specialfunctions.cpp

/*************************************************************************
Copyright (c) Sergey Bochkanov (ALGLIB project).

>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#include "stdafx.h"
#include "specialfunctions.h"

// disable some irrelevant warnings
#if (AE_COMPILER==AE_MSVC)
#pragma warning(disable:4100)
#pragma warning(disable:4127)
#pragma warning(disable:4702)
#pragma warning(disable:4996)
#endif
using namespace std;

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS IMPLEMENTATION OF C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{


/*************************************************************************
Gamma function

Input parameters:
    X   -   argument

Domain:
    0 < X < 171.6
    -170 < X < 0, X is not an integer.

Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE    -170,-33      20000       2.3e-15     3.3e-16
    IEEE     -33,  33     20000       9.4e-16     2.2e-16
    IEEE      33, 171.6   20000       2.3e-15     3.2e-16

Cephes Math Library Release 2.8:  June, 2000
Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double gammafunction(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::gammafunction(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Natural logarithm of gamma function

Input parameters:
    X       -   argument

Result:
    logarithm of the absolute value of the Gamma(X).

Output parameters:
    SgnGam  -   sign(Gamma(X))

Domain:
    0 < X < 2.55e305
    -2.55e305 < X < 0, X is not an integer.

ACCURACY:
arithmetic      domain        # trials     peak         rms
   IEEE    0, 3                 28000     5.4e-16     1.1e-16
   IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.

The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
   IEEE    -200, -4             10000     4.8e-16     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double lngamma(const double x, double &sgngam)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::lngamma(x, &sgngam, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Error function

The integral is

                          x
                           -
                2         | |          2
  erf(x)  =  --------     |    exp( - t  ) dt.
             sqrt(pi)   | |
                         -
                          0

For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,1         30000       3.7e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunction(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::errorfunction(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complementary error function

 1 - erf(x) =

                          inf.
                            -
                 2         | |          2
  erfc(x)  =  --------     |    exp( - t  ) dt
              sqrt(pi)   | |
                          -
                           x


For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,26.6417   30000       5.7e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunctionc(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::errorfunctionc(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Normal distribution function

Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:

                           x
                            -
                  1        | |          2
   ndtr(x)  = ---------    |    exp( - t /2 ) dt
              sqrt(2pi)  | |
                          -
                         -inf.

            =  ( 1 + erf(z) ) / 2
            =  erfc(z) / 2

where z = x/sqrt(2). Computation is via the functions
erf and erfc.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -13,0        30000       3.4e-14     6.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double normaldistribution(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::normaldistribution(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of the error function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double inverf(const double e)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::inverf(e, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of Normal distribution function

Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.


For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) );  then the approximation is
x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2).  For larger arguments,
w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

ACCURACY:

                     Relative error:
arithmetic   domain        # trials      peak         rms
   IEEE     0.125, 1        20000       7.2e-16     1.3e-16
   IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double invnormaldistribution(const double y0)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invnormaldistribution(y0, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Incomplete gamma integral

The function is defined by

                          x
                           -
                  1       | |  -t  a-1
 igam(a,x)  =   -----     |   e   t   dt.
                 -      | |
                | (a)    -
                          0


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30       200000       3.6e-14     2.9e-15
   IEEE      0,100      300000       9.9e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegamma(const double a, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::incompletegamma(a, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complemented incomplete gamma integral

The function is defined by


 igamc(a,x)   =   1 - igam(a,x)

                           inf.
                             -
                    1       | |  -t  a-1
              =   -----     |   e   t   dt.
                   -      | |
                  | (a)    -
                            x


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

Tested at random a, x.
               a         x                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
   IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegammac(const double a, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::incompletegammac(a, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of complemented imcomplete gamma integral

Given p, the function finds x such that

 igamc( a, x ) = p.

Starting with the approximate value

        3
 x = a t

 where

 t = 1 - d - ndtri(p) sqrt(d)

and

 d = 1/9a,

the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.

ACCURACY:

Tested at random a, p in the intervals indicated.

               a        p                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
   IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
   IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletegammac(const double a, const double y0)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invincompletegammac(a, y0, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Airy function

Solution of the differential equation

y"(x) = xy.

The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).

Evaluation is by power series summation for small x,
by rational minimax approximations for large x.



ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.

Arithmetic  domain   function  # trials      peak         rms
IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void airy(const double x, double &ai, double &aip, double &bi, double &bip)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::airy(x, &ai, &aip, &bi, &bip, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of order zero

Returns Bessel function of order zero of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:


       2         2
(w - r  ) (w - r  ) P (w) / Q (w)
      1         2    3       8

           2
where w = x  and the two r's are zeros of the function.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.

ACCURACY:

                     Absolute error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       60000       4.2e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj0(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselj0(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of order one

Returns Bessel function of order one of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       2.6e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj1(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselj1(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The ratio of jn(x) to j0(x) is computed by backward
recurrence.  First the ratio jn/jn-1 is found by a
continued fraction expansion.  Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.

If n = 0 or 1 the routine for j0 or j1 is called
directly.

ACCURACY:

                     Absolute error:
arithmetic   range      # trials      peak         rms
   IEEE      0, 30        5000       4.4e-16     7.9e-17


Not suitable for large n or x. Use jv() (fractional order) instead.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseljn(const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besseljn(n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of the second kind, order zero

Returns Bessel function of the second kind, of order
zero, of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
  y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.



ACCURACY:

 Absolute error, when y0(x) < 1; else relative error:

arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.3e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely0(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::bessely0(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of second kind of order one

Returns Bessel function of the second kind of order one
of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       1.0e-15     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely1(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::bessely1(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Bessel function of second kind of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().

If n = 0 or 1 the routine for y0 or y1 is called
directly.

ACCURACY:
                     Absolute error, except relative
                     when y > 1:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       3.4e-15     4.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselyn(const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselyn(n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Modified Bessel function of order zero

Returns modified Bessel function of order zero of the
argument.

The function is defined as i0(x) = j0( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       5.8e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli0(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besseli0(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Modified Bessel function of order one

Returns modified Bessel function of order one of the
argument.

The function is defined as i1(x) = -i j1( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.9e-15     2.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli1(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besseli1(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Modified Bessel function, second kind, order zero

Returns modified Bessel function of the second kind
of order zero of the argument.

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

Tested at 2000 random points between 0 and 8.  Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk0(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselk0(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Modified Bessel function, second kind, order one

Computes the modified Bessel function of the second kind
of order one of the argument.

The range is partitioned into the two intervals [0,2] and
(2, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk1(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselk1(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Modified Bessel function, second kind, integer order

Returns modified Bessel function of the second kind
of order n of the argument.

The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity).  An ascending power series is used in the
low range, and an asymptotic expansion in the high range.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        90000       1.8e-8      3.0e-10

Error is high only near the crossover point x = 9.55
between the two expansions used.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*************************************************************************/
double besselkn(const ae_int_t nn, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::besselkn(nn, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Beta function


                  -     -
                 | (a) | (b)
beta( a, b )  =  -----------.
                    -
                   | (a+b)

For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,30       30000       8.1e-14     1.1e-14

Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
*************************************************************************/
double beta(const double a, const double b)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::beta(a, b, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Incomplete beta integral

Returns incomplete beta integral of the arguments, evaluated
from zero to x.  The function is defined as

                 x
    -            -
   | (a+b)      | |  a-1     b-1
 -----------    |   t   (1-t)   dt.
  -     -     | |
 | (a) | (b)   -
                0

The domain of definition is 0 <= x <= 1.  In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation

   1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.

ACCURACY:

Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
                                       Relative error
arithmetic   domain     # trials      peak         rms
   IEEE      0,5         10000       6.9e-15     4.5e-16
   IEEE      0,85       250000       2.2e-13     1.7e-14
   IEEE      0,1000      30000       5.3e-12     6.3e-13
   IEEE      0,10000    250000       9.3e-11     7.1e-12
   IEEE      0,100000    10000       8.7e-10     4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletebeta(const double a, const double b, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::incompletebeta(a, b, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of imcomplete beta integral

Given y, the function finds x such that

 incbet( a, b, x ) = y .

The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.


ACCURACY:

                     Relative error:
               x     a,b
arithmetic   domain  domain  # trials    peak       rms
   IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
   IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
   IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
With a and b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
   IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
With a = .5, b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletebeta(const double a, const double b, const double y)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invincompletebeta(a, b, y, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Binomial distribution

Returns the sum of the terms 0 through k of the Binomial
probability density:

  k
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=0

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p), with p between 0 and 1.

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      4.3e-15     2.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialdistribution(const ae_int_t k, const ae_int_t n, const double p)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::binomialdistribution(k, n, p, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complemented binomial distribution

Returns the sum of the terms k+1 through n of the Binomial
probability density:

  n
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=k+1

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      6.7e-15     8.2e-16
 For p between 0 and .001:
   IEEE     0,100       100000      1.5e-13     2.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialcdistribution(const ae_int_t k, const ae_int_t n, const double p)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::binomialcdistribution(k, n, p, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse binomial distribution

Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.

This is accomplished using the inverse beta integral
function and the relation

1 - p = incbi( n-k, k+1, y ).

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      2.3e-14     6.4e-16
   IEEE     0,10000     100000      6.6e-12     1.2e-13
 For p between 10^-6 and 0.001:
   IEEE     0,100       100000      2.0e-12     1.3e-14
   IEEE     0,10000     100000      1.5e-12     3.2e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invbinomialdistribution(const ae_int_t k, const ae_int_t n, const double y)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invbinomialdistribution(k, n, y, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Calculation of the value of the Chebyshev polynomials of the
first and second kinds.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument, -1 <= x <= 1

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevcalculate(const ae_int_t r, const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::chebyshevcalculate(r, n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Summation of Chebyshev polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
or
    c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
depending on the R.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevsum(const real_1d_array &c, const ae_int_t r, const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::chebyshevsum(const_cast<alglib_impl::ae_vector*>(c.c_ptr()), r, n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void chebyshevcoefficients(const ae_int_t n, real_1d_array &c)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::chebyshevcoefficients(n, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Conversion of a series of Chebyshev polynomials to a power series.

Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
B[0] + B[1]*X + ... + B[N]*X^N.

Input parameters:
    A   -   Chebyshev series coefficients
    N   -   degree, N>=0

Output parameters
    B   -   power series coefficients
*************************************************************************/
void fromchebyshev(const real_1d_array &a, const ae_int_t n, real_1d_array &b)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::fromchebyshev(const_cast<alglib_impl::ae_vector*>(a.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(b.c_ptr()), &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Chi-square distribution

Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.


                                  x
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  0

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquaredistribution(const double v, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::chisquaredistribution(v, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complemented Chi-square distribution

Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:

                                 inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquarecdistribution(const double v, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::chisquarecdistribution(v, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of complemented Chi-square distribution

Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.

This is accomplished using the inverse gamma integral
function and the relation

   x/2 = igami( df/2, y );

ACCURACY:

See inverse incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double invchisquaredistribution(const double v, const double y)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invchisquaredistribution(v, y, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Dawson's Integral

Approximates the integral

                            x
                            -
                     2     | |        2
 dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
                         | |
                          -
                          0

Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,10        10000       6.9e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double dawsonintegral(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::dawsonintegral(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

using the approximation

    P(x)  -  log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralk(const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::ellipticintegralk(m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

where m = 1 - m1, using the approximation

    P(x)  -  log x Q(x).

The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.

K(0) = pi/2.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Àëãîðèòì âçÿò èç áèáëèîòåêè Cephes
*************************************************************************/
double ellipticintegralkhighprecision(const double m1)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::ellipticintegralkhighprecision(m1, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Incomplete elliptic integral of the first kind F(phi|m)

Approximates the integral



               phi
                -
               | |
               |           dt
F(phi_\m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.




ACCURACY:

Tested at random points with m in [0, 1] and phi as indicated.

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10       200000      7.4e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegralk(const double phi, const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::incompleteellipticintegralk(phi, m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complete elliptic integral of the second kind

Approximates the integral


           pi/2
            -
           | |                 2
E(m)  =    |    sqrt( 1 - m sin t ) dt
         | |
          -
           0

using the approximation

     P(x)  -  x log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0, 1       10000       2.1e-16     7.3e-17

Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegrale(const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::ellipticintegrale(m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Incomplete elliptic integral of the second kind

Approximates the integral


               phi
                -
               | |
               |                   2
E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
               |
             | |
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.

ACCURACY:

Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10      150000       3.3e-15     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegrale(const double phi, const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::incompleteellipticintegrale(phi, m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Exponential integral Ei(x)

              x
               -     t
              | |   e
   Ei(x) =   -|-   ---  dt .
            | |     t
             -
            -inf

Not defined for x <= 0.
See also expn.c.



ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,100       50000      8.6e-16     1.3e-16

Cephes Math Library Release 2.8:  May, 1999
Copyright 1999 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralei(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::exponentialintegralei(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Exponential integral En(x)

Evaluates the exponential integral

                inf.
                  -
                 | |   -xt
                 |    e
     E (x)  =    |    ----  dt.
      n          |      n
               | |     t
                -
                 1


Both n and x must be nonnegative.

The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       10000       1.7e-15     3.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 2000 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralen(const double x, const ae_int_t n)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::exponentialintegralen(x, n, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
F distribution

Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density).  This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula

P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).


The arguments a and b are greater than zero, and x is
nonnegative.

ACCURACY:

Tested at random points (a,b,x).

               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
   IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
   IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
   IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fdistribution(const ae_int_t a, const ae_int_t b, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::fdistribution(a, b, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complemented F distribution

Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).


                     inf.
                      -
             1       | |  a-1      b-1
1-P(x)  =  ------    |   t    (1-t)    dt
           B(a,b)  | |
                    -
                     x


The incomplete beta integral is used, according to the
formula

P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).


ACCURACY:

Tested at random points (a,b,x) in the indicated intervals.
               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
   IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
   IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
   IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fcdistribution(const ae_int_t a, const ae_int_t b, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::fcdistribution(a, b, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse of complemented F distribution

Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.

This is accomplished using the inverse beta integral
function and the relations

     z = incbi( df2/2, df1/2, p )
     x = df2 (1-z) / (df1 z).

Note: the following relations hold for the inverse of
the uncomplemented F distribution:

     z = incbi( df1/2, df2/2, p )
     x = df2 z / (df1 (1-z)).

ACCURACY:

Tested at random points (a,b,p).

             a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between .001 and 1:
   IEEE     1,100       100000      8.3e-15     4.7e-16
   IEEE     1,10000     100000      2.1e-11     1.4e-13
 For p between 10^-6 and 10^-3:
   IEEE     1,100        50000      1.3e-12     8.4e-15
   IEEE     1,10000      50000      3.0e-12     4.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invfdistribution(const ae_int_t a, const ae_int_t b, const double y)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invfdistribution(a, b, y, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Fresnel integral

Evaluates the Fresnel integrals

          x
          -
         | |
C(x) =   |   cos(pi/2 t**2) dt,
       | |
        -
         0

          x
          -
         | |
S(x) =   |   sin(pi/2 t**2) dt.
       | |
        -
         0


The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that

C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )



ACCURACY:

 Relative error.

Arithmetic  function   domain     # trials      peak         rms
  IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
  IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void fresnelintegral(const double x, double &c, double &s)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::fresnelintegral(x, &c, &s, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Calculation of the value of the Hermite polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial Hn at x
*************************************************************************/
double hermitecalculate(const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::hermitecalculate(n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Summation of Hermite polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial at x
*************************************************************************/
double hermitesum(const real_1d_array &c, const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::hermitesum(const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void hermitecoefficients(const ae_int_t n, real_1d_array &c)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::hermitecoefficients(n, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Jacobian Elliptic Functions

Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.

These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).

Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1.  In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.

ACCURACY:

Tested at random points with u between 0 and 10, m between
0 and 1.

           Absolute error (* = relative error):
arithmetic   function   # trials      peak         rms
   IEEE      phi         10000       9.2e-16*    1.4e-16*
   IEEE      sn          50000       4.1e-15     4.6e-16
   IEEE      cn          40000       3.6e-15     4.4e-16
   IEEE      dn          10000       1.3e-12     1.8e-14

 Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void jacobianellipticfunctions(const double u, const double m, double &sn, double &cn, double &dn, double &ph)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::jacobianellipticfunctions(u, m, &sn, &cn, &dn, &ph, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Calculation of the value of the Laguerre polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial Ln at x
*************************************************************************/
double laguerrecalculate(const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::laguerrecalculate(n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Summation of Laguerre polynomials using Clenshaw’s recurrence formula.

This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial at x
*************************************************************************/
double laguerresum(const real_1d_array &c, const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::laguerresum(const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void laguerrecoefficients(const ae_int_t n, real_1d_array &c)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::laguerrecoefficients(n, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Calculation of the value of the Legendre polynomial Pn.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial Pn at x
*************************************************************************/
double legendrecalculate(const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::legendrecalculate(n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Summation of Legendre polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial at x
*************************************************************************/
double legendresum(const real_1d_array &c, const ae_int_t n, const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::legendresum(const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void legendrecoefficients(const ae_int_t n, real_1d_array &c)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::legendrecoefficients(n, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Poisson distribution

Returns the sum of the first k+1 terms of the Poisson
distribution:

  k         j
  --   -m  m
  >   e    --
  --       j!
 j=0

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation

y = pdtr( k, m ) = igamc( k+1, m ).

The arguments must both be positive.
ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissondistribution(const ae_int_t k, const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::poissondistribution(k, m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Complemented Poisson distribution

Returns the sum of the terms k+1 to infinity of the Poisson
distribution:

 inf.       j
  --   -m  m
  >   e    --
  --       j!
 j=k+1

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula

y = pdtrc( k, m ) = igam( k+1, m ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissoncdistribution(const ae_int_t k, const double m)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::poissoncdistribution(k, m, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Inverse Poisson distribution

Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.

This is accomplished using the inverse gamma integral
function and the relation

   m = igami( k+1, y ).

ACCURACY:

See inverse incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invpoissondistribution(const ae_int_t k, const double y)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invpoissondistribution(k, y, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Psi (digamma) function

             d      -
  psi(x)  =  -- ln | (x)
             dx

is the logarithmic derivative of the gamma function.
For integer x,
                  n-1
                   -
psi(n) = -EUL  +   >  1/k.
                   -
                  k=1

This formula is used for 0 < n <= 10.  If x is negative, it
is transformed to a positive argument by the reflection
formula  psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence  psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:

                          inf.   B
                           -      2k
psi(x) = log(x) - 1/2x -   >   -------
                           -        2k
                          k=1   2k x

where the B2k are Bernoulli numbers.

ACCURACY:
   Relative error (except absolute when |psi| < 1):
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       1.3e-15     1.4e-16
   IEEE      -30,0       40000       1.5e-15     2.2e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double psi(const double x)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::psi(x, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Student's t distribution

Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:

                                     t
                                     -
                                    | |
             -                      |         2   -(k+1)/2
            | ( (k+1)/2 )           |  (     x   )
      ----------------------        |  ( 1 + --- )        dx
                    -               |  (      k  )
      sqrt( k pi ) | ( k/2 )        |
                                  | |
                                   -
                                  -inf.

Relation to incomplete beta integral:

       1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
       z = k/(k + t**2).

For t < -2, this is the method of computation.  For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.

ACCURACY:

Tested at random 1 <= k <= 25.  The "domain" refers to t.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -100,-2      50000       5.9e-15     1.4e-15
   IEEE     -2,100      500000       2.7e-15     4.9e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double studenttdistribution(const ae_int_t k, const double t)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::studenttdistribution(k, t, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Functional inverse of Student's t distribution

Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.

ACCURACY:

Tested at random 1 <= k <= 100.  The "domain" refers to p:
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE    .001,.999     25000       5.7e-15     8.0e-16
   IEEE    10^-6,.001    25000       2.0e-12     2.9e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invstudenttdistribution(const ae_int_t k, const double p)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        double result = alglib_impl::invstudenttdistribution(k, p, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return *(reinterpret_cast<double*>(&result));
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Sine and cosine integrals

Evaluates the integrals

                         x
                         -
                        |  cos t - 1
  Ci(x) = eul + ln x +  |  --------- dt,
                        |      t
                       -
                        0
            x
            -
           |  sin t
  Si(x) =  |  ----- dt
           |    t
          -
           0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that

Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)


ACCURACY:
   Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic   function   # trials      peak         rms
   IEEE        Si        30000       4.4e-16     7.3e-17
   IEEE        Ci        30000       6.9e-16     5.1e-17

Cephes Math Library Release 2.1:  January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
*************************************************************************/
void sinecosineintegrals(const double x, double &si, double &ci)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::sinecosineintegrals(x, &si, &ci, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}

/*************************************************************************
Hyperbolic sine and cosine integrals

Approximates the integrals

                           x
                           -
                          | |   cosh t - 1
  Chi(x) = eul + ln x +   |    -----------  dt,
                        | |          t
                         -
                         0

              x
              -
             | |  sinh t
  Shi(x) =   |    ------  dt
           | |       t
            -
            0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.


ACCURACY:

Test interval 0 to 88.
                     Relative error:
arithmetic   function  # trials      peak         rms
   IEEE         Shi      30000       6.9e-16     1.6e-16
       Absolute error, except relative when |Chi| > 1:
   IEEE         Chi      30000       8.4e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void hyperbolicsinecosineintegrals(const double x, double &shi, double &chi)
{
    alglib_impl::ae_state _alglib_env_state;
    alglib_impl::ae_state_init(&_alglib_env_state);
    try
    {
        alglib_impl::hyperbolicsinecosineintegrals(x, &shi, &chi, &_alglib_env_state);
        alglib_impl::ae_state_clear(&_alglib_env_state);
        return;
    }
    catch(alglib_impl::ae_error_type)
    {
        throw ap_error(_alglib_env_state.error_msg);
    }
}
}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS IMPLEMENTATION OF COMPUTATIONAL CORE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
static double gammafunc_gammastirf(double x, ae_state *_state);








static void bessel_besselmfirstcheb(double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state);
static void bessel_besselmnextcheb(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state);
static void bessel_besselm1firstcheb(double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state);
static void bessel_besselm1nextcheb(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state);
static void bessel_besselasympt0(double x,
     double* pzero,
     double* qzero,
     ae_state *_state);
static void bessel_besselasympt1(double x,
     double* pzero,
     double* qzero,
     ae_state *_state);




static double ibetaf_incompletebetafe(double a,
     double b,
     double x,
     double big,
     double biginv,
     ae_state *_state);
static double ibetaf_incompletebetafe2(double a,
     double b,
     double x,
     double big,
     double biginv,
     ae_state *_state);
static double ibetaf_incompletebetaps(double a,
     double b,
     double x,
     double maxgam,
     ae_state *_state);
































static void trigintegrals_chebiterationshichi(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state);





/*************************************************************************
Gamma function

Input parameters:
    X   -   argument

Domain:
    0 < X < 171.6
    -170 < X < 0, X is not an integer.

Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE    -170,-33      20000       2.3e-15     3.3e-16
    IEEE     -33,  33     20000       9.4e-16     2.2e-16
    IEEE      33, 171.6   20000       2.3e-15     3.2e-16

Cephes Math Library Release 2.8:  June, 2000
Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double gammafunction(double x, ae_state *_state)
{
#ifndef ALGLIB_INTERCEPTS_SPECFUNCS
    double p;
    double pp;
    double q;
    double qq;
    double z;
    ae_int_t i;
    double sgngam;
    double result;


    sgngam = 1;
    q = ae_fabs(x, _state);
    if( ae_fp_greater(q,33.0) )
    {
        if( ae_fp_less(x,0.0) )
        {
            p = ae_ifloor(q, _state);
            i = ae_round(p, _state);
            if( i%2==0 )
            {
                sgngam = -1;
            }
            z = q-p;
            if( ae_fp_greater(z,0.5) )
            {
                p = p+1;
                z = q-p;
            }
            z = q*ae_sin(ae_pi*z, _state);
            z = ae_fabs(z, _state);
            z = ae_pi/(z*gammafunc_gammastirf(q, _state));
        }
        else
        {
            z = gammafunc_gammastirf(x, _state);
        }
        result = sgngam*z;
        return result;
    }
    z = 1;
    while(ae_fp_greater_eq(x,3))
    {
        x = x-1;
        z = z*x;
    }
    while(ae_fp_less(x,0))
    {
        if( ae_fp_greater(x,-0.000000001) )
        {
            result = z/((1+0.5772156649015329*x)*x);
            return result;
        }
        z = z/x;
        x = x+1;
    }
    while(ae_fp_less(x,2))
    {
        if( ae_fp_less(x,0.000000001) )
        {
            result = z/((1+0.5772156649015329*x)*x);
            return result;
        }
        z = z/x;
        x = x+1.0;
    }
    if( ae_fp_eq(x,2) )
    {
        result = z;
        return result;
    }
    x = x-2.0;
    pp = 1.60119522476751861407E-4;
    pp = 1.19135147006586384913E-3+x*pp;
    pp = 1.04213797561761569935E-2+x*pp;
    pp = 4.76367800457137231464E-2+x*pp;
    pp = 2.07448227648435975150E-1+x*pp;
    pp = 4.94214826801497100753E-1+x*pp;
    pp = 9.99999999999999996796E-1+x*pp;
    qq = -2.31581873324120129819E-5;
    qq = 5.39605580493303397842E-4+x*qq;
    qq = -4.45641913851797240494E-3+x*qq;
    qq = 1.18139785222060435552E-2+x*qq;
    qq = 3.58236398605498653373E-2+x*qq;
    qq = -2.34591795718243348568E-1+x*qq;
    qq = 7.14304917030273074085E-2+x*qq;
    qq = 1.00000000000000000320+x*qq;
    result = z*pp/qq;
    return result;
#else
    return _ialglib_i_gammafunction(x);
#endif
}


/*************************************************************************
Natural logarithm of gamma function

Input parameters:
    X       -   argument

Result:
    logarithm of the absolute value of the Gamma(X).

Output parameters:
    SgnGam  -   sign(Gamma(X))

Domain:
    0 < X < 2.55e305
    -2.55e305 < X < 0, X is not an integer.

ACCURACY:
arithmetic      domain        # trials     peak         rms
   IEEE    0, 3                 28000     5.4e-16     1.1e-16
   IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.

The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
   IEEE    -200, -4             10000     4.8e-16     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double lngamma(double x, double* sgngam, ae_state *_state)
{
#ifndef ALGLIB_INTERCEPTS_SPECFUNCS
    double a;
    double b;
    double c;
    double p;
    double q;
    double u;
    double w;
    double z;
    ae_int_t i;
    double logpi;
    double ls2pi;
    double tmp;
    double result;

    *sgngam = 0;

    *sgngam = 1;
    logpi = 1.14472988584940017414;
    ls2pi = 0.91893853320467274178;
    if( ae_fp_less(x,-34.0) )
    {
        q = -x;
        w = lngamma(q, &tmp, _state);
        p = ae_ifloor(q, _state);
        i = ae_round(p, _state);
        if( i%2==0 )
        {
            *sgngam = -1;
        }
        else
        {
            *sgngam = 1;
        }
        z = q-p;
        if( ae_fp_greater(z,0.5) )
        {
            p = p+1;
            z = p-q;
        }
        z = q*ae_sin(ae_pi*z, _state);
        result = logpi-ae_log(z, _state)-w;
        return result;
    }
    if( ae_fp_less(x,13) )
    {
        z = 1;
        p = 0;
        u = x;
        while(ae_fp_greater_eq(u,3))
        {
            p = p-1;
            u = x+p;
            z = z*u;
        }
        while(ae_fp_less(u,2))
        {
            z = z/u;
            p = p+1;
            u = x+p;
        }
        if( ae_fp_less(z,0) )
        {
            *sgngam = -1;
            z = -z;
        }
        else
        {
            *sgngam = 1;
        }
        if( ae_fp_eq(u,2) )
        {
            result = ae_log(z, _state);
            return result;
        }
        p = p-2;
        x = x+p;
        b = -1378.25152569120859100;
        b = -38801.6315134637840924+x*b;
        b = -331612.992738871184744+x*b;
        b = -1162370.97492762307383+x*b;
        b = -1721737.00820839662146+x*b;
        b = -853555.664245765465627+x*b;
        c = 1;
        c = -351.815701436523470549+x*c;
        c = -17064.2106651881159223+x*c;
        c = -220528.590553854454839+x*c;
        c = -1139334.44367982507207+x*c;
        c = -2532523.07177582951285+x*c;
        c = -2018891.41433532773231+x*c;
        p = x*b/c;
        result = ae_log(z, _state)+p;
        return result;
    }
    q = (x-0.5)*ae_log(x, _state)-x+ls2pi;
    if( ae_fp_greater(x,100000000) )
    {
        result = q;
        return result;
    }
    p = 1/(x*x);
    if( ae_fp_greater_eq(x,1000.0) )
    {
        q = q+((7.9365079365079365079365*0.0001*p-2.7777777777777777777778*0.001)*p+0.0833333333333333333333)/x;
    }
    else
    {
        a = 8.11614167470508450300*0.0001;
        a = -5.95061904284301438324*0.0001+p*a;
        a = 7.93650340457716943945*0.0001+p*a;
        a = -2.77777777730099687205*0.001+p*a;
        a = 8.33333333333331927722*0.01+p*a;
        q = q+a/x;
    }
    result = q;
    return result;
#else
    return _ialglib_i_lngamma(x, sgngam);
#endif
}


static double gammafunc_gammastirf(double x, ae_state *_state)
{
    double y;
    double w;
    double v;
    double stir;
    double result;


    w = 1/x;
    stir = 7.87311395793093628397E-4;
    stir = -2.29549961613378126380E-4+w*stir;
    stir = -2.68132617805781232825E-3+w*stir;
    stir = 3.47222221605458667310E-3+w*stir;
    stir = 8.33333333333482257126E-2+w*stir;
    w = 1+w*stir;
    y = ae_exp(x, _state);
    if( ae_fp_greater(x,143.01608) )
    {
        v = ae_pow(x, 0.5*x-0.25, _state);
        y = v*(v/y);
    }
    else
    {
        y = ae_pow(x, x-0.5, _state)/y;
    }
    result = 2.50662827463100050242*y*w;
    return result;
}




/*************************************************************************
Error function

The integral is

                          x
                           -
                2         | |          2
  erf(x)  =  --------     |    exp( - t  ) dt.
             sqrt(pi)   | |
                         -
                          0

For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,1         30000       3.7e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunction(double x, ae_state *_state)
{
    double xsq;
    double s;
    double p;
    double q;
    double result;


    s = ae_sign(x, _state);
    x = ae_fabs(x, _state);
    if( ae_fp_less(x,0.5) )
    {
        xsq = x*x;
        p = 0.007547728033418631287834;
        p = -0.288805137207594084924010+xsq*p;
        p = 14.3383842191748205576712+xsq*p;
        p = 38.0140318123903008244444+xsq*p;
        p = 3017.82788536507577809226+xsq*p;
        p = 7404.07142710151470082064+xsq*p;
        p = 80437.3630960840172832162+xsq*p;
        q = 0.0;
        q = 1.00000000000000000000000+xsq*q;
        q = 38.0190713951939403753468+xsq*q;
        q = 658.070155459240506326937+xsq*q;
        q = 6379.60017324428279487120+xsq*q;
        q = 34216.5257924628539769006+xsq*q;
        q = 80437.3630960840172826266+xsq*q;
        result = s*1.1283791670955125738961589031*x*p/q;
        return result;
    }
    if( ae_fp_greater_eq(x,10) )
    {
        result = s;
        return result;
    }
    result = s*(1-errorfunctionc(x, _state));
    return result;
}


/*************************************************************************
Complementary error function

 1 - erf(x) =

                          inf.
                            -
                 2         | |          2
  erfc(x)  =  --------     |    exp( - t  ) dt
              sqrt(pi)   | |
                          -
                           x


For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,26.6417   30000       5.7e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunctionc(double x, ae_state *_state)
{
    double p;
    double q;
    double result;


    if( ae_fp_less(x,0) )
    {
        result = 2-errorfunctionc(-x, _state);
        return result;
    }
    if( ae_fp_less(x,0.5) )
    {
        result = 1.0-errorfunction(x, _state);
        return result;
    }
    if( ae_fp_greater_eq(x,10) )
    {
        result = 0;
        return result;
    }
    p = 0.0;
    p = 0.5641877825507397413087057563+x*p;
    p = 9.675807882987265400604202961+x*p;
    p = 77.08161730368428609781633646+x*p;
    p = 368.5196154710010637133875746+x*p;
    p = 1143.262070703886173606073338+x*p;
    p = 2320.439590251635247384768711+x*p;
    p = 2898.0293292167655611275846+x*p;
    p = 1826.3348842295112592168999+x*p;
    q = 1.0;
    q = 17.14980943627607849376131193+x*q;
    q = 137.1255960500622202878443578+x*q;
    q = 661.7361207107653469211984771+x*q;
    q = 2094.384367789539593790281779+x*q;
    q = 4429.612803883682726711528526+x*q;
    q = 6089.5424232724435504633068+x*q;
    q = 4958.82756472114071495438422+x*q;
    q = 1826.3348842295112595576438+x*q;
    result = ae_exp(-ae_sqr(x, _state), _state)*p/q;
    return result;
}


/*************************************************************************
Normal distribution function

Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:

                           x
                            -
                  1        | |          2
   ndtr(x)  = ---------    |    exp( - t /2 ) dt
              sqrt(2pi)  | |
                          -
                         -inf.

            =  ( 1 + erf(z) ) / 2
            =  erfc(z) / 2

where z = x/sqrt(2). Computation is via the functions
erf and erfc.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -13,0        30000       3.4e-14     6.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double normaldistribution(double x, ae_state *_state)
{
    double result;


    result = 0.5*(errorfunction(x/1.41421356237309504880, _state)+1);
    return result;
}


/*************************************************************************
Inverse of the error function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double inverf(double e, ae_state *_state)
{
    double result;


    result = invnormaldistribution(0.5*(e+1), _state)/ae_sqrt(2, _state);
    return result;
}


/*************************************************************************
Inverse of Normal distribution function

Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.


For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) );  then the approximation is
x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2).  For larger arguments,
w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

ACCURACY:

                     Relative error:
arithmetic   domain        # trials      peak         rms
   IEEE     0.125, 1        20000       7.2e-16     1.3e-16
   IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double invnormaldistribution(double y0, ae_state *_state)
{
    double expm2;
    double s2pi;
    double x;
    double y;
    double z;
    double y2;
    double x0;
    double x1;
    ae_int_t code;
    double p0;
    double q0;
    double p1;
    double q1;
    double p2;
    double q2;
    double result;


    expm2 = 0.13533528323661269189;
    s2pi = 2.50662827463100050242;
    if( ae_fp_less_eq(y0,0) )
    {
        result = -ae_maxrealnumber;
        return result;
    }
    if( ae_fp_greater_eq(y0,1) )
    {
        result = ae_maxrealnumber;
        return result;
    }
    code = 1;
    y = y0;
    if( ae_fp_greater(y,1.0-expm2) )
    {
        y = 1.0-y;
        code = 0;
    }
    if( ae_fp_greater(y,expm2) )
    {
        y = y-0.5;
        y2 = y*y;
        p0 = -59.9633501014107895267;
        p0 = 98.0010754185999661536+y2*p0;
        p0 = -56.6762857469070293439+y2*p0;
        p0 = 13.9312609387279679503+y2*p0;
        p0 = -1.23916583867381258016+y2*p0;
        q0 = 1;
        q0 = 1.95448858338141759834+y2*q0;
        q0 = 4.67627912898881538453+y2*q0;
        q0 = 86.3602421390890590575+y2*q0;
        q0 = -225.462687854119370527+y2*q0;
        q0 = 200.260212380060660359+y2*q0;
        q0 = -82.0372256168333339912+y2*q0;
        q0 = 15.9056225126211695515+y2*q0;
        q0 = -1.18331621121330003142+y2*q0;
        x = y+y*y2*p0/q0;
        x = x*s2pi;
        result = x;
        return result;
    }
    x = ae_sqrt(-2.0*ae_log(y, _state), _state);
    x0 = x-ae_log(x, _state)/x;
    z = 1.0/x;
    if( ae_fp_less(x,8.0) )
    {
        p1 = 4.05544892305962419923;
        p1 = 31.5251094599893866154+z*p1;
        p1 = 57.1628192246421288162+z*p1;
        p1 = 44.0805073893200834700+z*p1;
        p1 = 14.6849561928858024014+z*p1;
        p1 = 2.18663306850790267539+z*p1;
        p1 = -1.40256079171354495875*0.1+z*p1;
        p1 = -3.50424626827848203418*0.01+z*p1;
        p1 = -8.57456785154685413611*0.0001+z*p1;
        q1 = 1;
        q1 = 15.7799883256466749731+z*q1;
        q1 = 45.3907635128879210584+z*q1;
        q1 = 41.3172038254672030440+z*q1;
        q1 = 15.0425385692907503408+z*q1;
        q1 = 2.50464946208309415979+z*q1;
        q1 = -1.42182922854787788574*0.1+z*q1;
        q1 = -3.80806407691578277194*0.01+z*q1;
        q1 = -9.33259480895457427372*0.0001+z*q1;
        x1 = z*p1/q1;
    }
    else
    {
        p2 = 3.23774891776946035970;
        p2 = 6.91522889068984211695+z*p2;
        p2 = 3.93881025292474443415+z*p2;
        p2 = 1.33303460815807542389+z*p2;
        p2 = 2.01485389549179081538*0.1+z*p2;
        p2 = 1.23716634817820021358*0.01+z*p2;
        p2 = 3.01581553508235416007*0.0001+z*p2;
        p2 = 2.65806974686737550832*0.000001+z*p2;
        p2 = 6.23974539184983293730*0.000000001+z*p2;
        q2 = 1;
        q2 = 6.02427039364742014255+z*q2;
        q2 = 3.67983563856160859403+z*q2;
        q2 = 1.37702099489081330271+z*q2;
        q2 = 2.16236993594496635890*0.1+z*q2;
        q2 = 1.34204006088543189037*0.01+z*q2;
        q2 = 3.28014464682127739104*0.0001+z*q2;
        q2 = 2.89247864745380683936*0.000001+z*q2;
        q2 = 6.79019408009981274425*0.000000001+z*q2;
        x1 = z*p2/q2;
    }
    x = x0-x1;
    if( code!=0 )
    {
        x = -x;
    }
    result = x;
    return result;
}




/*************************************************************************
Incomplete gamma integral

The function is defined by

                          x
                           -
                  1       | |  -t  a-1
 igam(a,x)  =   -----     |   e   t   dt.
                 -      | |
                | (a)    -
                          0


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30       200000       3.6e-14     2.9e-15
   IEEE      0,100      300000       9.9e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegamma(double a, double x, ae_state *_state)
{
    double igammaepsilon;
    double ans;
    double ax;
    double c;
    double r;
    double tmp;
    double result;


    igammaepsilon = 0.000000000000001;
    if( ae_fp_less_eq(x,0)||ae_fp_less_eq(a,0) )
    {
        result = 0;
        return result;
    }
    if( ae_fp_greater(x,1)&&ae_fp_greater(x,a) )
    {
        result = 1-incompletegammac(a, x, _state);
        return result;
    }
    ax = a*ae_log(x, _state)-x-lngamma(a, &tmp, _state);
    if( ae_fp_less(ax,-709.78271289338399) )
    {
        result = 0;
        return result;
    }
    ax = ae_exp(ax, _state);
    r = a;
    c = 1;
    ans = 1;
    do
    {
        r = r+1;
        c = c*x/r;
        ans = ans+c;
    }
    while(ae_fp_greater(c/ans,igammaepsilon));
    result = ans*ax/a;
    return result;
}


/*************************************************************************
Complemented incomplete gamma integral

The function is defined by


 igamc(a,x)   =   1 - igam(a,x)

                           inf.
                             -
                    1       | |  -t  a-1
              =   -----     |   e   t   dt.
                   -      | |
                  | (a)    -
                            x


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

Tested at random a, x.
               a         x                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
   IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegammac(double a, double x, ae_state *_state)
{
    double igammaepsilon;
    double igammabignumber;
    double igammabignumberinv;
    double ans;
    double ax;
    double c;
    double yc;
    double r;
    double t;
    double y;
    double z;
    double pk;
    double pkm1;
    double pkm2;
    double qk;
    double qkm1;
    double qkm2;
    double tmp;
    double result;


    igammaepsilon = 0.000000000000001;
    igammabignumber = 4503599627370496.0;
    igammabignumberinv = 2.22044604925031308085*0.0000000000000001;
    if( ae_fp_less_eq(x,0)||ae_fp_less_eq(a,0) )
    {
        result = 1;
        return result;
    }
    if( ae_fp_less(x,1)||ae_fp_less(x,a) )
    {
        result = 1-incompletegamma(a, x, _state);
        return result;
    }
    ax = a*ae_log(x, _state)-x-lngamma(a, &tmp, _state);
    if( ae_fp_less(ax,-709.78271289338399) )
    {
        result = 0;
        return result;
    }
    ax = ae_exp(ax, _state);
    y = 1-a;
    z = x+y+1;
    c = 0;
    pkm2 = 1;
    qkm2 = x;
    pkm1 = x+1;
    qkm1 = z*x;
    ans = pkm1/qkm1;
    do
    {
        c = c+1;
        y = y+1;
        z = z+2;
        yc = y*c;
        pk = pkm1*z-pkm2*yc;
        qk = qkm1*z-qkm2*yc;
        if( ae_fp_neq(qk,0) )
        {
            r = pk/qk;
            t = ae_fabs((ans-r)/r, _state);
            ans = r;
        }
        else
        {
            t = 1;
        }
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
        if( ae_fp_greater(ae_fabs(pk, _state),igammabignumber) )
        {
            pkm2 = pkm2*igammabignumberinv;
            pkm1 = pkm1*igammabignumberinv;
            qkm2 = qkm2*igammabignumberinv;
            qkm1 = qkm1*igammabignumberinv;
        }
    }
    while(ae_fp_greater(t,igammaepsilon));
    result = ans*ax;
    return result;
}


/*************************************************************************
Inverse of complemented imcomplete gamma integral

Given p, the function finds x such that

 igamc( a, x ) = p.

Starting with the approximate value

        3
 x = a t

 where

 t = 1 - d - ndtri(p) sqrt(d)

and

 d = 1/9a,

the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.

ACCURACY:

Tested at random a, p in the intervals indicated.

               a        p                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
   IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
   IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletegammac(double a, double y0, ae_state *_state)
{
    double igammaepsilon;
    double iinvgammabignumber;
    double x0;
    double x1;
    double x;
    double yl;
    double yh;
    double y;
    double d;
    double lgm;
    double dithresh;
    ae_int_t i;
    ae_int_t dir;
    double tmp;
    double result;


    igammaepsilon = 0.000000000000001;
    iinvgammabignumber = 4503599627370496.0;
    x0 = iinvgammabignumber;
    yl = 0;
    x1 = 0;
    yh = 1;
    dithresh = 5*igammaepsilon;
    d = 1/(9*a);
    y = 1-d-invnormaldistribution(y0, _state)*ae_sqrt(d, _state);
    x = a*y*y*y;
    lgm = lngamma(a, &tmp, _state);
    i = 0;
    while(i<10)
    {
        if( ae_fp_greater(x,x0)||ae_fp_less(x,x1) )
        {
            d = 0.0625;
            break;
        }
        y = incompletegammac(a, x, _state);
        if( ae_fp_less(y,yl)||ae_fp_greater(y,yh) )
        {
            d = 0.0625;
            break;
        }
        if( ae_fp_less(y,y0) )
        {
            x0 = x;
            yl = y;
        }
        else
        {
            x1 = x;
            yh = y;
        }
        d = (a-1)*ae_log(x, _state)-x-lgm;
        if( ae_fp_less(d,-709.78271289338399) )
        {
            d = 0.0625;
            break;
        }
        d = -ae_exp(d, _state);
        d = (y-y0)/d;
        if( ae_fp_less(ae_fabs(d/x, _state),igammaepsilon) )
        {
            result = x;
            return result;
        }
        x = x-d;
        i = i+1;
    }
    if( ae_fp_eq(x0,iinvgammabignumber) )
    {
        if( ae_fp_less_eq(x,0) )
        {
            x = 1;
        }
        while(ae_fp_eq(x0,iinvgammabignumber))
        {
            x = (1+d)*x;
            y = incompletegammac(a, x, _state);
            if( ae_fp_less(y,y0) )
            {
                x0 = x;
                yl = y;
                break;
            }
            d = d+d;
        }
    }
    d = 0.5;
    dir = 0;
    i = 0;
    while(i<400)
    {
        x = x1+d*(x0-x1);
        y = incompletegammac(a, x, _state);
        lgm = (x0-x1)/(x1+x0);
        if( ae_fp_less(ae_fabs(lgm, _state),dithresh) )
        {
            break;
        }
        lgm = (y-y0)/y0;
        if( ae_fp_less(ae_fabs(lgm, _state),dithresh) )
        {
            break;
        }
        if( ae_fp_less_eq(x,0.0) )
        {
            break;
        }
        if( ae_fp_greater_eq(y,y0) )
        {
            x1 = x;
            yh = y;
            if( dir<0 )
            {
                dir = 0;
                d = 0.5;
            }
            else
            {
                if( dir>1 )
                {
                    d = 0.5*d+0.5;
                }
                else
                {
                    d = (y0-yl)/(yh-yl);
                }
            }
            dir = dir+1;
        }
        else
        {
            x0 = x;
            yl = y;
            if( dir>0 )
            {
                dir = 0;
                d = 0.5;
            }
            else
            {
                if( dir<-1 )
                {
                    d = 0.5*d;
                }
                else
                {
                    d = (y0-yl)/(yh-yl);
                }
            }
            dir = dir-1;
        }
        i = i+1;
    }
    result = x;
    return result;
}




/*************************************************************************
Airy function

Solution of the differential equation

y"(x) = xy.

The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).

Evaluation is by power series summation for small x,
by rational minimax approximations for large x.



ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.

