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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Eugene Brevdo <ebrevdo@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_SPECIAL_FUNCTIONS_H
#define EIGEN_SPECIAL_FUNCTIONS_H
namespace Eigen {
namespace internal {
// Parts of this code are based on the Cephes Math Library.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
//
// Permission has been kindly provided by the original author
// to incorporate the Cephes software into the Eigen codebase:
//
// From: Stephen Moshier
// To: Eugene Brevdo
// Subject: Re: Permission to wrap several cephes functions in Eigen
//
// Hello Eugene,
//
// Thank you for writing.
//
// If your licensing is similar to BSD, the formal way that has been
// handled is simply to add a statement to the effect that you are incorporating
// the Cephes software by permission of the author.
//
// Good luck with your project,
// Steve
/****************************************************************************
* Implementation of lgamma, requires C++11/C99 *
****************************************************************************/
template <typename Scalar>
struct lgamma_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template <typename Scalar>
struct lgamma_retval {
typedef Scalar type;
};
#if EIGEN_HAS_C99_MATH
// Since glibc 2.19
#if defined(__GLIBC__) && ((__GLIBC__>=2 && __GLIBC_MINOR__ >= 19) || __GLIBC__>2) \
&& (defined(_DEFAULT_SOURCE) || defined(_BSD_SOURCE) || defined(_SVID_SOURCE))
#define EIGEN_HAS_LGAMMA_R
#endif
// Glibc versions before 2.19
#if defined(__GLIBC__) && ((__GLIBC__==2 && __GLIBC_MINOR__ < 19) || __GLIBC__<2) \
&& (defined(_BSD_SOURCE) || defined(_SVID_SOURCE))
#define EIGEN_HAS_LGAMMA_R
#endif
template <>
struct lgamma_impl<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(float x) {
#if !defined(EIGEN_GPU_COMPILE_PHASE) && defined (EIGEN_HAS_LGAMMA_R) && !defined(__APPLE__)
int dummy;
return ::lgammaf_r(x, &dummy);
#elif defined(SYCL_DEVICE_ONLY)
return cl::sycl::lgamma(x);
#else
return ::lgammaf(x);
#endif
}
};
template <>
struct lgamma_impl<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(double x) {
#if !defined(EIGEN_GPU_COMPILE_PHASE) && defined(EIGEN_HAS_LGAMMA_R) && !defined(__APPLE__)
int dummy;
return ::lgamma_r(x, &dummy);
#elif defined(SYCL_DEVICE_ONLY)
return cl::sycl::lgamma(x);
#else
return ::lgamma(x);
#endif
}
};
#undef EIGEN_HAS_LGAMMA_R
#endif
/****************************************************************************
* Implementation of digamma (psi), based on Cephes *
****************************************************************************/
template <typename Scalar>
struct digamma_retval {
typedef Scalar type;
};
/*
*
* Polynomial evaluation helper for the Psi (digamma) function.
*
* digamma_impl_maybe_poly::run(s) evaluates the asymptotic Psi expansion for
* input Scalar s, assuming s is above 10.0.
*
* If s is above a certain threshold for the given Scalar type, zero
* is returned. Otherwise the polynomial is evaluated with enough
* coefficients for results matching Scalar machine precision.
*
*
*/
template <typename Scalar>
struct digamma_impl_maybe_poly {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template <>
struct digamma_impl_maybe_poly<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(const float s) {
const float A[] = {
-4.16666666666666666667E-3f,
3.96825396825396825397E-3f,
-8.33333333333333333333E-3f,
8.33333333333333333333E-2f
};
float z;
if (s < 1.0e8f) {
z = 1.0f / (s * s);
return z * internal::ppolevl<float, 3>::run(z, A);
} else return 0.0f;
}
};
template <>
struct digamma_impl_maybe_poly<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(const double s) {
const double A[] = {
8.33333333333333333333E-2,
-2.10927960927960927961E-2,
7.57575757575757575758E-3,
-4.16666666666666666667E-3,
3.96825396825396825397E-3,
-8.33333333333333333333E-3,
8.33333333333333333333E-2
};
double z;
if (s < 1.0e17) {
z = 1.0 / (s * s);
return z * internal::ppolevl<double, 6>::run(z, A);
}
else return 0.0;
}
};
template <typename Scalar>
struct digamma_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar x) {
/*
*
* Psi (digamma) function (modified for Eigen)
*
*
* SYNOPSIS:
*
* double x, y, psi();
*
* y = psi( x );
*
*
* DESCRIPTION:
*
* 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
*
* 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 (float):
* 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
*
* ACCURACY (double):
* Absolute error, relative when |psi| > 1 :
* arithmetic domain # trials peak rms
* IEEE -33,0 30000 8.2e-7 1.2e-7
* IEEE 0,33 100000 7.3e-7 7.7e-8
*
* ERROR MESSAGES:
* message condition value returned
* psi singularity x integer <=0 INFINITY
*/
Scalar p, q, nz, s, w, y;
bool negative = false;
const Scalar maxnum = NumTraits<Scalar>::infinity();
const Scalar m_pi = Scalar(EIGEN_PI);
const Scalar zero = Scalar(0);
const Scalar one = Scalar(1);
const Scalar half = Scalar(0.5);
nz = zero;
if (x <= zero) {
negative = true;
q = x;
p = numext::floor(q);
if (p == q) {
return maxnum;
}
/* Remove the zeros of tan(m_pi x)
* by subtracting the nearest integer from x
*/
nz = q - p;
if (nz != half) {
if (nz > half) {
p += one;
nz = q - p;
}
nz = m_pi / numext::tan(m_pi * nz);
}
else {
nz = zero;
}
x = one - x;
}
/* use the recurrence psi(x+1) = psi(x) + 1/x. */
s = x;
w = zero;
while (s < Scalar(10)) {
w += one / s;
s += one;
}
y = digamma_impl_maybe_poly<Scalar>::run(s);
y = numext::log(s) - (half / s) - y - w;
return (negative) ? y - nz : y;
}
};
/****************************************************************************
* Implementation of erf, requires C++11/C99 *
****************************************************************************/
/** \internal \returns the error function of \a a (coeff-wise)
Doesn't do anything fancy, just a 13/8-degree rational interpolant which
is accurate up to a couple of ulp in the range [-4, 4], outside of which
fl(erf(x)) = +/-1.
