Add 'third_party/abseil_cpp/' from commit '768eb2ca28'

git-subtree-dir: third_party/abseil_cpp
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git-subtree-split: 768eb2ca28

third_party/abseil_cpp/absl/random/internal/chi_square.cc

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+// 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/chi_square.h"
+
+#include <cmath>
+
+#include "absl/random/internal/distribution_test_util.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
+
+// Use Horner's method to evaluate a polynomial.
+template <typename T, unsigned N>
+inline T EvaluatePolynomial(T x, const T (&poly)[N]) {
+#if !defined(__EMSCRIPTEN__)
+  using std::fma;
+#endif
+  T p = poly[N - 1];
+  for (unsigned i = 2; i <= N; i++) {
+    p = fma(p, x, poly[N - i]);
+  }
+  return p;
+}
+
+static constexpr int kLargeDOF = 150;
+
+// Returns the probability of a normal z-value.
+//
+// Adapted from the POZ function in:
+//     Ibbetson D, Algorithm 209
+//     Collected Algorithms of the CACM 1963 p. 616
+//
+double POZ(double z) {
+  static constexpr double kP1[] = {
+      0.797884560593,  -0.531923007300, 0.319152932694,
+      -0.151968751364, 0.059054035642,  -0.019198292004,
+      0.005198775019,  -0.001075204047, 0.000124818987,
+  };
+  static constexpr double kP2[] = {
+      0.999936657524,  0.000535310849,  -0.002141268741, 0.005353579108,
+      -0.009279453341, 0.011630447319,  -0.010557625006, 0.006549791214,
+      -0.002034254874, -0.000794620820, 0.001390604284,  -0.000676904986,
+      -0.000019538132, 0.000152529290,  -0.000045255659,
+  };
+
+  const double kZMax = 6.0;  // Maximum meaningful z-value.
+  if (z == 0.0) {
+    return 0.5;
+  }
+  double x;
+  double y = 0.5 * std::fabs(z);
+  if (y >= (kZMax * 0.5)) {
+    x = 1.0;
+  } else if (y < 1.0) {
+    double w = y * y;
+    x = EvaluatePolynomial(w, kP1) * y * 2.0;
+  } else {
+    y -= 2.0;
+    x = EvaluatePolynomial(y, kP2);
+  }
+  return z > 0.0 ? ((x + 1.0) * 0.5) : ((1.0 - x) * 0.5);
+}
+
+// Approximates the survival function of the normal distribution.
+//
+// Algorithm 26.2.18, from:
+// [Abramowitz and Stegun, Handbook of Mathematical Functions,p.932]
+// http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
+//
+double normal_survival(double z) {
+  // Maybe replace with the alternate formulation.
+  // 0.5 * erfc((x - mean)/(sqrt(2) * sigma))
+  static constexpr double kR[] = {
+      1.0, 0.196854, 0.115194, 0.000344, 0.019527,
+  };
+  double r = EvaluatePolynomial(z, kR);
+  r *= r;
+  return 0.5 / (r * r);
+}
+
+}  // namespace
+
+// Calculates the critical chi-square value given degrees-of-freedom and a
+// p-value, usually using bisection. Also known by the name CRITCHI.
+double ChiSquareValue(int dof, double p) {
+  static constexpr double kChiEpsilon =
+      0.000001;  // Accuracy of the approximation.
+  static constexpr double kChiMax =
+      99999.0;  // Maximum chi-squared value.
+
+  const double p_value = 1.0 - p;
+  if (dof < 1 || p_value > 1.0) {
+    return 0.0;
+  }
+
+  if (dof > kLargeDOF) {
+    // For large degrees of freedom, use the normal approximation by
+    //     Wilson, E. B. and Hilferty, M. M. (1931)
+    //                     chi^2 - mean
+    //                Z = --------------
+    //                        stddev
+    const double z = InverseNormalSurvival(p_value);
+    const double mean = 1 - 2.0 / (9 * dof);