Arithmetic  domain   function  # trials      peak         rms
IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void airy(double x,
     double* ai,
     double* aip,
     double* bi,
     double* bip,
     ae_state *_state)
{
    double z;
    double zz;
    double t;
    double f;
    double g;
    double uf;
    double ug;
    double k;
    double zeta;
    double theta;
    ae_int_t domflg;
    double c1;
    double c2;
    double sqrt3;
    double sqpii;
    double afn;
    double afd;
    double agn;
    double agd;
    double apfn;
    double apfd;
    double apgn;
    double apgd;
    double an;
    double ad;
    double apn;
    double apd;
    double bn16;
    double bd16;
    double bppn;
    double bppd;

    *ai = 0;
    *aip = 0;
    *bi = 0;
    *bip = 0;

    sqpii = 5.64189583547756286948E-1;
    c1 = 0.35502805388781723926;
    c2 = 0.258819403792806798405;
    sqrt3 = 1.732050807568877293527;
    domflg = 0;
    if( ae_fp_greater(x,25.77) )
    {
        *ai = 0;
        *aip = 0;
        *bi = ae_maxrealnumber;
        *bip = ae_maxrealnumber;
        return;
    }
    if( ae_fp_less(x,-2.09) )
    {
        domflg = 15;
        t = ae_sqrt(-x, _state);
        zeta = -2.0*x*t/3.0;
        t = ae_sqrt(t, _state);
        k = sqpii/t;
        z = 1.0/zeta;
        zz = z*z;
        afn = -1.31696323418331795333E-1;
        afn = afn*zz-6.26456544431912369773E-1;
        afn = afn*zz-6.93158036036933542233E-1;
        afn = afn*zz-2.79779981545119124951E-1;
        afn = afn*zz-4.91900132609500318020E-2;
        afn = afn*zz-4.06265923594885404393E-3;
        afn = afn*zz-1.59276496239262096340E-4;
        afn = afn*zz-2.77649108155232920844E-6;
        afn = afn*zz-1.67787698489114633780E-8;
        afd = 1.00000000000000000000E0;
        afd = afd*zz+1.33560420706553243746E1;
        afd = afd*zz+3.26825032795224613948E1;
        afd = afd*zz+2.67367040941499554804E1;
        afd = afd*zz+9.18707402907259625840E0;
        afd = afd*zz+1.47529146771666414581E0;
        afd = afd*zz+1.15687173795188044134E-1;
        afd = afd*zz+4.40291641615211203805E-3;
        afd = afd*zz+7.54720348287414296618E-5;
        afd = afd*zz+4.51850092970580378464E-7;
        uf = 1.0+zz*afn/afd;
        agn = 1.97339932091685679179E-2;
        agn = agn*zz+3.91103029615688277255E-1;
        agn = agn*zz+1.06579897599595591108E0;
        agn = agn*zz+9.39169229816650230044E-1;
        agn = agn*zz+3.51465656105547619242E-1;
        agn = agn*zz+6.33888919628925490927E-2;
        agn = agn*zz+5.85804113048388458567E-3;
        agn = agn*zz+2.82851600836737019778E-4;
        agn = agn*zz+6.98793669997260967291E-6;
        agn = agn*zz+8.11789239554389293311E-8;
        agn = agn*zz+3.41551784765923618484E-10;
        agd = 1.00000000000000000000E0;
        agd = agd*zz+9.30892908077441974853E0;
        agd = agd*zz+1.98352928718312140417E1;
        agd = agd*zz+1.55646628932864612953E1;
        agd = agd*zz+5.47686069422975497931E0;
        agd = agd*zz+9.54293611618961883998E-1;
        agd = agd*zz+8.64580826352392193095E-2;
        agd = agd*zz+4.12656523824222607191E-3;
        agd = agd*zz+1.01259085116509135510E-4;
        agd = agd*zz+1.17166733214413521882E-6;
        agd = agd*zz+4.91834570062930015649E-9;
        ug = z*agn/agd;
        theta = zeta+0.25*ae_pi;
        f = ae_sin(theta, _state);
        g = ae_cos(theta, _state);
        *ai = k*(f*uf-g*ug);
        *bi = k*(g*uf+f*ug);
        apfn = 1.85365624022535566142E-1;
        apfn = apfn*zz+8.86712188052584095637E-1;
        apfn = apfn*zz+9.87391981747398547272E-1;
        apfn = apfn*zz+4.01241082318003734092E-1;
        apfn = apfn*zz+7.10304926289631174579E-2;
        apfn = apfn*zz+5.90618657995661810071E-3;
        apfn = apfn*zz+2.33051409401776799569E-4;
        apfn = apfn*zz+4.08718778289035454598E-6;
        apfn = apfn*zz+2.48379932900442457853E-8;
        apfd = 1.00000000000000000000E0;
        apfd = apfd*zz+1.47345854687502542552E1;
        apfd = apfd*zz+3.75423933435489594466E1;
        apfd = apfd*zz+3.14657751203046424330E1;
        apfd = apfd*zz+1.09969125207298778536E1;
        apfd = apfd*zz+1.78885054766999417817E0;
        apfd = apfd*zz+1.41733275753662636873E-1;
        apfd = apfd*zz+5.44066067017226003627E-3;
        apfd = apfd*zz+9.39421290654511171663E-5;
        apfd = apfd*zz+5.65978713036027009243E-7;
        uf = 1.0+zz*apfn/apfd;
        apgn = -3.55615429033082288335E-2;
        apgn = apgn*zz-6.37311518129435504426E-1;
        apgn = apgn*zz-1.70856738884312371053E0;
        apgn = apgn*zz-1.50221872117316635393E0;
        apgn = apgn*zz-5.63606665822102676611E-1;
        apgn = apgn*zz-1.02101031120216891789E-1;
        apgn = apgn*zz-9.48396695961445269093E-3;
        apgn = apgn*zz-4.60325307486780994357E-4;
        apgn = apgn*zz-1.14300836484517375919E-5;
        apgn = apgn*zz-1.33415518685547420648E-7;
        apgn = apgn*zz-5.63803833958893494476E-10;
        apgd = 1.00000000000000000000E0;
        apgd = apgd*zz+9.85865801696130355144E0;
        apgd = apgd*zz+2.16401867356585941885E1;
        apgd = apgd*zz+1.73130776389749389525E1;
        apgd = apgd*zz+6.17872175280828766327E0;
        apgd = apgd*zz+1.08848694396321495475E0;
        apgd = apgd*zz+9.95005543440888479402E-2;
        apgd = apgd*zz+4.78468199683886610842E-3;
        apgd = apgd*zz+1.18159633322838625562E-4;
        apgd = apgd*zz+1.37480673554219441465E-6;
        apgd = apgd*zz+5.79912514929147598821E-9;
        ug = z*apgn/apgd;
        k = sqpii*t;
        *aip = -k*(g*uf+f*ug);
        *bip = k*(f*uf-g*ug);
        return;
    }
    if( ae_fp_greater_eq(x,2.09) )
    {
        domflg = 5;
        t = ae_sqrt(x, _state);
        zeta = 2.0*x*t/3.0;
        g = ae_exp(zeta, _state);
        t = ae_sqrt(t, _state);
        k = 2.0*t*g;
        z = 1.0/zeta;
        an = 3.46538101525629032477E-1;
        an = an*z+1.20075952739645805542E1;
        an = an*z+7.62796053615234516538E1;
        an = an*z+1.68089224934630576269E2;
        an = an*z+1.59756391350164413639E2;
        an = an*z+7.05360906840444183113E1;
        an = an*z+1.40264691163389668864E1;
        an = an*z+9.99999999999999995305E-1;
        ad = 5.67594532638770212846E-1;
        ad = ad*z+1.47562562584847203173E1;
        ad = ad*z+8.45138970141474626562E1;
        ad = ad*z+1.77318088145400459522E2;
        ad = ad*z+1.64234692871529701831E2;
        ad = ad*z+7.14778400825575695274E1;
        ad = ad*z+1.40959135607834029598E1;
        ad = ad*z+1.00000000000000000470E0;
        f = an/ad;
        *ai = sqpii*f/k;
        k = -0.5*sqpii*t/g;
        apn = 6.13759184814035759225E-1;
        apn = apn*z+1.47454670787755323881E1;
        apn = apn*z+8.20584123476060982430E1;
        apn = apn*z+1.71184781360976385540E2;
        apn = apn*z+1.59317847137141783523E2;
        apn = apn*z+6.99778599330103016170E1;
        apn = apn*z+1.39470856980481566958E1;
        apn = apn*z+1.00000000000000000550E0;
        apd = 3.34203677749736953049E-1;
        apd = apd*z+1.11810297306158156705E1;
        apd = apd*z+7.11727352147859965283E1;
        apd = apd*z+1.58778084372838313640E2;
        apd = apd*z+1.53206427475809220834E2;
        apd = apd*z+6.86752304592780337944E1;
        apd = apd*z+1.38498634758259442477E1;
        apd = apd*z+9.99999999999999994502E-1;
        f = apn/apd;
        *aip = f*k;
        if( ae_fp_greater(x,8.3203353) )
        {
            bn16 = -2.53240795869364152689E-1;
            bn16 = bn16*z+5.75285167332467384228E-1;
            bn16 = bn16*z-3.29907036873225371650E-1;
            bn16 = bn16*z+6.44404068948199951727E-2;
            bn16 = bn16*z-3.82519546641336734394E-3;
            bd16 = 1.00000000000000000000E0;
            bd16 = bd16*z-7.15685095054035237902E0;
            bd16 = bd16*z+1.06039580715664694291E1;
            bd16 = bd16*z-5.23246636471251500874E0;
            bd16 = bd16*z+9.57395864378383833152E-1;
            bd16 = bd16*z-5.50828147163549611107E-2;
            f = z*bn16/bd16;
            k = sqpii*g;
            *bi = k*(1.0+f)/t;
            bppn = 4.65461162774651610328E-1;
            bppn = bppn*z-1.08992173800493920734E0;
            bppn = bppn*z+6.38800117371827987759E-1;
            bppn = bppn*z-1.26844349553102907034E-1;
            bppn = bppn*z+7.62487844342109852105E-3;
            bppd = 1.00000000000000000000E0;
            bppd = bppd*z-8.70622787633159124240E0;
            bppd = bppd*z+1.38993162704553213172E1;
            bppd = bppd*z-7.14116144616431159572E0;
            bppd = bppd*z+1.34008595960680518666E0;
            bppd = bppd*z-7.84273211323341930448E-2;
            f = z*bppn/bppd;
            *bip = k*t*(1.0+f);
            return;
        }
    }
    f = 1.0;
    g = x;
    t = 1.0;
    uf = 1.0;
    ug = x;
    k = 1.0;
    z = x*x*x;
    while(ae_fp_greater(t,ae_machineepsilon))
    {
        uf = uf*z;
        k = k+1.0;
        uf = uf/k;
        ug = ug*z;
        k = k+1.0;
        ug = ug/k;
        uf = uf/k;
        f = f+uf;
        k = k+1.0;
        ug = ug/k;
        g = g+ug;
        t = ae_fabs(uf/f, _state);
    }
    uf = c1*f;
    ug = c2*g;
    if( domflg%2==0 )
    {
        *ai = uf-ug;
    }
    if( domflg/2%2==0 )
    {
        *bi = sqrt3*(uf+ug);
    }
    k = 4.0;
    uf = x*x/2.0;
    ug = z/3.0;
    f = uf;
    g = 1.0+ug;
    uf = uf/3.0;
    t = 1.0;
    while(ae_fp_greater(t,ae_machineepsilon))
    {
        uf = uf*z;
        ug = ug/k;
        k = k+1.0;
        ug = ug*z;
        uf = uf/k;
        f = f+uf;
        k = k+1.0;
        ug = ug/k;
        uf = uf/k;
        g = g+ug;
        k = k+1.0;
        t = ae_fabs(ug/g, _state);
    }
    uf = c1*f;
    ug = c2*g;
    if( domflg/4%2==0 )
    {
        *aip = uf-ug;
    }
    if( domflg/8%2==0 )
    {
        *bip = sqrt3*(uf+ug);
    }
}




/*************************************************************************
Bessel function of order zero

Returns Bessel function of order zero of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:


       2         2
(w - r  ) (w - r  ) P (w) / Q (w)
      1         2    3       8

           2
where w = x  and the two r's are zeros of the function.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.

ACCURACY:

                     Absolute error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       60000       4.2e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj0(double x, ae_state *_state)
{
    double xsq;
    double nn;
    double pzero;
    double qzero;
    double p1;
    double q1;
    double result;


    if( ae_fp_less(x,0) )
    {
        x = -x;
    }
    if( ae_fp_greater(x,8.0) )
    {
        bessel_besselasympt0(x, &pzero, &qzero, _state);
        nn = x-ae_pi/4;
        result = ae_sqrt(2/ae_pi/x, _state)*(pzero*ae_cos(nn, _state)-qzero*ae_sin(nn, _state));
        return result;
    }
    xsq = ae_sqr(x, _state);
    p1 = 26857.86856980014981415848441;
    p1 = -40504123.71833132706360663322+xsq*p1;
    p1 = 25071582855.36881945555156435+xsq*p1;
    p1 = -8085222034853.793871199468171+xsq*p1;
    p1 = 1434354939140344.111664316553+xsq*p1;
    p1 = -136762035308817138.6865416609+xsq*p1;
    p1 = 6382059341072356562.289432465+xsq*p1;
    p1 = -117915762910761053603.8440800+xsq*p1;
    p1 = 493378725179413356181.6813446+xsq*p1;
    q1 = 1.0;
    q1 = 1363.063652328970604442810507+xsq*q1;
    q1 = 1114636.098462985378182402543+xsq*q1;
    q1 = 669998767.2982239671814028660+xsq*q1;
    q1 = 312304311494.1213172572469442+xsq*q1;
    q1 = 112775673967979.8507056031594+xsq*q1;
    q1 = 30246356167094626.98627330784+xsq*q1;
    q1 = 5428918384092285160.200195092+xsq*q1;
    q1 = 493378725179413356211.3278438+xsq*q1;
    result = p1/q1;
    return result;
}


/*************************************************************************
Bessel function of order one

Returns Bessel function of order one of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       2.6e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj1(double x, ae_state *_state)
{
    double s;
    double xsq;
    double nn;
    double pzero;
    double qzero;
    double p1;
    double q1;
    double result;


    s = ae_sign(x, _state);
    if( ae_fp_less(x,0) )
    {
        x = -x;
    }
    if( ae_fp_greater(x,8.0) )
    {
        bessel_besselasympt1(x, &pzero, &qzero, _state);
        nn = x-3*ae_pi/4;
        result = ae_sqrt(2/ae_pi/x, _state)*(pzero*ae_cos(nn, _state)-qzero*ae_sin(nn, _state));
        if( ae_fp_less(s,0) )
        {
            result = -result;
        }
        return result;
    }
    xsq = ae_sqr(x, _state);
    p1 = 2701.122710892323414856790990;
    p1 = -4695753.530642995859767162166+xsq*p1;
    p1 = 3413234182.301700539091292655+xsq*p1;
    p1 = -1322983480332.126453125473247+xsq*p1;
    p1 = 290879526383477.5409737601689+xsq*p1;
    p1 = -35888175699101060.50743641413+xsq*p1;
    p1 = 2316433580634002297.931815435+xsq*p1;
    p1 = -66721065689249162980.20941484+xsq*p1;
    p1 = 581199354001606143928.050809+xsq*p1;
    q1 = 1.0;
    q1 = 1606.931573481487801970916749+xsq*q1;
    q1 = 1501793.594998585505921097578+xsq*q1;
    q1 = 1013863514.358673989967045588+xsq*q1;
    q1 = 524371026216.7649715406728642+xsq*q1;
    q1 = 208166122130760.7351240184229+xsq*q1;
    q1 = 60920613989175217.46105196863+xsq*q1;
    q1 = 11857707121903209998.37113348+xsq*q1;
    q1 = 1162398708003212287858.529400+xsq*q1;
    result = s*x*p1/q1;
    return result;
}


/*************************************************************************
Bessel function of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The ratio of jn(x) to j0(x) is computed by backward
recurrence.  First the ratio jn/jn-1 is found by a
continued fraction expansion.  Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.

If n = 0 or 1 the routine for j0 or j1 is called
directly.

ACCURACY:

                     Absolute error:
arithmetic   range      # trials      peak         rms
   IEEE      0, 30        5000       4.4e-16     7.9e-17


Not suitable for large n or x. Use jv() (fractional order) instead.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseljn(ae_int_t n, double x, ae_state *_state)
{
    double pkm2;
    double pkm1;
    double pk;
    double xk;
    double r;
    double ans;
    ae_int_t k;
    ae_int_t sg;
    double result;


    if( n<0 )
    {
        n = -n;
        if( n%2==0 )
        {
            sg = 1;
        }
        else
        {
            sg = -1;
        }
    }
    else
    {
        sg = 1;
    }
    if( ae_fp_less(x,0) )
    {
        if( n%2!=0 )
        {
            sg = -sg;
        }
        x = -x;
    }
    if( n==0 )
    {
        result = sg*besselj0(x, _state);
        return result;
    }
    if( n==1 )
    {
        result = sg*besselj1(x, _state);
        return result;
    }
    if( n==2 )
    {
        if( ae_fp_eq(x,0) )
        {
            result = 0;
        }
        else
        {
            result = sg*(2.0*besselj1(x, _state)/x-besselj0(x, _state));
        }
        return result;
    }
    if( ae_fp_less(x,ae_machineepsilon) )
    {
        result = 0;
        return result;
    }
    k = 53;
    pk = 2*(n+k);
    ans = pk;
    xk = x*x;
    do
    {
        pk = pk-2.0;
        ans = pk-xk/ans;
        k = k-1;
    }
    while(k!=0);
    ans = x/ans;
    pk = 1.0;
    pkm1 = 1.0/ans;
    k = n-1;
    r = 2*k;
    do
    {
        pkm2 = (pkm1*r-pk*x)/x;
        pk = pkm1;
        pkm1 = pkm2;
        r = r-2.0;
        k = k-1;
    }
    while(k!=0);
    if( ae_fp_greater(ae_fabs(pk, _state),ae_fabs(pkm1, _state)) )
    {
        ans = besselj1(x, _state)/pk;
    }
    else
    {
        ans = besselj0(x, _state)/pkm1;
    }
    result = sg*ans;
    return result;
}


/*************************************************************************
Bessel function of the second kind, order zero

Returns Bessel function of the second kind, of order
zero, of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
  y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.



ACCURACY:

 Absolute error, when y0(x) < 1; else relative error:

arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.3e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely0(double x, ae_state *_state)
{
    double nn;
    double xsq;
    double pzero;
    double qzero;
    double p4;
    double q4;
    double result;


    if( ae_fp_greater(x,8.0) )
    {
        bessel_besselasympt0(x, &pzero, &qzero, _state);
        nn = x-ae_pi/4;
        result = ae_sqrt(2/ae_pi/x, _state)*(pzero*ae_sin(nn, _state)+qzero*ae_cos(nn, _state));
        return result;
    }
    xsq = ae_sqr(x, _state);
    p4 = -41370.35497933148554125235152;
    p4 = 59152134.65686889654273830069+xsq*p4;
    p4 = -34363712229.79040378171030138+xsq*p4;
    p4 = 10255208596863.94284509167421+xsq*p4;
    p4 = -1648605817185729.473122082537+xsq*p4;
    p4 = 137562431639934407.8571335453+xsq*p4;
    p4 = -5247065581112764941.297350814+xsq*p4;
    p4 = 65874732757195549259.99402049+xsq*p4;
    p4 = -27502866786291095837.01933175+xsq*p4;
    q4 = 1.0;
    q4 = 1282.452772478993804176329391+xsq*q4;
    q4 = 1001702.641288906265666651753+xsq*q4;
    q4 = 579512264.0700729537480087915+xsq*q4;
    q4 = 261306575504.1081249568482092+xsq*q4;
    q4 = 91620380340751.85262489147968+xsq*q4;
    q4 = 23928830434997818.57439356652+xsq*q4;
    q4 = 4192417043410839973.904769661+xsq*q4;
    q4 = 372645883898616588198.9980+xsq*q4;
    result = p4/q4+2/ae_pi*besselj0(x, _state)*ae_log(x, _state);
    return result;
}


/*************************************************************************
Bessel function of second kind of order one

Returns Bessel function of the second kind of order one
of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       1.0e-15     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely1(double x, ae_state *_state)
{
    double nn;
    double xsq;
    double pzero;
    double qzero;
    double p4;
    double q4;
    double result;


    if( ae_fp_greater(x,8.0) )
    {
        bessel_besselasympt1(x, &pzero, &qzero, _state);
        nn = x-3*ae_pi/4;
        result = ae_sqrt(2/ae_pi/x, _state)*(pzero*ae_sin(nn, _state)+qzero*ae_cos(nn, _state));
        return result;
    }
    xsq = ae_sqr(x, _state);
    p4 = -2108847.540133123652824139923;
    p4 = 3639488548.124002058278999428+xsq*p4;
    p4 = -2580681702194.450950541426399+xsq*p4;
    p4 = 956993023992168.3481121552788+xsq*p4;
    p4 = -196588746272214065.8820322248+xsq*p4;
    p4 = 21931073399177975921.11427556+xsq*p4;
    p4 = -1212297555414509577913.561535+xsq*p4;
    p4 = 26554738314348543268942.48968+xsq*p4;
    p4 = -99637534243069222259967.44354+xsq*p4;
    q4 = 1.0;
    q4 = 1612.361029677000859332072312+xsq*q4;
    q4 = 1563282.754899580604737366452+xsq*q4;
    q4 = 1128686837.169442121732366891+xsq*q4;
    q4 = 646534088126.5275571961681500+xsq*q4;
    q4 = 297663212564727.6729292742282+xsq*q4;
    q4 = 108225825940881955.2553850180+xsq*q4;
    q4 = 29549879358971486742.90758119+xsq*q4;
    q4 = 5435310377188854170800.653097+xsq*q4;
    q4 = 508206736694124324531442.4152+xsq*q4;
    result = x*p4/q4+2/ae_pi*(besselj1(x, _state)*ae_log(x, _state)-1/x);
    return result;
}


/*************************************************************************
Bessel function of second kind of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().

If n = 0 or 1 the routine for y0 or y1 is called
directly.

ACCURACY:
                     Absolute error, except relative
                     when y > 1:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       3.4e-15     4.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselyn(ae_int_t n, double x, ae_state *_state)
{
    ae_int_t i;
    double a;
    double b;
    double tmp;
    double s;
    double result;


    s = 1;
    if( n<0 )
    {
        n = -n;
        if( n%2!=0 )
        {
            s = -1;
        }
    }
    if( n==0 )
    {
        result = bessely0(x, _state);
        return result;
    }
    if( n==1 )
    {
        result = s*bessely1(x, _state);
        return result;
    }
    a = bessely0(x, _state);
    b = bessely1(x, _state);
    for(i=1; i<=n-1; i++)
    {
        tmp = b;
        b = 2*i/x*b-a;
        a = tmp;
    }
    result = s*b;
    return result;
}


/*************************************************************************
Modified Bessel function of order zero

Returns modified Bessel function of order zero of the
argument.

