234 lines
7.0 KiB
C++
234 lines
7.0 KiB
C++
// Copyright 2017 The Abseil Authors.
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//
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// Licensed under the Apache License, Version 2.0 (the "License");
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// you may not use this file except in compliance with the License.
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// You may obtain a copy of the License at
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//
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// https://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the License is distributed on an "AS IS" BASIS,
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// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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// See the License for the specific language governing permissions and
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// limitations under the License.
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#include "absl/random/internal/chi_square.h"
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#include <cmath>
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#include "absl/random/internal/distribution_test_util.h"
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namespace absl {
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ABSL_NAMESPACE_BEGIN
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namespace random_internal {
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namespace {
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#if defined(__EMSCRIPTEN__)
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// Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
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inline double fma(double x, double y, double z) {
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return (x * y) + z;
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}
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#endif
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// Use Horner's method to evaluate a polynomial.
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template <typename T, unsigned N>
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inline T EvaluatePolynomial(T x, const T (&poly)[N]) {
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#if !defined(__EMSCRIPTEN__)
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using std::fma;
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#endif
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T p = poly[N - 1];
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for (unsigned i = 2; i <= N; i++) {
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p = fma(p, x, poly[N - i]);
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}
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return p;
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}
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static constexpr int kLargeDOF = 150;
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// Returns the probability of a normal z-value.
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//
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// Adapted from the POZ function in:
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// Ibbetson D, Algorithm 209
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// Collected Algorithms of the CACM 1963 p. 616
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//
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double POZ(double z) {
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static constexpr double kP1[] = {
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0.797884560593, -0.531923007300, 0.319152932694,
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-0.151968751364, 0.059054035642, -0.019198292004,
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0.005198775019, -0.001075204047, 0.000124818987,
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};
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static constexpr double kP2[] = {
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0.999936657524, 0.000535310849, -0.002141268741, 0.005353579108,
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-0.009279453341, 0.011630447319, -0.010557625006, 0.006549791214,
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-0.002034254874, -0.000794620820, 0.001390604284, -0.000676904986,
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-0.000019538132, 0.000152529290, -0.000045255659,
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};
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const double kZMax = 6.0; // Maximum meaningful z-value.
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if (z == 0.0) {
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return 0.5;
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}
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double x;
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double y = 0.5 * std::fabs(z);
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if (y >= (kZMax * 0.5)) {
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x = 1.0;
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} else if (y < 1.0) {
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double w = y * y;
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x = EvaluatePolynomial(w, kP1) * y * 2.0;
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} else {
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y -= 2.0;
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x = EvaluatePolynomial(y, kP2);
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}
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return z > 0.0 ? ((x + 1.0) * 0.5) : ((1.0 - x) * 0.5);
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}
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// Approximates the survival function of the normal distribution.
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//
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// Algorithm 26.2.18, from:
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// [Abramowitz and Stegun, Handbook of Mathematical Functions,p.932]
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// http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
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//
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double normal_survival(double z) {
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// Maybe replace with the alternate formulation.
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// 0.5 * erfc((x - mean)/(sqrt(2) * sigma))
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static constexpr double kR[] = {
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1.0, 0.196854, 0.115194, 0.000344, 0.019527,
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};
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double r = EvaluatePolynomial(z, kR);
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r *= r;
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return 0.5 / (r * r);
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}
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} // namespace
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// Calculates the critical chi-square value given degrees-of-freedom and a
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// p-value, usually using bisection. Also known by the name CRITCHI.
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double ChiSquareValue(int dof, double p) {
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static constexpr double kChiEpsilon =
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0.000001; // Accuracy of the approximation.
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static constexpr double kChiMax =
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99999.0; // Maximum chi-squared value.