This implementation works on both scalars and Ts.
*/
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_fast_erf_float(const T& a_x) {
// Clamp the inputs to the range [-4, 4] since anything outside
// this range is +/-1.0f in single-precision.
const T plus_4 = pset1<T>(4.f);
const T minus_4 = pset1<T>(-4.f);
const T x = pmax(pmin(a_x, plus_4), minus_4);
// The monomial coefficients of the numerator polynomial (odd).
const T alpha_1 = pset1<T>(-1.60960333262415e-02f);
const T alpha_3 = pset1<T>(-2.95459980854025e-03f);
const T alpha_5 = pset1<T>(-7.34990630326855e-04f);
const T alpha_7 = pset1<T>(-5.69250639462346e-05f);
const T alpha_9 = pset1<T>(-2.10102402082508e-06f);
const T alpha_11 = pset1<T>(2.77068142495902e-08f);
const T alpha_13 = pset1<T>(-2.72614225801306e-10f);
// The monomial coefficients of the denominator polynomial (even).
const T beta_0 = pset1<T>(-1.42647390514189e-02f);
const T beta_2 = pset1<T>(-7.37332916720468e-03f);
const T beta_4 = pset1<T>(-1.68282697438203e-03f);
const T beta_6 = pset1<T>(-2.13374055278905e-04f);
const T beta_8 = pset1<T>(-1.45660718464996e-05f);
// Since the polynomials are odd/even, we need x^2.
const T x2 = pmul(x, x);
// Evaluate the numerator polynomial p.
T p = pmadd(x2, alpha_13, alpha_11);
p = pmadd(x2, p, alpha_9);
p = pmadd(x2, p, alpha_7);
p = pmadd(x2, p, alpha_5);
p = pmadd(x2, p, alpha_3);
p = pmadd(x2, p, alpha_1);
p = pmul(x, p);
// Evaluate the denominator polynomial p.
T q = pmadd(x2, beta_8, beta_6);
q = pmadd(x2, q, beta_4);
q = pmadd(x2, q, beta_2);
q = pmadd(x2, q, beta_0);
// Divide the numerator by the denominator.
return pdiv(p, q);
}
template <typename T>
struct erf_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE T run(const T x) {
return generic_fast_erf_float(x);
}
};
template <typename Scalar>
struct erf_retval {
typedef Scalar type;
};
#if EIGEN_HAS_C99_MATH
template <>
struct erf_impl<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(float x) {
#if defined(SYCL_DEVICE_ONLY)
return cl::sycl::erf(x);
#else
return generic_fast_erf_float(x);
#endif
}
};
template <>
struct erf_impl<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(double x) {
#if defined(SYCL_DEVICE_ONLY)
return cl::sycl::erf(x);
#else
return ::erf(x);
#endif
}
};
#endif // EIGEN_HAS_C99_MATH
/***************************************************************************
* Implementation of erfc, requires C++11/C99 *
****************************************************************************/
template <typename Scalar>
struct erfc_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template <typename Scalar>
struct erfc_retval {
typedef Scalar type;
};
#if EIGEN_HAS_C99_MATH
template <>
struct erfc_impl<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float run(const float x) {
#if defined(SYCL_DEVICE_ONLY)
return cl::sycl::erfc(x);
#else
return ::erfcf(x);
#endif
}
};
template <>
struct erfc_impl<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double run(const double x) {
#if defined(SYCL_DEVICE_ONLY)
return cl::sycl::erfc(x);
#else
return ::erfc(x);
#endif
}
};
#endif // EIGEN_HAS_C99_MATH
/***************************************************************************
* Implementation of ndtri. *
****************************************************************************/
/* Inverse of Normal distribution function (modified for Eigen).
*
*
* SYNOPSIS:
*
* double x, y, ndtri();
*
* x = ndtri( y );
*
*
*
* DESCRIPTION:
*
* 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
* DEC 0.125, 1 5500 9.5e-17 2.1e-17
* DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17
* IEEE 0.125, 1 20000 7.2e-16 1.3e-16
* IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtri domain x <= 0 -MAXNUM
* ndtri domain x >= 1 MAXNUM
*
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1985, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
// TODO: Add a cheaper approximation for float.
template<typename T>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T flipsign(
const T& should_flipsign, const T& x) {
const T sign_mask = pset1<T>(-0.0);
T sign_bit = pand<T>(should_flipsign, sign_mask);
return pxor<T>(sign_bit, x);
}
template<>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double flipsign<double>(
const double& should_flipsign, const double& x) {
return should_flipsign == 0 ? x : -x;
}
template<>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float flipsign<float>(
const float& should_flipsign, const float& x) {
return should_flipsign == 0 ? x : -x;
}
// We split this computation in to two so that in the scalar path
// only one branch is evaluated (due to our template specialization of pselect
// being an if statement.)
template <typename T, typename ScalarType>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_ndtri_gt_exp_neg_two(const T& b) {
const ScalarType p0[] = {
ScalarType(-5.99633501014107895267e1),
ScalarType(9.80010754185999661536e1),
ScalarType(-5.66762857469070293439e1),
ScalarType(1.39312609387279679503e1),
ScalarType(-1.23916583867381258016e0)
};
const ScalarType q0[] = {
ScalarType(1.0),
ScalarType(1.95448858338141759834e0),
ScalarType(4.67627912898881538453e0),
ScalarType(8.63602421390890590575e1),
ScalarType(-2.25462687854119370527e2),
ScalarType(2.00260212380060660359e2),
ScalarType(-8.20372256168333339912e1),
ScalarType(1.59056225126211695515e1),
ScalarType(-1.18331621121330003142e0)
};
const T sqrt2pi = pset1<T>(ScalarType(2.50662827463100050242e0));
const T half = pset1<T>(ScalarType(0.5));
T c, c2, ndtri_gt_exp_neg_two;
c = psub(b, half);
c2 = pmul(c, c);
ndtri_gt_exp_neg_two = pmadd(c, pmul(
c2, pdiv(
internal::ppolevl<T, 4>::run(c2, p0),
internal::ppolevl<T, 8>::run(c2, q0))), c);
return pmul(ndtri_gt_exp_neg_two, sqrt2pi);
}
template <typename T, typename ScalarType>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_ndtri_lt_exp_neg_two(
const T& b, const T& should_flipsign) {
/* Approximation for interval z = sqrt(-2 log a ) between 2 and 8
* i.e., a between exp(-2) = .135 and exp(-32) = 1.27e-14.