The function is defined as i0(x) = j0( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       5.8e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli0(double x, ae_state *_state)
{
    double y;
    double v;
    double z;
    double b0;
    double b1;
    double b2;
    double result;


    if( ae_fp_less(x,0) )
    {
        x = -x;
    }
    if( ae_fp_less_eq(x,8.0) )
    {
        y = x/2.0-2.0;
        bessel_besselmfirstcheb(-4.41534164647933937950E-18, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 3.33079451882223809783E-17, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -2.43127984654795469359E-16, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.71539128555513303061E-15, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -1.16853328779934516808E-14, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 7.67618549860493561688E-14, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -4.85644678311192946090E-13, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 2.95505266312963983461E-12, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -1.72682629144155570723E-11, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 9.67580903537323691224E-11, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -5.18979560163526290666E-10, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 2.65982372468238665035E-9, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -1.30002500998624804212E-8, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 6.04699502254191894932E-8, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -2.67079385394061173391E-7, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.11738753912010371815E-6, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -4.41673835845875056359E-6, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.64484480707288970893E-5, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -5.75419501008210370398E-5, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.88502885095841655729E-4, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -5.76375574538582365885E-4, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.63947561694133579842E-3, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -4.32430999505057594430E-3, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.05464603945949983183E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -2.37374148058994688156E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 4.93052842396707084878E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -9.49010970480476444210E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.71620901522208775349E-1, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -3.04682672343198398683E-1, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 6.76795274409476084995E-1, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        result = ae_exp(x, _state)*v;
        return result;
    }
    z = 32.0/x-2.0;
    bessel_besselmfirstcheb(-7.23318048787475395456E-18, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -4.83050448594418207126E-18, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 4.46562142029675999901E-17, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 3.46122286769746109310E-17, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -2.82762398051658348494E-16, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -3.42548561967721913462E-16, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 1.77256013305652638360E-15, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 3.81168066935262242075E-15, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -9.55484669882830764870E-15, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -4.15056934728722208663E-14, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 1.54008621752140982691E-14, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 3.85277838274214270114E-13, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 7.18012445138366623367E-13, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -1.79417853150680611778E-12, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -1.32158118404477131188E-11, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, -3.14991652796324136454E-11, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 1.18891471078464383424E-11, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 4.94060238822496958910E-10, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 3.39623202570838634515E-9, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 2.26666899049817806459E-8, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 2.04891858946906374183E-7, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 2.89137052083475648297E-6, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 6.88975834691682398426E-5, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 3.36911647825569408990E-3, &b0, &b1, &b2, _state);
    bessel_besselmnextcheb(z, 8.04490411014108831608E-1, &b0, &b1, &b2, _state);
    v = 0.5*(b0-b2);
    result = ae_exp(x, _state)*v/ae_sqrt(x, _state);
    return result;
}


/*************************************************************************
Modified Bessel function of order one

Returns modified Bessel function of order one of the
argument.

The function is defined as i1(x) = -i j1( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.9e-15     2.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli1(double x, ae_state *_state)
{
    double y;
    double z;
    double v;
    double b0;
    double b1;
    double b2;
    double result;


    z = ae_fabs(x, _state);
    if( ae_fp_less_eq(z,8.0) )
    {
        y = z/2.0-2.0;
        bessel_besselm1firstcheb(2.77791411276104639959E-18, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.11142121435816608115E-17, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.55363195773620046921E-16, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.10559694773538630805E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 7.60068429473540693410E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -5.04218550472791168711E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.22379336594557470981E-13, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.98397439776494371520E-12, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.17361862988909016308E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -6.66348972350202774223E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.62559028155211703701E-10, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.88724975172282928790E-9, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 9.38153738649577178388E-9, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -4.44505912879632808065E-8, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.00329475355213526229E-7, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -8.56872026469545474066E-7, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.47025130813767847674E-6, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.32731636560394358279E-5, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 4.78156510755005422638E-5, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.61760815825896745588E-4, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 5.12285956168575772895E-4, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.51357245063125314899E-3, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 4.15642294431288815669E-3, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.05640848946261981558E-2, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.47264490306265168283E-2, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -5.29459812080949914269E-2, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.02643658689847095384E-1, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.76416518357834055153E-1, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.52587186443633654823E-1, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        z = v*z*ae_exp(z, _state);
    }
    else
    {
        y = 32.0/z-2.0;
        bessel_besselm1firstcheb(7.51729631084210481353E-18, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 4.41434832307170791151E-18, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -4.65030536848935832153E-17, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.20952592199342395980E-17, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.96262899764595013876E-16, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.30820231092092828324E-16, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.88035477551078244854E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.81440307243700780478E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.04202769841288027642E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 4.27244001671195135429E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.10154184277266431302E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -4.08355111109219731823E-13, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -7.19855177624590851209E-13, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.03562854414708950722E-12, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.41258074366137813316E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.25260358301548823856E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.89749581235054123450E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -5.58974346219658380687E-10, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.83538038596423702205E-9, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.63146884688951950684E-8, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.51223623787020892529E-7, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.88256480887769039346E-6, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.10588938762623716291E-4, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -9.76109749136146840777E-3, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 7.78576235018280120474E-1, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        z = v*ae_exp(z, _state)/ae_sqrt(z, _state);
    }
    if( ae_fp_less(x,0) )
    {
        z = -z;
    }
    result = z;
    return result;
}


/*************************************************************************
Modified Bessel function, second kind, order zero

Returns modified Bessel function of the second kind
of order zero of the argument.

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

Tested at 2000 random points between 0 and 8.  Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk0(double x, ae_state *_state)
{
    double y;
    double z;
    double v;
    double b0;
    double b1;
    double b2;
    double result;


    ae_assert(ae_fp_greater(x,0), "Domain error in BesselK0: x<=0", _state);
    if( ae_fp_less_eq(x,2) )
    {
        y = x*x-2.0;
        bessel_besselmfirstcheb(1.37446543561352307156E-16, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 4.25981614279661018399E-14, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.03496952576338420167E-11, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.90451637722020886025E-9, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 2.53479107902614945675E-7, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 2.28621210311945178607E-5, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 1.26461541144692592338E-3, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 3.59799365153615016266E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, 3.44289899924628486886E-1, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(y, -5.35327393233902768720E-1, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        v = v-ae_log(0.5*x, _state)*besseli0(x, _state);
    }
    else
    {
        z = 8.0/x-2.0;
        bessel_besselmfirstcheb(5.30043377268626276149E-18, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.64758043015242134646E-17, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 5.21039150503902756861E-17, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.67823109680541210385E-16, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 5.51205597852431940784E-16, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.84859337734377901440E-15, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 6.34007647740507060557E-15, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -2.22751332699166985548E-14, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 8.03289077536357521100E-14, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -2.98009692317273043925E-13, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 1.14034058820847496303E-12, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -4.51459788337394416547E-12, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 1.85594911495471785253E-11, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -7.95748924447710747776E-11, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 3.57739728140030116597E-10, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.69753450938905987466E-9, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 8.57403401741422608519E-9, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -4.66048989768794782956E-8, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 2.76681363944501510342E-7, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.83175552271911948767E-6, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 1.39498137188764993662E-5, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -1.28495495816278026384E-4, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 1.56988388573005337491E-3, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, -3.14481013119645005427E-2, &b0, &b1, &b2, _state);
        bessel_besselmnextcheb(z, 2.44030308206595545468E0, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        v = v*ae_exp(-x, _state)/ae_sqrt(x, _state);
    }
    result = v;
    return result;
}


/*************************************************************************
Modified Bessel function, second kind, order one

Computes the modified Bessel function of the second kind
of order one of the argument.

The range is partitioned into the two intervals [0,2] and
(2, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk1(double x, ae_state *_state)
{
    double y;
    double z;
    double v;
    double b0;
    double b1;
    double b2;
    double result;


    z = 0.5*x;
    ae_assert(ae_fp_greater(z,0), "Domain error in K1", _state);
    if( ae_fp_less_eq(x,2) )
    {
        y = x*x-2.0;
        bessel_besselm1firstcheb(-7.02386347938628759343E-18, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.42744985051936593393E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -6.66690169419932900609E-13, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.41148839263352776110E-10, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.21338763073472585583E-8, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.43340614156596823496E-6, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.73028895751305206302E-4, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -6.97572385963986435018E-3, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.22611180822657148235E-1, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.53155960776544875667E-1, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.52530022733894777053E0, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        result = ae_log(z, _state)*besseli1(x, _state)+v/x;
    }
    else
    {
        y = 8.0/x-2.0;
        bessel_besselm1firstcheb(-5.75674448366501715755E-18, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.79405087314755922667E-17, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -5.68946255844285935196E-17, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.83809354436663880070E-16, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -6.05704724837331885336E-16, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.03870316562433424052E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -7.01983709041831346144E-15, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.47715442448130437068E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -8.97670518232499435011E-14, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 3.34841966607842919884E-13, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.28917396095102890680E-12, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 5.13963967348173025100E-12, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.12996783842756842877E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 9.21831518760500529508E-11, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -4.19035475934189648750E-10, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.01504975519703286596E-9, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.03457624656780970260E-8, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 5.74108412545004946722E-8, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -3.50196060308781257119E-7, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.40648494783721712015E-6, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -1.93619797416608296024E-5, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.95215518471351631108E-4, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, -2.85781685962277938680E-3, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 1.03923736576817238437E-1, &b0, &b1, &b2, _state);
        bessel_besselm1nextcheb(y, 2.72062619048444266945E0, &b0, &b1, &b2, _state);
        v = 0.5*(b0-b2);
        result = ae_exp(-x, _state)*v/ae_sqrt(x, _state);
    }
    return result;
}


/*************************************************************************
Modified Bessel function, second kind, integer order

Returns modified Bessel function of the second kind
of order n of the argument.

The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity).  An ascending power series is used in the
low range, and an asymptotic expansion in the high range.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        90000       1.8e-8      3.0e-10

Error is high only near the crossover point x = 9.55
between the two expansions used.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*************************************************************************/
double besselkn(ae_int_t nn, double x, ae_state *_state)
{
    double k;
    double kf;
    double nk1f;
    double nkf;
    double zn;
    double t;
    double s;
    double z0;
    double z;
    double ans;
    double fn;
    double pn;
    double pk;
    double zmn;
    double tlg;
    double tox;
    ae_int_t i;
    ae_int_t n;
    double eul;
    double result;


    eul = 5.772156649015328606065e-1;
    if( nn<0 )
    {
        n = -nn;
    }
    else
    {
        n = nn;
    }
    ae_assert(n<=31, "Overflow in BesselKN", _state);
    ae_assert(ae_fp_greater(x,0), "Domain error in BesselKN", _state);
    if( ae_fp_less_eq(x,9.55) )
    {
        ans = 0.0;
        z0 = 0.25*x*x;
        fn = 1.0;
        pn = 0.0;
        zmn = 1.0;
        tox = 2.0/x;
        if( n>0 )
        {
            pn = -eul;
            k = 1.0;
            for(i=1; i<=n-1; i++)
            {
                pn = pn+1.0/k;
                k = k+1.0;
                fn = fn*k;
            }
            zmn = tox;
            if( n==1 )
            {
                ans = 1.0/x;
            }
            else
            {
                nk1f = fn/n;
                kf = 1.0;
                s = nk1f;
                z = -z0;
                zn = 1.0;
                for(i=1; i<=n-1; i++)
                {
                    nk1f = nk1f/(n-i);
                    kf = kf*i;
                    zn = zn*z;
                    t = nk1f*zn/kf;
                    s = s+t;
                    ae_assert(ae_fp_greater(ae_maxrealnumber-ae_fabs(t, _state),ae_fabs(s, _state)), "Overflow in BesselKN", _state);
                    ae_assert(!(ae_fp_greater(tox,1.0)&&ae_fp_less(ae_maxrealnumber/tox,zmn)), "Overflow in BesselKN", _state);
                    zmn = zmn*tox;
                }
                s = s*0.5;
                t = ae_fabs(s, _state);
                ae_assert(!(ae_fp_greater(zmn,1.0)&&ae_fp_less(ae_maxrealnumber/zmn,t)), "Overflow in BesselKN", _state);
                ae_assert(!(ae_fp_greater(t,1.0)&&ae_fp_less(ae_maxrealnumber/t,zmn)), "Overflow in BesselKN", _state);
                ans = s*zmn;
            }
        }
        tlg = 2.0*ae_log(0.5*x, _state);
        pk = -eul;
        if( n==0 )
        {
            pn = pk;
            t = 1.0;
        }
        else
        {
            pn = pn+1.0/n;
            t = 1.0/fn;
        }
        s = (pk+pn-tlg)*t;
        k = 1.0;
        do
        {
            t = t*(z0/(k*(k+n)));
            pk = pk+1.0/k;
            pn = pn+1.0/(k+n);
            s = s+(pk+pn-tlg)*t;
            k = k+1.0;
        }
        while(ae_fp_greater(ae_fabs(t/s, _state),ae_machineepsilon));
        s = 0.5*s/zmn;
        if( n%2!=0 )
        {
            s = -s;
        }
        ans = ans+s;
        result = ans;
        return result;
    }
    if( ae_fp_greater(x,ae_log(ae_maxrealnumber, _state)) )
    {
        result = 0;
        return result;
    }
    k = n;
    pn = 4.0*k*k;
    pk = 1.0;
    z0 = 8.0*x;
    fn = 1.0;
    t = 1.0;
    s = t;
    nkf = ae_maxrealnumber;
    i = 0;
    do
    {
        z = pn-pk*pk;
        t = t*z/(fn*z0);
        nk1f = ae_fabs(t, _state);
        if( i>=n&&ae_fp_greater(nk1f,nkf) )
        {
            break;
        }
        nkf = nk1f;
        s = s+t;
        fn = fn+1.0;
        pk = pk+2.0;
        i = i+1;
    }
    while(ae_fp_greater(ae_fabs(t/s, _state),ae_machineepsilon));
    result = ae_exp(-x, _state)*ae_sqrt(ae_pi/(2.0*x), _state)*s;
    return result;
}


/*************************************************************************
Internal subroutine

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
static void bessel_besselmfirstcheb(double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state)
{


    *b0 = c;
    *b1 = 0.0;
    *b2 = 0.0;
}


/*************************************************************************
Internal subroutine

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
static void bessel_besselmnextcheb(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state)
{


    *b2 = *b1;
    *b1 = *b0;
    *b0 = x*(*b1)-(*b2)+c;
}


/*************************************************************************
Internal subroutine

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
static void bessel_besselm1firstcheb(double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state)
{


    *b0 = c;
    *b1 = 0.0;
    *b2 = 0.0;
}


/*************************************************************************
Internal subroutine

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
static void bessel_besselm1nextcheb(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state)
{


    *b2 = *b1;
    *b1 = *b0;
    *b0 = x*(*b1)-(*b2)+c;
}


static void bessel_besselasympt0(double x,
     double* pzero,
     double* qzero,
     ae_state *_state)
{
    double xsq;
    double p2;
    double q2;
    double p3;
    double q3;

    *pzero = 0;
    *qzero = 0;

    xsq = 64.0/(x*x);
    p2 = 0.0;
    p2 = 2485.271928957404011288128951+xsq*p2;
    p2 = 153982.6532623911470917825993+xsq*p2;
    p2 = 2016135.283049983642487182349+xsq*p2;
    p2 = 8413041.456550439208464315611+xsq*p2;
    p2 = 12332384.76817638145232406055+xsq*p2;
    p2 = 5393485.083869438325262122897+xsq*p2;
    q2 = 1.0;
    q2 = 2615.700736920839685159081813+xsq*q2;
    q2 = 156001.7276940030940592769933+xsq*q2;
    q2 = 2025066.801570134013891035236+xsq*q2;
    q2 = 8426449.050629797331554404810+xsq*q2;
    q2 = 12338310.22786324960844856182+xsq*q2;
    q2 = 5393485.083869438325560444960+xsq*q2;
    p3 = -0.0;
    p3 = -4.887199395841261531199129300+xsq*p3;
    p3 = -226.2630641933704113967255053+xsq*p3;
    p3 = -2365.956170779108192723612816+xsq*p3;
    p3 = -8239.066313485606568803548860+xsq*p3;
    p3 = -10381.41698748464093880530341+xsq*p3;
    p3 = -3984.617357595222463506790588+xsq*p3;
    q3 = 1.0;
    q3 = 408.7714673983499223402830260+xsq*q3;
    q3 = 15704.89191515395519392882766+xsq*q3;
    q3 = 156021.3206679291652539287109+xsq*q3;
    q3 = 533291.3634216897168722255057+xsq*q3;
    q3 = 666745.4239319826986004038103+xsq*q3;
    q3 = 255015.5108860942382983170882+xsq*q3;
    *pzero = p2/q2;
    *qzero = 8*p3/q3/x;
}


static void bessel_besselasympt1(double x,
     double* pzero,
     double* qzero,
     ae_state *_state)
{
    double xsq;
    double p2;
    double q2;
    double p3;
    double q3;

    *pzero = 0;
    *qzero = 0;

    xsq = 64.0/(x*x);
    p2 = -1611.616644324610116477412898;
    p2 = -109824.0554345934672737413139+xsq*p2;
    p2 = -1523529.351181137383255105722+xsq*p2;
    p2 = -6603373.248364939109255245434+xsq*p2;
    p2 = -9942246.505077641195658377899+xsq*p2;
    p2 = -4435757.816794127857114720794+xsq*p2;
    q2 = 1.0;
    q2 = -1455.009440190496182453565068+xsq*q2;
    q2 = -107263.8599110382011903063867+xsq*q2;
    q2 = -1511809.506634160881644546358+xsq*q2;
    q2 = -6585339.479723087072826915069+xsq*q2;
    q2 = -9934124.389934585658967556309+xsq*q2;
    q2 = -4435757.816794127856828016962+xsq*q2;
    p3 = 35.26513384663603218592175580;
    p3 = 1706.375429020768002061283546+xsq*p3;
    p3 = 18494.26287322386679652009819+xsq*p3;
    p3 = 66178.83658127083517939992166+xsq*p3;
    p3 = 85145.16067533570196555001171+xsq*p3;
    p3 = 33220.91340985722351859704442+xsq*p3;
    q3 = 1.0;
    q3 = 863.8367769604990967475517183+xsq*q3;
    q3 = 37890.22974577220264142952256+xsq*q3;
    q3 = 400294.4358226697511708610813+xsq*q3;
    q3 = 1419460.669603720892855755253+xsq*q3;
    q3 = 1819458.042243997298924553839+xsq*q3;
    q3 = 708712.8194102874357377502472+xsq*q3;
    *pzero = p2/q2;
    *qzero = 8*p3/q3/x;
}




/*************************************************************************
Beta function


                  -     -
                 | (a) | (b)
beta( a, b )  =  -----------.
                    -
                   | (a+b)

For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,30       30000       8.1e-14     1.1e-14

Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
*************************************************************************/
double beta(double a, double b, ae_state *_state)
{
    double y;
    double sg;
    double s;
    double result;


    sg = 1;
    ae_assert(ae_fp_greater(a,0)||ae_fp_neq(a,ae_ifloor(a, _state)), "Overflow in Beta", _state);
    ae_assert(ae_fp_greater(b,0)||ae_fp_neq(b,ae_ifloor(b, _state)), "Overflow in Beta", _state);
    y = a+b;
    if( ae_fp_greater(ae_fabs(y, _state),171.624376956302725) )
    {
        y = lngamma(y, &s, _state);
        sg = sg*s;
        y = lngamma(b, &s, _state)-y;
        sg = sg*s;
        y = lngamma(a, &s, _state)+y;
        sg = sg*s;
        ae_assert(ae_fp_less_eq(y,ae_log(ae_maxrealnumber, _state)), "Overflow in Beta", _state);
        result = sg*ae_exp(y, _state);
        return result;
    }
    y = gammafunction(y, _state);
    ae_assert(ae_fp_neq(y,0), "Overflow in Beta", _state);
    if( ae_fp_greater(a,b) )
    {
        y = gammafunction(a, _state)/y;
        y = y*gammafunction(b, _state);
    }
    else
    {
        y = gammafunction(b, _state)/y;
        y = y*gammafunction(a, _state);
    }
    result = y;
    return result;
}




/*************************************************************************
Incomplete beta integral

Returns incomplete beta integral of the arguments, evaluated
from zero to x.  The function is defined as

                 x
    -            -
   | (a+b)      | |  a-1     b-1
 -----------    |   t   (1-t)   dt.
  -     -     | |
 | (a) | (b)   -
                0

The domain of definition is 0 <= x <= 1.  In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation

   1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.

ACCURACY:

Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
                                       Relative error
arithmetic   domain     # trials      peak         rms
   IEEE      0,5         10000       6.9e-15     4.5e-16
   IEEE      0,85       250000       2.2e-13     1.7e-14
   IEEE      0,1000      30000       5.3e-12     6.3e-13
   IEEE      0,10000    250000       9.3e-11     7.1e-12
   IEEE      0,100000    10000       8.7e-10     4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletebeta(double a, double b, double x, ae_state *_state)
{
    double t;
    double xc;
    double w;
    double y;
    ae_int_t flag;
    double sg;
    double big;
    double biginv;
    double maxgam;
    double minlog;
    double maxlog;
    double result;


    big = 4.503599627370496e15;
    biginv = 2.22044604925031308085e-16;
    maxgam = 171.624376956302725;
    minlog = ae_log(ae_minrealnumber, _state);
    maxlog = ae_log(ae_maxrealnumber, _state);
    ae_assert(ae_fp_greater(a,0)&&ae_fp_greater(b,0), "Domain error in IncompleteBeta", _state);
    ae_assert(ae_fp_greater_eq(x,0)&&ae_fp_less_eq(x,1), "Domain error in IncompleteBeta", _state);
    if( ae_fp_eq(x,0) )
    {
        result = 0;
        return result;
    }
    if( ae_fp_eq(x,1) )
    {
        result = 1;
        return result;
    }
    flag = 0;
    if( ae_fp_less_eq(b*x,1.0)&&ae_fp_less_eq(x,0.95) )
    {
        result = ibetaf_incompletebetaps(a, b, x, maxgam, _state);
        return result;
    }
    w = 1.0-x;
    if( ae_fp_greater(x,a/(a+b)) )
    {
        flag = 1;
        t = a;
        a = b;
        b = t;
        xc = x;
        x = w;
    }
    else
    {
        xc = w;
    }
    if( (flag==1&&ae_fp_less_eq(b*x,1.0))&&ae_fp_less_eq(x,0.95) )
    {
        t = ibetaf_incompletebetaps(a, b, x, maxgam, _state);
        if( ae_fp_less_eq(t,ae_machineepsilon) )
        {
            result = 1.0-ae_machineepsilon;
        }
        else
        {
            result = 1.0-t;
        }
        return result;
    }
    y = x*(a+b-2.0)-(a-1.0);
    if( ae_fp_less(y,0.0) )
    {
        w = ibetaf_incompletebetafe(a, b, x, big, biginv, _state);
    }
    else
    {
        w = ibetaf_incompletebetafe2(a, b, x, big, biginv, _state)/xc;
    }
    y = a*ae_log(x, _state);
    t = b*ae_log(xc, _state);
    if( (ae_fp_less(a+b,maxgam)&&ae_fp_less(ae_fabs(y, _state),maxlog))&&ae_fp_less(ae_fabs(t, _state),maxlog) )
    {
        t = ae_pow(xc, b, _state);
        t = t*ae_pow(x, a, _state);
        t = t/a;
        t = t*w;
        t = t*(gammafunction(a+b, _state)/(gammafunction(a, _state)*gammafunction(b, _state)));
        if( flag==1 )
        {
            if( ae_fp_less_eq(t,ae_machineepsilon) )
            {
                result = 1.0-ae_machineepsilon;
            }
            else
            {
                result = 1.0-t;
            }
        }
        else
        {
            result = t;
        }
        return result;
    }
    y = y+t+lngamma(a+b, &sg, _state)-lngamma(a, &sg, _state)-lngamma(b, &sg, _state);
    y = y+ae_log(w/a, _state);
    if( ae_fp_less(y,minlog) )
    {
        t = 0.0;
    }
    else
    {
        t = ae_exp(y, _state);
    }
    if( flag==1 )
    {
        if( ae_fp_less_eq(t,ae_machineepsilon) )
        {
            t = 1.0-ae_machineepsilon;
        }
        else
        {
            t = 1.0-t;
        }
    }
    result = t;
    return result;
}


/*************************************************************************
Inverse of imcomplete beta integral

Given y, the function finds x such that

 incbet( a, b, x ) = y .

