wk + numertmz[3] * reimtmzm2);
257 tmp[4 * i + 3] = -f * (numertmz[3] * wk - numertmz[2] * reimtmzm2);
258 }
259 if (N & 1) {
260 // we may have missed one element in the last loop; if so, process
261 // it now...
262 const T wk = tmp[2 * N - 2], f = tmp[2 * N - 1];
263 tmp[2 * (N - 1) + 0] = -f * (numertmz[0] * wk + numertmz[1] * reimtmzm2);
264 tmp[2 * (N - 1) + 1] = -f * (numertmz[1] * wk - numertmz[0] * reimtmzm2);
265 }
266 T sumre = (-a[0] / znorm) * (numerarr[0] * zre + numerarr[1] * zim);
267 T sumim = (-a[0] / znorm) * (numerarr[1] * zre - numerarr[0] * zim);
268 for (unsigned i = 0; i < N; ++i) {
269 sumre += tmp[2 * i + 0];
270 sumim += tmp[2 * i + 1];
271 }
272 // END fully vectorisable code
273#endif
274 // prepare the result
275 if (negimz) {
276 // use erfc(-z) = 2 - erfc(z) to get good accuracy for
277 // Im(z) < 0: 2 / exp(z^2) - w(z)
278 const T z2im = -T(2) * zre * zim;
279 const T z2re = -(zre + zim) * (zre - zim);
280 T ez2re = z2re, ez2im = z2im;
281 faddeeva_impl::cexp(ez2re, ez2im);
282 return std::complex<T>(T(2) * ez2re + sumim / twosqrtpi,
283 T(2) * ez2im - sumre / twosqrtpi);
284 } else {
285 return std::complex<T>(-sumim / twosqrtpi, sumre / twosqrtpi);
286 }
287 }
288
289 static const double npi24[24] = { // precomputed values n * pi
290 0.00000000000000000e+00, 3.14159265358979324e+00, 6.28318530717958648e+00,
291 9.42477796076937972e+00, 1.25663706143591730e+01, 1.57079632679489662e+01,
292 1.88495559215387594e+01, 2.19911485751285527e+01, 2.51327412287183459e+01,
293 2.82743338823081391e+01, 3.14159265358979324e+01, 3.45575191894877256e+01,
294 3.76991118430775189e+01, 4.08407044966673121e+01, 4.39822971502571053e+01,
295 4.71238898038468986e+01, 5.02654824574366918e+01, 5.34070751110264851e+01,
296 5.65486677646162783e+01, 5.96902604182060715e+01, 6.28318530717958648e+01,
297 6.59734457253856580e+01, 6.91150383789754512e+01, 7.22566310325652445e+01,
298 };
299 static const double a24[24] = { // precomputed Fourier coefficient prefactors
300 2.95408975150919338e-01, 2.75840233292177084e-01, 2.24573955224615866e-01,
301 1.59414938273911723e-01, 9.86657664154541891e-02, 5.32441407876394120e-02,
302 2.50521500053936484e-02, 1.02774656705395362e-02, 3.67616433284484706e-03,
303 1.14649364124223317e-03, 3.11757015046197600e-04, 7.39143342960301488e-05,
304 1.52794934280083635e-05, 2.75395660822107093e-06, 4.32785878190124505e-07,
305 5.93003040874588103e-08, 7.08449030774820423e-09, 7.37952063581678038e-10,
306 6.70217160600200763e-11, 5.30726516347079017e-12, 3.66432411346763916e-13,
307 2.20589494494103134e-14, 1.15782686262855879e-15, 5.29871142946730482e-17,
308 };
309 static const double taylorarr24[24 * 12] = {
310 // real part imaginary part, low order coefficients last
311 // nsing = 0
312 0.00000000000000000e-00, 3.00901111225470020e-01,
313 5.00000000000000000e-01, 0.00000000000000000e-00,
314 0.00000000000000000e-00, -7.52252778063675049e-01,
315 -1.00000000000000000e-00, 0.00000000000000000e-00,
316 0.00000000000000000e-00, 1.12837916709551257e+00,
317 1.00000000000000000e-00, 0.00000000000000000e-00,
318 // nsing = 1
319 -2.22423508493755319e-01, 1.87966717746229718e-01,
320 3.41805419240637628e-01, 3.42752593807919263e-01,
321 4.66574321730757753e-01, -5.59649213591058097e-01,
322 -8.05759710273191021e-01, -5.38989366115424093e-01,
323 -4.88914083733395200e-01, 9.80580906465856792e-01,
324 9.33757118080975970e-01, 2.82273885115127769e-01,