*/
const ScalarType p1[] = {
ScalarType(4.05544892305962419923e0),
ScalarType(3.15251094599893866154e1),
ScalarType(5.71628192246421288162e1),
ScalarType(4.40805073893200834700e1),
ScalarType(1.46849561928858024014e1),
ScalarType(2.18663306850790267539e0),
ScalarType(-1.40256079171354495875e-1),
ScalarType(-3.50424626827848203418e-2),
ScalarType(-8.57456785154685413611e-4)
};
const ScalarType q1[] = {
ScalarType(1.0),
ScalarType(1.57799883256466749731e1),
ScalarType(4.53907635128879210584e1),
ScalarType(4.13172038254672030440e1),
ScalarType(1.50425385692907503408e1),
ScalarType(2.50464946208309415979e0),
ScalarType(-1.42182922854787788574e-1),
ScalarType(-3.80806407691578277194e-2),
ScalarType(-9.33259480895457427372e-4)
};
/* Approximation for interval z = sqrt(-2 log a ) between 8 and 64
* i.e., a between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
*/
const ScalarType p2[] = {
ScalarType(3.23774891776946035970e0),
ScalarType(6.91522889068984211695e0),
ScalarType(3.93881025292474443415e0),
ScalarType(1.33303460815807542389e0),
ScalarType(2.01485389549179081538e-1),
ScalarType(1.23716634817820021358e-2),
ScalarType(3.01581553508235416007e-4),
ScalarType(2.65806974686737550832e-6),
ScalarType(6.23974539184983293730e-9)
};
const ScalarType q2[] = {
ScalarType(1.0),
ScalarType(6.02427039364742014255e0),
ScalarType(3.67983563856160859403e0),
ScalarType(1.37702099489081330271e0),
ScalarType(2.16236993594496635890e-1),
ScalarType(1.34204006088543189037e-2),
ScalarType(3.28014464682127739104e-4),
ScalarType(2.89247864745380683936e-6),
ScalarType(6.79019408009981274425e-9)
};
const T eight = pset1<T>(ScalarType(8.0));
const T one = pset1<T>(ScalarType(1));
const T neg_two = pset1<T>(ScalarType(-2));
T x, x0, x1, z;
x = psqrt(pmul(neg_two, plog(b)));
x0 = psub(x, pdiv(plog(x), x));
z = pdiv(one, x);
x1 = pmul(
z, pselect(
pcmp_lt(x, eight),
pdiv(internal::ppolevl<T, 8>::run(z, p1),
internal::ppolevl<T, 8>::run(z, q1)),
pdiv(internal::ppolevl<T, 8>::run(z, p2),
internal::ppolevl<T, 8>::run(z, q2))));
return flipsign(should_flipsign, psub(x0, x1));
}
template <typename T, typename ScalarType>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
T generic_ndtri(const T& a) {
const T maxnum = pset1<T>(NumTraits<ScalarType>::infinity());
const T neg_maxnum = pset1<T>(-NumTraits<ScalarType>::infinity());
const T zero = pset1<T>(ScalarType(0));
const T one = pset1<T>(ScalarType(1));
// exp(-2)
const T exp_neg_two = pset1<T>(ScalarType(0.13533528323661269189));
T b, ndtri, should_flipsign;
should_flipsign = pcmp_le(a, psub(one, exp_neg_two));
b = pselect(should_flipsign, a, psub(one, a));
ndtri = pselect(
pcmp_lt(exp_neg_two, b),
generic_ndtri_gt_exp_neg_two<T, ScalarType>(b),
generic_ndtri_lt_exp_neg_two<T, ScalarType>(b, should_flipsign));
return pselect(
pcmp_le(a, zero), neg_maxnum,
pselect(pcmp_le(one, a), maxnum, ndtri));
}
template <typename Scalar>
struct ndtri_retval {
typedef Scalar type;
};
#if !EIGEN_HAS_C99_MATH
template <typename Scalar>
struct ndtri_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
# else
template <typename Scalar>
struct ndtri_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
return generic_ndtri<Scalar, Scalar>(x);
}
};
#endif // EIGEN_HAS_C99_MATH
/**************************************************************************************************************
* Implementation of igammac (complemented incomplete gamma integral), based on Cephes but requires C++11/C99 *
**************************************************************************************************************/
template <typename Scalar>
struct igammac_retval {
typedef Scalar type;
};
// NOTE: cephes_helper is also used to implement zeta
template <typename Scalar>
struct cephes_helper {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar machep() { assert(false && "machep not supported for this type"); return 0.0; }
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar big() { assert(false && "big not supported for this type"); return 0.0; }
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar biginv() { assert(false && "biginv not supported for this type"); return 0.0; }
};
template <>
struct cephes_helper<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float machep() {
return NumTraits<float>::epsilon() / 2; // 1.0 - machep == 1.0
}
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float big() {
// use epsneg (1.0 - epsneg == 1.0)
return 1.0f / (NumTraits<float>::epsilon() / 2);
}
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float biginv() {
// epsneg
return machep();
}
};
template <>
struct cephes_helper<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double machep() {
return NumTraits<double>::epsilon() / 2; // 1.0 - machep == 1.0
}
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double big() {
return 1.0 / NumTraits<double>::epsilon();
}
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double biginv() {
// inverse of eps
return NumTraits<double>::epsilon();
}
};
enum IgammaComputationMode { VALUE, DERIVATIVE, SAMPLE_DERIVATIVE };
template <typename Scalar>
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar main_igamma_term(Scalar a, Scalar x) {
/* Compute x**a * exp(-x) / gamma(a) */
Scalar logax = a * numext::log(x) - x - lgamma_impl<Scalar>::run(a);
if (logax < -numext::log(NumTraits<Scalar>::highest()) ||
// Assuming x and a aren't Nan.
