Creation of LTS branch "lts_2019_08_08"20190808

  - 9ee91d3e430fb33a4590486573792eb0fa146c2d Export of internal Abseil changes by Abseil Team <absl-team@google.com>
  - 8efba58a3b656e9b41fb0471ae6453425a61c520 Export of internal Abseil changes by Abseil Team <absl-team@google.com>
  - b49b8d16b67ec6912899684b732e6367f258cfdb Export of internal Abseil changes by Abseil Team <absl-team@google.com>
  - 67222ffc4c83d918ce8395aa61769eeb77df4c4d Export of internal Abseil changes by Abseil Team <absl-team@google.com>
  - c5c4db4f5191fe5e76cbf68dcc71fb28702f7d2b Export of internal Abseil changes by Abseil Team <absl-team@google.com>
  - 14550beb3b7b97195e483fb74b5efb906395c31e Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 52e88ee56b72cf32bc66534d942c7398ce481331 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 36d37ab992038f52276ca66b9da80c1cf0f57dc2 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - ad1485c8986246b2ae9105e512738d0e97aec887 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - f3840bc5e33ce4932e35986cf3718450c6f02af2 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 278b26058c036833a4f7f3047d3f4d9296527f87 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - c6c3c1b498e4ee939b24be59cae29d59c3863be8 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 44efe96dfca674a17b45ca53fc77fb69f1e29bf4 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 3c98fcc0461bd2a4b9c149d4748a7373a225cf4b Merge pull request #340 from jtsylve/macos_cxx17_fix by Matt Calabrese <38107210+mattcalabrese-google@users.noreply.github.com>
  - 74d91756c11bc22f9b0108b94da9326f7f9e376f Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - e6b050212c859fbaf67abac76105da10ec348274 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - c964fcffac27bd4a9ff67fe393410dd1146ef8b8 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 72e09a54d993b192db32be14c65adf7e9bd08c31 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - d65e19dfcd8697076f68598c0131c6930cdcd74d Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 5162fc83d2f3b79a9753ed59594c43966afdd37a Merge pull request #336 from shields/patch-2 by Shaindel Schwartz <31392632+shaindelschwartz@users.noreply.github.com>
  - 0389f7bf58fa41f35b3ad60be61d32f31e4f8ed6 Merge pull request #335 from shields/patch-1 by Shaindel Schwartz <31392632+shaindelschwartz@users.noreply.github.com>
  - e9324d926a9189e222741fce6e676f0944661a72 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 43ef2148c0936ebf7cb4be6b19927a9d9d145b8f Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - a13d3df2b3ba68aeead92e2d078fba0510d55024 Merge pull request #323 from gosnik/master by Gennadiy Rozental <rogeeff@google.com>
  - 310a11865c97c5cdcc42a4ee2c2e3578423afe69 Merge pull request #324 from RasPat1/patch-1 by Gennadiy Rozental <rogeeff@google.com>
  - 8f11724067248acc330b4d1f12f0c76d03f2cfb1 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - b1dd425423380126f6441ce4fbb6f8f6c75b793a Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 361cb8a9db2f2130442389fd80593255be26d681 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 0238ab0a831f179518c1a814f9584e99da2d75a3 Merge pull request #321 from christoph-cullmann/c4245_fix... by Xiaoyi Zhang <zhangxy988@gmail.com>
  - 61c9bf3e3e1c28a4aa6d7f1be4b37fd473bb5529 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - bc9101f9982391019521161a36179b52555ed212 Merge pull request #320 from christoph-cullmann/master by Xiaoyi Zhang <zhangxy988@gmail.com>
  - 2f76a9bf50046e396138cc8eeb3cdc17b7a5ac24 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 4adaf5490921f13028b55018c9f550277de5aebb Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 27c30ec671cb7b5ba84c4e79feff7fd0b0ac6338 Avoid undefined behavior when nullptr is passed to memcpy... by Roman Gershman <romange@gmail.com>
  - ce65f5ac3cbf897bb5e3de1a51d80fd00866abaa Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - a18fc7461e7409c2ad64e28537261db1e02e76fa Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 8a394b19c149cab50534b04c5e21d42bc2217a7d Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - daf381e8535a1f1f1b8a75966a74e7cca63dee89 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - fa00c321073c7ea40a4fc3dfc8a06309eae3d025 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 436ba6c4a0ea3a06eca6e055f9c8d296bf3bae12 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 0cbdc774b97f7e80ab60dbe2ed4eaca3b2e33fc8 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 27c2f6e2f3b5929fbd322b0f0ca392eb02efd9f8 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - aa468ad75539619b47979911297efbb629c52e44 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - cd86d0d20ab167c33b23d3875db68d1d4bad3a3b Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 33841c5c963aa9c3f096ef8e6c1e71624b941940 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - ca3f87560a0eef716195cadf66dc6b938a579ec6 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - d902eb869bcfacc1bad14933ed9af4bed006d481 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - a02f62f456f2c4a7ecf2be3104fe0c6e16fbad9a Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 0b545b460141b882b244a1efcef7621d59278160 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - dbae8764fbd429bf7d7745e24bcf73962177a7c0 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 044da8a29c923506af0f0b46bc46f43c1e1300b5 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 6cc6ac44e065b9e8975fadfd6ccb99cbcf89aac4 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 666fc1266bccfd8e6eaaa084e7b42580bb8eb199 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 93dfcf74cb5fccae3da07897d8613ae6cab958a0 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 2c8421e1c6cef0da9e8a20b01c15256ec9ec116d Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 5b65c4af5107176555b23a638e5947686410ac1f Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - eab2078b53c9e3d9d240135c09d27e3393acb50a Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 253eb7416421661873afbaa33828a850db978541 [CMake] Set correct flags for clang-cl (#278) by Loo Rong Jie <loorongjie@gmail.com>