+    const double variance = 2.0 / (9 * dof);
+    // Cannot use this method if the variance is 0.
+    if (variance != 0) {
+      return std::pow(z * std::sqrt(variance) + mean, 3.0) * dof;
+    }
+  }
+
+  if (p_value <= 0.0) return kChiMax;
+
+  // Otherwise search for the p value by bisection
+  double min_chisq = 0.0;
+  double max_chisq = kChiMax;
+  double current = dof / std::sqrt(p_value);
+  while ((max_chisq - min_chisq) > kChiEpsilon) {
+    if (ChiSquarePValue(current, dof) < p_value) {
+      max_chisq = current;
+    } else {
+      min_chisq = current;
+    }
+    current = (max_chisq + min_chisq) * 0.5;
+  }
+  return current;
+}
+
+// Calculates the p-value (probability) of a given chi-square value
+// and degrees of freedom.
+//
+// Adapted from the POCHISQ function from:
+//     Hill, I. D. and Pike, M. C.  Algorithm 299
+//     Collected Algorithms of the CACM 1963 p. 243
+//
+double ChiSquarePValue(double chi_square, int dof) {
+  static constexpr double kLogSqrtPi =
+      0.5723649429247000870717135;  // Log[Sqrt[Pi]]
+  static constexpr double kInverseSqrtPi =
+      0.5641895835477562869480795;  // 1/(Sqrt[Pi])
+
+  // For large degrees of freedom, use the normal approximation by
+  //     Wilson, E. B. and Hilferty, M. M. (1931)
+  // Via Wikipedia:
+  //   By the Central Limit Theorem, because the chi-square distribution is the
+  //   sum of k independent random variables with finite mean and variance, it
+  //   converges to a normal distribution for large k.
+  if (dof > kLargeDOF) {
+    // Re-scale everything.
+    const double chi_square_scaled = std::pow(chi_square / dof, 1.0 / 3);
+    const double mean = 1 - 2.0 / (9 * dof);
+    const double variance = 2.0 / (9 * dof);
+    // If variance is 0, this method cannot be used.
+    if (variance != 0) {
+      const double z = (chi_square_scaled - mean) / std::sqrt(variance);
+      if (z > 0) {
+        return normal_survival(z);
+      } else if (z < 0) {
+        return 1.0 - normal_survival(-z);
+      } else {
+        return 0.5;
+      }
+    }
+  }
+
+  // The chi square function is >= 0 for any degrees of freedom.
+  // In other words, probability that the chi square function >= 0 is 1.
+  if (chi_square <= 0.0) return 1.0;
+
+  // If the degrees of freedom is zero, the chi square function is always 0 by
+  // definition. In other words, the probability that the chi square function
+  // is > 0 is zero (chi square values <= 0 have been filtered above).
+  if (dof < 1) return 0;
+
+  auto capped_exp = [](double x) { return x < -20 ? 0.0 : std::exp(x); };
+  static constexpr double kBigX = 20;
+
+  double a = 0.5 * chi_square;
+  const bool even = !(dof & 1);  // True if dof is an even number.
+  const double y = capped_exp(-a);
+  double s = even ? y : (2.0 * POZ(-std::sqrt(chi_square)));
+
+  if (dof <= 2) {
+    return s;
+  }
+
+  chi_square = 0.5 * (dof - 1.0);
+  double z = (even ? 1.0 : 0.5);
+  if (a > kBigX) {
+    double e = (even ? 0.0 : kLogSqrtPi);
+    double c = std::log(a);
+    while (z <= chi_square) {
+      e = std::log(z) + e;
+      s += capped_exp(c * z - a - e);
+      z += 1.0;
+    }
+    return s;
+  }
+
+  double e = (even ? 1.0 : (kInverseSqrtPi / std::sqrt(a)));
+  double c = 0.0;
+  while (z <= chi_square) {
+    e = e * (a / z);
+    c = c + e;
+    z += 1.0;
+  }
+  return c * y + s;
+}
+
+}  // namespace random_internal
+ABSL_NAMESPACE_END
+}  // namespace absl