18 static const double kSqrt2 = 1.41421356237309515;
72 MATH_ERROR_MSG(
"crystalball_cdf",
"CrystalBall cdf not defined for n <=1");
73 return std::numeric_limits<double>::quiet_NaN();
76 double abs_alpha = std::abs(alpha);
77 double C =
n/abs_alpha * 1./(
n-1.) *
std::exp(-alpha*alpha/2.);
79 double totIntegral =
sigma*(
C+D);
82 return (alpha > 0) ? 1. - integral/totIntegral : integral/totIntegral;
87 MATH_ERROR_MSG(
"crystalball_cdf_c",
"CrystalBall cdf not defined for n <=1");
88 return std::numeric_limits<double>::quiet_NaN();
90 double abs_alpha = std::abs(alpha);
91 double C =
n/abs_alpha * 1./(
n-1.) *
std::exp(-alpha*alpha/2.);
93 double totIntegral =
sigma*(
C+D);
96 return (alpha > 0) ? integral/totIntegral : 1. - (integral/totIntegral);
107 if (
sigma == 0)
return 0;
110 MATH_ERROR_MSG(
"crystalball_integral",
"CrystalBall function not defined at alpha=0");
113 bool useLog = (
n == 1.0);
114 if (
n<=0)
MATH_WARN_MSG(
"crystalball_integral",
"No physical meaning when n<=0");
116 double z = (
x-mean)/
sigma;
117 if (alpha < 0 ) z = -z;
119 double abs_alpha = std::abs(alpha);
128 const double oneoversqrt2 = 1./
sqrt(2.);
132 double B =
n/abs_alpha - abs_alpha;
135 double C = (
n/abs_alpha) * (1./(
n-1)) *
std::exp(-alpha*alpha/2.);
150 return sigma * (intgaus + intpow);
156 if ((
x-x0) < 0)
return 1.0;
163 if ((
x-x0) < 0)
return 0.0;
172 if (
n < 0 ||
m < 0)
return std::numeric_limits<double>::quiet_NaN();
174 double z =
m/(
m +
n*(
x-x0));
187 return std::numeric_limits<double>::quiet_NaN();
189 double z =
n*(
x-x0)/(
m +
n*(
x-x0));
191 if (z > 0.9 &&
n > 1 &&
m > 1)
204 double gamma_cdf(
double x,
double alpha,
double theta,
double x0)
245 double sign = (p>0) ? 1. : -1;
253 double sign = (p>0) ? 1. : -1;
260 if ((
x-x0) <
a)
return 1.0;
261 else if ((
x-x0) >=
b)
return 0.0;
262 else return (
b-(
x-x0))/(
b-
a);
268 if ((
x-x0) <
a)
return 0.0;
269 else if ((
x-x0) >=
b)
return 1.0;
270 else return ((
x-x0)-
a)/(
b-
a);
279 double a = (double)
n + 1.0;
288 double a = (double)
n + 1.0;
297 if ( k >=
n)
return 0;
298 double a = (double) k + 1.0;
299 double b = (double)
n - k;
308 if ( k >=
n)
return 1.0;
310 double a = (double) k + 1.0;
311 double b = (double)
n - k;
321 if (p < 0 || p > 1)
return 0;
330 if (
n < 0)
return 0;
331 if ( p < 0 || p > 1)
return 0;
343 static double p1[5] = {0.2514091491e+0,-0.6250580444e-1, 0.1458381230e-1,-0.2108817737e-2, 0.7411247290e-3};
344 static double q1[5] = {1.0 ,-0.5571175625e-2, 0.6225310236e-1,-0.3137378427e-2, 0.1931496439e-2};
346 static double p2[4] = {0.2868328584e+0, 0.3564363231e+0, 0.1523518695e+0, 0.2251304883e-1};
347 static double q2[4] = {1.0 , 0.6191136137e+0, 0.1720721448e+0, 0.2278594771e-1};
349 static double p3[4] = {0.2868329066e+0, 0.3003828436e+0, 0.9950951941e-1, 0.8733827185e-2};
350 static double q3[4] = {1.0 , 0.4237190502e+0, 0.1095631512e+0, 0.8693851567e-2};
352 static double p4[4] = {0.1000351630e+1, 0.4503592498e+1, 0.1085883880e+2, 0.7536052269e+1};
353 static double q4[4] = {1.0 , 0.5539969678e+1, 0.1933581111e+2, 0.2721321508e+2};
355 static double p5[4] = {0.1000006517e+1, 0.4909414111e+2, 0.8505544753e+2, 0.1532153455e+3};