(numext::isnan)(logax)) {
return Scalar(0);
}
return numext::exp(logax);
}
template <typename Scalar, IgammaComputationMode mode>
EIGEN_DEVICE_FUNC
int igamma_num_iterations() {
/* Returns the maximum number of internal iterations for igamma computation.
*/
if (mode == VALUE) {
return 2000;
}
if (internal::is_same<Scalar, float>::value) {
return 200;
} else if (internal::is_same<Scalar, double>::value) {
return 500;
} else {
return 2000;
}
}
template <typename Scalar, IgammaComputationMode mode>
struct igammac_cf_impl {
/* Computes igamc(a, x) or derivative (depending on the mode)
* using the continued fraction expansion of the complementary
* incomplete Gamma function.
*
* Preconditions:
* a > 0
* x >= 1
* x >= a
*/
EIGEN_DEVICE_FUNC
static Scalar run(Scalar a, Scalar x) {
const Scalar zero = 0;
const Scalar one = 1;
const Scalar two = 2;
const Scalar machep = cephes_helper<Scalar>::machep();
const Scalar big = cephes_helper<Scalar>::big();
const Scalar biginv = cephes_helper<Scalar>::biginv();
if ((numext::isinf)(x)) {
return zero;
}
Scalar ax = main_igamma_term<Scalar>(a, x);
// This is independent of mode. If this value is zero,
// then the function value is zero. If the function value is zero,
// then we are in a neighborhood where the function value evalutes to zero,
// so the derivative is zero.
if (ax == zero) {
return zero;
}
// continued fraction
Scalar y = one - a;
Scalar z = x + y + one;
Scalar c = zero;
Scalar pkm2 = one;
Scalar qkm2 = x;
Scalar pkm1 = x + one;
Scalar qkm1 = z * x;
Scalar ans = pkm1 / qkm1;
Scalar dpkm2_da = zero;
Scalar dqkm2_da = zero;
Scalar dpkm1_da = zero;
Scalar dqkm1_da = -x;
Scalar dans_da = (dpkm1_da - ans * dqkm1_da) / qkm1;
for (int i = 0; i < igamma_num_iterations<Scalar, mode>(); i++) {
c += one;
y += one;
z += two;
Scalar yc = y * c;
Scalar pk = pkm1 * z - pkm2 * yc;
Scalar qk = qkm1 * z - qkm2 * yc;
Scalar dpk_da = dpkm1_da * z - pkm1 - dpkm2_da * yc + pkm2 * c;
Scalar dqk_da = dqkm1_da * z - qkm1 - dqkm2_da * yc + qkm2 * c;
if (qk != zero) {
Scalar ans_prev = ans;
ans = pk / qk;
Scalar dans_da_prev = dans_da;
dans_da = (dpk_da - ans * dqk_da) / qk;
if (mode == VALUE) {
if (numext::abs(ans_prev - ans) <= machep * numext::abs(ans)) {
break;
}
} else {
if (numext::abs(dans_da - dans_da_prev) <= machep) {
break;
}
}
}
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
dpkm2_da = dpkm1_da;
dpkm1_da = dpk_da;
dqkm2_da = dqkm1_da;
dqkm1_da = dqk_da;
if (numext::abs(pk) > big) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
dpkm2_da *= biginv;
dpkm1_da *= biginv;
dqkm2_da *= biginv;
dqkm1_da *= biginv;
}
}
/* Compute x**a * exp(-x) / gamma(a) */
Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a);
Scalar dax_da = ax * dlogax_da;
switch (mode) {
case VALUE:
return ans * ax;
case DERIVATIVE:
return ans * dax_da + dans_da * ax;
case SAMPLE_DERIVATIVE:
default: // this is needed to suppress clang warning
return -(dans_da + ans * dlogax_da) * x;
}
}
};
template <typename Scalar, IgammaComputationMode mode>
struct igamma_series_impl {
/* Computes igam(a, x) or its derivative (depending on the mode)
* using the series expansion of the incomplete Gamma function.
*
* Preconditions:
* x > 0
* a > 0
* !(x > 1 && x > a)
*/
EIGEN_DEVICE_FUNC
static Scalar run(Scalar a, Scalar x) {
const Scalar zero = 0;
const Scalar one = 1;
const Scalar machep = cephes_helper<Scalar>::machep();
Scalar ax = main_igamma_term<Scalar>(a, x);
// This is independent of mode. If this value is zero,
// then the function value is zero. If the function value is zero,
// then we are in a neighborhood where the function value evalutes to zero,
// so the derivative is zero.