The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.


ACCURACY:

                     Relative error:
               x     a,b
arithmetic   domain  domain  # trials    peak       rms
   IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
   IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
   IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
With a and b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
   IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
With a = .5, b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletebeta(double a, double b, double y, ae_state *_state)
{
    double aaa;
    double bbb;
    double y0;
    double d;
    double yyy;
    double x;
    double x0;
    double x1;
    double lgm;
    double yp;
    double di;
    double dithresh;
    double yl;
    double yh;
    double xt;
    ae_int_t i;
    ae_int_t rflg;
    ae_int_t dir;
    ae_int_t nflg;
    double s;
    ae_int_t mainlooppos;
    ae_int_t ihalve;
    ae_int_t ihalvecycle;
    ae_int_t newt;
    ae_int_t newtcycle;
    ae_int_t breaknewtcycle;
    ae_int_t breakihalvecycle;
    double result;


    i = 0;
    ae_assert(ae_fp_greater_eq(y,0)&&ae_fp_less_eq(y,1), "Domain error in InvIncompleteBeta", _state);
    
    /*
     * special cases
     */
    if( ae_fp_eq(y,0) )
    {
        result = 0;
        return result;
    }
    if( ae_fp_eq(y,1.0) )
    {
        result = 1;
        return result;
    }
    
    /*
     * these initializations are not really necessary,
     * but without them compiler complains about 'possibly uninitialized variables'.
     */
    dithresh = 0;
    rflg = 0;
    aaa = 0;
    bbb = 0;
    y0 = 0;
    x = 0;
    yyy = 0;
    lgm = 0;
    dir = 0;
    di = 0;
    
    /*
     * normal initializations
     */
    x0 = 0.0;
    yl = 0.0;
    x1 = 1.0;
    yh = 1.0;
    nflg = 0;
    mainlooppos = 0;
    ihalve = 1;
    ihalvecycle = 2;
    newt = 3;
    newtcycle = 4;
    breaknewtcycle = 5;
    breakihalvecycle = 6;
    
    /*
     * main loop
     */
    for(;;)
    {
        
        /*
         * start
         */
        if( mainlooppos==0 )
        {
            if( ae_fp_less_eq(a,1.0)||ae_fp_less_eq(b,1.0) )
            {
                dithresh = 1.0e-6;
                rflg = 0;
                aaa = a;
                bbb = b;
                y0 = y;
                x = aaa/(aaa+bbb);
                yyy = incompletebeta(aaa, bbb, x, _state);
                mainlooppos = ihalve;
                continue;
            }
            else
            {
                dithresh = 1.0e-4;
            }
            yp = -invnormaldistribution(y, _state);
            if( ae_fp_greater(y,0.5) )
            {
                rflg = 1;
                aaa = b;
                bbb = a;
                y0 = 1.0-y;
                yp = -yp;
            }
            else
            {
                rflg = 0;
                aaa = a;
                bbb = b;
                y0 = y;
            }
            lgm = (yp*yp-3.0)/6.0;
            x = 2.0/(1.0/(2.0*aaa-1.0)+1.0/(2.0*bbb-1.0));
            d = yp*ae_sqrt(x+lgm, _state)/x-(1.0/(2.0*bbb-1.0)-1.0/(2.0*aaa-1.0))*(lgm+5.0/6.0-2.0/(3.0*x));
            d = 2.0*d;
            if( ae_fp_less(d,ae_log(ae_minrealnumber, _state)) )
            {
                x = 0;
                break;
            }
            x = aaa/(aaa+bbb*ae_exp(d, _state));
            yyy = incompletebeta(aaa, bbb, x, _state);
            yp = (yyy-y0)/y0;
            if( ae_fp_less(ae_fabs(yp, _state),0.2) )
            {
                mainlooppos = newt;
                continue;
            }
            mainlooppos = ihalve;
            continue;
        }
        
        /*
         * ihalve
         */
        if( mainlooppos==ihalve )
        {
            dir = 0;
            di = 0.5;
            i = 0;
            mainlooppos = ihalvecycle;
            continue;
        }
        
        /*
         * ihalvecycle
         */
        if( mainlooppos==ihalvecycle )
        {
            if( i<=99 )
            {
                if( i!=0 )
                {
                    x = x0+di*(x1-x0);
                    if( ae_fp_eq(x,1.0) )
                    {
                        x = 1.0-ae_machineepsilon;
                    }
                    if( ae_fp_eq(x,0.0) )
                    {
                        di = 0.5;
                        x = x0+di*(x1-x0);
                        if( ae_fp_eq(x,0.0) )
                        {
                            break;
                        }
                    }
                    yyy = incompletebeta(aaa, bbb, x, _state);
                    yp = (x1-x0)/(x1+x0);
                    if( ae_fp_less(ae_fabs(yp, _state),dithresh) )
                    {
                        mainlooppos = newt;
                        continue;
                    }
                    yp = (yyy-y0)/y0;
                    if( ae_fp_less(ae_fabs(yp, _state),dithresh) )
                    {
                        mainlooppos = newt;
                        continue;
                    }
                }
                if( ae_fp_less(yyy,y0) )
                {
                    x0 = x;
                    yl = yyy;
                    if( dir<0 )
                    {
                        dir = 0;
                        di = 0.5;
                    }
                    else
                    {
                        if( dir>3 )
                        {
                            di = 1.0-(1.0-di)*(1.0-di);
                        }
                        else
                        {
                            if( dir>1 )
                            {
                                di = 0.5*di+0.5;
                            }
                            else
                            {
                                di = (y0-yyy)/(yh-yl);
                            }
                        }
                    }
                    dir = dir+1;
                    if( ae_fp_greater(x0,0.75) )
                    {
                        if( rflg==1 )
                        {
                            rflg = 0;
                            aaa = a;
                            bbb = b;
                            y0 = y;
                        }
                        else
                        {
                            rflg = 1;
                            aaa = b;
                            bbb = a;
                            y0 = 1.0-y;
                        }
                        x = 1.0-x;
                        yyy = incompletebeta(aaa, bbb, x, _state);
                        x0 = 0.0;
                        yl = 0.0;
                        x1 = 1.0;
                        yh = 1.0;
                        mainlooppos = ihalve;
                        continue;
                    }
                }
                else
                {
                    x1 = x;
                    if( rflg==1&&ae_fp_less(x1,ae_machineepsilon) )
                    {
                        x = 0.0;
                        break;
                    }
                    yh = yyy;
                    if( dir>0 )
                    {
                        dir = 0;
                        di = 0.5;
                    }
                    else
                    {
                        if( dir<-3 )
                        {
                            di = di*di;
                        }
                        else
                        {
                            if( dir<-1 )
                            {
                                di = 0.5*di;
                            }
                            else
                            {
                                di = (yyy-y0)/(yh-yl);
                            }
                        }
                    }
                    dir = dir-1;
                }
                i = i+1;
                mainlooppos = ihalvecycle;
                continue;
            }
            else
            {
                mainlooppos = breakihalvecycle;
                continue;
            }
        }
        
        /*
         * breakihalvecycle
         */
        if( mainlooppos==breakihalvecycle )
        {
            if( ae_fp_greater_eq(x0,1.0) )
            {
                x = 1.0-ae_machineepsilon;
                break;
            }
            if( ae_fp_less_eq(x,0.0) )
            {
                x = 0.0;
                break;
            }
            mainlooppos = newt;
            continue;
        }
        
        /*
         * newt
         */
        if( mainlooppos==newt )
        {
            if( nflg!=0 )
            {
                break;
            }
            nflg = 1;
            lgm = lngamma(aaa+bbb, &s, _state)-lngamma(aaa, &s, _state)-lngamma(bbb, &s, _state);
            i = 0;
            mainlooppos = newtcycle;
            continue;
        }
        
        /*
         * newtcycle
         */
        if( mainlooppos==newtcycle )
        {
            if( i<=7 )
            {
                if( i!=0 )
                {
                    yyy = incompletebeta(aaa, bbb, x, _state);
                }
                if( ae_fp_less(yyy,yl) )
                {
                    x = x0;
                    yyy = yl;
                }
                else
                {
                    if( ae_fp_greater(yyy,yh) )
                    {
                        x = x1;
                        yyy = yh;
                    }
                    else
                    {
                        if( ae_fp_less(yyy,y0) )
                        {
                            x0 = x;
                            yl = yyy;
                        }
                        else
                        {
                            x1 = x;
                            yh = yyy;
                        }
                    }
                }
                if( ae_fp_eq(x,1.0)||ae_fp_eq(x,0.0) )
                {
                    mainlooppos = breaknewtcycle;
                    continue;
                }
                d = (aaa-1.0)*ae_log(x, _state)+(bbb-1.0)*ae_log(1.0-x, _state)+lgm;
                if( ae_fp_less(d,ae_log(ae_minrealnumber, _state)) )
                {
                    break;
                }
                if( ae_fp_greater(d,ae_log(ae_maxrealnumber, _state)) )
                {
                    mainlooppos = breaknewtcycle;
                    continue;
                }
                d = ae_exp(d, _state);
                d = (yyy-y0)/d;
                xt = x-d;
                if( ae_fp_less_eq(xt,x0) )
                {
                    yyy = (x-x0)/(x1-x0);
                    xt = x0+0.5*yyy*(x-x0);
                    if( ae_fp_less_eq(xt,0.0) )
                    {
                        mainlooppos = breaknewtcycle;
                        continue;
                    }
                }
                if( ae_fp_greater_eq(xt,x1) )
                {
                    yyy = (x1-x)/(x1-x0);
                    xt = x1-0.5*yyy*(x1-x);
                    if( ae_fp_greater_eq(xt,1.0) )
                    {
                        mainlooppos = breaknewtcycle;
                        continue;
                    }
                }
                x = xt;
                if( ae_fp_less(ae_fabs(d/x, _state),128.0*ae_machineepsilon) )
                {
                    break;
                }
                i = i+1;
                mainlooppos = newtcycle;
                continue;
            }
            else
            {
                mainlooppos = breaknewtcycle;
                continue;
            }
        }
        
        /*
         * breaknewtcycle
         */
        if( mainlooppos==breaknewtcycle )
        {
            dithresh = 256.0*ae_machineepsilon;
            mainlooppos = ihalve;
            continue;
        }
    }
    
    /*
     * done
     */
    if( rflg!=0 )
    {
        if( ae_fp_less_eq(x,ae_machineepsilon) )
        {
            x = 1.0-ae_machineepsilon;
        }
        else
        {
            x = 1.0-x;
        }
    }
    result = x;
    return result;
}


/*************************************************************************
Continued fraction expansion #1 for incomplete beta integral

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
static double ibetaf_incompletebetafe(double a,
     double b,
     double x,
     double big,
     double biginv,
     ae_state *_state)
{
    double xk;
    double pk;
    double pkm1;
    double pkm2;
    double qk;
    double qkm1;
    double qkm2;
    double k1;
    double k2;
    double k3;
    double k4;
    double k5;
    double k6;
    double k7;
    double k8;
    double r;
    double t;
    double ans;
    double thresh;
    ae_int_t n;
    double result;


    k1 = a;
    k2 = a+b;
    k3 = a;
    k4 = a+1.0;
    k5 = 1.0;
    k6 = b-1.0;
    k7 = k4;
    k8 = a+2.0;
    pkm2 = 0.0;
    qkm2 = 1.0;
    pkm1 = 1.0;
    qkm1 = 1.0;
    ans = 1.0;
    r = 1.0;
    n = 0;
    thresh = 3.0*ae_machineepsilon;
    do
    {
        xk = -x*k1*k2/(k3*k4);
        pk = pkm1+pkm2*xk;
        qk = qkm1+qkm2*xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
        xk = x*k5*k6/(k7*k8);
        pk = pkm1+pkm2*xk;
        qk = qkm1+qkm2*xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
        if( ae_fp_neq(qk,0) )
        {
            r = pk/qk;
        }
        if( ae_fp_neq(r,0) )
        {
            t = ae_fabs((ans-r)/r, _state);
            ans = r;
        }
        else
        {
            t = 1.0;
        }
        if( ae_fp_less(t,thresh) )
        {
            break;
        }
        k1 = k1+1.0;
        k2 = k2+1.0;
        k3 = k3+2.0;
        k4 = k4+2.0;
        k5 = k5+1.0;
        k6 = k6-1.0;
        k7 = k7+2.0;
        k8 = k8+2.0;
        if( ae_fp_greater(ae_fabs(qk, _state)+ae_fabs(pk, _state),big) )
        {
            pkm2 = pkm2*biginv;
            pkm1 = pkm1*biginv;
            qkm2 = qkm2*biginv;
            qkm1 = qkm1*biginv;
        }
        if( ae_fp_less(ae_fabs(qk, _state),biginv)||ae_fp_less(ae_fabs(pk, _state),biginv) )
        {
            pkm2 = pkm2*big;
            pkm1 = pkm1*big;
            qkm2 = qkm2*big;
            qkm1 = qkm1*big;
        }
        n = n+1;
    }
    while(n!=300);
    result = ans;
    return result;
}


/*************************************************************************
Continued fraction expansion #2
for incomplete beta integral

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
static double ibetaf_incompletebetafe2(double a,
     double b,
     double x,
     double big,
     double biginv,
     ae_state *_state)
{
    double xk;
    double pk;
    double pkm1;
    double pkm2;
    double qk;
    double qkm1;
    double qkm2;
    double k1;
    double k2;
    double k3;
    double k4;
    double k5;
    double k6;
    double k7;
    double k8;
    double r;
    double t;
    double ans;
    double z;
    double thresh;
    ae_int_t n;
    double result;


    k1 = a;
    k2 = b-1.0;
    k3 = a;
    k4 = a+1.0;
    k5 = 1.0;
    k6 = a+b;
    k7 = a+1.0;
    k8 = a+2.0;
    pkm2 = 0.0;
    qkm2 = 1.0;
    pkm1 = 1.0;
    qkm1 = 1.0;
    z = x/(1.0-x);
    ans = 1.0;
    r = 1.0;
    n = 0;
    thresh = 3.0*ae_machineepsilon;
    do
    {
        xk = -z*k1*k2/(k3*k4);
        pk = pkm1+pkm2*xk;
        qk = qkm1+qkm2*xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
        xk = z*k5*k6/(k7*k8);
        pk = pkm1+pkm2*xk;
        qk = qkm1+qkm2*xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
        if( ae_fp_neq(qk,0) )
        {
            r = pk/qk;
        }
        if( ae_fp_neq(r,0) )
        {
            t = ae_fabs((ans-r)/r, _state);
            ans = r;
        }
        else
        {
            t = 1.0;
        }
        if( ae_fp_less(t,thresh) )
        {
            break;
        }
        k1 = k1+1.0;
        k2 = k2-1.0;
        k3 = k3+2.0;
        k4 = k4+2.0;
        k5 = k5+1.0;
        k6 = k6+1.0;
        k7 = k7+2.0;
        k8 = k8+2.0;
        if( ae_fp_greater(ae_fabs(qk, _state)+ae_fabs(pk, _state),big) )
        {
            pkm2 = pkm2*biginv;
            pkm1 = pkm1*biginv;
            qkm2 = qkm2*biginv;
            qkm1 = qkm1*biginv;
        }
        if( ae_fp_less(ae_fabs(qk, _state),biginv)||ae_fp_less(ae_fabs(pk, _state),biginv) )
        {
            pkm2 = pkm2*big;
            pkm1 = pkm1*big;
            qkm2 = qkm2*big;
            qkm1 = qkm1*big;
        }
        n = n+1;
    }
    while(n!=300);
    result = ans;
    return result;
}


/*************************************************************************
Power series for incomplete beta integral.
Use when b*x is small and x not too close to 1.

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
static double ibetaf_incompletebetaps(double a,
     double b,
     double x,
     double maxgam,
     ae_state *_state)
{
    double s;
    double t;
    double u;
    double v;
    double n;
    double t1;
    double z;
    double ai;
    double sg;
    double result;


    ai = 1.0/a;
    u = (1.0-b)*x;
    v = u/(a+1.0);
    t1 = v;
    t = u;
    n = 2.0;
    s = 0.0;
    z = ae_machineepsilon*ai;
    while(ae_fp_greater(ae_fabs(v, _state),z))
    {
        u = (n-b)*x/n;
        t = t*u;
        v = t/(a+n);
        s = s+v;
        n = n+1.0;
    }
    s = s+t1;
    s = s+ai;
    u = a*ae_log(x, _state);
    if( ae_fp_less(a+b,maxgam)&&ae_fp_less(ae_fabs(u, _state),ae_log(ae_maxrealnumber, _state)) )
    {
        t = gammafunction(a+b, _state)/(gammafunction(a, _state)*gammafunction(b, _state));
        s = s*t*ae_pow(x, a, _state);
    }
    else
    {
        t = lngamma(a+b, &sg, _state)-lngamma(a, &sg, _state)-lngamma(b, &sg, _state)+u+ae_log(s, _state);
        if( ae_fp_less(t,ae_log(ae_minrealnumber, _state)) )
        {
            s = 0.0;
        }
        else
        {
            s = ae_exp(t, _state);
        }
    }
    result = s;
    return result;
}




/*************************************************************************
Binomial distribution

Returns the sum of the terms 0 through k of the Binomial
probability density:

  k
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=0

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p), with p between 0 and 1.

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      4.3e-15     2.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialdistribution(ae_int_t k,
     ae_int_t n,
     double p,
     ae_state *_state)
{
    double dk;
    double dn;
    double result;


    ae_assert(ae_fp_greater_eq(p,0)&&ae_fp_less_eq(p,1), "Domain error in BinomialDistribution", _state);
    ae_assert(k>=-1&&k<=n, "Domain error in BinomialDistribution", _state);
    if( k==-1 )
    {
        result = 0;
        return result;
    }
    if( k==n )
    {
        result = 1;
        return result;
    }
    dn = n-k;
    if( k==0 )
    {
        dk = ae_pow(1.0-p, dn, _state);
    }
    else
    {
        dk = k+1;
        dk = incompletebeta(dn, dk, 1.0-p, _state);
    }
    result = dk;
    return result;
}


/*************************************************************************
Complemented binomial distribution

Returns the sum of the terms k+1 through n of the Binomial
probability density:

  n
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=k+1

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      6.7e-15     8.2e-16
 For p between 0 and .001:
   IEEE     0,100       100000      1.5e-13     2.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialcdistribution(ae_int_t k,
     ae_int_t n,
     double p,
     ae_state *_state)
{
    double dk;
    double dn;
    double result;


    ae_assert(ae_fp_greater_eq(p,0)&&ae_fp_less_eq(p,1), "Domain error in BinomialDistributionC", _state);
    ae_assert(k>=-1&&k<=n, "Domain error in BinomialDistributionC", _state);
    if( k==-1 )
    {
        result = 1;
        return result;
    }
    if( k==n )
    {
        result = 0;
        return result;
    }
    dn = n-k;
    if( k==0 )
    {
        if( ae_fp_less(p,0.01) )
        {
            dk = -nuexpm1(dn*nulog1p(-p, _state), _state);
        }
        else
        {
            dk = 1.0-ae_pow(1.0-p, dn, _state);
        }
    }
    else
    {
        dk = k+1;
        dk = incompletebeta(dk, dn, p, _state);
    }
    result = dk;
    return result;
}


/*************************************************************************
Inverse binomial distribution

Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.

This is accomplished using the inverse beta integral
function and the relation

1 - p = incbi( n-k, k+1, y ).

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      2.3e-14     6.4e-16
   IEEE     0,10000     100000      6.6e-12     1.2e-13
 For p between 10^-6 and 0.001:
   IEEE     0,100       100000      2.0e-12     1.3e-14
   IEEE     0,10000     100000      1.5e-12     3.2e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invbinomialdistribution(ae_int_t k,
     ae_int_t n,
     double y,
     ae_state *_state)
{
    double dk;
    double dn;
    double p;
    double result;


    ae_assert(k>=0&&k<n, "Domain error in InvBinomialDistribution", _state);
    dn = n-k;
    if( k==0 )
    {
        if( ae_fp_greater(y,0.8) )
        {
            p = -nuexpm1(nulog1p(y-1.0, _state)/dn, _state);
        }
        else
        {
            p = 1.0-ae_pow(y, 1.0/dn, _state);
        }
    }
    else
    {
        dk = k+1;
        p = incompletebeta(dn, dk, 0.5, _state);
        if( ae_fp_greater(p,0.5) )
        {
            p = invincompletebeta(dk, dn, 1.0-y, _state);
        }
        else
        {
            p = 1.0-invincompletebeta(dn, dk, y, _state);
        }
    }
    result = p;
    return result;
}




/*************************************************************************
Calculation of the value of the Chebyshev polynomials of the
first and second kinds.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument, -1 <= x <= 1

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevcalculate(ae_int_t r,
     ae_int_t n,
     double x,
     ae_state *_state)
{
    ae_int_t i;
    double a;
    double b;
    double result;


    result = 0;
    
    /*
     * Prepare A and B
     */
    if( r==1 )
    {
        a = 1;
        b = x;
    }
    else
    {
        a = 1;
        b = 2*x;
    }
    
    /*
     * Special cases: N=0 or N=1
     */
    if( n==0 )
    {
        result = a;
        return result;
    }
    if( n==1 )
    {
        result = b;
        return result;
    }
    
    /*
     * General case: N>=2
     */
    for(i=2; i<=n; i++)
    {
        result = 2*x*b-a;
        a = b;
        b = result;
    }
    return result;
}


/*************************************************************************
Summation of Chebyshev polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
or
    c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
depending on the R.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevsum(/* Real    */ ae_vector* c,
     ae_int_t r,
     ae_int_t n,
     double x,
     ae_state *_state)
{
    double b1;
    double b2;
    ae_int_t i;
    double result;


    b1 = 0;
    b2 = 0;
    for(i=n; i>=1; i--)
    {
        result = 2*x*b1-b2+c->ptr.p_double[i];
        b2 = b1;
        b1 = result;
    }
    if( r==1 )
    {
        result = -b2+x*b1+c->ptr.p_double[0];
    }
    else
    {
        result = -b2+2*x*b1+c->ptr.p_double[0];
    }
    return result;
}


/*************************************************************************
Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void chebyshevcoefficients(ae_int_t n,
     /* Real    */ ae_vector* c,
     ae_state *_state)
{
    ae_int_t i;

    ae_vector_clear(c);

    ae_vector_set_length(c, n+1, _state);
    for(i=0; i<=n; i++)
    {
        c->ptr.p_double[i] = 0;
    }
    if( n==0||n==1 )
    {
        c->ptr.p_double[n] = 1;
    }
    else
    {
        c->ptr.p_double[n] = ae_exp((n-1)*ae_log(2, _state), _state);
        for(i=0; i<=n/2-1; i++)
        {
            c->ptr.p_double[n-2*(i+1)] = -c->ptr.p_double[n-2*i]*(n-2*i)*(n-2*i-1)/4/(i+1)/(n-i-1);
        }
    }
}


/*************************************************************************
Conversion of a series of Chebyshev polynomials to a power series.

Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
B[0] + B[1]*X + ... + B[N]*X^N.

Input parameters:
    A   -   Chebyshev series coefficients
    N   -   degree, N>=0
    
Output parameters
    B   -   power series coefficients
*************************************************************************/
void fromchebyshev(/* Real    */ ae_vector* a,
     ae_int_t n,
     /* Real    */ ae_vector* b,
     ae_state *_state)
{
    ae_int_t i;
    ae_int_t k;
    double e;
    double d;

    ae_vector_clear(b);

    ae_vector_set_length(b, n+1, _state);
    for(i=0; i<=n; i++)
    {
        b->ptr.p_double[i] = 0;
    }
    d = 0;
    i = 0;
    do
    {
        k = i;
        do
        {
            e = b->ptr.p_double[k];
            b->ptr.p_double[k] = 0;
            if( i<=1&&k==i )
            {
                b->ptr.p_double[k] = 1;
            }
            else
            {
                if( i!=0 )
                {
                    b->ptr.p_double[k] = 2*d;
                }
                if( k>i+1 )
                {
                    b->ptr.p_double[k] = b->ptr.p_double[k]-b->ptr.p_double[k-2];
                }
            }
            d = e;
            k = k+1;
        }
        while(k<=n);
        d = b->ptr.p_double[i];
        e = 0;
        k = i;
        while(k<=n)
        {
            e = e+b->ptr.p_double[k]*a->ptr.p_double[k];
            k = k+2;
        }
        b->ptr.p_double[i] = e;
        i = i+1;
    }
    while(i<=n);
}




/*************************************************************************
Chi-square distribution

Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.


                                  x
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  0

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquaredistribution(double v, double x, ae_state *_state)
{
    double result;


    ae_assert(ae_fp_greater_eq(x,0)&&ae_fp_greater_eq(v,1), "Domain error in ChiSquareDistribution", _state);
    result = incompletegamma(v/2.0, x/2.0, _state);
    return result;
}


/*************************************************************************
Complemented Chi-square distribution

Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:

                                 inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquarecdistribution(double v, double x, ae_state *_state)
{
    double result;


    ae_assert(ae_fp_greater_eq(x,0)&&ae_fp_greater_eq(v,1), "Domain error in ChiSquareDistributionC", _state);
    result = incompletegammac(v/2.0, x/2.0, _state);
    return result;
}


/*************************************************************************
Inverse of complemented Chi-square distribution

Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.

This is accomplished using the inverse gamma integral
function and the relation

   x/2 = igami( df/2, y );

ACCURACY:

See inverse incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double invchisquaredistribution(double v, double y, ae_state *_state)
{
    double result;


    ae_assert((ae_fp_greater_eq(y,0)&&ae_fp_less_eq(y,1))&&ae_fp_greater_eq(v,1), "Domain error in InvChiSquareDistribution", _state);
    result = 2*invincompletegammac(0.5*v, y, _state);
    return result;
}




/*************************************************************************
Dawson's Integral

Approximates the integral

                            x
                            -
                     2     | |        2
 dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
                         | |
                          -
                          0

Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,10        10000       6.9e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double dawsonintegral(double x, ae_state *_state)
{
    double x2;
    double y;
    ae_int_t sg;
    double an;
    double ad;
    double bn;
    double bd;
    double cn;
    double cd;
    double result;


    sg = 1;
    if( ae_fp_less(x,0) )
    {
        sg = -1;
        x = -x;
    }
    if( ae_fp_less(x,3.25) )
    {
        x2 = x*x;
        an = 1.13681498971755972054E-11;
        an = an*x2+8.49262267667473811108E-10;
        an = an*x2+1.94434204175553054283E-8;
        an = an*x2+9.53151741254484363489E-7;
        an = an*x2+3.07828309874913200438E-6;
        an = an*x2+3.52513368520288738649E-4;
        an = an*x2+(-8.50149846724410912031E-4);
        an = an*x2+4.22618223005546594270E-2;
        an = an*x2+(-9.17480371773452345351E-2);
        an = an*x2+9.99999999999999994612E-1;
        ad = 2.40372073066762605484E-11;
        ad = ad*x2+1.48864681368493396752E-9;
        ad = ad*x2+5.21265281010541664570E-8;
        ad = ad*x2+1.27258478273186970203E-6;
        ad = ad*x2+2.32490249820789513991E-5;
        ad = ad*x2+3.25524741826057911661E-4;
        ad = ad*x2+3.48805814657162590916E-3;
        ad = ad*x2+2.79448531198828973716E-2;
        ad = ad*x2+1.58874241960120565368E-1;
        ad = ad*x2+5.74918629489320327824E-1;
        ad = ad*x2+1.00000000000000000539E0;
        y = x*an/ad;
        result = sg*y;
        return result;
    }
    x2 = 1.0/(x*x);
    if( ae_fp_less(x,6.25) )
    {
        bn = 5.08955156417900903354E-1;
        bn = bn*x2-2.44754418142697847934E-1;
        bn = bn*x2+9.41512335303534411857E-2;
        bn = bn*x2-2.18711255142039025206E-2;
        bn = bn*x2+3.66207612329569181322E-3;
        bn = bn*x2-4.23209114460388756528E-4;
        bn = bn*x2+3.59641304793896631888E-5;
        bn = bn*x2-2.14640351719968974225E-6;
        bn = bn*x2+9.10010780076391431042E-8;
        bn = bn*x2-2.40274520828250956942E-9;
        bn = bn*x2+3.59233385440928410398E-11;
        bd = 1.00000000000000000000E0;
        bd = bd*x2-6.31839869873368190192E-1;
        bd = bd*x2+2.36706788228248691528E-1;
        bd = bd*x2-5.31806367003223277662E-2;
        bd = bd*x2+8.48041718586295374409E-3;
        bd = bd*x2-9.47996768486665330168E-4;
        bd = bd*x2+7.81025592944552338085E-5;
        bd = bd*x2-4.55875153252442634831E-6;
        bd = bd*x2+1.89100358111421846170E-7;
        bd = bd*x2-4.91324691331920606875E-9;
        bd = bd*x2+7.18466403235734541950E-11;
        y = 1.0/x+x2*bn/(bd*x);
        result = sg*0.5*y;
        return result;
    }
    if( ae_fp_greater(x,1.0E9) )
    {
        result = sg*0.5/x;
        return result;
    }
    cn = -5.90592860534773254987E-1;
    cn = cn*x2+6.29235242724368800674E-1;
    cn = cn*x2-1.72858975380388136411E-1;
    cn = cn*x2+1.64837047825189632310E-2;
    cn = cn*x2-4.86827613020462700845E-4;
    cd = 1.00000000000000000000E0;
    cd = cd*x2-2.69820057197544900361E0;
    cd = cd*x2+1.73270799045947845857E0;
    cd = cd*x2-3.93708582281939493482E-1;
    cd = cd*x2+3.44278924041233391079E-2;
    cd = cd*x2-9.73655226040941223894E-4;
    y = 1.0/x+x2*cn/(cd*x);
    result = sg*0.5*y;
    return result;
}




/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

using the approximation

    P(x)  -  log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralk(double m, ae_state *_state)
{
    double result;


    result = ellipticintegralkhighprecision(1.0-m, _state);
    return result;
}


/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

where m = 1 - m1, using the approximation

    P(x)  -  log x Q(x).

The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.

K(0) = pi/2.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Àëãîðèòì âçÿò èç áèáëèîòåêè Cephes
*************************************************************************/
double ellipticintegralkhighprecision(double m1, ae_state *_state)
{
    double p;
    double q;
    double result;


    if( ae_fp_less_eq(m1,ae_machineepsilon) )
    {
        result = 1.3862943611198906188E0-0.5*ae_log(m1, _state);
    }
    else
    {
        p = 1.37982864606273237150E-4;
        p = p*m1+2.28025724005875567385E-3;
        p = p*m1+7.97404013220415179367E-3;
        p = p*m1+9.85821379021226008714E-3;
        p = p*m1+6.87489687449949877925E-3;
        p = p*m1+6.18901033637687613229E-3;
        p = p*m1+8.79078273952743772254E-3;
        p = p*m1+1.49380448916805252718E-2;
        p = p*m1+3.08851465246711995998E-2;
        p = p*m1+9.65735902811690126535E-2;
        p = p*m1+1.38629436111989062502E0;
        q = 2.94078955048598507511E-5;
        q = q*m1+9.14184723865917226571E-4;
        q = q*m1+5.94058303753167793257E-3;
        q = q*m1+1.54850516649762399335E-2;
        q = q*m1+2.39089602715924892727E-2;
        q = q*m1+3.01204715227604046988E-2;
        q = q*m1+3.73774314173823228969E-2;
        q = q*m1+4.88280347570998239232E-2;
        q = q*m1+7.03124996963957469739E-2;
        q = q*m1+1.24999999999870820058E-1;
        q = q*m1+4.99999999999999999821E-1;
        result = p-q*ae_log(m1, _state);
    }
    return result;
}


/*************************************************************************
Incomplete elliptic integral of the first kind F(phi|m)

Approximates the integral



               phi
                -
               | |
               |           dt
F(phi_\m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.




ACCURACY:

Tested at random points with m in [0, 1] and phi as indicated.

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10       200000      7.4e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegralk(double phi, double m, ae_state *_state)
{
    double a;
    double b;
    double c;
    double e;
    double temp;
    double pio2;
    double t;
    double k;
    ae_int_t d;
    ae_int_t md;
    ae_int_t s;
    ae_int_t npio2;
    double result;


    pio2 = 1.57079632679489661923;
    if( ae_fp_eq(m,0) )
    {
        result = phi;
        return result;
    }
    a = 1-m;
    if( ae_fp_eq(a,0) )
    {
        result = ae_log(ae_tan(0.5*(pio2+phi), _state), _state);
        return result;
    }
    npio2 = ae_ifloor(phi/pio2, _state);
    if( npio2%2!=0 )
    {
        npio2 = npio2+1;
    }
    if( npio2!=0 )
    {
        k = ellipticintegralk(1-a, _state);
        phi = phi-npio2*pio2;
    }
    else
    {
        k = 0;
    }
    if( ae_fp_less(phi,0) )
    {
        phi = -phi;
        s = -1;
    }
    else
    {
        s = 0;
    }
    b = ae_sqrt(a, _state);
    t = ae_tan(phi, _state);
    if( ae_fp_greater(ae_fabs(t, _state),10) )
    {
        e = 1.0/(b*t);
        if( ae_fp_less(ae_fabs(e, _state),10) )
        {
            e = ae_atan(e, _state);
            if( npio2==0 )
            {
                k = ellipticintegralk(1-a, _state);
            }
            temp = k-incompleteellipticintegralk(e, m, _state);
            if( s<0 )
            {
                temp = -temp;
            }
            result = temp+npio2*k;
            return result;
        }
    }
    a = 1.0;
    c = ae_sqrt(m, _state);
    d = 1;
    md = 0;
    while(ae_fp_greater(ae_fabs(c/a, _state),ae_machineepsilon))
    {
        temp = b/a;
        phi = phi+ae_atan(t*temp, _state)+md*ae_pi;
        md = ae_trunc((phi+pio2)/ae_pi, _state);
        t = t*(1.0+temp)/(1.0-temp*t*t);
        c = 0.5*(a-b);
        temp = ae_sqrt(a*b, _state);
        a = 0.5*(a+b);
        b = temp;
        d = d+d;
    }
    temp = (ae_atan(t, _state)+md*ae_pi)/(d*a);
    if( s<0 )
    {
        temp = -temp;
    }
    result = temp+npio2*k;
    return result;
}


/*************************************************************************
Complete elliptic integral of the second kind

Approximates the integral


           pi/2
            -
           | |                 2
E(m)  =    |    sqrt( 1 - m sin t ) dt
         | |
          -
           0

using the approximation

     P(x)  -  x log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0, 1       10000       2.1e-16     7.3e-17

Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegrale(double m, ae_state *_state)
{
    double p;
    double q;
    double result;


    ae_assert(ae_fp_greater_eq(m,0)&&ae_fp_less_eq(m,1), "Domain error in EllipticIntegralE: m<0 or m>1", _state);
    m = 1-m;
    if( ae_fp_eq(m,0) )
    {
        result = 1;
        return result;
    }
    p = 1.53552577301013293365E-4;
    p = p*m+2.50888492163602060990E-3;
    p = p*m+8.68786816565889628429E-3;
    p = p*m+1.07350949056076193403E-2;
    p = p*m+7.77395492516787092951E-3;
    p = p*m+7.58395289413514708519E-3;
    p = p*m+1.15688436810574127319E-2;
    p = p*m+2.18317996015557253103E-2;
    p = p*m+5.68051945617860553470E-2;
    p = p*m+4.43147180560990850618E-1;
    p = p*m+1.00000000000000000299E0;
    q = 3.27954898576485872656E-5;
    q = q*m+1.00962792679356715133E-3;
    q = q*m+6.50609489976927491433E-3;
    q = q*m+1.68862163993311317300E-2;
    q = q*m+2.61769742454493659583E-2;
    q = q*m+3.34833904888224918614E-2;
    q = q*m+4.27180926518931511717E-2;
    q = q*m+5.85936634471101055642E-2;
    q = q*m+9.37499997197644278445E-2;
    q = q*m+2.49999999999888314361E-1;
    result = p-q*m*ae_log(m, _state);
    return result;
}


/*************************************************************************
Incomplete elliptic integral of the second kind

Approximates the integral


               phi
                -
               | |
               |                   2
E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
               |
             | |
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.

ACCURACY:

Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10      150000       3.3e-15     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegrale(double phi, double m, ae_state *_state)
{
    double pio2;
    double a;
    double b;
    double c;
    double e;
    double temp;
    double lphi;
    double t;
    double ebig;
    ae_int_t d;
    ae_int_t md;
    ae_int_t npio2;
    ae_int_t s;
    double result;


    pio2 = 1.57079632679489661923;
    if( ae_fp_eq(m,0) )
    {
        result = phi;
        return result;
    }
    lphi = phi;
    npio2 = ae_ifloor(lphi/pio2, _state);
    if( npio2%2!=0 )
    {
        npio2 = npio2+1;
    }
    lphi = lphi-npio2*pio2;
    if( ae_fp_less(lphi,0) )
    {
        lphi = -lphi;
        s = -1;
    }
    else
    {
        s = 1;
    }
    a = 1.0-m;
    ebig = ellipticintegrale(m, _state);
    if( ae_fp_eq(a,0) )
    {
        temp = ae_sin(lphi, _state);
        if( s<0 )
        {
            temp = -temp;
        }
        result = temp+npio2*ebig;
        return result;
    }
    t = ae_tan(lphi, _state);
    b = ae_sqrt(a, _state);
    
    /*
     * Thanks to Brian Fitzgerald <fitzgb@mml0.meche.rpi.edu>
     * for pointing out an instability near odd multiples of pi/2
     */
    if( ae_fp_greater(ae_fabs(t, _state),10) )
    {
        
        /*
         * Transform the amplitude
         */
        e = 1.0/(b*t);
        
        /*
         * ... but avoid multiple recursions.
         */
        if( ae_fp_less(ae_fabs(e, _state),10) )
        {
            e = ae_atan(e, _state);
            temp = ebig+m*ae_sin(lphi, _state)*ae_sin(e, _state)-incompleteellipticintegrale(e, m, _state);
            if( s<0 )
            {
                temp = -temp;
            }
            result = temp+npio2*ebig;
            return result;
        }
    }
    c = ae_sqrt(m, _state);
    a = 1.0;
    d = 1;
    e = 0.0;
    md = 0;
    while(ae_fp_greater(ae_fabs(c/a, _state),ae_machineepsilon))
    {
        temp = b/a;
        lphi = lphi+ae_atan(t*temp, _state)+md*ae_pi;
        md = ae_trunc((lphi+pio2)/ae_pi, _state);
        t = t*(1.0+temp)/(1.0-temp*t*t);
        c = 0.5*(a-b);
        temp = ae_sqrt(a*b, _state);
        a = 0.5*(a+b);
        b = temp;
        d = d+d;
        e = e+c*ae_sin(lphi, _state);
    }
    temp = ebig/ellipticintegralk(m, _state);
    temp = temp*((ae_atan(t, _state)+md*ae_pi)/(d*a));
    temp = temp+e;
    if( s<0 )
    {
        temp = -temp;
    }
    result = temp+npio2*ebig;
    return result;
}




/*************************************************************************
Exponential integral Ei(x)

              x
               -     t
              | |   e
   Ei(x) =   -|-   ---  dt .
            | |     t
             -
            -inf

Not defined for x <= 0.
See also expn.c.



ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,100       50000      8.6e-16     1.3e-16

Cephes Math Library Release 2.8:  May, 1999
Copyright 1999 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralei(double x, ae_state *_state)
{
    double eul;
    double f;
    double f1;
    double f2;
    double w;
    double result;


    eul = 0.5772156649015328606065;
    if( ae_fp_less_eq(x,0) )
    {
        result = 0;
        return result;
    }
    if( ae_fp_less(x,2) )
    {
        f1 = -5.350447357812542947283;
        f1 = f1*x+218.5049168816613393830;
        f1 = f1*x-4176.572384826693777058;
        f1 = f1*x+55411.76756393557601232;
        f1 = f1*x-331338.1331178144034309;
        f1 = f1*x+1592627.163384945414220;
        f2 = 1.000000000000000000000;
        f2 = f2*x-52.50547959112862969197;
        f2 = f2*x+1259.616186786790571525;
        f2 = f2*x-17565.49581973534652631;
        f2 = f2*x+149306.2117002725991967;
        f2 = f2*x-729494.9239640527645655;
        f2 = f2*x+1592627.163384945429726;
        f = f1/f2;
        result = eul+ae_log(x, _state)+x*f;
        return result;
    }
    if( ae_fp_less(x,4) )
    {
        w = 1/x;
        f1 = 1.981808503259689673238E-2;
        f1 = f1*w-1.271645625984917501326;
        f1 = f1*w-2.088160335681228318920;
        f1 = f1*w+2.755544509187936721172;
        f1 = f1*w-4.409507048701600257171E-1;
        f1 = f1*w+4.665623805935891391017E-2;
        f1 = f1*w-1.545042679673485262580E-3;
        f1 = f1*w+7.059980605299617478514E-5;
        f2 = 1.000000000000000000000;
        f2 = f2*w+1.476498670914921440652;
        f2 = f2*w+5.629177174822436244827E-1;
        f2 = f2*w+1.699017897879307263248E-1;
        f2 = f2*w+2.291647179034212017463E-2;
        f2 = f2*w+4.450150439728752875043E-3;
        f2 = f2*w+1.727439612206521482874E-4;
        f2 = f2*w+3.953167195549672482304E-5;
        f = f1/f2;
        result = ae_exp(x, _state)*w*(1+w*f);
        return result;
    }
    if( ae_fp_less(x,8) )
    {
        w = 1/x;
        f1 = -1.373215375871208729803;
        f1 = f1*w-7.084559133740838761406E-1;
        f1 = f1*w+1.580806855547941010501;
        f1 = f1*w-2.601500427425622944234E-1;
        f1 = f1*w+2.994674694113713763365E-2;
        f1 = f1*w-1.038086040188744005513E-3;
        f1 = f1*w+4.371064420753005429514E-5;
        f1 = f1*w+2.141783679522602903795E-6;
        f2 = 1.000000000000000000000;
        f2 = f2*w+8.585231423622028380768E-1;
        f2 = f2*w+4.483285822873995129957E-1;
        f2 = f2*w+7.687932158124475434091E-2;
        f2 = f2*w+2.449868241021887685904E-2;
        f2 = f2*w+8.832165941927796567926E-4;
        f2 = f2*w+4.590952299511353531215E-4;
        f2 = f2*w+(-4.729848351866523044863E-6);
        f2 = f2*w+2.665195537390710170105E-6;
        f = f1/f2;
        result = ae_exp(x, _state)*w*(1+w*f);
        return result;
    }
    if( ae_fp_less(x,16) )
    {
        w = 1/x;
        f1 = -2.106934601691916512584;
        f1 = f1*w+1.732733869664688041885;
        f1 = f1*w-2.423619178935841904839E-1;
        f1 = f1*w+2.322724180937565842585E-2;
        f1 = f1*w+2.372880440493179832059E-4;
        f1 = f1*w-8.343219561192552752335E-5;
        f1 = f1*w+1.363408795605250394881E-5;
        f1 = f1*w-3.655412321999253963714E-7;
        f1 = f1*w+1.464941733975961318456E-8;
        f1 = f1*w+6.176407863710360207074E-10;
        f2 = 1.000000000000000000000;
        f2 = f2*w-2.298062239901678075778E-1;
        f2 = f2*w+1.105077041474037862347E-1;
        f2 = f2*w-1.566542966630792353556E-2;
        f2 = f2*w+2.761106850817352773874E-3;
        f2 = f2*w-2.089148012284048449115E-4;
        f2 = f2*w+1.708528938807675304186E-5;
        f2 = f2*w-4.459311796356686423199E-7;
        f2 = f2*w+1.394634930353847498145E-8;
        f2 = f2*w+6.150865933977338354138E-10;
        f = f1/f2;
        result = ae_exp(x, _state)*w*(1+w*f);
        return result;
    }
    if( ae_fp_less(x,32) )
    {
        w = 1/x;
        f1 = -2.458119367674020323359E-1;
        f1 = f1*w-1.483382253322077687183E-1;
        f1 = f1*w+7.248291795735551591813E-2;
        f1 = f1*w-1.348315687380940523823E-2;
        f1 = f1*w+1.342775069788636972294E-3;
        f1 = f1*w-7.942465637159712264564E-5;
        f1 = f1*w+2.644179518984235952241E-6;
        f1 = f1*w-4.239473659313765177195E-8;
        f2 = 1.000000000000000000000;
        f2 = f2*w-1.044225908443871106315E-1;
        f2 = f2*w-2.676453128101402655055E-1;
        f2 = f2*w+9.695000254621984627876E-2;
        f2 = f2*w-1.601745692712991078208E-2;
        f2 = f2*w+1.496414899205908021882E-3;
        f2 = f2*w-8.462452563778485013756E-5;
        f2 = f2*w+2.728938403476726394024E-6;
        f2 = f2*w-4.239462431819542051337E-8;
        f = f1/f2;
        result = ae_exp(x, _state)*w*(1+w*f);
        return result;
    }
    if( ae_fp_less(x,64) )
    {
        w = 1/x;
        f1 = 1.212561118105456670844E-1;
        f1 = f1*w-5.823133179043894485122E-1;
        f1 = f1*w+2.348887314557016779211E-1;
        f1 = f1*w-3.040034318113248237280E-2;
        f1 = f1*w+1.510082146865190661777E-3;
        f1 = f1*w-2.523137095499571377122E-5;
        f2 = 1.000000000000000000000;
        f2 = f2*w-1.002252150365854016662;
        f2 = f2*w+2.928709694872224144953E-1;
        f2 = f2*w-3.337004338674007801307E-2;
        f2 = f2*w+1.560544881127388842819E-3;
        f2 = f2*w-2.523137093603234562648E-5;
        f = f1/f2;
        result = ae_exp(x, _state)*w*(1+w*f);
        return result;
    }
    w = 1/x;
    f1 = -7.657847078286127362028E-1;
    f1 = f1*w+6.886192415566705051750E-1;
    f1 = f1*w-2.132598113545206124553E-1;
    f1 = f1*w+3.346107552384193813594E-2;
    f1 = f1*w-3.076541477344756050249E-3;
    f1 = f1*w+1.747119316454907477380E-4;
    f1 = f1*w-6.103711682274170530369E-6;
    f1 = f1*w+1.218032765428652199087E-7;
    f1 = f1*w-1.086076102793290233007E-9;
    f2 = 1.000000000000000000000;
    f2 = f2*w-1.888802868662308731041;
    f2 = f2*w+1.066691687211408896850;
    f2 = f2*w-2.751915982306380647738E-1;
    f2 = f2*w+3.930852688233823569726E-2;
    f2 = f2*w-3.414684558602365085394E-3;
    f2 = f2*w+1.866844370703555398195E-4;
    f2 = f2*w-6.345146083130515357861E-6;
    f2 = f2*w+1.239754287483206878024E-7;
    f2 = f2*w-1.086076102793126632978E-9;
    f = f1/f2;
    result = ae_exp(x, _state)*w*(1+w*f);
    return result;
}


/*************************************************************************
Exponential integral En(x)

Evaluates the exponential integral

                inf.
                  -
                 | |   -xt
                 |    e
     E (x)  =    |    ----  dt.
      n          |      n
               | |     t
                -
                 1


Both n and x must be nonnegative.