325 // nsing = 2
326 -2.60522586513312894e-01, -4.26259455096092786e-02,
327 1.36549702008863349e-03, 4.39243227763478846e-01,
328 6.50591493715480700e-01, -1.23422352472779046e-01,
329 -3.43379903564271318e-01, -8.13862662890748911e-01,
330 -7.96093943501906645e-01, 6.11271022503935772e-01,
331 7.60213717643090957e-01, 4.93801903948967945e-01,
332 // nsing = 3
333 -1.18249853727020186e-01, -1.90471659765411376e-01,
334 -2.59044664869706839e-01, 2.69333898502392004e-01,
335 4.99077838344125714e-01, 2.64644800189075006e-01,
336 1.26114512111568737e-01, -7.46519337025968199e-01,
337 -8.47666863706379907e-01, 1.89347715957263646e-01,
338 5.39641485816297176e-01, 5.97805988669631615e-01,
339 // nsing = 4
340 4.94825297066481491e-02, -1.71428212158876197e-01,
341 -2.97766677111471585e-01, 1.60773286596649656e-02,
342 1.88114210832460682e-01, 4.11734391195006462e-01,
343 3.98540613293909842e-01, -4.63321903522162715e-01,
344 -6.99522070542463639e-01, -1.32412024008354582e-01,
345 3.33997185986131785e-01, 6.01983450812696742e-01,
346 // nsing = 5
347 1.18367078448232332e-01, -6.09533063579086850e-02,
348 -1.74762998833038991e-01, -1.39098099222000187e-01,
349 -6.71534655984154549e-02, 3.34462251996496680e-01,
350 4.37429678577360024e-01, -1.59613865629038012e-01,
351 -4.71863911886034656e-01, -2.92759316465055762e-01,
352 1.80238737704018306e-01, 5.42834914744283253e-01,
353 // nsing = 6
354 8.87698096005701290e-02, 2.84339354980994902e-02,
355 -3.18943083830766399e-02, -1.53946887977045862e-01,
356 -1.71825061547624858e-01, 1.70734367410600348e-01,
357 3.33690792296469441e-01, 3.97048587678703930e-02,
358 -2.66422678503135697e-01, -3.18469797424381480e-01,
359 8.48049724711137773e-02, 4.60546329221462864e-01,
360 // nsing = 7
361 2.99767046276705077e-02, 5.34659695701718247e-02,
362 4.53131030251822568e-02, -9.37915401977138648e-02,
363 -1.57982359988083777e-01, 3.82170507060760740e-02,
364 1.98891589845251706e-01, 1.17546677047049354e-01,
365 -1.27514335237079297e-01, -2.72741112680307074e-01,
366 3.47906344595283767e-02, 3.82277517244493224e-01,
367 // nsing = 8
368 -7.35922494437203395e-03, 3.72011290318534610e-02,
369 5.66783220847204687e-02, -3.21015398169199501e-02,
370 -1.00308737825172555e-01, -2.57695148077963515e-02,
371 9.67294850588435368e-02, 1.18174625238337507e-01,
372 -5.21266530264988508e-02, -2.08850084114630861e-01,
373 1.24443217440050976e-02, 3.19239968065752286e-01,
374 // nsing = 9
375 -1.66126772808035320e-02, 1.46180329587665321e-02,
376 3.85927576915247303e-02, 1.18910471133003227e-03,
377 -4.94003498320899806e-02, -3.93468443660139110e-02,
378 3.92113167048952835e-02, 9.03306084789976219e-02,
379 -1.82889636251263500e-02, -1.53816215444915245e-01,
380 3.88103861995563741e-03, 2.72090310854550347e-01,
381 // nsing = 10
382 -1.21245068916826880e-02, 1.59080224420074489e-03,
383 1.91116222508366035e-02, 1.05879549199053302e-02,
384 -1.97228428219695318e-02, -3.16962067712639397e-02,
385 1.34110372628315158e-02, 6.18045654429108837e-02,
386 -5.52574921865441838e-03, -1.14259663804569455e-01,
387 1.05534036292203489e-03, 2.37326534898818288e-01,
388 // nsing = 11
389 -5.96835002183177493e-03, -2.42594931567031205e-03,
390 7.44753817476594184e-03, 9.33450807578394386e-03,
391 -6.52649522783026481e-03, -2.08165802069352019e-02,
392 3.89988065678848650e-03, 4.12784313451549132e-02,
393 -1.44110721106127920e-03, -8.76484782997757425e-02,
