absl/random/internal/distribution_test_util.cc - third_party/abseil-cpp - Git at Google

 // Copyright 2017 The Abseil Authors.
 //
 // Licensed under the Apache License, Version 2.0 (the "License");
 // you may not use this file except in compliance with the License.
 // You may obtain a copy of the License at
 //
 //      https://www.apache.org/licenses/LICENSE-2.0
 //
 // Unless required by applicable law or agreed to in writing, software
 // distributed under the License is distributed on an "AS IS" BASIS,
 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 // See the License for the specific language governing permissions and
 // limitations under the License.

 #include "absl/random/internal/distribution_test_util.h"

 #include <cassert>
 #include <cmath>
 #include <string>
 #include <vector>

 #include "absl/base/internal/raw_logging.h"
 #include "absl/base/macros.h"
 #include "absl/strings/str_cat.h"
 #include "absl/strings/str_format.h"

 namespace absl {
 ABSL_NAMESPACE_BEGIN
 namespace random_internal {
 namespace {

 #if defined(__EMSCRIPTEN__)
 // Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
 inline double fma(double x, double y, double z) { return (x * y) + z; }
 #endif

 }  // namespace

 DistributionMoments ComputeDistributionMoments(
     absl::Span<const double> data_points) {
   DistributionMoments result;

   // Compute m1
   for (double x : data_points) {
     result.n++;
     result.mean += x;
   }
   result.mean /= static_cast<double>(result.n);

   // Compute m2, m3, m4
   for (double x : data_points) {
     double v = x - result.mean;
     result.variance += v * v;
     result.skewness += v * v * v;
     result.kurtosis += v * v * v * v;
   }
   result.variance /= static_cast<double>(result.n - 1);

   result.skewness /= static_cast<double>(result.n);
   result.skewness /= std::pow(result.variance, 1.5);

   result.kurtosis /= static_cast<double>(result.n);
   result.kurtosis /= std::pow(result.variance, 2.0);
   return result;

   // When validating the min/max count, the following confidence intervals may
   // be of use:
   // 3.291 * stddev = 99.9% CI
   // 2.576 * stddev = 99% CI
   // 1.96 * stddev  = 95% CI
   // 1.65 * stddev  = 90% CI
 }

 std::ostream& operator<<(std::ostream& os, const DistributionMoments& moments) {
   return os << absl::StrFormat("mean=%f, stddev=%f, skewness=%f, kurtosis=%f",
                                moments.mean, std::sqrt(moments.variance),
                                moments.skewness, moments.kurtosis);
 }

 double InverseNormalSurvival(double x) {
   // inv_sf(u) = -sqrt(2) * erfinv(2u-1)
   static constexpr double kSqrt2 = 1.4142135623730950488;
   return -kSqrt2 * absl::random_internal::erfinv(2 * x - 1.0);
 }

 bool Near(absl::string_view msg, double actual, double expected, double bound) {
   assert(bound > 0.0);
   double delta = fabs(expected - actual);
   if (delta < bound) {
     return true;
   }

   std::string formatted = absl::StrCat(
       msg, " actual=", actual, " expected=", expected, " err=", delta / bound);
   ABSL_RAW_LOG(INFO, "%s", formatted.c_str());
   return false;
 }

 // TODO(absl-team): Replace with an "ABSL_HAVE_SPECIAL_MATH" and try
 // to use std::beta().  As of this writing P0226R1 is not implemented
 // in libc++: http://libcxx.llvm.org/cxx1z_status.html
 double beta(double p, double q) {
   // Beta(x, y) = Gamma(x) * Gamma(y) / Gamma(x+y)
   double lbeta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
   return std::exp(lbeta);
 }

 // Approximation to inverse of the Error Function in double precision.
 // (http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf)
 double erfinv(double x) {
 #if !defined(__EMSCRIPTEN__)
   using std::fma;
 #endif

