// Boost.Polygon library detail/voronoi_robust_fpt.hpp header file
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// Copyright Andrii Sydorchuk 2010-2012.
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or copy at
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// http://www.boost.org/LICENSE_1_0.txt)
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// See http://www.boost.org for updates, documentation, and revision history.
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#ifndef BOOST_POLYGON_DETAIL_VORONOI_ROBUST_FPT
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#define BOOST_POLYGON_DETAIL_VORONOI_ROBUST_FPT
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#include <algorithm>
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#include <cmath>
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// Geometry predicates with floating-point variables usually require
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// high-precision predicates to retrieve the correct result.
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// Epsilon robust predicates give the result within some epsilon relative
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// error, but are a lot faster than high-precision predicates.
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// To make algorithm robust and efficient epsilon robust predicates are
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// used at the first step. In case of the undefined result high-precision
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// arithmetic is used to produce required robustness. This approach
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// requires exact computation of epsilon intervals within which epsilon
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// robust predicates have undefined value.
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// There are two ways to measure an error of floating-point calculations:
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// relative error and ULPs (units in the last place).
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// Let EPS be machine epsilon, then next inequalities have place:
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// 1 EPS <= 1 ULP <= 2 EPS (1), 0.5 ULP <= 1 EPS <= 1 ULP (2).
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// ULPs are good for measuring rounding errors and comparing values.
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// Relative errors are good for computation of general relative
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// error of formulas or expressions. So to calculate epsilon
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// interval within which epsilon robust predicates have undefined result
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// next schema is used:
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// 1) Compute rounding errors of initial variables using ULPs;
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// 2) Transform ULPs to epsilons using upper bound of the (1);
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// 3) Compute relative error of the formula using epsilon arithmetic;
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// 4) Transform epsilon to ULPs using upper bound of the (2);
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// In case two values are inside undefined ULP range use high-precision
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// arithmetic to produce the correct result, else output the result.
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// Look at almost_equal function to see how two floating-point variables
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// are checked to fit in the ULP range.
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// If A has relative error of r(A) and B has relative error of r(B) then:
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// 1) r(A + B) <= max(r(A), r(B)), for A * B >= 0;
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// 2) r(A - B) <= B*r(A)+A*r(B)/(A-B), for A * B >= 0;
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// 2) r(A * B) <= r(A) + r(B);
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// 3) r(A / B) <= r(A) + r(B);
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// In addition rounding error should be added, that is always equal to
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// 0.5 ULP or at most 1 epsilon. As you might see from the above formulas
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// subtraction relative error may be extremely large, that's why
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// epsilon robust comparator class is used to store floating point values
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// and compute subtraction as the final step of the evaluation.
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// For further information about relative errors and ULPs try this link:
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// http://docs.sun.com/source/806-3568/ncg_goldberg.html
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namespace boost {
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namespace polygon {
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namespace detail {
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template <typename T>
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T get_sqrt(const T& that) {
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return (std::sqrt)(that);
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}
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template <typename T>
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bool is_pos(const T& that) {
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return that > 0;
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}
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template <typename T>
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bool is_neg(const T& that) {
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return that < 0;
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}
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template <typename T>
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bool is_zero(const T& that) {
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return that == 0;
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}
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template <typename _fpt>
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class robust_fpt {
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public:
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typedef _fpt floating_point_type;
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typedef _fpt relative_error_type;
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// Rounding error is at most 1 EPS.
