// 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. #ifndef ABSL_RANDOM_GAUSSIAN_DISTRIBUTION_H_ #define ABSL_RANDOM_GAUSSIAN_DISTRIBUTION_H_ // absl::gaussian_distribution implements the Ziggurat algorithm // for generating random gaussian numbers. // // Implementation based on "The Ziggurat Method for Generating Random Variables" // by George Marsaglia and Wai Wan Tsang: http://www.jstatsoft.org/v05/i08/ // #include <cmath> #include <cstdint> #include <istream> #include <limits> #include <type_traits> #include "absl/base/config.h" #include "absl/random/internal/fast_uniform_bits.h" #include "absl/random/internal/generate_real.h" #include "absl/random/internal/iostream_state_saver.h" namespace absl { ABSL_NAMESPACE_BEGIN namespace random_internal { // absl::gaussian_distribution_base implements the underlying ziggurat algorithm // using the ziggurat tables generated by the gaussian_distribution_gentables // binary. // // The specific algorithm has some of the improvements suggested by the // 2005 paper, "An Improved Ziggurat Method to Generate Normal Random Samples", // Jurgen A Doornik. (https://www.doornik.com/research/ziggurat.pdf) class ABSL_DLL gaussian_distribution_base { public: template <typename URBG> inline double zignor(URBG& g); // NOLINT(runtime/references) private: friend class TableGenerator; template <typename URBG> inline double zignor_fallback(URBG& g, // NOLINT(runtime/references) bool neg); // Constants used for the gaussian distribution. static constexpr double kR = 3.442619855899; // Start of the tail. static constexpr double kRInv = 0.29047645161474317; // ~= (1.0 / kR) . static constexpr double kV = 9.91256303526217e-3; static constexpr uint64_t kMask = 0x07f; // The ziggurat tables store the pdf(f) and inverse-pdf(x) for equal-area // points on one-half of the normal distribution, where the pdf function, // pdf = e ^ (-1/2 *x^2), assumes that the mean = 0 & stddev = 1. // // These tables are just over 2kb in size; larger tables might improve the // distributions, but also lead to more cache pollution. // // x = {3.71308, 3.44261, 3.22308, ..., 0} // f = {0.00101, 0.00266, 0.00554, ..., 1} struct Tables { double x[kMask + 2]; double f[kMask + 2]; }; static const Tables zg_; random_internal::FastUniformBits<uint64_t> fast_u64_; }; } // namespace random_internal // absl::gaussian_distribution: // Generates a number conforming to a Gaussian distribution. template <typename RealType = double> class gaussian_distribution : random_internal::gaussian_distribution_base { public: using result_type = RealType; class param_type { public: using distribution_type = gaussian_distribution; explicit param_type(result_type mean = 0, result_type stddev = 1) : mean_(mean), stddev_(stddev) {} // Returns the mean distribution parameter. The mean specifies the location // of the peak. The default value is 0.0. result_type mean() const { return mean_; } // Returns the deviation distribution parameter. The default value is 1.0. result_type stddev() const { return stddev_; } friend bool operator==(const param_type& a, const param_type& b) { return a.mean_ == b.mean_ && a.stddev_ == b.stddev_; } friend bool operator!=(const param_type& a, const param_type& b) { return !(a == b); } private: result_type mean_; result_type stddev_; static_assert( std::is_floating_point<RealType>::value, "Class-template absl::gaussian_distribution<> must be parameterized " "using a floating-point type."); }; gaussian_distribution() : gaussian_distribution(0) {} explicit gaussian_distribution(result_type mean, result_type stddev = 1) : param_(mean, stddev) {} explicit gaussian_distribution(const param_type& p) : param_(p) {} void reset() {} // Generating functions template <typename URBG> result_type operator()(URBG& g) { // NOLINT(runtime/references) return (*this)(g, param_); } template <typename URBG> result_type operator()(URBG& g, // NOLINT(runtime/references) const param_type& p); param_type param() const { return param_; } void param(const param_type& p) { param_ = p; } result_type(min)() const { return -std::numeric_limits<result_type>::infinity(); } result_type(max)() const { return std::numeric_limits<result_type>::infinity(); } result_type mean() const { return param_.mean(); } result_type stddev() const { return param_.stddev(); } friend bool operator==(const gaussian_distribution& a, const gaussian_distribution& b) { return a.param_ == b.param_; } friend bool operator!