   double w = 0.0;
   double p = 0.0;
   w = -std::log((1.0 - x) * (1.0 + x));
   if (w < 6.250000) {
     w = w - 3.125000;
     p = -3.6444120640178196996e-21;
     p = fma(p, w, -1.685059138182016589e-19);
     p = fma(p, w, 1.2858480715256400167e-18);
     p = fma(p, w, 1.115787767802518096e-17);
     p = fma(p, w, -1.333171662854620906e-16);
     p = fma(p, w, 2.0972767875968561637e-17);
     p = fma(p, w, 6.6376381343583238325e-15);
     p = fma(p, w, -4.0545662729752068639e-14);
     p = fma(p, w, -8.1519341976054721522e-14);
     p = fma(p, w, 2.6335093153082322977e-12);
     p = fma(p, w, -1.2975133253453532498e-11);
     p = fma(p, w, -5.4154120542946279317e-11);
     p = fma(p, w, 1.051212273321532285e-09);
     p = fma(p, w, -4.1126339803469836976e-09);
     p = fma(p, w, -2.9070369957882005086e-08);
     p = fma(p, w, 4.2347877827932403518e-07);
     p = fma(p, w, -1.3654692000834678645e-06);
     p = fma(p, w, -1.3882523362786468719e-05);
     p = fma(p, w, 0.0001867342080340571352);
     p = fma(p, w, -0.00074070253416626697512);
     p = fma(p, w, -0.0060336708714301490533);
     p = fma(p, w, 0.24015818242558961693);
     p = fma(p, w, 1.6536545626831027356);
   } else if (w < 16.000000) {
     w = std::sqrt(w) - 3.250000;
     p = 2.2137376921775787049e-09;
     p = fma(p, w, 9.0756561938885390979e-08);
     p = fma(p, w, -2.7517406297064545428e-07);
     p = fma(p, w, 1.8239629214389227755e-08);
     p = fma(p, w, 1.5027403968909827627e-06);
     p = fma(p, w, -4.013867526981545969e-06);
     p = fma(p, w, 2.9234449089955446044e-06);
     p = fma(p, w, 1.2475304481671778723e-05);
     p = fma(p, w, -4.7318229009055733981e-05);
     p = fma(p, w, 6.8284851459573175448e-05);
     p = fma(p, w, 2.4031110387097893999e-05);
     p = fma(p, w, -0.0003550375203628474796);
     p = fma(p, w, 0.00095328937973738049703);
     p = fma(p, w, -0.0016882755560235047313);
     p = fma(p, w, 0.0024914420961078508066);
     p = fma(p, w, -0.0037512085075692412107);
     p = fma(p, w, 0.005370914553590063617);
     p = fma(p, w, 1.0052589676941592334);
     p = fma(p, w, 3.0838856104922207635);
   } else {
     w = std::sqrt(w) - 5.000000;
     p = -2.7109920616438573243e-11;
     p = fma(p, w, -2.5556418169965252055e-10);
     p = fma(p, w, 1.5076572693500548083e-09);
     p = fma(p, w, -3.7894654401267369937e-09);
     p = fma(p, w, 7.6157012080783393804e-09);
     p = fma(p, w, -1.4960026627149240478e-08);
     p = fma(p, w, 2.9147953450901080826e-08);
     p = fma(p, w, -6.7711997758452339498e-08);
     p = fma(p, w, 2.2900482228026654717e-07);
     p = fma(p, w, -9.9298272942317002539e-07);
     p = fma(p, w, 4.5260625972231537039e-06);
     p = fma(p, w, -1.9681778105531670567e-05);
     p = fma(p, w, 7.5995277030017761139e-05);
     p = fma(p, w, -0.00021503011930044477347);
     p = fma(p, w, -0.00013871931833623122026);
     p = fma(p, w, 1.0103004648645343977);
     p = fma(p, w, 4.8499064014085844221);
   }
   return p * x;
 }

 namespace {

 // Direct implementation of AS63, BETAIN()
 // https://www.jstor.org/stable/2346797?seq=3#page_scan_tab_contents.
 //
 // BETAIN(x, p, q, beta)
 //  x:     the value of the upper limit x.
 //  p:     the value of the parameter p.
 //  q:     the value of the parameter q.
 //  beta:  the value of ln B(p, q)
 //
 double BetaIncompleteImpl(const double x, const double p, const double q,
                           const double beta) {
   if (p < (p + q) * x) {
     // Incomplete beta function is symmetrical, so return the complement.
     return 1. - BetaIncompleteImpl(1.0 - x, q, p, beta);
   }

   double psq = p + q;
   const double kErr = 1e-14;
   const double xc = 1. - x;
   const double pre =
       std::exp(p * std::log(x) + (q - 1.) * std::log(xc) - beta) / p;

   double term = 1.;
   double ai = 1.;
   double result = 1.;
   int ns = static_cast<int>(q + xc * psq);