394 2.50210184908121337e-04, 2.11131066219336647e-01,
395 // nsing = 12
396 -2.24505212235034193e-03, -2.38114524227619446e-03,
397 2.36375918970809340e-03, 5.97324040603806266e-03,
398 -1.81333819936645381e-03, -1.28126250720444051e-02,
399 9.69251586187208358e-04, 2.83055679874589732e-02,
400 -3.24986363596307374e-04, -6.97056268370209313e-02,
401 5.17231862038123061e-05, 1.90681117197597520e-01,
402 // nsing = 13
403 -6.76887607549779069e-04, -1.48589685249767064e-03,
404 6.22548369472046953e-04, 3.43871156746448680e-03,
405 -4.26557147166379929e-04, -7.98854145009655400e-03,
406 2.06644460919535524e-04, 2.03107152586353217e-02,
407 -6.34563929410856987e-05, -5.71425144910115832e-02,
408 9.32252179140502456e-06, 1.74167663785025829e-01,
409 // nsing = 14
410 -1.67596437777156162e-04, -8.05384193869903178e-04,
411 1.37627277777023791e-04, 1.97652692602724093e-03,
412 -8.54392244879459717e-05, -5.23088906415977167e-03,
413 3.78965577556493513e-05, 1.52191559129376333e-02,
414 -1.07393019498185646e-05, -4.79347862153366295e-02,
415 1.46503970628861795e-06, 1.60471011683477685e-01,
416 // nsing = 15
417 -3.45715760630978778e-05, -4.31089554210205493e-04,
418 2.57350138106549737e-05, 1.19449262097417514e-03,
419 -1.46322227517372253e-05, -3.61303766799909378e-03,
420 5.99057675687392260e-06, 1.17993805017130890e-02,
421 -1.57660578509526722e-06, -4.09165023743669707e-02,
422 2.00739683204152177e-07, 1.48879348585662670e-01,
423 // nsing = 16
424 -5.99735188857573424e-06, -2.42949218855805052e-04,
425 4.09249090936269722e-06, 7.67400152727128171e-04,
426 -2.14920611287648034e-06, -2.60710519575546230e-03,
427 8.17591694958640978e-07, 9.38581640137393053e-03,
428 -2.00910914042737743e-07, -3.54045580123653803e-02,
429 2.39819738182594508e-08, 1.38916449405613711e-01,
430 // nsing = 17
431 -8.80708505155966658e-07, -1.46479474515521504e-04,
432 5.55693207391871904e-07, 5.19165587844615415e-04,
433 -2.71391142598826750e-07, -1.94439427580099576e-03,
434 9.64641799864928425e-08, 7.61536975207357980e-03,
435 -2.22357616069432967e-08, -3.09762939485679078e-02,
436 2.49806920458212581e-09, 1.30247401712293206e-01,
437 // nsing = 18
438 -1.10007111030476390e-07, -9.35886150886691786e-05,
439 6.46244096997824390e-08, 3.65267193418479043e-04,
440 -2.95175785569292542e-08, -1.48730955943961081e-03,
441 9.84949251974795537e-09, 6.27824679148707177e-03,
442 -2.13827217704781576e-09, -2.73545766571797965e-02,
443 2.26877724435352177e-10, 1.22627158810895267e-01,
444 // nsing = 19
445 -1.17302439957657553e-08, -6.24890956722053332e-05,
446 6.45231881609786173e-09, 2.64799907072561543e-04,
447 -2.76943921343331654e-09, -1.16094187847598385e-03,
448 8.71074689656480749e-10, 5.24514377390761210e-03,
449 -1.78730768958639407e-10, -2.43489203319091538e-02,
450 1.79658223341365988e-11, 1.15870972518909888e-01,
451 // nsing = 20
452 -1.07084502471985403e-09, -4.31515421260633319e-05,
453 5.54152563270547927e-10, 1.96606443937168357e-04,
454 -2.24423474431542338e-10, -9.21550077887211094e-04,
455 6.67734377376211580e-11, 4.43201203646827019e-03,
456 -1.29896907717633162e-11, -2.18236356404862774e-02,
457 1.24042409733678516e-12, 1.09836276968151848e-01,
458 // nsing = 21
459 -8.38816525569060600e-11, -3.06091807093959821e-05,
460 4.10033961556230842e-11, 1.48895624771753491e-04,
461 -1.57238128435253905e-11, -7.42073499862065649e-04,
462 4.43938379112418832e-12, 3.78197089773957382e-03,