356 static double q5[4] = {1.0 , 0.5009928881e+2, 0.1399819104e+3, 0.4200002909e+3};
358 static double p6[4] = {0.1000000983e+1, 0.1329868456e+3, 0.9162149244e+3,-0.9605054274e+3};
359 static double q6[4] = {1.0 , 0.1339887843e+3, 0.1055990413e+4, 0.5532224619e+3};
361 static double a1[4] = {0 ,-0.4583333333e+0, 0.6675347222e+0,-0.1641741416e+1};
362 static double a2[4] = {0 , 1.0 ,-0.4227843351e+0,-0.2043403138e+1};
364 double v = (
x - x0)/xi;
377 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*
v)*
v)*
v)*
v);
380 lan = (p2[0]+(p2[1]+(p2[2]+p2[3]*
v)*
v)*
v)/(q2[0]+(q2[1]+(q2[2]+q2[3]*
v)*
v)*
v);
383 lan = (p3[0]+(p3[1]+(p3[2]+p3[3]*
v)*
v)*
v)/(q3[0]+(q3[1]+(q3[2]+q3[3]*
v)*
v)*
v);
388 lan = (p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/(q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
393 lan = (p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/(q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
398 lan = (p6[0]+(p6[1]+(p6[2]+p6[3]*u)*u)*u)/(q6[0]+(q6[1]+(q6[2]+q6[3]*u)*u)*u);
403 lan = 1-(a2[1]+(a2[2]+a2[3]*u)*u)*u;
414 static double p1[5] = {-0.8949374280E+0, 0.4631783434E+0,-0.4053332915E-1,
415 0.1580075560E-1,-0.3423874194E-2};
416 static double q1[5] = { 1.0 , 0.1002930749E+0, 0.3575271633E-1,
417 -0.1915882099E-2, 0.4811072364E-4};
418 static double p2[5] = {-0.8933384046E+0, 0.1161296496E+0, 0.1200082940E+0,
419 0.2185699725E-1, 0.2128892058E-2};
420 static double q2[5] = { 1.0 , 0.4935531886E+0, 0.1066347067E+0,
421 0.1250161833E-1, 0.5494243254E-3};
422 static double p3[5] = {-0.8933322067E+0, 0.2339544896E+0, 0.8257653222E-1,
423 0.1411226998E-1, 0.2892240953E-3};
424 static double q3[5] = { 1.0 , 0.3616538408E+0, 0.6628026743E-1,
425 0.4839298984E-2, 0.5248310361E-4};
426 static double p4[4] = { 0.9358419425E+0, 0.6716831438E+2,-0.6765069077E+3,
428 static double q4[4] = { 1.0 , 0.7752562854E+2,-0.5637811998E+3,
430 static double p5[4] = { 0.9489335583E+0, 0.5561246706E+3, 0.3208274617E+5,
432 static double q5[4] = { 1.0 , 0.6028275940E+3, 0.3716962017E+5,
434 static double a0[6] = {-0.4227843351E+0,-0.1544313298E+0, 0.4227843351E+0,
435 0.3276496874E+1, 0.2043403138E+1,-0.8681296500E+1};
436 static double a1[4] = { 0, -0.4583333333E+0, 0.6675347222E+0,
438 static double a2[5] = { 0, -0.1958333333E+1, 0.5563368056E+1,
439 -0.2111352961E+2, 0.1006946266E+3};
441 double v = (
x-x0)/xi;
446 xm1lan =
v-u*(1+(a2[1]+(a2[2]+(a2[3]+a2[4]*u)*u)*u)*u)/
447 (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
451 xm1lan = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*
v)*
v)*
v)*
v)/
452 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*
v)*
v)*
v)*
v);
456 xm1lan = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*
v)*
v)*
v)*
v)/
457 (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*
v)*
v)*
v)*
v);
461 xm1lan = (p3[0]+(p3[1]+(p3[2]+(p3[3]+p3[4]*
v)*
v)*
v)*
v)/
462 (q3[0]+(q3[1]+(q3[2]+(q3[3]+q3[4]*
v)*
v)*
v)*
v);
467 xm1lan =
std::log(
v)*(p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/
468 (q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
473 xm1lan =