   // Use the soper reduction forumla.
   double rx = (ns == 0) ? x : x / xc;
   double temp = q - ai;
   for (;;) {
     term = term * temp * rx / (p + ai);
     result = result + term;
     temp = std::fabs(term);
     if (temp < kErr && temp < kErr * result) {
       return result * pre;
     }
     ai = ai + 1.;
     --ns;
     if (ns >= 0) {
       temp = q - ai;
       if (ns == 0) {
         rx = x;
       }
     } else {
       temp = psq;
       psq = psq + 1.;
     }
   }

   // NOTE: See also TOMS Alogrithm 708.
   // http://www.netlib.org/toms/index.html
   //
   // NOTE: The NWSC library also includes BRATIO / ISUBX (p87)
   // https://archive.org/details/DTIC_ADA261511/page/n75
 }

 // Direct implementation of AS109, XINBTA(p, q, beta, alpha)
 // https://www.jstor.org/stable/2346798?read-now=1&seq=4#page_scan_tab_contents
 // https://www.jstor.org/stable/2346887?seq=1#page_scan_tab_contents
 //
 // XINBTA(p, q, beta, alhpa)
 //  p:     the value of the parameter p.
 //  q:     the value of the parameter q.
 //  beta:  the value of ln B(p, q)
 //  alpha: the value of the lower tail area.
 //
 double BetaIncompleteInvImpl(const double p, const double q, const double beta,
                              const double alpha) {
   if (alpha < 0.5) {
     // Inverse Incomplete beta function is symmetrical, return the complement.
     return 1. - BetaIncompleteInvImpl(q, p, beta, 1. - alpha);
   }
   const double kErr = 1e-14;
   double value = kErr;

   // Compute the initial estimate.
   {
     double r = std::sqrt(-std::log(alpha * alpha));
     double y =
         r - fma(r, 0.27061, 2.30753) / fma(r, fma(r, 0.04481, 0.99229), 1.0);
     if (p > 1. && q > 1.) {
       r = (y * y - 3.) / 6.;
       double s = 1. / (p + p - 1.);
       double t = 1. / (q + q - 1.);
       double h = 2. / s + t;
       double w =
           y * std::sqrt(h + r) / h - (t - s) * (r + 5. / 6. - t / (3. * h));
       value = p / (p + q * std::exp(w + w));
     } else {
       r = q + q;
       double t = 1.0 / (9. * q);
       double u = 1.0 - t + y * std::sqrt(t);
       t = r * (u * u * u);
       if (t <= 0) {
         value = 1.0 - std::exp((std::log((1.0 - alpha) * q) + beta) / q);
       } else {
         t = (4.0 * p + r - 2.0) / t;
         if (t <= 1) {
           value = std::exp((std::log(alpha * p) + beta) / p);
         } else {
           value = 1.0 - 2.0 / (t + 1.0);
         }
       }
     }
   }

   // Solve for x using a modified newton-raphson method using the function
   // BetaIncomplete.
   {
     value = std::max(value, kErr);
     value = std::min(value, 1.0 - kErr);

     const double r = 1.0 - p;
     const double t = 1.0 - q;
     double y;
     double yprev = 0;
     double sq = 1;
     double prev = 1;
     for (;;) {
       if (value < 0 || value > 1.0) {
         // Error case; value went infinite.
         return std::numeric_limits<double>::infinity();
       } else if (value == 0 || value == 1) {
         y = value;
       } else {
         y = BetaIncompleteImpl(value, p, q, beta);
         if (!std::isfinite(y)) {
           return y;
         }
       }
       y = (y - alpha) *
           std::exp(beta + r * std::log(value) + t * std::log(1.0 - value));
       if (y * yprev <= 0) {
         prev = std::max(sq, std::numeric_limits<double>::min());
       }
       double g = 1.0;
       for (;;) {
         const double adj = g * y;
         const double adj_sq = adj * adj;
         if (adj_sq >= prev) {
           g = g / 3.0;
           continue;
         }
         const double tx = value - adj;
         if (tx < 0 || tx > 1) {
           g = g / 3.0;
           continue;
         }
         if (prev < kErr) {
           return value;
         }
         if (y * y < kErr) {
           return value;
         }
         if (tx == value) {
           return value;
         }
         if (tx == 0 || tx == 1) {
           g = g / 3.0;
           continue;
         }
         value = tx;
         yprev = y;
         break;
       }
     }
   }