463 -8.21067867869285873e-13, -1.96793607299577220e-02,
464 7.46725770201828754e-14, 1.04410965521273064e-01,
465 // nsing = 22
466 -5.64848922712870507e-12, -2.22021942382507691e-05,
467 2.61729537775838587e-12, 1.14683068921649992e-04,
468 -9.53316139085394895e-13, -6.05021573565916914e-04,
469 2.56116039498542220e-13, 3.25530796858307225e-03,
470 -4.51482829896525004e-14, -1.78416955716514289e-02,
471 3.91940313268087086e-15, 9.95054815464739996e-02,
472 // nsing = 23
473 -3.27482357793897640e-13, -1.64138890390689871e-05,
474 1.44278798346454523e-13, 8.96362542918265398e-05,
475 -5.00524303437266481e-14, -4.98699756861136127e-04,
476 1.28274026095767213e-14, 2.82359118537843949e-03,
477 -2.16009593993917109e-15, -1.62538825704327487e-02,
478 1.79368667683853708e-16, 9.50473084594884184e-02
479 };
480
481 const double npi11[11] = { // precomputed values n * pi
482 0.00000000000000000e+00, 3.14159265358979324e+00, 6.28318530717958648e+00,
483 9.42477796076937972e+00, 1.25663706143591730e+01, 1.57079632679489662e+01,
484 1.88495559215387594e+01, 2.19911485751285527e+01, 2.51327412287183459e+01,
485 2.82743338823081391e+01, 3.14159265358979324e+01
486 };
487 const double a11[11] = { // precomputed Fourier coefficient prefactors
488 4.43113462726379007e-01, 3.79788034073635143e-01, 2.39122407410867584e-01,
489 1.10599187402169792e-01, 3.75782250080904725e-02, 9.37936104296856288e-03,
490 1.71974046186334976e-03, 2.31635559000523461e-04, 2.29192401420125452e-05,
491 1.66589592139340077e-06, 8.89504561311882155e-08
492 };
493 const double taylorarr11[11 * 6] = {
494 // real part imaginary part, low order coefficients last
495 // nsing = 0
496 -1.00000000000000000e+00, 0.00000000000000000e+00,
497 0.00000000000000000e-01, 1.12837916709551257e+00,
498 1.00000000000000000e+00, 0.00000000000000000e+00,
499 // nsing = 1
500 -5.92741768247463996e-01, -7.19914991991294310e-01,
501 -6.73156763521649944e-01, 8.14025039279059577e-01,
502 8.57089811121701143e-01, 4.00248106586639754e-01,
503 // nsing = 2
504 1.26114512111568737e-01, -7.46519337025968199e-01,
505 -8.47666863706379907e-01, 1.89347715957263646e-01,
506 5.39641485816297176e-01, 5.97805988669631615e-01,
507 // nsing = 3
508 4.43238482668529408e-01, -3.03563167310638372e-01,
509 -5.88095866853990048e-01, -2.32638360700858412e-01,
510 2.49595637924601714e-01, 5.77633779156009340e-01,
511 // nsing = 4
512 3.33690792296469441e-01, 3.97048587678703930e-02,
513 -2.66422678503135697e-01, -3.18469797424381480e-01,
514 8.48049724711137773e-02, 4.60546329221462864e-01,
515 // nsing = 5
516 1.42043544696751869e-01, 1.24094227867032671e-01,
517 -8.31224229982140323e-02, -2.40766729258442100e-01,
518 2.11669512031059302e-02, 3.48650139549945097e-01,
519 // nsing = 6
520 3.92113167048952835e-02, 9.03306084789976219e-02,
521 -1.82889636251263500e-02, -1.53816215444915245e-01,
522 3.88103861995563741e-03, 2.72090310854550347e-01,
523 // nsing = 7
524 7.37741897722738503e-03, 5.04625223970221539e-02,
525 -2.87394336989990770e-03, -9.96122819257496929e-02,
526 5.22745478269428248e-04, 2.23361039070072101e-01,
527 // nsing = 8
528 9.69251586187208358e-04, 2.83055679874589732e-02,
529 -3.24986363596307374e-04, -6.97056268370209313e-02,
530 5.17231862038123061e-05, 1.90681117197597520e-01,
531 // nsing = 9
532 9.01625563468897100e-05, 1.74961124275657019e-02,
533 -2.65745127697337342e-05, -5.22070356354932341e-02,