The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       10000       1.7e-15     3.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 2000 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralen(double x, ae_int_t n, ae_state *_state)
{
    double r;
    double t;
    double yk;
    double xk;
    double pk;
    double pkm1;
    double pkm2;
    double qk;
    double qkm1;
    double qkm2;
    double psi;
    double z;
    ae_int_t i;
    ae_int_t k;
    double big;
    double eul;
    double result;


    eul = 0.57721566490153286060;
    big = 1.44115188075855872*ae_pow(10, 17, _state);
    if( ((n<0||ae_fp_less(x,0))||ae_fp_greater(x,170))||(ae_fp_eq(x,0)&&n<2) )
    {
        result = -1;
        return result;
    }
    if( ae_fp_eq(x,0) )
    {
        result = (double)1/(double)(n-1);
        return result;
    }
    if( n==0 )
    {
        result = ae_exp(-x, _state)/x;
        return result;
    }
    if( n>5000 )
    {
        xk = x+n;
        yk = 1/(xk*xk);
        t = n;
        result = yk*t*(6*x*x-8*t*x+t*t);
        result = yk*(result+t*(t-2.0*x));
        result = yk*(result+t);
        result = (result+1)*ae_exp(-x, _state)/xk;
        return result;
    }
    if( ae_fp_less_eq(x,1) )
    {
        psi = -eul-ae_log(x, _state);
        for(i=1; i<=n-1; i++)
        {
            psi = psi+(double)1/(double)i;
        }
        z = -x;
        xk = 0;
        yk = 1;
        pk = 1-n;
        if( n==1 )
        {
            result = 0.0;
        }
        else
        {
            result = 1.0/pk;
        }
        do
        {
            xk = xk+1;
            yk = yk*z/xk;
            pk = pk+1;
            if( ae_fp_neq(pk,0) )
            {
                result = result+yk/pk;
            }
            if( ae_fp_neq(result,0) )
            {
                t = ae_fabs(yk/result, _state);
            }
            else
            {
                t = 1;
            }
        }
        while(ae_fp_greater_eq(t,ae_machineepsilon));
        t = 1;
        for(i=1; i<=n-1; i++)
        {
            t = t*z/i;
        }
        result = psi*t-result;
        return result;
    }
    else
    {
        k = 1;
        pkm2 = 1;
        qkm2 = x;
        pkm1 = 1.0;
        qkm1 = x+n;
        result = pkm1/qkm1;
        do
        {
            k = k+1;
            if( k%2==1 )
            {
                yk = 1;
                xk = n+(double)(k-1)/(double)2;
            }
            else
            {
                yk = x;
                xk = (double)k/(double)2;
            }
            pk = pkm1*yk+pkm2*xk;
            qk = qkm1*yk+qkm2*xk;
            if( ae_fp_neq(qk,0) )
            {
                r = pk/qk;
                t = ae_fabs((result-r)/r, _state);
                result = r;
            }
            else
            {
                t = 1;
            }
            pkm2 = pkm1;
            pkm1 = pk;
            qkm2 = qkm1;
            qkm1 = qk;
            if( ae_fp_greater(ae_fabs(pk, _state),big) )
            {
                pkm2 = pkm2/big;
                pkm1 = pkm1/big;
                qkm2 = qkm2/big;
                qkm1 = qkm1/big;
            }
        }
        while(ae_fp_greater_eq(t,ae_machineepsilon));
        result = result*ae_exp(-x, _state);
    }
    return result;
}




/*************************************************************************
F distribution

Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density).  This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula

P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).


The arguments a and b are greater than zero, and x is
nonnegative.

ACCURACY:

Tested at random points (a,b,x).

               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
   IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
   IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
   IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state)
{
    double w;
    double result;


    ae_assert((a>=1&&b>=1)&&ae_fp_greater_eq(x,0), "Domain error in FDistribution", _state);
    w = a*x;
    w = w/(b+w);
    result = incompletebeta(0.5*a, 0.5*b, w, _state);
    return result;
}


/*************************************************************************
Complemented F distribution

Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).


                     inf.
                      -
             1       | |  a-1      b-1
1-P(x)  =  ------    |   t    (1-t)    dt
           B(a,b)  | |
                    -
                     x


The incomplete beta integral is used, according to the
formula

P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).


ACCURACY:

Tested at random points (a,b,x) in the indicated intervals.
               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
   IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
   IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
   IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fcdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state)
{
    double w;
    double result;


    ae_assert((a>=1&&b>=1)&&ae_fp_greater_eq(x,0), "Domain error in FCDistribution", _state);
    w = b/(b+a*x);
    result = incompletebeta(0.5*b, 0.5*a, w, _state);
    return result;
}


/*************************************************************************
Inverse of complemented F distribution

Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.

This is accomplished using the inverse beta integral
function and the relations

     z = incbi( df2/2, df1/2, p )
     x = df2 (1-z) / (df1 z).

Note: the following relations hold for the inverse of
the uncomplemented F distribution:

     z = incbi( df1/2, df2/2, p )
     x = df2 z / (df1 (1-z)).

ACCURACY:

Tested at random points (a,b,p).

             a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between .001 and 1:
   IEEE     1,100       100000      8.3e-15     4.7e-16
   IEEE     1,10000     100000      2.1e-11     1.4e-13
 For p between 10^-6 and 10^-3:
   IEEE     1,100        50000      1.3e-12     8.4e-15
   IEEE     1,10000      50000      3.0e-12     4.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invfdistribution(ae_int_t a,
     ae_int_t b,
     double y,
     ae_state *_state)
{
    double w;
    double result;


    ae_assert(((a>=1&&b>=1)&&ae_fp_greater(y,0))&&ae_fp_less_eq(y,1), "Domain error in InvFDistribution", _state);
    
    /*
     * Compute probability for x = 0.5
     */
    w = incompletebeta(0.5*b, 0.5*a, 0.5, _state);
    
    /*
     * If that is greater than y, then the solution w < .5
     * Otherwise, solve at 1-y to remove cancellation in (b - b*w)
     */
    if( ae_fp_greater(w,y)||ae_fp_less(y,0.001) )
    {
        w = invincompletebeta(0.5*b, 0.5*a, y, _state);
        result = (b-b*w)/(a*w);
    }
    else
    {
        w = invincompletebeta(0.5*a, 0.5*b, 1.0-y, _state);
        result = b*w/(a*(1.0-w));
    }
    return result;
}




/*************************************************************************
Fresnel integral

Evaluates the Fresnel integrals

          x
          -
         | |
C(x) =   |   cos(pi/2 t**2) dt,
       | |
        -
         0

          x
          -
         | |
S(x) =   |   sin(pi/2 t**2) dt.
       | |
        -
         0


The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that

C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )



ACCURACY:

 Relative error.

Arithmetic  function   domain     # trials      peak         rms
  IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
  IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void fresnelintegral(double x, double* c, double* s, ae_state *_state)
{
    double xxa;
    double f;
    double g;
    double cc;
    double ss;
    double t;
    double u;
    double x2;
    double sn;
    double sd;
    double cn;
    double cd;
    double fn;
    double fd;
    double gn;
    double gd;
    double mpi;
    double mpio2;


    mpi = 3.14159265358979323846;
    mpio2 = 1.57079632679489661923;
    xxa = x;
    x = ae_fabs(xxa, _state);
    x2 = x*x;
    if( ae_fp_less(x2,2.5625) )
    {
        t = x2*x2;
        sn = -2.99181919401019853726E3;
        sn = sn*t+7.08840045257738576863E5;
        sn = sn*t-6.29741486205862506537E7;
        sn = sn*t+2.54890880573376359104E9;
        sn = sn*t-4.42979518059697779103E10;
        sn = sn*t+3.18016297876567817986E11;
        sd = 1.00000000000000000000E0;
        sd = sd*t+2.81376268889994315696E2;
        sd = sd*t+4.55847810806532581675E4;
        sd = sd*t+5.17343888770096400730E6;
        sd = sd*t+4.19320245898111231129E8;
        sd = sd*t+2.24411795645340920940E10;
        sd = sd*t+6.07366389490084639049E11;
        cn = -4.98843114573573548651E-8;
        cn = cn*t+9.50428062829859605134E-6;
        cn = cn*t-6.45191435683965050962E-4;
        cn = cn*t+1.88843319396703850064E-2;
        cn = cn*t-2.05525900955013891793E-1;
        cn = cn*t+9.99999999999999998822E-1;
        cd = 3.99982968972495980367E-12;
        cd = cd*t+9.15439215774657478799E-10;
        cd = cd*t+1.25001862479598821474E-7;
        cd = cd*t+1.22262789024179030997E-5;
        cd = cd*t+8.68029542941784300606E-4;
        cd = cd*t+4.12142090722199792936E-2;
        cd = cd*t+1.00000000000000000118E0;
        *s = ae_sign(xxa, _state)*x*x2*sn/sd;
        *c = ae_sign(xxa, _state)*x*cn/cd;
        return;
    }
    if( ae_fp_greater(x,36974.0) )
    {
        *c = ae_sign(xxa, _state)*0.5;
        *s = ae_sign(xxa, _state)*0.5;
        return;
    }
    x2 = x*x;
    t = mpi*x2;
    u = 1/(t*t);
    t = 1/t;
    fn = 4.21543555043677546506E-1;
    fn = fn*u+1.43407919780758885261E-1;
    fn = fn*u+1.15220955073585758835E-2;
    fn = fn*u+3.45017939782574027900E-4;
    fn = fn*u+4.63613749287867322088E-6;
    fn = fn*u+3.05568983790257605827E-8;
    fn = fn*u+1.02304514164907233465E-10;
    fn = fn*u+1.72010743268161828879E-13;
    fn = fn*u+1.34283276233062758925E-16;
    fn = fn*u+3.76329711269987889006E-20;
    fd = 1.00000000000000000000E0;
    fd = fd*u+7.51586398353378947175E-1;
    fd = fd*u+1.16888925859191382142E-1;
    fd = fd*u+6.44051526508858611005E-3;
    fd = fd*u+1.55934409164153020873E-4;
    fd = fd*u+1.84627567348930545870E-6;
    fd = fd*u+1.12699224763999035261E-8;
    fd = fd*u+3.60140029589371370404E-11;
    fd = fd*u+5.88754533621578410010E-14;
    fd = fd*u+4.52001434074129701496E-17;
    fd = fd*u+1.25443237090011264384E-20;
    gn = 5.04442073643383265887E-1;
    gn = gn*u+1.97102833525523411709E-1;
    gn = gn*u+1.87648584092575249293E-2;
    gn = gn*u+6.84079380915393090172E-4;
    gn = gn*u+1.15138826111884280931E-5;
    gn = gn*u+9.82852443688422223854E-8;
    gn = gn*u+4.45344415861750144738E-10;
    gn = gn*u+1.08268041139020870318E-12;
    gn = gn*u+1.37555460633261799868E-15;
    gn = gn*u+8.36354435630677421531E-19;
    gn = gn*u+1.86958710162783235106E-22;
    gd = 1.00000000000000000000E0;
    gd = gd*u+1.47495759925128324529E0;
    gd = gd*u+3.37748989120019970451E-1;
    gd = gd*u+2.53603741420338795122E-2;
    gd = gd*u+8.14679107184306179049E-4;
    gd = gd*u+1.27545075667729118702E-5;
    gd = gd*u+1.04314589657571990585E-7;
    gd = gd*u+4.60680728146520428211E-10;
    gd = gd*u+1.10273215066240270757E-12;
    gd = gd*u+1.38796531259578871258E-15;
    gd = gd*u+8.39158816283118707363E-19;
    gd = gd*u+1.86958710162783236342E-22;
    f = 1-u*fn/fd;
    g = t*gn/gd;
    t = mpio2*x2;
    cc = ae_cos(t, _state);
    ss = ae_sin(t, _state);
    t = mpi*x;
    *c = 0.5+(f*ss-g*cc)/t;
    *s = 0.5-(f*cc+g*ss)/t;
    *c = *c*ae_sign(xxa, _state);
    *s = *s*ae_sign(xxa, _state);
}




/*************************************************************************
Calculation of the value of the Hermite polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial Hn at x
*************************************************************************/
double hermitecalculate(ae_int_t n, double x, ae_state *_state)
{
    ae_int_t i;
    double a;
    double b;
    double result;


    result = 0;
    
    /*
     * Prepare A and B
     */
    a = 1;
    b = 2*x;
    
    /*
     * Special cases: N=0 or N=1
     */
    if( n==0 )
    {
        result = a;
        return result;
    }
    if( n==1 )
    {
        result = b;
        return result;
    }
    
    /*
     * General case: N>=2
     */
    for(i=2; i<=n; i++)
    {
        result = 2*x*b-2*(i-1)*a;
        a = b;
        b = result;
    }
    return result;
}


/*************************************************************************
Summation of Hermite polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial at x
*************************************************************************/
double hermitesum(/* Real    */ ae_vector* c,
     ae_int_t n,
     double x,
     ae_state *_state)
{
    double b1;
    double b2;
    ae_int_t i;
    double result;


    b1 = 0;
    b2 = 0;
    result = 0;
    for(i=n; i>=0; i--)
    {
        result = 2*(x*b1-(i+1)*b2)+c->ptr.p_double[i];
        b2 = b1;
        b1 = result;
    }
    return result;
}


/*************************************************************************
Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void hermitecoefficients(ae_int_t n,
     /* Real    */ ae_vector* c,
     ae_state *_state)
{
    ae_int_t i;

    ae_vector_clear(c);

    ae_vector_set_length(c, n+1, _state);
    for(i=0; i<=n; i++)
    {
        c->ptr.p_double[i] = 0;
    }
    c->ptr.p_double[n] = ae_exp(n*ae_log(2, _state), _state);
    for(i=0; i<=n/2-1; i++)
    {
        c->ptr.p_double[n-2*(i+1)] = -c->ptr.p_double[n-2*i]*(n-2*i)*(n-2*i-1)/4/(i+1);
    }
}




/*************************************************************************
Jacobian Elliptic Functions

Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.

These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).

Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1.  In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.

ACCURACY:

Tested at random points with u between 0 and 10, m between
0 and 1.

           Absolute error (* = relative error):
arithmetic   function   # trials      peak         rms
   IEEE      phi         10000       9.2e-16*    1.4e-16*
   IEEE      sn          50000       4.1e-15     4.6e-16
   IEEE      cn          40000       3.6e-15     4.4e-16
   IEEE      dn          10000       1.3e-12     1.8e-14

 Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void jacobianellipticfunctions(double u,
     double m,
     double* sn,
     double* cn,
     double* dn,
     double* ph,
     ae_state *_state)
{
    ae_frame _frame_block;
    double ai;
    double b;
    double phi;
    double t;
    double twon;
    ae_vector a;
    ae_vector c;
    ae_int_t i;

    ae_frame_make(_state, &_frame_block);
    *sn = 0;
    *cn = 0;
    *dn = 0;
    *ph = 0;
    ae_vector_init(&a, 0, DT_REAL, _state, ae_true);
    ae_vector_init(&c, 0, DT_REAL, _state, ae_true);

    ae_assert(ae_fp_greater_eq(m,0)&&ae_fp_less_eq(m,1), "Domain error in JacobianEllipticFunctions: m<0 or m>1", _state);
    ae_vector_set_length(&a, 8+1, _state);
    ae_vector_set_length(&c, 8+1, _state);
    if( ae_fp_less(m,1.0e-9) )
    {
        t = ae_sin(u, _state);
        b = ae_cos(u, _state);
        ai = 0.25*m*(u-t*b);
        *sn = t-ai*b;
        *cn = b+ai*t;
        *ph = u-ai;
        *dn = 1.0-0.5*m*t*t;
        ae_frame_leave(_state);
        return;
    }
    if( ae_fp_greater_eq(m,0.9999999999) )
    {
        ai = 0.25*(1.0-m);
        b = ae_cosh(u, _state);
        t = ae_tanh(u, _state);
        phi = 1.0/b;
        twon = b*ae_sinh(u, _state);
        *sn = t+ai*(twon-u)/(b*b);
        *ph = 2.0*ae_atan(ae_exp(u, _state), _state)-1.57079632679489661923+ai*(twon-u)/b;
        ai = ai*t*phi;
        *cn = phi-ai*(twon-u);
        *dn = phi+ai*(twon+u);
        ae_frame_leave(_state);
        return;
    }
    a.ptr.p_double[0] = 1.0;
    b = ae_sqrt(1.0-m, _state);
    c.ptr.p_double[0] = ae_sqrt(m, _state);
    twon = 1.0;
    i = 0;
    while(ae_fp_greater(ae_fabs(c.ptr.p_double[i]/a.ptr.p_double[i], _state),ae_machineepsilon))
    {
        if( i>7 )
        {
            ae_assert(ae_false, "Overflow in JacobianEllipticFunctions", _state);
            break;
        }
        ai = a.ptr.p_double[i];
        i = i+1;
        c.ptr.p_double[i] = 0.5*(ai-b);
        t = ae_sqrt(ai*b, _state);
        a.ptr.p_double[i] = 0.5*(ai+b);
        b = t;
        twon = twon*2.0;
    }
    phi = twon*a.ptr.p_double[i]*u;
    do
    {
        t = c.ptr.p_double[i]*ae_sin(phi, _state)/a.ptr.p_double[i];
        b = phi;
        phi = (ae_asin(t, _state)+phi)/2.0;
        i = i-1;
    }
    while(i!=0);
    *sn = ae_sin(phi, _state);
    t = ae_cos(phi, _state);
    *cn = t;
    *dn = t/ae_cos(phi-b, _state);
    *ph = phi;
    ae_frame_leave(_state);
}




/*************************************************************************
Calculation of the value of the Laguerre polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial Ln at x
*************************************************************************/
double laguerrecalculate(ae_int_t n, double x, ae_state *_state)
{
    double a;
    double b;
    double i;
    double result;


    result = 1;
    a = 1;
    b = 1-x;
    if( n==1 )
    {
        result = b;
    }
    i = 2;
    while(ae_fp_less_eq(i,n))
    {
        result = ((2*i-1-x)*b-(i-1)*a)/i;
        a = b;
        b = result;
        i = i+1;
    }
    return result;
}


/*************************************************************************
Summation of Laguerre polynomials using Clenshaw’s recurrence formula.

This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial at x
*************************************************************************/
double laguerresum(/* Real    */ ae_vector* c,
     ae_int_t n,
     double x,
     ae_state *_state)
{
    double b1;
    double b2;
    ae_int_t i;
    double result;


    b1 = 0;
    b2 = 0;
    result = 0;
    for(i=n; i>=0; i--)
    {
        result = (2*i+1-x)*b1/(i+1)-(i+1)*b2/(i+2)+c->ptr.p_double[i];
        b2 = b1;
        b1 = result;
    }
    return result;
}


/*************************************************************************
Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void laguerrecoefficients(ae_int_t n,
     /* Real    */ ae_vector* c,
     ae_state *_state)
{
    ae_int_t i;

    ae_vector_clear(c);

    ae_vector_set_length(c, n+1, _state);
    c->ptr.p_double[0] = 1;
    for(i=0; i<=n-1; i++)
    {
        c->ptr.p_double[i+1] = -c->ptr.p_double[i]*(n-i)/(i+1)/(i+1);
    }
}




/*************************************************************************
Calculation of the value of the Legendre polynomial Pn.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial Pn at x
*************************************************************************/
double legendrecalculate(ae_int_t n, double x, ae_state *_state)
{
    double a;
    double b;
    ae_int_t i;
    double result;


    result = 1;
    a = 1;
    b = x;
    if( n==0 )
    {
        result = a;
        return result;
    }
    if( n==1 )
    {
        result = b;
        return result;
    }
    for(i=2; i<=n; i++)
    {
        result = ((2*i-1)*x*b-(i-1)*a)/i;
        a = b;
        b = result;
    }
    return result;
}


/*************************************************************************
Summation of Legendre polynomials using Clenshaw’s recurrence formula.