   // NOTES: See also: Asymptotic inversion of the incomplete beta function.
   // https://core.ac.uk/download/pdf/82140723.pdf
   //
   // NOTE: See the Boost library documentation as well:
   // https://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html
 }

 }  // namespace

 double BetaIncomplete(const double x, const double p, const double q) {
   // Error cases.
   if (p < 0 || q < 0 || x < 0 || x > 1.0) {
     return std::numeric_limits<double>::infinity();
   }
   if (x == 0 || x == 1) {
     return x;
   }
   // ln(Beta(p, q))
   double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
   return BetaIncompleteImpl(x, p, q, beta);
 }

 double BetaIncompleteInv(const double p, const double q, const double alpha) {
   // Error cases.
   if (p < 0 || q < 0 || alpha < 0 || alpha > 1.0) {
     return std::numeric_limits<double>::infinity();
   }
   if (alpha == 0 || alpha == 1) {
     return alpha;
   }
   // ln(Beta(p, q))
   double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
   return BetaIncompleteInvImpl(p, q, beta, alpha);
 }

 // Given `num_trials` trials each with probability `p` of success, the
 // probability of no failures is `p^k`. To ensure the probability of a failure
 // is no more than `p_fail`, it must be that `p^k == 1 - p_fail`. This function
 // computes `p` from that equation.
 double RequiredSuccessProbability(const double p_fail, const int num_trials) {
   double p = std::exp(std::log(1.0 - p_fail) / static_cast<double>(num_trials));
   ABSL_ASSERT(p > 0);
   return p;
 }

 double ZScore(double expected_mean, const DistributionMoments& moments) {
   return (moments.mean - expected_mean) /
          (std::sqrt(moments.variance) /
           std::sqrt(static_cast<double>(moments.n)));
 }

 double MaxErrorTolerance(double acceptance_probability) {
   double one_sided_pvalue = 0.5 * (1.0 - acceptance_probability);
   const double max_err = InverseNormalSurvival(one_sided_pvalue);
   ABSL_ASSERT(max_err > 0);
   return max_err;
 }

 }  // namespace random_internal
 ABSL_NAMESPACE_END
 }  // namespace absl
	// Copyright 2017 The Abseil Authors.
	//
	// Licensed under the Apache License, Version 2.0 (the "License");
	// you may not use this file except in compliance with the License.
	// You may obtain a copy of the License at
	//
	// https://www.apache.org/licenses/LICENSE-2.0
	//
	// Unless required by applicable law or agreed to in writing, software
	// distributed under the License is distributed on an "AS IS" BASIS,
	// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
	// See the License for the specific language governing permissions and
	// limitations under the License.

	#include "absl/random/internal/distribution_test_util.h"

	#include <cassert>
	#include <cmath>
	#include <string>
	#include <vector>

	#include "absl/base/internal/raw_logging.h"
	#include "absl/base/macros.h"
	#include "absl/strings/str_cat.h"
	#include "absl/strings/str_format.h"

	namespace absl {
	ABSL_NAMESPACE_BEGIN
	namespace random_internal {
	namespace {

	#if defined(__EMSCRIPTEN__)
	// Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
	inline double fma(double x, double y, double z) { return (x * y) + z; }
	#endif

	} // namespace

	DistributionMoments ComputeDistributionMoments(
	absl::Span<const double> data_points) {
	DistributionMoments result;

	// Compute m1
	for (double x : data_points) {
	result.n++;
	result.mean += x;
	}
	result.mean /= static_cast<double>(result.n);

	// Compute m2, m3, m4
	for (double x : data_points) {
	double v = x - result.mean;
	result.variance += v * v;
	result.skewness += v * v * v;
	result.kurtosis += v * v * v * v;
	}
	result.variance /= static_cast<double>(result.n - 1);

	result.skewness /= static_cast<double>(result.n);
	result.skewness /= std::pow(result.variance, 1.5);

	result.kurtosis /= static_cast<double>(result.n);
	result.kurtosis /= std::pow(result.variance, 2.0);
	return result;