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enum {
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ROUNDING_ERROR = 1
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};
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robust_fpt() : fpv_(0.0), re_(0.0) {}
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explicit robust_fpt(floating_point_type fpv) :
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fpv_(fpv), re_(0.0) {}
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robust_fpt(floating_point_type fpv, relative_error_type error) :
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fpv_(fpv), re_(error) {}
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floating_point_type fpv() const { return fpv_; }
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relative_error_type re() const { return re_; }
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relative_error_type ulp() const { return re_; }
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bool has_pos_value() const {
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return is_pos(fpv_);
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}
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bool has_neg_value() const {
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return is_neg(fpv_);
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}
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bool has_zero_value() const {
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return is_zero(fpv_);
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}
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robust_fpt operator-() const {
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return robust_fpt(-fpv_, re_);
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}
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robust_fpt& operator+=(const robust_fpt& that) {
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floating_point_type fpv = this->fpv_ + that.fpv_;
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if ((!is_neg(this->fpv_) && !is_neg(that.fpv_)) ||
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(!is_pos(this->fpv_) && !is_pos(that.fpv_))) {
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this->re_ = (std::max)(this->re_, that.re_) + ROUNDING_ERROR;
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} else {
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floating_point_type temp =
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(this->fpv_ * this->re_ - that.fpv_ * that.re_) / fpv;
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if (is_neg(temp))
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temp = -temp;
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this->re_ = temp + ROUNDING_ERROR;
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}
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this->fpv_ = fpv;
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return *this;
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}
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robust_fpt& operator-=(const robust_fpt& that) {
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floating_point_type fpv = this->fpv_ - that.fpv_;
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if ((!is_neg(this->fpv_) && !is_pos(that.fpv_)) ||
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(!is_pos(this->fpv_) && !is_neg(that.fpv_))) {
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this->re_ = (std::max)(this->re_, that.re_) + ROUNDING_ERROR;
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} else {
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floating_point_type temp =
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(this->fpv_ * this->re_ + that.fpv_ * that.re_) / fpv;
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if (is_neg(temp))
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temp = -temp;
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this->re_ = temp + ROUNDING_ERROR;
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}
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this->fpv_ = fpv;
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return *this;
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}
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robust_fpt& operator*=(const robust_fpt& that) {
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this->re_ += that.re_ + ROUNDING_ERROR;
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this->fpv_ *= that.fpv_;
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return *this;
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}
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robust_fpt& operator/=(const robust_fpt& that) {
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this->re_ += that.re_ + ROUNDING_ERROR;
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this->fpv_ /= that.fpv_;
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return *this;
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}
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robust_fpt operator+(const robust_fpt& that) const {
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floating_point_type fpv = this->fpv_ + that.fpv_;
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relative_error_type re;
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if ((!is_neg(this->fpv_) && !is_neg(that.fpv_)) ||
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(!is_pos(this->fpv_) && !is_pos(that.fpv_))) {
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re = (std::max)(this->re_, that.re_) + ROUNDING_ERROR;
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} else {
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floating_point_type temp =
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(this->fpv_ * this->re_ - that.fpv_ * that.re_) / fpv;
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if (is_neg(temp))
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temp = -temp;
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re = temp + ROUNDING_ERROR;
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}
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return robust_fpt(fpv, re);
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}
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robust_fpt operator-(const robust_fpt& that) const {
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floating_point_type fpv = this->fpv_ - that.fpv_;
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relative_error_type re;
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if ((!is_neg(this->fpv_) && !is_pos(that.fpv_)) ||
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(!is_pos(this->fpv_) && !is_neg(that.fpv_))) {
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re = (std::max)(this->re_, that.re_) + ROUNDING_ERROR;
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} else {
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floating_point_type temp =
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(this->fpv_ * this->re_ + that.fpv_ * that.re_) / fpv;
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if (is_neg(temp))
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temp = -temp;
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re = temp + ROUNDING_ERROR;
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}
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return robust_fpt(fpv, re);
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}
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robust_fpt operator*(const robust_fpt& that) const {
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floating_point_type fpv = this->fpv_ * that.fpv_;
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relative_error_type re = this->re_ + that.re_ + ROUNDING_ERROR;
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return robust_fpt(fpv, re);
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}
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robust_fpt operator/(const robust_fpt& that) const {
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floating_point_type fpv = this->fpv_ / that.fpv_;
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relative_error_type re = this->re_ + that.re_ + ROUNDING_ERROR;
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return robust_fpt(fpv, re);
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}
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robust_fpt sqrt() const {
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return robust_fpt(get_sqrt(fpv_),
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re_ * static_cast<relative_error_type>(0.5) +
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ROUNDING_ERROR);
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}
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private:
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floating_point_type fpv_;
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relative_error_type re_;
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};
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template <typename T>
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robust_fpt<T> get_sqrt(const robust_fpt<T>& that) {
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return that.sqrt();
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}
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template <typename T>
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bool is_pos(const robust_fpt<T>& that) {
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return that.has_pos_value();
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}
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template <typename T>
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bool is_neg(const robust_fpt<T>& that) {
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return that.has_neg_value();
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}
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template <typename T>
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bool is_zero(const robust_fpt<T>& that) {
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return that.has_zero_value();
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}
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// robust_dif consists of two not negative values: value1 and value2.