if (ax == zero) {
return zero;
}
ax /= a;
/* power series */
Scalar r = a;
Scalar c = one;
Scalar ans = one;
Scalar dc_da = zero;
Scalar dans_da = zero;
for (int i = 0; i < igamma_num_iterations<Scalar, mode>(); i++) {
r += one;
Scalar term = x / r;
Scalar dterm_da = -x / (r * r);
dc_da = term * dc_da + dterm_da * c;
dans_da += dc_da;
c *= term;
ans += c;
if (mode == VALUE) {
if (c <= machep * ans) {
break;
}
} else {
if (numext::abs(dc_da) <= machep * numext::abs(dans_da)) {
break;
}
}
}
Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a + one);
Scalar dax_da = ax * dlogax_da;
switch (mode) {
case VALUE:
return ans * ax;
case DERIVATIVE:
return ans * dax_da + dans_da * ax;
case SAMPLE_DERIVATIVE:
default: // this is needed to suppress clang warning
return -(dans_da + ans * dlogax_da) * x / a;
}
}
};
#if !EIGEN_HAS_C99_MATH
template <typename Scalar>
struct igammac_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar a, Scalar x) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
#else
template <typename Scalar>
struct igammac_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar a, Scalar x) {
/* igamc()
*
* Incomplete gamma integral (modified for Eigen)
*
*
*
* SYNOPSIS:
*
* double a, x, y, igamc();
*
* y = igamc( a, x );
*
* DESCRIPTION:
*
* 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 (float):
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 30000 7.8e-6 5.9e-7
*
*
* ACCURACY (double):
*
* 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 2.2: June, 1992
Copyright 1985, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
const Scalar zero = 0;
const Scalar one = 1;
const Scalar nan = NumTraits<Scalar>::quiet_NaN();
if ((x < zero) || (a <= zero)) {
// domain error
return nan;
}
if ((numext::isnan)(a) || (numext::isnan)(x)) { // propagate nans
return nan;
}
if ((x < one) || (x < a)) {
return (one - igamma_series_impl<Scalar, VALUE>::run(a, x));
}
return igammac_cf_impl<Scalar, VALUE>::run(a, x);
}
};
#endif // EIGEN_HAS_C99_MATH
/************************************************************************************************
* Implementation of igamma (incomplete gamma integral), based on Cephes but requires C++11/C99 *
************************************************************************************************/
#if !EIGEN_HAS_C99_MATH
template <typename Scalar, IgammaComputationMode mode>
struct igamma_generic_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar x) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
#else
template <typename Scalar, IgammaComputationMode mode>
struct igamma_generic_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar a, Scalar x) {
/* Depending on the mode, returns
* - VALUE: incomplete Gamma function igamma(a, x)
* - DERIVATIVE: derivative of incomplete Gamma function d/da igamma(a, x)
* - SAMPLE_DERIVATIVE: implicit derivative of a Gamma random variable
* x ~ Gamma(x | a, 1), dx/da = -1 / Gamma(x | a, 1) * d igamma(a, x) / dx
*
* Derivatives are implemented by forward-mode differentiation.
*/
const Scalar zero = 0;
const Scalar one = 1;
const Scalar nan = NumTraits<Scalar>::quiet_NaN();
if (x == zero) return zero;
if ((x < zero) || (a <= zero)) { // domain error
return nan;
}
if ((numext::isnan)(a) || (numext::isnan)(x)) { // propagate nans
return nan;
}
if ((x > one) && (x > a)) {
Scalar ret = igammac_cf_impl<Scalar, mode>::run(a, x);
if (mode == VALUE) {
return one - ret;
} else {
return -ret;
}
}
return igamma_series_impl<Scalar, mode>::run(a, x);
}
};
#endif // EIGEN_HAS_C99_MATH
template <typename Scalar>
struct igamma_retval {
typedef Scalar type;
};
template <typename Scalar>
struct igamma_impl : igamma_generic_impl<Scalar, VALUE> {
/* igam()
* Incomplete gamma integral.
*
* The CDF of Gamma(a, 1) random variable at the point x.
*
* Accuracy estimation. For each a in [10^-2, 10^-1...10^3] we sample
* 50 Gamma random variables x ~ Gamma(x | a, 1), a total of 300 points.
* The ground truth is computed by mpmath. Mean absolute error:
* float: 1.26713e-05
* double: 2.33606e-12
*
* Cephes documentation below.
*
* SYNOPSIS:
*
* double a, x, y, igam();
*
* y = igam( a, x );
*
* DESCRIPTION:
*
* 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 (double):
*
* 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
*
*
* ACCURACY (float):
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 20000 7.8e-6 5.9e-7
*
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1985, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* left tail of incomplete gamma function:
*
* inf. k
* a -x - x
* x e > ----------
* - -
* k=0 | (a+k+1)
*
*/
};
template <typename Scalar>
struct igamma_der_a_retval : igamma_retval<Scalar> {};
template <typename Scalar>
struct igamma_der_a_impl : igamma_generic_impl<Scalar, DERIVATIVE> {
/* Derivative of the incomplete Gamma function with respect to a.
*
* Computes d/da igamma(a, x) by forward differentiation of the igamma code.
*
* Accuracy estimation. For each a in [10^-2, 10^-1...10^3] we sample
* 50 Gamma random variables x ~ Gamma(x | a, 1), a total of 300 points.
* The ground truth is computed by mpmath. Mean absolute error:
* float: 6.17992e-07
* double: 4.60453e-12
*
* Reference:
* R. Moore. "Algorithm AS 187: Derivatives of the incomplete gamma
* integral". Journal of the Royal Statistical Society. 1982
*/
};
template <typename Scalar>
struct gamma_sample_der_alpha_retval : igamma_retval<Scalar> {};
template <typename Scalar>
struct gamma_sample_der_alpha_impl
: igamma_generic_impl<Scalar, SAMPLE_DERIVATIVE> {
/* Derivative of a Gamma random variable sample with respect to alpha.
*
* Consider a sample of a Gamma random variable with the concentration
* parameter alpha: sample ~ Gamma(alpha, 1). The reparameterization
* derivative that we want to compute is dsample / dalpha =
* d igammainv(alpha, u) / dalpha, where u = igamma(alpha, sample).
* However, this formula is numerically unstable and expensive, so instead
* we use implicit differentiation:
*
* igamma(alpha, sample) = u, where u ~ Uniform(0, 1).
* Apply d / dalpha to both sides:
* d igamma(alpha, sample) / dalpha
* + d igamma(alpha, sample) / dsample * dsample/dalpha = 0
* d igamma(alpha, sample) / dalpha
* + Gamma(sample | alpha, 1) dsample / dalpha = 0
* dsample/dalpha = - (d igamma(alpha, sample) / dalpha)
* / Gamma(sample | alpha, 1)
*
* Here Gamma(sample | alpha, 1) is the PDF of the Gamma distribution
* (note that the derivative of the CDF w.r.t. sample is the PDF).
* See the reference below for more details.
*
* The derivative of igamma(alpha, sample) is computed by forward
* differentiation of the igamma code. Division by the Gamma PDF is performed
* in the same code, increasing the accuracy and speed due to cancellation
* of some terms.