This routine calculates
    c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial at x
*************************************************************************/
double legendresum(/* Real    */ ae_vector* c,
     ae_int_t n,
     double x,
     ae_state *_state)
{
    double b1;
    double b2;
    ae_int_t i;
    double result;


    b1 = 0;
    b2 = 0;
    result = 0;
    for(i=n; i>=0; i--)
    {
        result = (2*i+1)*x*b1/(i+1)-(i+1)*b2/(i+2)+c->ptr.p_double[i];
        b2 = b1;
        b1 = result;
    }
    return result;
}


/*************************************************************************
Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void legendrecoefficients(ae_int_t n,
     /* Real    */ ae_vector* c,
     ae_state *_state)
{
    ae_int_t i;

    ae_vector_clear(c);

    ae_vector_set_length(c, n+1, _state);
    for(i=0; i<=n; i++)
    {
        c->ptr.p_double[i] = 0;
    }
    c->ptr.p_double[n] = 1;
    for(i=1; i<=n; i++)
    {
        c->ptr.p_double[n] = c->ptr.p_double[n]*(n+i)/2/i;
    }
    for(i=0; i<=n/2-1; i++)
    {
        c->ptr.p_double[n-2*(i+1)] = -c->ptr.p_double[n-2*i]*(n-2*i)*(n-2*i-1)/2/(i+1)/(2*(n-i)-1);
    }
}




/*************************************************************************
Poisson distribution

Returns the sum of the first k+1 terms of the Poisson
distribution:

  k         j
  --   -m  m
  >   e    --
  --       j!
 j=0

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation

y = pdtr( k, m ) = igamc( k+1, m ).

The arguments must both be positive.
ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissondistribution(ae_int_t k, double m, ae_state *_state)
{
    double result;


    ae_assert(k>=0&&ae_fp_greater(m,0), "Domain error in PoissonDistribution", _state);
    result = incompletegammac(k+1, m, _state);
    return result;
}


/*************************************************************************
Complemented Poisson distribution

Returns the sum of the terms k+1 to infinity of the Poisson
distribution:

 inf.       j
  --   -m  m
  >   e    --
  --       j!
 j=k+1

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula

y = pdtrc( k, m ) = igam( k+1, m ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissoncdistribution(ae_int_t k, double m, ae_state *_state)
{
    double result;


    ae_assert(k>=0&&ae_fp_greater(m,0), "Domain error in PoissonDistributionC", _state);
    result = incompletegamma(k+1, m, _state);
    return result;
}


/*************************************************************************
Inverse Poisson distribution

Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.

This is accomplished using the inverse gamma integral
function and the relation

   m = igami( k+1, y ).

ACCURACY:

See inverse incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invpoissondistribution(ae_int_t k, double y, ae_state *_state)
{
    double result;


    ae_assert((k>=0&&ae_fp_greater_eq(y,0))&&ae_fp_less(y,1), "Domain error in InvPoissonDistribution", _state);
    result = invincompletegammac(k+1, y, _state);
    return result;
}




/*************************************************************************
Psi (digamma) function

             d      -
  psi(x)  =  -- ln | (x)
             dx

is the logarithmic derivative of the gamma function.
For integer x,
                  n-1
                   -
psi(n) = -EUL  +   >  1/k.
                   -
                  k=1

This formula is used for 0 < n <= 10.  If x is negative, it
is transformed to a positive argument by the reflection
formula  psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence  psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:

                          inf.   B
                           -      2k
psi(x) = log(x) - 1/2x -   >   -------
                           -        2k
                          k=1   2k x

where the B2k are Bernoulli numbers.

ACCURACY:
   Relative error (except absolute when |psi| < 1):
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       1.3e-15     1.4e-16
   IEEE      -30,0       40000       1.5e-15     2.2e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double psi(double x, ae_state *_state)
{
    double p;
    double q;
    double nz;
    double s;
    double w;
    double y;
    double z;
    double polv;
    ae_int_t i;
    ae_int_t n;
    ae_int_t negative;
    double result;


    negative = 0;
    nz = 0.0;
    if( ae_fp_less_eq(x,0) )
    {
        negative = 1;
        q = x;
        p = ae_ifloor(q, _state);
        if( ae_fp_eq(p,q) )
        {
            ae_assert(ae_false, "Singularity in Psi(x)", _state);
            result = ae_maxrealnumber;
            return result;
        }
        nz = q-p;
        if( ae_fp_neq(nz,0.5) )
        {
            if( ae_fp_greater(nz,0.5) )
            {
                p = p+1.0;
                nz = q-p;
            }
            nz = ae_pi/ae_tan(ae_pi*nz, _state);
        }
        else
        {
            nz = 0.0;
        }
        x = 1.0-x;
    }
    if( ae_fp_less_eq(x,10.0)&&ae_fp_eq(x,ae_ifloor(x, _state)) )
    {
        y = 0.0;
        n = ae_ifloor(x, _state);
        for(i=1; i<=n-1; i++)
        {
            w = i;
            y = y+1.0/w;
        }
        y = y-0.57721566490153286061;
    }
    else
    {
        s = x;
        w = 0.0;
        while(ae_fp_less(s,10.0))
        {
            w = w+1.0/s;
            s = s+1.0;
        }
        if( ae_fp_less(s,1.0E17) )
        {
            z = 1.0/(s*s);
            polv = 8.33333333333333333333E-2;
            polv = polv*z-2.10927960927960927961E-2;
            polv = polv*z+7.57575757575757575758E-3;
            polv = polv*z-4.16666666666666666667E-3;
            polv = polv*z+3.96825396825396825397E-3;
            polv = polv*z-8.33333333333333333333E-3;
            polv = polv*z+8.33333333333333333333E-2;
            y = z*polv;
        }
        else
        {
            y = 0.0;
        }
        y = ae_log(s, _state)-0.5/s-y-w;
    }
    if( negative!=0 )
    {
        y = y-nz;
    }
    result = y;
    return result;
}




/*************************************************************************
Student's t distribution

Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:

                                     t
                                     -
                                    | |
             -                      |         2   -(k+1)/2
            | ( (k+1)/2 )           |  (     x   )
      ----------------------        |  ( 1 + --- )        dx
                    -               |  (      k  )
      sqrt( k pi ) | ( k/2 )        |
                                  | |
                                   -
                                  -inf.

Relation to incomplete beta integral:

       1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
       z = k/(k + t**2).

For t < -2, this is the method of computation.  For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.

ACCURACY:

Tested at random 1 <= k <= 25.  The "domain" refers to t.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -100,-2      50000       5.9e-15     1.4e-15
   IEEE     -2,100      500000       2.7e-15     4.9e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double studenttdistribution(ae_int_t k, double t, ae_state *_state)
{
    double x;
    double rk;
    double z;
    double f;
    double tz;
    double p;
    double xsqk;
    ae_int_t j;
    double result;


    ae_assert(k>0, "Domain error in StudentTDistribution", _state);
    if( ae_fp_eq(t,0) )
    {
        result = 0.5;
        return result;
    }
    if( ae_fp_less(t,-2.0) )
    {
        rk = k;
        z = rk/(rk+t*t);
        result = 0.5*incompletebeta(0.5*rk, 0.5, z, _state);
        return result;
    }
    if( ae_fp_less(t,0) )
    {
        x = -t;
    }
    else
    {
        x = t;
    }
    rk = k;
    z = 1.0+x*x/rk;
    if( k%2!=0 )
    {
        xsqk = x/ae_sqrt(rk, _state);
        p = ae_atan(xsqk, _state);
        if( k>1 )
        {
            f = 1.0;
            tz = 1.0;
            j = 3;
            while(j<=k-2&&ae_fp_greater(tz/f,ae_machineepsilon))
            {
                tz = tz*((j-1)/(z*j));
                f = f+tz;
                j = j+2;
            }
            p = p+f*xsqk/z;
        }
        p = p*2.0/ae_pi;
    }
    else
    {
        f = 1.0;
        tz = 1.0;
        j = 2;
        while(j<=k-2&&ae_fp_greater(tz/f,ae_machineepsilon))
        {
            tz = tz*((j-1)/(z*j));
            f = f+tz;
            j = j+2;
        }
        p = f*x/ae_sqrt(z*rk, _state);
    }
    if( ae_fp_less(t,0) )
    {
        p = -p;
    }
    result = 0.5+0.5*p;
    return result;
}


/*************************************************************************
Functional inverse of Student's t distribution

Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.

ACCURACY:

Tested at random 1 <= k <= 100.  The "domain" refers to p:
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE    .001,.999     25000       5.7e-15     8.0e-16
   IEEE    10^-6,.001    25000       2.0e-12     2.9e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invstudenttdistribution(ae_int_t k, double p, ae_state *_state)
{
    double t;
    double rk;
    double z;
    ae_int_t rflg;
    double result;


    ae_assert((k>0&&ae_fp_greater(p,0))&&ae_fp_less(p,1), "Domain error in InvStudentTDistribution", _state);
    rk = k;
    if( ae_fp_greater(p,0.25)&&ae_fp_less(p,0.75) )
    {
        if( ae_fp_eq(p,0.5) )
        {
            result = 0;
            return result;
        }
        z = 1.0-2.0*p;
        z = invincompletebeta(0.5, 0.5*rk, ae_fabs(z, _state), _state);
        t = ae_sqrt(rk*z/(1.0-z), _state);
        if( ae_fp_less(p,0.5) )
        {
            t = -t;
        }
        result = t;
        return result;
    }
    rflg = -1;
    if( ae_fp_greater_eq(p,0.5) )
    {
        p = 1.0-p;
        rflg = 1;
    }
    z = invincompletebeta(0.5*rk, 0.5, 2.0*p, _state);
    if( ae_fp_less(ae_maxrealnumber*z,rk) )
    {
        result = rflg*ae_maxrealnumber;
        return result;
    }
    t = ae_sqrt(rk/z-rk, _state);
    result = rflg*t;
    return result;
}




/*************************************************************************
Sine and cosine integrals

Evaluates the integrals

                         x
                         -
                        |  cos t - 1
  Ci(x) = eul + ln x +  |  --------- dt,
                        |      t
                       -
                        0
            x
            -
           |  sin t
  Si(x) =  |  ----- dt
           |    t
          -
           0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that

Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)


ACCURACY:
   Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic   function   # trials      peak         rms
   IEEE        Si        30000       4.4e-16     7.3e-17
   IEEE        Ci        30000       6.9e-16     5.1e-17

Cephes Math Library Release 2.1:  January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
*************************************************************************/
void sinecosineintegrals(double x,
     double* si,
     double* ci,
     ae_state *_state)
{
    double z;
    double c;
    double s;
    double f;
    double g;
    ae_int_t sg;
    double sn;
    double sd;
    double cn;
    double cd;
    double fn;
    double fd;
    double gn;
    double gd;

    *si = 0;
    *ci = 0;

    if( ae_fp_less(x,0) )
    {
        sg = -1;
        x = -x;
    }
    else
    {
        sg = 0;
    }
    if( ae_fp_eq(x,0) )
    {
        *si = 0;
        *ci = -ae_maxrealnumber;
        return;
    }
    if( ae_fp_greater(x,1.0E9) )
    {
        *si = 1.570796326794896619-ae_cos(x, _state)/x;
        *ci = ae_sin(x, _state)/x;
        return;
    }
    if( ae_fp_less_eq(x,4) )
    {
        z = x*x;
        sn = -8.39167827910303881427E-11;
        sn = sn*z+4.62591714427012837309E-8;
        sn = sn*z-9.75759303843632795789E-6;
        sn = sn*z+9.76945438170435310816E-4;
        sn = sn*z-4.13470316229406538752E-2;
        sn = sn*z+1.00000000000000000302E0;
        sd = 2.03269266195951942049E-12;
        sd = sd*z+1.27997891179943299903E-9;
        sd = sd*z+4.41827842801218905784E-7;
        sd = sd*z+9.96412122043875552487E-5;
        sd = sd*z+1.42085239326149893930E-2;
        sd = sd*z+9.99999999999999996984E-1;
        s = x*sn/sd;
        cn = 2.02524002389102268789E-11;
        cn = cn*z-1.35249504915790756375E-8;
        cn = cn*z+3.59325051419993077021E-6;
        cn = cn*z-4.74007206873407909465E-4;
        cn = cn*z+2.89159652607555242092E-2;
        cn = cn*z-1.00000000000000000080E0;
        cd = 4.07746040061880559506E-12;
        cd = cd*z+3.06780997581887812692E-9;
        cd = cd*z+1.23210355685883423679E-6;
        cd = cd*z+3.17442024775032769882E-4;
        cd = cd*z+5.10028056236446052392E-2;
        cd = cd*z+4.00000000000000000080E0;
        c = z*cn/cd;
        if( sg!=0 )
        {
            s = -s;
        }
        *si = s;
        *ci = 0.57721566490153286061+ae_log(x, _state)+c;
        return;
    }
    s = ae_sin(x, _state);
    c = ae_cos(x, _state);
    z = 1.0/(x*x);
    if( ae_fp_less(x,8) )
    {
        fn = 4.23612862892216586994E0;
        fn = fn*z+5.45937717161812843388E0;
        fn = fn*z+1.62083287701538329132E0;
        fn = fn*z+1.67006611831323023771E-1;
        fn = fn*z+6.81020132472518137426E-3;
        fn = fn*z+1.08936580650328664411E-4;
        fn = fn*z+5.48900223421373614008E-7;
        fd = 1.00000000000000000000E0;
        fd = fd*z+8.16496634205391016773E0;
        fd = fd*z+7.30828822505564552187E0;
        fd = fd*z+1.86792257950184183883E0;
        fd = fd*z+1.78792052963149907262E-1;
        fd = fd*z+7.01710668322789753610E-3;
        fd = fd*z+1.10034357153915731354E-4;
        fd = fd*z+5.48900252756255700982E-7;
        f = fn/(x*fd);
        gn = 8.71001698973114191777E-2;
        gn = gn*z+6.11379109952219284151E-1;
        gn = gn*z+3.97180296392337498885E-1;
        gn = gn*z+7.48527737628469092119E-2;
        gn = gn*z+5.38868681462177273157E-3;
        gn = gn*z+1.61999794598934024525E-4;
        gn = gn*z+1.97963874140963632189E-6;
        gn = gn*z+7.82579040744090311069E-9;
        gd = 1.00000000000000000000E0;
        gd = gd*z+1.64402202413355338886E0;
        gd = gd*z+6.66296701268987968381E-1;
        gd = gd*z+9.88771761277688796203E-2;
        gd = gd*z+6.22396345441768420760E-3;
        gd = gd*z+1.73221081474177119497E-4;
        gd = gd*z+2.02659182086343991969E-6;
        gd = gd*z+7.82579218933534490868E-9;
        g = z*gn/gd;
    }
    else
    {
        fn = 4.55880873470465315206E-1;
        fn = fn*z+7.13715274100146711374E-1;
        fn = fn*z+1.60300158222319456320E-1;
        fn = fn*z+1.16064229408124407915E-2;
        fn = fn*z+3.49556442447859055605E-4;
        fn = fn*z+4.86215430826454749482E-6;
        fn = fn*z+3.20092790091004902806E-8;
        fn = fn*z+9.41779576128512936592E-11;
        fn = fn*z+9.70507110881952024631E-14;
        fd = 1.00000000000000000000E0;
        fd = fd*z+9.17463611873684053703E-1;
        fd = fd*z+1.78685545332074536321E-1;
        fd = fd*z+1.22253594771971293032E-2;
        fd = fd*z+3.58696481881851580297E-4;
        fd = fd*z+4.92435064317881464393E-6;
        fd = fd*z+3.21956939101046018377E-8;
        fd = fd*z+9.43720590350276732376E-11;
        fd = fd*z+9.70507110881952025725E-14;
        f = fn/(x*fd);
        gn = 6.97359953443276214934E-1;
        gn = gn*z+3.30410979305632063225E-1;
        gn = gn*z+3.84878767649974295920E-2;
        gn = gn*z+1.71718239052347903558E-3;
        gn = gn*z+3.48941165502279436777E-5;
        gn = gn*z+3.47131167084116673800E-7;
        gn = gn*z+1.70404452782044526189E-9;
        gn = gn*z+3.85945925430276600453E-12;
        gn = gn*z+3.14040098946363334640E-15;
        gd = 1.00000000000000000000E0;
        gd = gd*z+1.68548898811011640017E0;
        gd = gd*z+4.87852258695304967486E-1;
        gd = gd*z+4.67913194259625806320E-2;
        gd = gd*z+1.90284426674399523638E-3;
        gd = gd*z+3.68475504442561108162E-5;
        gd = gd*z+3.57043223443740838771E-7;
        gd = gd*z+1.72693748966316146736E-9;
        gd = gd*z+3.87830166023954706752E-12;
        gd = gd*z+3.14040098946363335242E-15;
        g = z*gn/gd;
    }
    *si = 1.570796326794896619-f*c-g*s;
    if( sg!=0 )
    {
        *si = -*si;
    }
    *ci = f*s-g*c;
}


/*************************************************************************
Hyperbolic sine and cosine integrals

Approximates the integrals

                           x
                           -
                          | |   cosh t - 1
  Chi(x) = eul + ln x +   |    -----------  dt,
                        | |          t
                         -
                         0

              x
              -
             | |  sinh t
  Shi(x) =   |    ------  dt
           | |       t
            -
            0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.


ACCURACY:

Test interval 0 to 88.
                     Relative error:
arithmetic   function  # trials      peak         rms
   IEEE         Shi      30000       6.9e-16     1.6e-16
       Absolute error, except relative when |Chi| > 1:
   IEEE         Chi      30000       8.4e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void hyperbolicsinecosineintegrals(double x,
     double* shi,
     double* chi,
     ae_state *_state)
{
    double k;
    double z;
    double c;
    double s;
    double a;
    ae_int_t sg;
    double b0;
    double b1;
    double b2;

    *shi = 0;
    *chi = 0;

    if( ae_fp_less(x,0) )
    {
        sg = -1;
        x = -x;
    }
    else
    {
        sg = 0;
    }
    if( ae_fp_eq(x,0) )
    {
        *shi = 0;
        *chi = -ae_maxrealnumber;
        return;
    }
    if( ae_fp_less(x,8.0) )
    {
        z = x*x;
        a = 1.0;
        s = 1.0;
        c = 0.0;
        k = 2.0;
        do
        {
            a = a*z/k;
            c = c+a/k;
            k = k+1.0;
            a = a/k;
            s = s+a/k;
            k = k+1.0;
        }
        while(ae_fp_greater_eq(ae_fabs(a/s, _state),ae_machineepsilon));
        s = s*x;
    }
    else
    {
        if( ae_fp_less(x,18.0) )
        {
            a = (576.0/x-52.0)/10.0;
            k = ae_exp(x, _state)/x;
            b0 = 1.83889230173399459482E-17;
            b1 = 0.0;
            trigintegrals_chebiterationshichi(a, -9.55485532279655569575E-17, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 2.04326105980879882648E-16, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.09896949074905343022E-15, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.31313534344092599234E-14, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 5.93976226264314278932E-14, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -3.47197010497749154755E-14, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.40059764613117131000E-12, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 9.49044626224223543299E-12, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.61596181145435454033E-11, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.77899784436430310321E-10, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.35455469767246947469E-9, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.03257121792819495123E-9, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -3.56699611114982536845E-8, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.44818877384267342057E-7, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 7.82018215184051295296E-7, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -5.39919118403805073710E-6, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -3.12458202168959833422E-5, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 8.90136741950727517826E-5, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 2.02558474743846862168E-3, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 2.96064440855633256972E-2, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.11847751047257036625E0, &b0, &b1, &b2, _state);
            s = k*0.5*(b0-b2);
            b0 = -8.12435385225864036372E-18;
            b1 = 0.0;
            trigintegrals_chebiterationshichi(a, 2.17586413290339214377E-17, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 5.22624394924072204667E-17, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -9.48812110591690559363E-16, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 5.35546311647465209166E-15, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.21009970113732918701E-14, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -6.00865178553447437951E-14, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 7.16339649156028587775E-13, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -2.93496072607599856104E-12, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.40359438136491256904E-12, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 8.76302288609054966081E-11, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -4.40092476213282340617E-10, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -1.87992075640569295479E-10, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.31458150989474594064E-8, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -4.75513930924765465590E-8, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -2.21775018801848880741E-7, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.94635531373272490962E-6, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 4.33505889257316408893E-6, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -6.13387001076494349496E-5, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, -3.13085477492997465138E-4, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 4.97164789823116062801E-4, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 2.64347496031374526641E-2, &b0, &b1, &b2, _state);
            trigintegrals_chebiterationshichi(a, 1.11446150876699213025E0, &b0, &b1, &b2, _state);
            c = k*0.5*(b0-b2);
        }
        else
        {
            if( ae_fp_less_eq(x,88.0) )
            {
                a = (6336.0/x-212.0)/70.0;
                k = ae_exp(x, _state)/x;
                b0 = -1.05311574154850938805E-17;
                b1 = 0.0;
                trigintegrals_chebiterationshichi(a, 2.62446095596355225821E-17, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 8.82090135625368160657E-17, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -3.38459811878103047136E-16, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -8.30608026366935789136E-16, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 3.93397875437050071776E-15, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.01765565969729044505E-14, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -4.21128170307640802703E-14, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -1.60818204519802480035E-13, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 3.34714954175994481761E-13, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 2.72600352129153073807E-12, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.66894954752839083608E-12, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -3.49278141024730899554E-11, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -1.58580661666482709598E-10, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -1.79289437183355633342E-10, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.76281629144264523277E-9, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.69050228879421288846E-8, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.25391771228487041649E-7, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.16229947068677338732E-6, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.61038260117376323993E-5, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 3.49810375601053973070E-4, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.28478065259647610779E-2, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.03665722588798326712E0, &b0, &b1, &b2, _state);
                s = k*0.5*(b0-b2);
                b0 = 8.06913408255155572081E-18;
                b1 = 0.0;
                trigintegrals_chebiterationshichi(a, -2.08074168180148170312E-17, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -5.98111329658272336816E-17, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 2.68533951085945765591E-16, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 4.52313941698904694774E-16, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -3.10734917335299464535E-15, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -4.42823207332531972288E-15, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 3.49639695410806959872E-14, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 6.63406731718911586609E-14, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -3.71902448093119218395E-13, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -1.27135418132338309016E-12, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 2.74851141935315395333E-12, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 2.33781843985453438400E-11, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 2.71436006377612442764E-11, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -2.56600180000355990529E-10, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -1.61021375163803438552E-9, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -4.72543064876271773512E-9, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, -3.00095178028681682282E-9, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 7.79387474390914922337E-8, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.06942765566401507066E-6, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.59503164802313196374E-5, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 3.49592575153777996871E-4, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.28475387530065247392E-2, &b0, &b1, &b2, _state);
                trigintegrals_chebiterationshichi(a, 1.03665693917934275131E0, &b0, &b1, &b2, _state);
                c = k*0.5*(b0-b2);
            }
            else
            {
                if( sg!=0 )
                {
                    *shi = -ae_maxrealnumber;
                }
                else
                {
                    *shi = ae_maxrealnumber;
                }
                *chi = ae_maxrealnumber;
                return;
            }
        }
    }
    if( sg!=0 )
    {
        s = -s;
    }
    *shi = s;
    *chi = 0.57721566490153286061+ae_log(x, _state)+c;
}


static void trigintegrals_chebiterationshichi(double x,
     double c,
     double* b0,
     double* b1,
     double* b2,
     ae_state *_state)
{


    *b2 = *b1;
    *b1 = *b0;
    *b0 = x*(*b1)-(*b2)+c;
}



}