<random>: Implement modified ziggurat algorithm for normal_distribution
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@@ -9,6 +9,7 @@ | |
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| #if _STL_COMPILER_PREPROCESSOR | ||
| #include <__msvc_int128.hpp> | ||
| #include <__msvc_random_ziggurat_tables.hpp> | ||
| #include <algorithm> | ||
| #include <cmath> | ||
| #include <cstdint> | ||
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@@ -2645,53 +2646,90 @@ public: | |
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| private: | ||
| template <class _Engine> | ||
| _NODISCARD result_type _Eval_std(_Engine& _Eng) { | ||
| // McFarland, C. D. (2015). A modified ziggurat algorithm for generating exponentially and normally distributed | ||
| // pseudorandom numbers. Journal of Statistical Computation and Simulation, 86(7), 1281-1294. | ||
| // https://doi.org/10.1080/00949655.2015.1060234 | ||
| using _Fty = conditional_t<numeric_limits<_Ty>::digits <= 24, float, double>; | ||
| constexpr const auto& _Tables = _Normal_distribution_tables<_Fty>::_Value; | ||
| using _Traits = remove_reference_t<decltype(_Tables)>; | ||
| _Rng_from_urng_v2<typename _Traits::_Uint_type, _Engine> _Generator(_Eng); | ||
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| const auto _Xbits = static_cast<typename _Traits::_Xtype>(_Generator._Get_all_bits()); | ||
| const auto _Regular_layer = static_cast<uint8_t>(_Xbits); | ||
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| if (_Regular_layer < _Tables._Layer_num - 1) { | ||
| // rectangular region of a middle layer | ||
| return static_cast<_Ty>(_Tables._Layer_widths[_Regular_layer + 1] * static_cast<_Fty>(_Xbits)); | ||
| } else { | ||
| const auto _Lbits = static_cast<typename _Traits::_Uint_type>(_Generator._Get_all_bits()); | ||
| const auto _Index = static_cast<uint8_t>(_Lbits); | ||
| const uint8_t _Irregular_layer = | ||
| _Lbits < _Tables._Alias_probabilities[_Index] ? _Index : _Tables._Alias_indices[_Index]; | ||
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| if (_Irregular_layer < _Tables._Layer_num) { | ||
| // tail of a non-bottommost layer | ||
| const _Fty _Xx0 = _Tables._Layer_widths[_Irregular_layer] * _Tables._Width_scale; | ||
| const _Fty _Xx1 = _Tables._Layer_widths[_Irregular_layer + 1] * _Tables._Width_scale; | ||
| const _Fty _Yy0 = _Tables._Layer_heights[_Irregular_layer]; | ||
| const _Fty _Yy1 = _Tables._Layer_heights[_Irregular_layer + 1]; | ||
| const _Fty _Xdiff = _Xx1 - _Xx0; | ||
| const _Fty _Ydiff = _Yy1 - _Yy0; | ||
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| for (;;) { | ||
| const _Fty _Dx = _Xdiff * _STD _Nrand_impl<_Fty>(_Eng); | ||
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Contributor
There was a problem hiding this comment. The upper bits of |
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| const _Fty _Dy = _Ydiff * _STD _Nrand_impl<_Fty>(_Eng); | ||
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| // _Dexp = -(_Xval^2 - _Xx0^2) / 2 | ||
| // f(_Xval) = exp(-_Xval^2 / 2) = _Yy0 * exp(_Dexp) | ||
| // _Yy0 >= d(f(_Xval))/d(_Dexp) = f(_Xval) >= _Yy1 | ||
| // _Dexp * _Yy0 <= f(_Xval) - _Yy0 <= _Dexp * _Yy1 | ||
| const _Fty _Dexp = -_Dx * (_Xx0 + _Dx * _Fty{0.5}); | ||
| if (_Dy > _Dexp * _Yy1) { | ||
| // _Yval > _Yy0 + _Dexp * _Yy1 >= f(_Xval), reject | ||
| continue; | ||
| } | ||
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| const _Fty _Xval = _Xx0 + _Dx; | ||
| if (_Dy < _Dexp * _Yy0) { | ||
| // _Yval < _Yy0 + _Dexp * _Yy0 <= f(_Xval), accept | ||
| return static_cast<_Ty>(_Xbits < 0 ? -_Xval : _Xval); | ||
| } | ||
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| const _Fty _Yval = _Yy0 + _Dy; | ||
| if (_Yval < _STD exp(_Xval * _Xval * _Fty{-0.5})) { | ||
| return static_cast<_Ty>(_Xbits < 0 ? -_Xval : _Xval); | ||
| } | ||
| } | ||
| } else if (_Irregular_layer == _Tables._Layer_num) { | ||
| // tail of the bottommost layer (infinite width) | ||
| constexpr _Fty _Tail = _Tables._Layer_widths[_Tables._Layer_num] * _Tables._Width_scale; | ||
| constexpr _Fty _Tail_reciprocal = _Fty{1.0} / _Tail; | ||
| for (;;) { | ||
| const _Fty _Xval = -_STD log(_Fty{1.0} - _STD _Nrand_impl<_Fty>(_Eng)) * _Tail_reciprocal; | ||
| const _Fty _Yval = _STD _Nrand_impl<_Fty>(_Eng); | ||
| if (_Yval < _STD exp(_Xval * _Xval * _Fty{-0.5})) { | ||
| const _Fty _Xorig = _Tail + _Xval; | ||
| return static_cast<_Ty>(_Xbits < 0 ? -_Xorig : _Xorig); | ||
| } | ||
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Contributor
There was a problem hiding this comment. Marsaglia's tail method might be faster if we don't have a fast exponential generator, as it only needs |
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| } | ||
| } else { | ||
| // rectangular region of the bottommost layer | ||
| return static_cast<_Ty>(_Tables._Layer_widths[_Tables._Layer_num] * static_cast<_Fty>(_Xbits)); | ||
| } | ||
| } | ||
| } | ||
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| template <class _Engine> | ||
| result_type _Eval(_Engine& _Eng, const param_type& _Par0) { | ||
| return _Eval_std(_Eng) * _Par0._Sigma + _Par0._Mean; | ||
| } | ||
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| param_type _Par; | ||
| // TRANSITION, ABI: _Valid must be initialized to false, and must be reset to false when reset() is called, but is | ||
| // unused by the current implementation | ||
| bool _Valid; | ||
| _Ty _Xx2; // TRANSITION, ABI: unused by the current implementation | ||
| }; | ||
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| _EXPORT_STD template <class _Ty = double> | ||
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In common cases (e.g., CVTSI2SD with ties-to-even) this conversion to floating point is slightly biased toward values where the significand ends in 0. It's also correlated with
_Layerbecause a tie only happens when the lower eight or eleven bits of_Xbitsare 1 followed by all 0s. Occurs below.I'm unsure how much effect this has, but if it doesn't affect the performance much, maybe it would be cleaner to mask or shift out the lower bits as they're used.
[Edit: I shouldn't say that round-to-even bias happens only with those value of the lower bits. I think it's sufficient but not necessary.]