=(const gaussian_distribution& a, const gaussian_distribution& b) { return a.param_ != b.param_; } private: param_type param_; }; // -------------------------------------------------------------------------- // Implementation details only below // -------------------------------------------------------------------------- template <typename RealType> template <typename URBG> typename gaussian_distribution<RealType>::result_type gaussian_distribution<RealType>::operator()( URBG& g, // NOLINT(runtime/references) const param_type& p) { return p.mean() + p.stddev() * static_cast<result_type>(zignor(g)); } template <typename CharT, typename Traits, typename RealType> std::basic_ostream<CharT, Traits>& operator<<( std::basic_ostream<CharT, Traits>& os, // NOLINT(runtime/references) const gaussian_distribution<RealType>& x) { auto saver = random_internal::make_ostream_state_saver(os); os.precision(random_internal::stream_precision_helper<RealType>::kPrecision); os << x.mean() << os.fill() << x.stddev(); return os; } template <typename CharT, typename Traits, typename RealType> std::basic_istream<CharT, Traits>& operator>>( std::basic_istream<CharT, Traits>& is, // NOLINT(runtime/references) gaussian_distribution<RealType>& x) { // NOLINT(runtime/references) using result_type = typename gaussian_distribution<RealType>::result_type; using param_type = typename gaussian_distribution<RealType>::param_type; auto saver = random_internal::make_istream_state_saver(is); auto mean = random_internal::read_floating_point<result_type>(is); if (is.fail()) return is; auto stddev = random_internal::read_floating_point<result_type>(is); if (!is.fail()) { x.param(param_type(mean, stddev)); } return is; } namespace random_internal { template <typename URBG> inline double gaussian_distribution_base::zignor_fallback(URBG& g, bool neg) { using random_internal::GeneratePositiveTag; using random_internal::GenerateRealFromBits; // This fallback path happens approximately 0.05% of the time. double x, y; do { // kRInv = 1/r, U(0, 1) x = kRInv * std::log(GenerateRealFromBits<double, GeneratePositiveTag, false>( fast_u64_(g))); y = -std::log( GenerateRealFromBits<double, GeneratePositiveTag, false>(fast_u64_(g))); } while ((y + y) < (x * x)); return neg ? (x - kR) : (kR - x); } template <typename URBG> inline double gaussian_distribution_base::zignor( URBG& g) { // NOLINT(runtime/references) using random_internal::GeneratePositiveTag; using random_internal::GenerateRealFromBits; using random_internal::GenerateSignedTag; while (true) { // We use a single uint64_t to generate both a double and a strip. // These bits are unused when the generated double is > 1/2^5. // This may introduce some bias from the duplicated low bits of small // values (those smaller than 1/2^5, which all end up on the left tail). uint64_t bits = fast_u64_(g); int i = static_cast<int>(bits & kMask); // pick a random strip double j = GenerateRealFromBits<double, GenerateSignedTag, false>( bits); // U(-1, 1) const double x = j * zg_.x[i]; // Retangular box. Handles >97% of all cases. // For any given box, this handles between 75% and 99% of values. // Equivalent to U(01) < (x[i+1] / x[i]), and when i == 0, ~93.5% if (std::abs(x) < zg_.x[i + 1]) { return x; } // i == 0: Base box. Sample using a ratio of uniforms. if (i == 0) { // This path happens about 0.05% of the time. return zignor_fallback(g, j < 0); } // i > 0: Wedge samples using precomputed values. double v = GenerateRealFromBits<double, GeneratePositiveTag, false>( fast_u64_(g)); // U(0, 1) if ((zg_.f[i + 1] + v * (zg_.f[i] - zg_.f[i + 1])) < std::exp(-0.5 * x * x)) { return x; } // The wedge was missed; reject the value and try again. } } } // namespace random_internal ABSL_NAMESPACE_END } // namespace absl #endif // ABSL_RANDOM_GAUSSIAN_DISTRIBUTION_H_