  - e75672f6afc7e8f23ee7b532e86d1b3b9be3984e Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - bf29470384a101b307873b26d358433138c857fc Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 6fd827124facd8336981e73218997f9e73029b4f Merge pull request #280 from chiumichael/master by Derek Mauro <761129+derekmauro@users.noreply.github.com>
  - 7c7754fb3ed9ffb57d35fe8658f3ba4d73a31e72 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 256be563447a315f2a7993ec669460ba475fa86a Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 88a152ae747c3c42dc9167d46c590929b048d436 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - c1cecb25a94c075725e9d2640f6b978a8f61957b Implement Span::first and Span::last from C++20 (#274) by Girts <girtsf@users.noreply.github.com>
  - 38b704384cd2f17590b3922b97744be0b43622c9 Changed HTTP URLs to HTTPS where possible (#270) by nik7273 <nik8470@gmail.com>
  - febc5ee6a92d0eb7dac1fceaa6c648cf6521b4dc Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 9fdf5e5b805412cb2a2e624d3e9a11588120465f Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 419f3184f8ebcdb23105295eadd2a569f3351eb9 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - b312c3cb53a0aad75a85ac2bf57c4a614fbd48d4 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 308ce31528a7edfa39f5f6d36142278a0ae1bf45 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 93d155bc4414f6c121bb1f19dba9fdb27c8943bc Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 426eaa4aa44e4580418bee46c1bd13911151bfb1 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 2901ec32a919311384d6ad4194e2d927c06831f7 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - d78310fe5a82f2e0e6e16509ef8079c8d7e4674e Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - a4cb1c8ba61531a63f9d309eea01ac1d43d8371d Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 540e2537b92cd4abfae6ceddfe24304345461f32 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 89ea0c5ff34aaa5855cfc7aa41f323b8a0ef0ede Merge pull request #255 from uilianries/hotfix/conan by ahedberg <ahedberg@google.com>
  - 5e0dcf72c64fae912184d2e0de87195fe8f0a425 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 0dffca4e36791c7beeda04da10460b534283948a Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 6b4201f9ef650637510a21b8d6cbcc3bee4a606f Fix GCC8 warnings by Boris Staletic <boris.staletic@gmail.com>
  - 0b1e6d417b414aad9282e32e8c49c719edeb63c1 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - efccc502606bed768e50a6cd5806d8eb13e4e935 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 5e6a78131f7bd5940218462c07d88cdefdd75dbe Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 5eea0f713c14ac17788b83e496f11903f8e2bbb0 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 66f9becbb98ecc083f4db349b4b1e0ca9de93b15 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 018b4db1d73ec8238e6dc4b17fd9e1fd7468d0ed Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 9449ae94397f2fd683851348e25ed8c93f75b3b9 Merge pull request #243 from ThomsonTan/FixIntrinsic by Alex Strelnikov <strel@google.com>
  - b16aeb6756bdab08cdf12d40baab5b51f7d15b16 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 7ffbe09f3d85504bd018783bbe1e2c12992fe47c Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 01b471d9f3ebef27f5aaca14b66509099fa8cd6c Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 7bd8f36c741c7cbe311611d7981bf38ba04c6fef Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 968a34ffdaadd7db062a9621dfbdf8b2d16e05af Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 3e2e9b5557e76d098de4b8a2a659125b98ca519b Merge pull request #231 from uilianries/feature/conan by Mark Barolak <mbxx@users.noreply.github.com>
  - 111ca7060a6ff50115ca85b59f6b5d8c8c5e9105 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 389ec3f906f018661a5308458d623d01f96d7b23 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 8fbcdb90952c57828c4a9c2f6d79fcd7cae9088f Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 455dc17ba1af9635f0b60155bc565bc572a1e722 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - f197d7c72a54064cfde5a2058f1513a4a0ee36fb Export of internal Abseil changes. by Abseil Team <absl-team@google.com>
  - 284378a71b32dfb3af4e3661f585e671d1b603a3 Export of internal Abseil changes. by Abseil Team <absl-team@google.com>

GitOrigin-RevId: 9ee91d3e430fb33a4590486573792eb0fa146c2d
Change-Id: Ia06e548bc106cc9d136f6c65714be6645317aced

1 files changed, 418 insertions, 0 deletions

diff --git a/absl/random/internal/distribution_test_util.cc b/absl/random/internal/distribution_test_util.cc
new file mode 100644
index 00000000..4fb4149d
--- /dev/null
+++ b/absl/random/internal/distribution_test_util.cc
@@ -0,0 +1,418 @@
+// 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 {
+inline namespace lts_2019_08_08 {
+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
+}  // inline namespace lts_2019_08_08
+}  // namespace absl