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// The resulting expression is equal to the value1 - value2.
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// Subtraction of a positive value is equivalent to the addition to value2
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// and subtraction of a negative value is equivalent to the addition to
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// value1. The structure implicitly avoids difference computation.
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template <typename T>
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class robust_dif {
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public:
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robust_dif() :
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positive_sum_(0),
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negative_sum_(0) {}
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explicit robust_dif(const T& value) :
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positive_sum_((value > 0)?value:0),
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negative_sum_((value < 0)?-value:0) {}
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robust_dif(const T& pos, const T& neg) :
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positive_sum_(pos),
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negative_sum_(neg) {}
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T dif() const {
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return positive_sum_ - negative_sum_;
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}
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T pos() const {
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return positive_sum_;
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}
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T neg() const {
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return negative_sum_;
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}
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robust_dif<T> operator-() const {
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return robust_dif(negative_sum_, positive_sum_);
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}
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robust_dif<T>& operator+=(const T& val) {
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if (!is_neg(val))
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positive_sum_ += val;
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else
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negative_sum_ -= val;
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return *this;
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}
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robust_dif<T>& operator+=(const robust_dif<T>& that) {
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positive_sum_ += that.positive_sum_;
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negative_sum_ += that.negative_sum_;
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return *this;
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}
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robust_dif<T>& operator-=(const T& val) {
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if (!is_neg(val))
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negative_sum_ += val;
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else
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positive_sum_ -= val;
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return *this;
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}
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robust_dif<T>& operator-=(const robust_dif<T>& that) {
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positive_sum_ += that.negative_sum_;
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negative_sum_ += that.positive_sum_;
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return *this;
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}
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robust_dif<T>& operator*=(const T& val) {
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if (!is_neg(val)) {
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positive_sum_ *= val;
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negative_sum_ *= val;
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} else {
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positive_sum_ *= -val;
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negative_sum_ *= -val;
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swap();
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}
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return *this;
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}
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robust_dif<T>& operator*=(const robust_dif<T>& that) {
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T positive_sum = this->positive_sum_ * that.positive_sum_ +
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this->negative_sum_ * that.negative_sum_;
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T negative_sum = this->positive_sum_ * that.negative_sum_ +
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this->negative_sum_ * that.positive_sum_;
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positive_sum_ = positive_sum;
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negative_sum_ = negative_sum;
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return *this;
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}
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robust_dif<T>& operator/=(const T& val) {
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if (!is_neg(val)) {
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positive_sum_ /= val;
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negative_sum_ /= val;
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} else {
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positive_sum_ /= -val;
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negative_sum_ /= -val;
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swap();
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}
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return *this;
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}
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private:
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void swap() {
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(std::swap)(positive_sum_, negative_sum_);
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}
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T positive_sum_;
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T negative_sum_;
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};
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template<typename T>
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robust_dif<T> operator+(const robust_dif<T>& lhs,
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const robust_dif<T>& rhs) {
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return robust_dif<T>(lhs.pos() + rhs.pos(), lhs.neg() + rhs.neg());
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}
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template<typename T>