	// When validating the min/max count, the following confidence intervals may
	// be of use:
	// 3.291 * stddev = 99.9% CI
	// 2.576 * stddev = 99% CI
	// 1.96 * stddev = 95% CI
	// 1.65 * stddev = 90% CI
	}

	std::ostream& operator<<(std::ostream& os, const DistributionMoments& moments) {
	return os << absl::StrFormat("mean=%f, stddev=%f, skewness=%f, kurtosis=%f",
	moments.mean, std::sqrt(moments.variance),
	moments.skewness, moments.kurtosis);
	}

	double InverseNormalSurvival(double x) {
	// inv_sf(u) = -sqrt(2) * erfinv(2u-1)
	static constexpr double kSqrt2 = 1.4142135623730950488;
	return -kSqrt2 * absl::random_internal::erfinv(2 * x - 1.0);
	}

	bool Near(absl::string_view msg, double actual, double expected, double bound) {
	assert(bound > 0.0);
	double delta = fabs(expected - actual);
	if (delta < bound) {
	return true;
	}

	std::string formatted = absl::StrCat(
	msg, " actual=", actual, " expected=", expected, " err=", delta / bound);
	ABSL_RAW_LOG(INFO, "%s", formatted.c_str());
	return false;
	}

	// TODO(absl-team): Replace with an "ABSL_HAVE_SPECIAL_MATH" and try
	// to use std::beta(). As of this writing P0226R1 is not implemented
	// in libc++: http://libcxx.llvm.org/cxx1z_status.html
	double beta(double p, double q) {
	// Beta(x, y) = Gamma(x) * Gamma(y) / Gamma(x+y)
	double lbeta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
	return std::exp(lbeta);
	}

	// Approximation to inverse of the Error Function in double precision.
	// (http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf)
	double erfinv(double x) {
	#if !defined(__EMSCRIPTEN__)
	using std::fma;
	#endif

	double w = 0.0;
	double p = 0.0;
	w = -std::log((1.0 - x) * (1.0 + x));
	if (w < 6.250000) {
	w = w - 3.125000;
	p = -3.6444120640178196996e-21;
	p = fma(p, w, -1.685059138182016589e-19);
	p = fma(p, w, 1.2858480715256400167e-18);
	p = fma(p, w, 1.115787767802518096e-17);
	p = fma(p, w, -1.333171662854620906e-16);
	p = fma(p, w, 2.0972767875968561637e-17);
	p = fma(p, w, 6.6376381343583238325e-15);
	p = fma(p, w, -4.0545662729752068639e-14);
	p = fma(p, w, -8.1519341976054721522e-14);
	p = fma(p, w, 2.6335093153082322977e-12);
	p = fma(p, w, -1.2975133253453532498e-11);
	p = fma(p, w, -5.4154120542946279317e-11);
	p = fma(p, w, 1.051212273321532285e-09);
	p = fma(p, w, -4.1126339803469836976e-09);
	p = fma(p, w, -2.9070369957882005086e-08);
	p = fma(p, w, 4.2347877827932403518e-07);
	p = fma(p, w, -1.3654692000834678645e-06);
	p = fma(p, w, -1.3882523362786468719e-05);
	p = fma(p, w, 0.0001867342080340571352);
	p = fma(p, w, -0.00074070253416626697512);
	p = fma(p, w, -0.0060336708714301490533);
	p = fma(p, w, 0.24015818242558961693);
	p = fma(p, w, 1.6536545626831027356);
	} else if (w < 16.000000) {
	w = std::sqrt(w) - 3.250000;
	p = 2.2137376921775787049e-09;
	p = fma(p, w, 9.0756561938885390979e-08);
	p = fma(p, w, -2.7517406297064545428e-07);
	p = fma(p, w, 1.8239629214389227755e-08);
	p = fma(p, w, 1.5027403968909827627e-06);
	p = fma(p, w, -4.013867526981545969e-06);
	p = fma(p, w, 2.9234449089955446044e-06);
	p = fma(p, w, 1.2475304481671778723e-05);
	p = fma(p, w, -4.7318229009055733981e-05);
	p = fma(p, w, 6.8284851459573175448e-05);
	p = fma(p, w, 2.4031110387097893999e-05);
	p = fma(p, w, -0.0003550375203628474796);
	p = fma(p, w, 0.00095328937973738049703);
	p = fma(p, w, -0.0016882755560235047313);
	p = fma(p, w, 0.0024914420961078508066);
	p = fma(p, w, -0.0037512085075692412107);
	p = fma(p, w, 0.005370914553590063617);
	p = fma(p, w, 1.0052589676941592334);
	p = fma(p, w, 3.0838856104922207635);
	} else {
	w = std::sqrt(w) - 5.000000;
	p = -2.7109920616438573243e-11;
	p = fma(p, w, -2.5556418169965252055e-10);
	p = fma(p, w, 1.5076572693500548083e-09);
	p = fma(p, w, -3.7894654401267369937e-09);
	p = fma(p, w, 7.6157012080783393804e-09);
	p = fma(p, w, -1.4960026627149240478e-08);
	p = fma(p, w, 2.9147953450901080826e-08);
	p = fma(p, w, -6.7711997758452339498e-08);
	p = fma(p, w, 2.2900482228026654717e-07);
	p = fma(p, w, -9.9298272942317002539e-07);
	p = fma(p, w, 4.5260625972231537039e-06);
	p = fma(p, w, -1.9681778105531670567e-05);
	p = fma(p, w, 7.5995277030017761139e-05);
	p = fma(p, w, -0.00021503011930044477347);
	p = fma(p, w, -0.00013871931833623122026);
	p = fma(p, w, 1.0103004648645343977);
	p = fma(p, w, 4.8499064014085844221);
	}
	return p * x;
	}