std::log(
v)*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
474 (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
481 xm1lan = (u+a0[0]+(-u+a0[1]+(a0[2]*u+a0[3]+(a0[4]*u+a0[5])*
v)*
v)*
v)/
482 (1-(1-(a0[2]+a0[4]*
v)*
v)*
v);
484 return xm1lan*xi + x0;
494 static double p1[5] = { 0.1169837582E+1,-0.4834874539E+0, 0.4383774644E+0,
495 0.3287175228E-2, 0.1879129206E-1};
496 static double q1[5] = { 1.0 , 0.1795154326E+0, 0.4612795899E-1,
497 0.2183459337E-2, 0.7226623623E-4};
498 static double p2[5] = { 0.1157939823E+1,-0.3842809495E+0, 0.3317532899E+0,
499 0.3547606781E-1, 0.6725645279E-2};
500 static double q2[5] = { 1.0 , 0.2916824021E+0, 0.5259853480E-1,
501 0.3840011061E-2, 0.9950324173E-4};
502 static double p3[4] = { 0.1178191282E+1, 0.1011623342E+2,-0.1285585291E+2,
504 static double q3[4] = { 1.0 , 0.8614160194E+1, 0.3118929630E+2,
506 static double p4[5] = { 0.1030763698E+1, 0.1216758660E+3, 0.1637431386E+4,
507 -0.2171466507E+4, 0.7010168358E+4};
508 static double q4[5] = { 1.0 , 0.1022487911E+3, 0.1377646350E+4,
509 0.3699184961E+4, 0.4251315610E+4};
510 static double p5[4] = { 0.1010084827E+1, 0.3944224824E+3, 0.1773025353E+5,
512 static double q5[4] = { 1.0 , 0.3605950254E+3, 0.1392784158E+5,
514 static double a0[7] = {-0.2043403138E+1,-0.8455686702E+0,-0.3088626596E+0,
515 0.5821346754E+1, 0.4227843351E+0, 0.6552993748E+1,
517 static double a1[4] = { 0. ,-0.4583333333E+0, 0.6675347222E+0,
519 static double a2[4] = {-0.1958333333E+1, 0.5563368056E+1,-0.2111352961E+2,
521 static double a3[4] = {-1.0 , 0.4458333333E+1,-0.2116753472E+2,
524 double v = (
x-x0)/xi;
530 (
v/u+a2[0]*
v+a3[0]+(a2[1]*
v+a3[1]+(a2[2]*
v+a3[2]+
531 (a2[3]*
v+a3[3])*u)*u)*u)/
532 (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
536 xm2lan = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*
v)*
v)*
v)*
v)/
537 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*
v)*
v)*
v)*
v);
541 xm2lan = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*
v)*
v)*
v)*
v)/
542 (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*
v)*
v)*
v)*
v);
547 xm2lan =
v*(p3[0]+(p3[1]+(p3[2]+p3[3]*u)*u)*u)/
548 (q3[0]+(q3[1]+(q3[2]+q3[3]*u)*u)*u);
553 xm2lan =
v*(p4[0]+(p4[1]+(p4[2]+(p4[3]+p4[4]*u)*u)*u)*u)/
554 (q4[0]+(q4[1]+(q4[2]+(q4[3]+q4[4]*u)*u)*u)*u);
559 xm2lan =
v*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
560 (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
565 v = 1/(u-u*(u+
log(u)-
v)/(u+1));
567 xm2lan = (1/
v+u*u+a0[0]+a0[1]*u+(-u*u+a0[2]*u+a0[3]+
568 (a0[4]*u*u+a0[5]*u+a0[6])*
v)*
v)/(1-(1-a0[4]*
v)*
v);
570 if (x0 == 0)
return xm2lan*xi*xi;
572 return xm2lan*xi*xi + (2*xm1lan-x0)*x0;
#define MATH_ERROR_MSG(loc, str)
#define MATH_WARN_MSG(loc, str)
double pow(double, double)
double poisson_cdf(unsigned int n, double mu)
Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisso...
double crystalball_integral(double x, double alpha, double n, double sigma, double x0=0)
Integral of the not-normalized Crystal Ball function.
double uniform_cdf(double x, double a, double b, double x0=0)
Cumulative distribution function of the uniform (flat) distribution (lower tail).