534 3.75952450449939411e-06, 1.67018782142871146e-01,
535 // nsing = 10
536 5.99057675687392260e-06, 1.17993805017130890e-02,
537 -1.57660578509526722e-06, -4.09165023743669707e-02,
538 2.00739683204152177e-07, 1.48879348585662670e-01
539 };
540}
541
542std::complex<double> RooMath::faddeeva(std::complex<double> z)
543{
544 return faddeeva_impl::faddeeva_smabmq_impl<double, 24, 6, 9>(
545 z.real(), z.imag(), 12., faddeeva_impl::a24,
547}
548
549std::complex<double> RooMath::faddeeva_fast(std::complex<double> z)
550{
551 return faddeeva_impl::faddeeva_smabmq_impl<double, 11, 3, 3>(
552 z.real(), z.imag(), 8., faddeeva_impl::a11,
554}
555
556std::complex<double> RooMath::erfc(const std::complex<double> z)
557{
558 double re = -z.real() * z.real() + z.imag() * z.imag();
559 double im = -2. * z.real() * z.imag();
560 faddeeva_impl::cexp(re, im);
561 return (z.real() >= 0.) ?
562 (std::complex<double>(re, im) *
563 faddeeva(std::complex<double>(-z.imag(), z.real()))) :
564 (2. - std::complex<double>(re, im) *
565 faddeeva(std::complex<double>(z.imag(), -z.real())));
566}
567
568std::complex<double> RooMath::erfc_fast(const std::complex<double> z)
569{
570 double re = -z.real() * z.real() + z.imag() * z.imag();
571 double im = -2. * z.real() * z.imag();
572 faddeeva_impl::cexp(re, im);
573 return (z.real() >= 0.) ?
574 (std::complex<double>(re, im) *
575 faddeeva_fast(std::complex<double>(-z.imag(), z.real()))) :
576 (2. - std::complex<double>(re, im) *
577 faddeeva_fast(std::complex<double>(z.imag(), -z.real())));
578}
579
580std::complex<double> RooMath::erf(const std::complex<double> z)
581{
582 double re = -z.real() * z.real() + z.imag() * z.imag();
583 double im = -2. * z.real() * z.imag();
584 faddeeva_impl::cexp(re, im);
585 return (z.real() >= 0.) ?
586 (1. - std::complex<double>(re, im) *
587 faddeeva(std::complex<double>(-z.imag(), z.real()))) :
588 (std::complex<double>(re, im) *
589 faddeeva(std::complex<double>(z.imag(), -z.real())) - 1.);
590}
591
592std::complex<double> RooMath::erf_fast(const std::complex<double> z)
593{
594 double re = -z.real() * z.real() + z.imag() * z.imag();
595 double im = -2. * z.real() * z.imag();
596 faddeeva_impl::cexp(re, im);
597 return (z.real() >= 0.) ?
598 (1. - std::complex<double>(re, im) *
599 faddeeva_fast(std::complex<double>(-z.imag(), z.real()))) :
600 (std::complex<double>(re, im) *
601 faddeeva_fast(std::complex<double>(z.imag(), -z.real())) - 1.);
602}
603
604
606{
607 // Interpolate array 'ya' with 'n' elements for 'x' (between 0 and 'n'-1)
608
609 // Int to Double conversion is faster via array lookup than type conversion!
610 static Double_t itod[20] = { 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0,
611 10.0,11.0,12.0,13.0,14.0,15.0,16.0,17.0,18.0,19.0} ;
612 int i,m,ns=1 ;
613 Double_t den,dif,dift/*,ho,hp,w*/,y,dy ;
614 Double_t c[20], d[20] ;
615
616 dif = fabs(x) ;
617 for(i=1 ; i<=n ; i++) {
618 if ((dift=fabs(x-itod[i-1]))<dif) {
619 ns=i ;
620 dif=dift ;
621 }
622 c[i] = ya[i-1] ;
623 d[i] = ya[i-1] ;
624 }
625
626 y=ya[--ns] ;
627 for(m=1 ; m<n; m++) {
628 for(i=1 ; i<=n-m ; i++) {
629 den=(c[i+1] - d[i])/itod[m] ;
630 d[i]=(x-itod[i+m-1])*den ;
631 c[i]=(x-itod[i-1])*den ;
632 }
633 dy = (2*ns)<(n-m) ? c[ns+1] : d[ns--] ;
634 y += dy ;
635 }
636 return y ;
637}
638
639
640
642{
643 // Interpolate array 'ya' with 'n' elements for 'xa'
644
645 // Int to Double conversion is faster via array lookup than type conversion!