*
* Accuracy estimation. For each alpha in [10^-2, 10^-1...10^3] we sample
* 50 Gamma random variables sample ~ Gamma(sample | alpha, 1), a total of 300
* points. The ground truth is computed by mpmath. Mean absolute error:
* float: 2.1686e-06
* double: 1.4774e-12
*
* Reference:
* M. Figurnov, S. Mohamed, A. Mnih "Implicit Reparameterization Gradients".
* 2018
*/
};
/*****************************************************************************
* Implementation of Riemann zeta function of two arguments, based on Cephes *
*****************************************************************************/
template <typename Scalar>
struct zeta_retval {
typedef Scalar type;
};
template <typename Scalar>
struct zeta_impl_series {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(const Scalar) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
template <>
struct zeta_impl_series<float> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE bool run(float& a, float& b, float& s, const float x, const float machep) {
int i = 0;
while(i < 9)
{
i += 1;
a += 1.0f;
b = numext::pow( a, -x );
s += b;
if( numext::abs(b/s) < machep )
return true;
}
//Return whether we are done
return false;
}
};
template <>
struct zeta_impl_series<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE bool run(double& a, double& b, double& s, const double x, const double machep) {
int i = 0;
while( (i < 9) || (a <= 9.0) )
{
i += 1;
a += 1.0;
b = numext::pow( a, -x );
s += b;
if( numext::abs(b/s) < machep )
return true;
}
//Return whether we are done
return false;
}
};
template <typename Scalar>
struct zeta_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar x, Scalar q) {
/* zeta.c
*
* Riemann zeta function of two arguments
*
*
*
* SYNOPSIS:
*
* double x, q, y, zeta();
*
* y = zeta( x, q );
*
*
*
* DESCRIPTION:
*
*
*
* inf.
* - -x
* zeta(x,q) = > (k+q)
* -
* k=0
*
* where x > 1 and q is not a negative integer or zero.
* The Euler-Maclaurin summation formula is used to obtain
* the expansion
*
* n
* - -x
* zeta(x,q) = > (k+q)
* -
* k=1
*
* 1-x inf. B x(x+1)...(x+2j)
* (n+q) 1 - 2j
* + --------- - ------- + > --------------------
* x-1 x - x+2j+1
* 2(n+q) j=1 (2j)! (n+q)
*
* where the B2j are Bernoulli numbers. Note that (see zetac.c)
* zeta(x,1) = zetac(x) + 1.
*
*
*
* ACCURACY:
*
* Relative error for single precision:
* arithmetic domain # trials peak rms
* IEEE 0,25 10000 6.9e-7 1.0e-7
*
* Large arguments may produce underflow in powf(), in which
* case the results are inaccurate.
*
* REFERENCE:
*
* Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
* Series, and Products, p. 1073; Academic Press, 1980.
*
*/
int i;
Scalar p, r, a, b, k, s, t, w;
const Scalar A[] = {
Scalar(12.0),
Scalar(-720.0),
Scalar(30240.0),
Scalar(-1209600.0),
Scalar(47900160.0),
Scalar(-1.8924375803183791606e9), /*1.307674368e12/691*/
Scalar(7.47242496e10),
Scalar(-2.950130727918164224e12), /*1.067062284288e16/3617*/
Scalar(1.1646782814350067249e14), /*5.109094217170944e18/43867*/
Scalar(-4.5979787224074726105e15), /*8.028576626982912e20/174611*/
Scalar(1.8152105401943546773e17), /*1.5511210043330985984e23/854513*/
Scalar(-7.1661652561756670113e18) /*1.6938241367317436694528e27/236364091*/
};
const Scalar maxnum = NumTraits<Scalar>::infinity();
const Scalar zero = 0.0, half = 0.5, one = 1.0;
const Scalar machep = cephes_helper<Scalar>::machep();
const Scalar nan = NumTraits<Scalar>::quiet_NaN();
if( x == one )
return maxnum;
if( x < one )
{
return nan;
}
if( q <= zero )
{
if(q == numext::floor(q))
{
return maxnum;
}
p = x;
r = numext::floor(p);
if (p != r)
return nan;
}
/* Permit negative q but continue sum until n+q > +9 .
* This case should be handled by a reflection formula.
* If q<0 and x is an integer, there is a relation to
* the polygamma function.
*/
s = numext::pow( q, -x );
a = q;
b = zero;
// Run the summation in a helper function that is specific to the floating precision
if (zeta_impl_series<Scalar>::run(a, b, s, x, machep)) {
return s;
}
w = a;
s += b*w/(x-one);
s -= half * b;
a = one;
k = zero;
for( i=0; i<12; i++ )
{
a *= x + k;
b /= w;
t = a*b/A[i];
s = s + t;
t = numext::abs(t/s);
if( t < machep ) {
break;
}
k += one;
a *= x + k;
b /= w;
k += one;
}
return s;
}
};
/****************************************************************************
* Implementation of polygamma function, requires C++11/C99 *
****************************************************************************/
template <typename Scalar>
struct polygamma_retval {
typedef Scalar type;
};
#if !EIGEN_HAS_C99_MATH
template <typename Scalar>
struct polygamma_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(Scalar n, Scalar x) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
#else
template <typename Scalar>
struct polygamma_impl {
EIGEN_DEVICE_FUNC
static Scalar run(Scalar n, Scalar x) {
Scalar zero = 0.0, one = 1.0;
Scalar nplus = n + one;
const Scalar nan = NumTraits<Scalar>::quiet_NaN();
// Check that n is an integer
if (numext::floor(n) != n) {
return nan;
}
// Just return the digamma function for n = 1
else if (n == zero) {
return digamma_impl<Scalar>::run(x);
}
// Use the same implementation as scipy
else {
Scalar factorial = numext::exp(lgamma_impl<Scalar>::run(nplus));
return numext::pow(-one, nplus) * factorial * zeta_impl<Scalar>::run(nplus, x);
}
}
};
#endif // EIGEN_HAS_C99_MATH
/************************************************************************************************
* Implementation of betainc (incomplete beta integral), based on Cephes but requires C++11/C99 *
************************************************************************************************/
template <typename Scalar>
struct betainc_retval {
typedef Scalar type;
};
#if !EIGEN_HAS_C99_MATH
template <typename Scalar>
struct betainc_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar b, Scalar x) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
#else
template <typename Scalar>
struct betainc_impl {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(Scalar, Scalar, Scalar) {
/* betaincf.c
*
* Incomplete beta integral
*
*
* SYNOPSIS:
*
* float a, b, x, y, betaincf();
*
* y = betaincf( a, b, x );
*
*
* DESCRIPTION:
*
* 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 - betainc( a, b, x ) = betainc( b, a, 1-x ).
