// 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_