double binomial_cdf_c(unsigned int k, double p, unsigned int n)
Complement of the cumulative distribution function of the Binomial distribution.
double lognormal_cdf(double x, double m, double s, double x0=0)
Cumulative distribution function of the lognormal distribution (lower tail).
double cauchy_cdf_c(double x, double b, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is a...
double fdistribution_cdf(double x, double n, double m, double x0=0)
Cumulative distribution function of the F-distribution (lower tail).
double binomial_cdf(unsigned int k, double p, unsigned int n)
Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binom...
double landau_cdf(double x, double xi=1, double x0=0)
Cumulative distribution function of the Landau distribution (lower tail).
double lognormal_cdf_c(double x, double m, double s, double x0=0)
Complement of the cumulative distribution function of the lognormal distribution (upper tail).
double uniform_cdf_c(double x, double a, double b, double x0=0)
Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail).
double fdistribution_cdf_c(double x, double n, double m, double x0=0)
Complement of the cumulative distribution function of the F-distribution (upper tail).
double crystalball_cdf_c(double x, double alpha, double n, double sigma, double x0=0)
Complement of the Cumulative distribution for the Crystal Ball distribution.
double normal_cdf_c(double x, double sigma=1, double x0=0)
Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail)...
double chisquared_cdf(double x, double r, double x0=0)
Cumulative distribution function of the distribution with degrees of freedom (lower tail).
double negative_binomial_cdf_c(unsigned int k, double p, double n)
Complement of the cumulative distribution function of the Negative Binomial distribution.
double gamma_cdf_c(double x, double alpha, double theta, double x0=0)
Complement of the cumulative distribution function of the gamma distribution (upper tail).
double beta_cdf(double x, double a, double b)
Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.
double crystalball_cdf(double x, double alpha, double n, double sigma, double x0=0)
Cumulative distribution for the Crystal Ball distribution function.
double negative_binomial_cdf(unsigned int k, double p, double n)
Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of ...
double breitwigner_cdf_c(double x, double gamma, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and ...
double normal_cdf(double x, double sigma=1, double x0=0)
Cumulative distribution function of the normal (Gaussian) distribution (lower tail).
double breitwigner_cdf(double x, double gamma, double x0=0)
Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (jus...
double exponential_cdf_c(double x, double lambda, double x0=0)
Complement of the cumulative distribution function of the exponential distribution (upper tail).
double tdistribution_cdf(double x, double r, double x0=0)
Cumulative distribution function of Student's t-distribution (lower tail).
double beta_cdf_c(double x, double a, double b)
Complement of the cumulative distribution function of the beta distribution.
double poisson_cdf_c(unsigned int n, double mu)
Complement of the cumulative distribution function of the Poisson distribution.
double chisquared_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of the distribution with degrees of freedom (upp...
double cauchy_cdf(double x, double b, double x0=0)
Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian dis...
double tdistribution_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of Student's t-distribution (upper tail).
double gamma_cdf(double x, double alpha, double theta, double x0=0)
Cumulative distribution function of the gamma distribution (lower tail).
double exponential_cdf(double x, double lambda, double x0=0)
Cumulative distribution function of the exponential distribution (lower tail).
double inc_gamma_c(double a, double x)
Calculates the normalized (regularized) upper incomplete gamma function (upper integral)
double inc_beta(double x, double a, double b)
Calculates the normalized (regularized) incomplete beta function.
double erfc(double x)
Complementary error function.
double inc_gamma(double a, double x)
Calculates the normalized (regularized) lower incomplete gamma function (lower integral)
double erf(double x)
Error function encountered in integrating the normal distribution.
double landau_xm1(double x, double xi=1, double x0=0)
First moment (mean) of the truncated Landau distribution.
double landau_xm2(double x, double xi=1, double x0=0)
Second moment of the truncated Landau distribution.
Namespace for new Math classes and functions.
double expm1(double x)
exp(x) -1 with error cancellation when x is small
VecExpr< UnaryOp< Sqrt< T >, VecExpr< A, T, D >, T >, T, D > sqrt(const VecExpr< A, T, D > &rhs)
double gaussian_cdf_c(double x, double sigma=1, double x0=0)
Alternative name for same function.
static const double kSqrt2
Namespace for new ROOT classes and functions.
static constexpr double s