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const double p_value = 1.0 - p;
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if (dof < 1 || p_value > 1.0) {
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return 0.0;
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}
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if (dof > kLargeDOF) {
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// For large degrees of freedom, use the normal approximation by
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// Wilson, E. B. and Hilferty, M. M. (1931)
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// chi^2 - mean
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// Z = --------------
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// stddev
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const double z = InverseNormalSurvival(p_value);
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const double mean = 1 - 2.0 / (9 * dof);
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const double variance = 2.0 / (9 * dof);
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// Cannot use this method if the variance is 0.
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if (variance != 0) {
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double term = z * std::sqrt(variance) + mean;
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return dof * (term * term * term);
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}
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}
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if (p_value <= 0.0) return kChiMax;
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// Otherwise search for the p value by bisection
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double min_chisq = 0.0;
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double max_chisq = kChiMax;
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double current = dof / std::sqrt(p_value);
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while ((max_chisq - min_chisq) > kChiEpsilon) {
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if (ChiSquarePValue(current, dof) < p_value) {
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max_chisq = current;
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} else {
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min_chisq = current;
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}
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current = (max_chisq + min_chisq) * 0.5;
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}
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return current;
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}
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// Calculates the p-value (probability) of a given chi-square value
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// and degrees of freedom.
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//
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// Adapted from the POCHISQ function from:
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// Hill, I. D. and Pike, M. C. Algorithm 299
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// Collected Algorithms of the CACM 1963 p. 243
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//
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double ChiSquarePValue(double chi_square, int dof) {
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static constexpr double kLogSqrtPi =
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0.5723649429247000870717135; // Log[Sqrt[Pi]]
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static constexpr double kInverseSqrtPi =
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0.5641895835477562869480795; // 1/(Sqrt[Pi])
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// For large degrees of freedom, use the normal approximation by
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// Wilson, E. B. and Hilferty, M. M. (1931)
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// Via Wikipedia:
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// By the Central Limit Theorem, because the chi-square distribution is the
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// sum of k independent random variables with finite mean and variance, it
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// converges to a normal distribution for large k.
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if (dof > kLargeDOF) {
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// Re-scale everything.
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const double chi_square_scaled = std::pow(chi_square / dof, 1.0 / 3);
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const double mean = 1 - 2.0 / (9 * dof);
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const double variance = 2.0 / (9 * dof);
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// If variance is 0, this method cannot be used.
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if (variance != 0) {
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const double z = (chi_square_scaled - mean) / std::sqrt(variance);
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if (z > 0) {
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return normal_survival(z);
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} else if (z < 0) {
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return 1.0 - normal_survival(-z);
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} else {
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return 0.5;
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}
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}
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}
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// The chi square function is >= 0 for any degrees of freedom.
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// In other words, probability that the chi square function >= 0 is 1.
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if (chi_square <= 0.0) return 1.0;
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// If the degrees of freedom is zero, the chi square function is always 0 by
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// definition. In other words, the probability that the chi square function
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// is > 0 is zero (chi square values <= 0 have been filtered above).
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if (dof < 1) return 0;
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auto capped_exp = [](double x) { return x < -20 ? 0.0 : std::exp(x); };
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static constexpr double kBigX = 20;
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double a = 0.5 * chi_square;
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const bool even = !(dof & 1); // True if dof is an even number.
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const double y = capped_exp(-a);
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double s = even ? y : (2.0 * POZ(-std::sqrt(chi_square)));
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if (dof <= 2) {
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return s;
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}
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chi_square = 0.5 * (dof - 1.0);
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double z = (even ? 1.0 : 0.5);
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if (a > kBigX) {
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double e = (even ? 0.0 : kLogSqrtPi);
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double c = std::log(a);
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while (z <= chi_square) {
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e = std::log(z) + e;
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s += capped_exp(c * z - a - e);
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z += 1.0;
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}
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return s;
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}
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double e = (even ? 1.0 : (kInverseSqrtPi / std::sqrt(a)));
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double c = 0.0;
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while (z <= chi_square) {
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e = e * (a / z);
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c = c + e;
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z += 1.0;
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}
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return c * y + s;
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}
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} // namespace random_internal
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ABSL_NAMESPACE_END
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} // namespace absl
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