	namespace {

	// Direct implementation of AS63, BETAIN()
	// https://www.jstor.org/stable/2346797?seq=3#page_scan_tab_contents.
	//
	// BETAIN(x, p, q, beta)
	// x: the value of the upper limit x.
	// p: the value of the parameter p.
	// q: the value of the parameter q.
	// beta: the value of ln B(p, q)
	//
	double BetaIncompleteImpl(const double x, const double p, const double q,
	const double beta) {
	if (p < (p + q) * x) {
	// Incomplete beta function is symmetrical, so return the complement.
	return 1. - BetaIncompleteImpl(1.0 - x, q, p, beta);
	}

	double psq = p + q;
	const double kErr = 1e-14;
	const double xc = 1. - x;
	const double pre =
	std::exp(p * std::log(x) + (q - 1.) * std::log(xc) - beta) / p;

	double term = 1.;
	double ai = 1.;
	double result = 1.;
	int ns = static_cast<int>(q + xc * psq);

	// Use the soper reduction forumla.
	double rx = (ns == 0) ? x : x / xc;
	double temp = q - ai;
	for (;;) {
	term = term * temp * rx / (p + ai);
	result = result + term;
	temp = std::fabs(term);
	if (temp < kErr && temp < kErr * result) {
	return result * pre;
	}
	ai = ai + 1.;
	--ns;
	if (ns >= 0) {
	temp = q - ai;
	if (ns == 0) {
	rx = x;
	}
	} else {
	temp = psq;
	psq = psq + 1.;
	}
	}

	// NOTE: See also TOMS Alogrithm 708.
	// http://www.netlib.org/toms/index.html
	//
	// NOTE: The NWSC library also includes BRATIO / ISUBX (p87)
	// https://archive.org/details/DTIC_ADA261511/page/n75
	}

	// Direct implementation of AS109, XINBTA(p, q, beta, alpha)
	// https://www.jstor.org/stable/2346798?read-now=1&seq=4#page_scan_tab_contents
	// https://www.jstor.org/stable/2346887?seq=1#page_scan_tab_contents
	//
	// XINBTA(p, q, beta, alhpa)
	// p: the value of the parameter p.
	// q: the value of the parameter q.
	// beta: the value of ln B(p, q)
	// alpha: the value of the lower tail area.
	//
	double BetaIncompleteInvImpl(const double p, const double q, const double beta,
	const double alpha) {
	if (alpha < 0.5) {
	// Inverse Incomplete beta function is symmetrical, return the complement.
	return 1. - BetaIncompleteInvImpl(q, p, beta, 1. - alpha);
	}
	const double kErr = 1e-14;
	double value = kErr;