646// static Double_t itod[20] = { 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0,
647// 10.0,11.0,12.0,13.0,14.0,15.0,16.0,17.0,18.0,19.0} ;
648 int i,m,ns=1 ;
649 Double_t den,dif,dift,ho,hp,w,y,dy ;
650 Double_t c[20], d[20] ;
651
652 dif = fabs(x-xa[0]) ;
653 for(i=1 ; i<=n ; i++) {
654 if ((dift=fabs(x-xa[i-1]))<dif) {
655 ns=i ;
656 dif=dift ;
657 }
658 c[i] = ya[i-1] ;
659 d[i] = ya[i-1] ;
660 }
661
662 y=ya[--ns] ;
663 for(m=1 ; m<n; m++) {
664 for(i=1 ; i<=n-m ; i++) {
665 ho=xa[i-1]-x ;
666 hp=xa[i-1+m]-x ;
667 w=c[i+1]-d[i] ;
668 den=ho-hp ;
669 if (den==0.) {
670 oocoutE((TObject*)0,Eval) << "RooMath::interpolate ERROR: zero distance between points not allowed" << endl ;
671 return 0 ;
672 }
673 den = w/den ;
674 d[i]=hp*den ;
675 c[i]=ho*den;
676 }
677 dy = (2*ns)<(n-m) ? c[ns+1] : d[ns--] ;
678 y += dy ;
679 }
680 return y ;
681}
#define d(i)
Definition: RSha256.hxx:102
#define f(i)
Definition: RSha256.hxx:104
#define c(i)
Definition: RSha256.hxx:101
#define e(i)
Definition: RSha256.hxx:103
#define oocoutE(o, a)
Definition: RooMsgService.h:48
int Int_t
Definition: RtypesCore.h:43
double Double_t
Definition: RtypesCore.h:57
#define N
double cos(double)
double sin(double)
double exp(double)
#define NCF(TN, I, C)
Definition: cfortran.h:897
static std::complex< double > erfc(const std::complex< double > z)
complex erfc function
Definition: RooMath.cxx:556
static std::complex< double > erf(const std::complex< double > z)
complex erf function
Definition: RooMath.cxx:580
static std::complex< double > faddeeva(std::complex< double > z)
evaluate Faddeeva function for complex argument
Definition: RooMath.cxx:542
static std::complex< double > faddeeva_fast(std::complex< double > z)
evaluate Faddeeva function for complex argument (fast version)
Definition: RooMath.cxx:549
static std::complex< double > erfc_fast(const std::complex< double > z)
complex erfc function (fast version)
Definition: RooMath.cxx:568
static std::complex< double > erf_fast(const std::complex< double > z)
complex erf function (fast version)
Definition: RooMath.cxx:592
static Double_t interpolate(Double_t yArr[], Int_t nOrder, Double_t x)
Definition: RooMath.cxx:605
Mother of all ROOT objects.
Definition: TObject.h:37
Double_t y[n]
Definition: legend1.C:17
Double_t x[n]
Definition: legend1.C:17
const Int_t n
Definition: legend1.C:16
double T(double x)
Definition: ChebyshevPol.h:34
VecExpr< UnaryOp< Fabs< T >, VecExpr< A, T, D >, T >, T, D > fabs(const VecExpr< A, T, D > &rhs)
static constexpr double ns
const double npi11[11]
Definition: RooMath.cxx:481
static std::complex< T > faddeeva_smabmq_impl(T zre, T zim, const T tm, const T(&a)[N], const T(&npi)[N], const T(&taylorarr)[N *NTAYLOR *2])
Definition: RooMath.cxx:119
static const double npi24[24]
Definition: RooMath.cxx:289
static const double a24[24]
Definition: RooMath.cxx:299
const double taylorarr11[11 *6]
Definition: RooMath.cxx:493
static void cexp(double &re, double &im)
Definition: RooMath.cxx:36
const double a11[11]
Definition: RooMath.cxx:487
static const double taylorarr24[24 *12]
Definition: RooMath.cxx:309
auto * m
Definition: textangle.C:8
auto * a
Definition: textangle.C:12