*
* The integral is evaluated by a continued fraction expansion.
* If a < 1, the function calls itself recursively after a
* transformation to increase a to a+1.
*
* ACCURACY (float):
*
* Tested at random points (a,b,x) with a and b in the indicated
* interval and x between 0 and 1.
*
* arithmetic domain # trials peak rms
* Relative error:
* IEEE 0,30 10000 3.7e-5 5.1e-6
* IEEE 0,100 10000 1.7e-4 2.5e-5
* The useful domain for relative error is limited by underflow
* of the single precision exponential function.
* Absolute error:
* IEEE 0,30 100000 2.2e-5 9.6e-7
* IEEE 0,100 10000 6.5e-5 3.7e-6
*
* Larger errors may occur for extreme ratios of a and b.
*
* ACCURACY (double):
* 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.
*
* ERROR MESSAGES:
* message condition value returned
* incbet domain x<0, x>1 nan
* incbet underflow nan
*/
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, Scalar>::value == false),
THIS_TYPE_IS_NOT_SUPPORTED);
return Scalar(0);
}
};
/* Continued fraction expansion #1 for incomplete beta integral (small_branch = True)
* Continued fraction expansion #2 for incomplete beta integral (small_branch = False)
*/
template <typename Scalar>
struct incbeta_cfe {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE Scalar run(Scalar a, Scalar b, Scalar x, bool small_branch) {
EIGEN_STATIC_ASSERT((internal::is_same<Scalar, float>::value ||
internal::is_same<Scalar, double>::value),
THIS_TYPE_IS_NOT_SUPPORTED);
const Scalar big = cephes_helper<Scalar>::big();
const Scalar machep = cephes_helper<Scalar>::machep();
const Scalar biginv = cephes_helper<Scalar>::biginv();
const Scalar zero = 0;
const Scalar one = 1;
const Scalar two = 2;
Scalar xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
Scalar k1, k2, k3, k4, k5, k6, k7, k8, k26update;
Scalar ans;
int n;
const int num_iters = (internal::is_same<Scalar, float>::value) ? 100 : 300;
const Scalar thresh =
(internal::is_same<Scalar, float>::value) ? machep : Scalar(3) * machep;
Scalar r = (internal::is_same<Scalar, float>::value) ? zero : one;
if (small_branch) {
k1 = a;
k2 = a + b;
k3 = a;
k4 = a + one;
k5 = one;
k6 = b - one;
k7 = k4;
k8 = a + two;
k26update = one;
} else {
k1 = a;
k2 = b - one;
k3 = a;
k4 = a + one;
k5 = one;
k6 = a + b;
k7 = a + one;
k8 = a + two;
k26update = -one;
x = x / (one - x);
}
pkm2 = zero;
qkm2 = one;
pkm1 = one;
qkm1 = one;
ans = one;
n = 0;
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 (qk != zero) {
r = pk / qk;
if (numext::abs(ans - r) < numext::abs(r) * thresh) {
return r;
}
ans = r;
}
k1 += one;
k2 += k26update;
k3 += two;
k4 += two;
k5 += one;
k6 -= k26update;
k7 += two;
k8 += two;
if ((numext::abs(qk) + numext::abs(pk)) > big) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
if ((numext::abs(qk) < biginv) || (numext::abs(pk) < biginv)) {
pkm2 *= big;
pkm1 *= big;
qkm2 *= big;
qkm1 *= big;
}
} while (++n < num_iters);
return ans;
}
};
/* Helper functions depending on the Scalar type */
template <typename Scalar>
struct betainc_helper {};
template <>
struct betainc_helper<float> {
/* Core implementation, assumes a large (> 1.0) */
EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE float incbsa(float aa, float bb,
float xx) {
float ans, a, b, t, x, onemx;
bool reversed_a_b = false;
onemx = 1.0f - xx;
/* see if x is greater than the mean */
if (xx > (aa / (aa + bb))) {
reversed_a_b = true;
a = bb;
b = aa;
t = xx;
x = onemx;
} else {
a = aa;
b = bb;
t = onemx;
x = xx;
}
/* Choose expansion for optimal convergence */
if (b > 10.0f) {
if (numext::abs(b * x / a) < 0.3f) {
t = betainc_helper<float>::incbps(a, b, x);
if (reversed_a_b) t = 1.0f - t;
return t;
}
}
ans = x * (a + b - 2.0f) / (a - 1.0f);
if (ans < 1.0f) {
ans = incbeta_cfe<float>::run(a, b, x, true /* small_branch */);
t = b * numext::log(t);
} else {
ans = incbeta_cfe<float>::run(a, b, x, false /* small_branch */);
t = (b - 1.0f) * numext::log(t);
}
t += a * numext::log(x) + lgamma_impl<float>::run(a + b) -
lgamma_impl<float>::run(a) - lgamma_impl<float>::run(b);
t += numext::log(ans / a);
t = numext::exp(t);
if (reversed_a_b) t = 1.0f - t;
return t;
}
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE float incbps(float a, float b, float x) {
float t, u, y, s;
const float machep = cephes_helper<float>::machep();
y = a * numext::log(x) + (b - 1.0f) * numext::log1p(-x) - numext::log(a);
y -= lgamma_impl<float>::run(a) + lgamma_impl<float>::run(b);
y += lgamma_impl<float>::run(a + b);
t = x / (1.0f - x);
s = 0.0f;
u = 1.0f;
do {
b -= 1.0f;
if (b == 0.0f) {
break;
}
a += 1.0f;
u *= t * b / a;
s += u;
} while (numext::abs(u) > machep);
return numext::exp(y) * (1.0f + s);
}
};
template <>
struct betainc_impl<float> {
EIGEN_DEVICE_FUNC
static float run(float a, float b, float x) {
const float nan = NumTraits<float>::quiet_NaN();
float ans, t;
if (a <= 0.0f) return nan;
if (b <= 0.0f) return nan;
if ((x <= 0.0f) || (x >= 1.0f)) {
if (x == 0.0f) return 0.0f;
if (x == 1.0f) return 1.0f;
// mtherr("betaincf", DOMAIN);
return nan;
}
/* transformation for small aa */
if (a <= 1.0f) {
ans = betainc_helper<float>::incbsa(a + 1.0f, b, x);
t = a * numext::log(x) + b * numext::log1p(-x) +
lgamma_impl<float>::run(a + b) - lgamma_impl<float>::run(a + 1.0f) -
lgamma_impl<float>::run(b);
return (ans + numext::exp(t));
} else {
return betainc_helper<float>::incbsa(a, b, x);
}
}
};
template <>
struct betainc_helper<double> {
EIGEN_DEVICE_FUNC
static EIGEN_STRONG_INLINE double incbps(double a, double b, double x) {
const double machep = cephes_helper<double>::machep();
double s, t, u, v, n, t1, z, ai;
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 = machep * ai;
while (numext::abs(v) > z) {
u = (n - b) * x / n;
t *= u;
v = t / (a + n);
s += v;
n += 1.0;
}
s += t1;
s += ai;
u = a * numext::log(x);
// TODO: gamma() is not directly implemented in Eigen.