	// Compute the initial estimate.
	{
	double r = std::sqrt(-std::log(alpha * alpha));
	double y =
	r - fma(r, 0.27061, 2.30753) / fma(r, fma(r, 0.04481, 0.99229), 1.0);
	if (p > 1. && q > 1.) {
	r = (y * y - 3.) / 6.;
	double s = 1. / (p + p - 1.);
	double t = 1. / (q + q - 1.);
	double h = 2. / s + t;
	double w =
	y * std::sqrt(h + r) / h - (t - s) * (r + 5. / 6. - t / (3. * h));
	value = p / (p + q * std::exp(w + w));
	} else {
	r = q + q;
	double t = 1.0 / (9. * q);
	double u = 1.0 - t + y * std::sqrt(t);
	t = r * (u * u * u);
	if (t <= 0) {
	value = 1.0 - std::exp((std::log((1.0 - alpha) * q) + beta) / q);
	} else {
	t = (4.0 * p + r - 2.0) / t;
	if (t <= 1) {
	value = std::exp((std::log(alpha * p) + beta) / p);
	} else {
	value = 1.0 - 2.0 / (t + 1.0);
	}
	}
	}
	}

	// Solve for x using a modified newton-raphson method using the function
	// BetaIncomplete.
	{
	value = std::max(value, kErr);
	value = std::min(value, 1.0 - kErr);

	const double r = 1.0 - p;
	const double t = 1.0 - q;
	double y;
	double yprev = 0;
	double sq = 1;
	double prev = 1;
	for (;;) {
	if (value < 0 \|\| value > 1.0) {
	// Error case; value went infinite.
	return std::numeric_limits<double>::infinity();
	} else if (value == 0 \|\| value == 1) {
	y = value;
	} else {
	y = BetaIncompleteImpl(value, p, q, beta);
	if (!std::isfinite(y)) {
	return y;
	}
	}
	y = (y - alpha) *
	std::exp(beta + r * std::log(value) + t * std::log(1.0 - value));
	if (y * yprev <= 0) {
	prev = std::max(sq, std::numeric_limits<double>::min());
	}
	double g = 1.0;
	for (;;) {
	const double adj = g * y;
	const double adj_sq = adj * adj;
	if (adj_sq >= prev) {
	g = g / 3.0;
	continue;
	}
	const double tx = value - adj;
	if (tx < 0 \|\| tx > 1) {
	g = g / 3.0;
	continue;
	}
	if (prev < kErr) {
	return value;
	}
	if (y * y < kErr) {
	return value;
	}
	if (tx == value) {
	return value;
	}
	if (tx == 0 \|\| tx == 1) {
	g = g / 3.0;
	continue;
	}
	value = tx;
	yprev = y;
	break;
	}
	}
	}

	// NOTES: See also: Asymptotic inversion of the incomplete beta function.
	// https://core.ac.uk/download/pdf/82140723.pdf
	//
	// NOTE: See the Boost library documentation as well:
	// https://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html
	}

	} // namespace

	double BetaIncomplete(const double x, const double p, const double q) {
	// Error cases.
	if (p < 0 \|\| q < 0 \|\| x < 0 \|\| x > 1.0) {
	return std::numeric_limits<double>::infinity();
	}
	if (x == 0 \|\| x == 1) {
	return x;
	}
	// ln(Beta(p, q))
	double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
	return BetaIncompleteImpl(x, p, q, beta);
	}

	double BetaIncompleteInv(const double p, const double q, const double alpha) {
	// Error cases.
	if (p < 0 \|\| q < 0 \|\| alpha < 0 \|\| alpha > 1.0) {
	return std::numeric_limits<double>::infinity();
	}
	if (alpha == 0 \|\| alpha == 1) {
	return alpha;
	}
	// ln(Beta(p, q))
	double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
	return BetaIncompleteInvImpl(p, q, beta, alpha);
	}

	// Given `num_trials` trials each with probability `p` of success, the
	// probability of no failures is `p^k`. To ensure the probability of a failure
	// is no more than `p_fail`, it must be that `p^k == 1 - p_fail`. This function
	// computes `p` from that equation.
	double RequiredSuccessProbability(const double p_fail, const int num_trials) {
	double p = std::exp(std::log(1.0 - p_fail) / static_cast<double>(num_trials));
	ABSL_ASSERT(p > 0);
	return p;
	}

	double ZScore(double expected_mean, const DistributionMoments& moments) {
	return (moments.mean - expected_mean) /
	(std::sqrt(moments.variance) /
	std::sqrt(static_cast<double>(moments.n)));
	}

	double MaxErrorTolerance(double acceptance_probability) {
	double one_sided_pvalue = 0.5 * (1.0 - acceptance_probability);
	const double max_err = InverseNormalSurvival(one_sided_pvalue);
	ABSL_ASSERT(max_err > 0);
	return max_err;
	}

	} // namespace random_internal
	ABSL_NAMESPACE_END
	} // namespace absl