/*
if ((a + b) < maxgam && numext::abs(u) < maxlog) {
t = gamma(a + b) / (gamma(a) * gamma(b));
s = s * t * pow(x, a);
}
*/
t = lgamma_impl<double>::run(a + b) - lgamma_impl<double>::run(a) -
lgamma_impl<double>::run(b) + u + numext::log(s);
return s = numext::exp(t);
}
};
template <>
struct betainc_impl<double> {
EIGEN_DEVICE_FUNC
static double run(double aa, double bb, double xx) {
const double nan = NumTraits<double>::quiet_NaN();
const double machep = cephes_helper<double>::machep();
// const double maxgam = 171.624376956302725;
double a, b, t, x, xc, w, y;
bool reversed_a_b = false;
if (aa <= 0.0 || bb <= 0.0) {
return nan; // goto domerr;
}
if ((xx <= 0.0) || (xx >= 1.0)) {
if (xx == 0.0) return (0.0);
if (xx == 1.0) return (1.0);
// mtherr("incbet", DOMAIN);
return nan;
}
if ((bb * xx) <= 1.0 && xx <= 0.95) {
return betainc_helper<double>::incbps(aa, bb, xx);
}
w = 1.0 - xx;
/* Reverse a and b if x is greater than the mean. */
if (xx > (aa / (aa + bb))) {
reversed_a_b = true;
a = bb;
b = aa;
xc = xx;
x = w;
} else {
a = aa;
b = bb;
xc = w;
x = xx;
}
if (reversed_a_b && (b * x) <= 1.0 && x <= 0.95) {
t = betainc_helper<double>::incbps(a, b, x);
if (t <= machep) {
t = 1.0 - machep;
} else {
t = 1.0 - t;
}
return t;
}
/* Choose expansion for better convergence. */
y = x * (a + b - 2.0) - (a - 1.0);
if (y < 0.0) {
w = incbeta_cfe<double>::run(a, b, x, true /* small_branch */);
} else {
w = incbeta_cfe<double>::run(a, b, x, false /* small_branch */) / xc;
}
/* Multiply w by the factor
a b _ _ _
x (1-x) | (a+b) / ( a | (a) | (b) ) . */
y = a * numext::log(x);
t = b * numext::log(xc);
// TODO: gamma is not directly implemented in Eigen.
/*
if ((a + b) < maxgam && numext::abs(y) < maxlog && numext::abs(t) < maxlog)
{
t = pow(xc, b);
t *= pow(x, a);
t /= a;
t *= w;
t *= gamma(a + b) / (gamma(a) * gamma(b));
} else {
*/
/* Resort to logarithms. */
y += t + lgamma_impl<double>::run(a + b) - lgamma_impl<double>::run(a) -
lgamma_impl<double>::run(b);
y += numext::log(w / a);
t = numext::exp(y);
/* } */
// done:
if (reversed_a_b) {
if (t <= machep) {
t = 1.0 - machep;
} else {
t = 1.0 - t;
}
}
return t;
}
};
#endif // EIGEN_HAS_C99_MATH
} // end namespace internal
namespace numext {
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(lgamma, Scalar)
lgamma(const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(lgamma, Scalar)::run(x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(digamma, Scalar)
digamma(const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(digamma, Scalar)::run(x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(zeta, Scalar)
zeta(const Scalar& x, const Scalar& q) {
return EIGEN_MATHFUNC_IMPL(zeta, Scalar)::run(x, q);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(polygamma, Scalar)
polygamma(const Scalar& n, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(polygamma, Scalar)::run(n, x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erf, Scalar)
erf(const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(erf, Scalar)::run(x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(erfc, Scalar)
erfc(const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(erfc, Scalar)::run(x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(ndtri, Scalar)
ndtri(const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(ndtri, Scalar)::run(x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igamma, Scalar)
igamma(const Scalar& a, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(igamma, Scalar)::run(a, x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igamma_der_a, Scalar)
igamma_der_a(const Scalar& a, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(igamma_der_a, Scalar)::run(a, x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(gamma_sample_der_alpha, Scalar)
gamma_sample_der_alpha(const Scalar& a, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(gamma_sample_der_alpha, Scalar)::run(a, x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(igammac, Scalar)
igammac(const Scalar& a, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(igammac, Scalar)::run(a, x);
}
template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(betainc, Scalar)
betainc(const Scalar& a, const Scalar& b, const Scalar& x) {
return EIGEN_MATHFUNC_IMPL(betainc, Scalar)::run(a, b, x);
}
} // end namespace numext
} // end namespace Eigen
#endif // EIGEN_SPECIAL_FUNCTIONS_H