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robust_dif<T> operator+(const robust_dif<T>& lhs, const T& rhs) {
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if (!is_neg(rhs)) {
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return robust_dif<T>(lhs.pos() + rhs, lhs.neg());
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} else {
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return robust_dif<T>(lhs.pos(), lhs.neg() - rhs);
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}
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}
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template<typename T>
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robust_dif<T> operator+(const T& lhs, const robust_dif<T>& rhs) {
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if (!is_neg(lhs)) {
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return robust_dif<T>(lhs + rhs.pos(), rhs.neg());
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} else {
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return robust_dif<T>(rhs.pos(), rhs.neg() - lhs);
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}
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}
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template<typename T>
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robust_dif<T> operator-(const robust_dif<T>& lhs,
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const robust_dif<T>& rhs) {
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return robust_dif<T>(lhs.pos() + rhs.neg(), lhs.neg() + rhs.pos());
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}
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template<typename T>
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robust_dif<T> operator-(const robust_dif<T>& lhs, const T& rhs) {
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if (!is_neg(rhs)) {
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return robust_dif<T>(lhs.pos(), lhs.neg() + rhs);
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} else {
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return robust_dif<T>(lhs.pos() - rhs, lhs.neg());
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}
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}
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template<typename T>
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robust_dif<T> operator-(const T& lhs, const robust_dif<T>& rhs) {
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if (!is_neg(lhs)) {
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return robust_dif<T>(lhs + rhs.neg(), rhs.pos());
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} else {
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return robust_dif<T>(rhs.neg(), rhs.pos() - lhs);
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}
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}
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template<typename T>
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robust_dif<T> operator*(const robust_dif<T>& lhs,
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const robust_dif<T>& rhs) {
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T res_pos = lhs.pos() * rhs.pos() + lhs.neg() * rhs.neg();
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T res_neg = lhs.pos() * rhs.neg() + lhs.neg() * rhs.pos();
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return robust_dif<T>(res_pos, res_neg);
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}
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template<typename T>
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robust_dif<T> operator*(const robust_dif<T>& lhs, const T& val) {
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if (!is_neg(val)) {
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return robust_dif<T>(lhs.pos() * val, lhs.neg() * val);
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} else {
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return robust_dif<T>(-lhs.neg() * val, -lhs.pos() * val);
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}
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}
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template<typename T>
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robust_dif<T> operator*(const T& val, const robust_dif<T>& rhs) {
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if (!is_neg(val)) {
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return robust_dif<T>(val * rhs.pos(), val * rhs.neg());
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} else {
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return robust_dif<T>(-val * rhs.neg(), -val * rhs.pos());
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}
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}
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template<typename T>
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robust_dif<T> operator/(const robust_dif<T>& lhs, const T& val) {
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if (!is_neg(val)) {
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return robust_dif<T>(lhs.pos() / val, lhs.neg() / val);
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} else {
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return robust_dif<T>(-lhs.neg() / val, -lhs.pos() / val);
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}
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}
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// Used to compute expressions that operate with sqrts with predefined
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// relative error. Evaluates expressions of the next type:
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// sum(i = 1 .. n)(A[i] * sqrt(B[i])), 1 <= n <= 4.
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template <typename _int, typename _fpt, typename _converter>
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class robust_sqrt_expr {
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public:
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enum MAX_RELATIVE_ERROR {
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MAX_RELATIVE_ERROR_EVAL1 = 4,
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MAX_RELATIVE_ERROR_EVAL2 = 7,
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MAX_RELATIVE_ERROR_EVAL3 = 16,
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MAX_RELATIVE_ERROR_EVAL4 = 25
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};
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// Evaluates expression (re = 4 EPS):
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// A[0] * sqrt(B[0]).
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_fpt eval1(_int* A, _int* B) {
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_fpt a = convert(A[0]);
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_fpt b = convert(B[0]);
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return a * get_sqrt(b);
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}
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// Evaluates expression (re = 7 EPS):
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// A[0] * sqrt(B[0]) + A[1] * sqrt(B[1]).
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_fpt eval2(_int* A, _int* B) {
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_fpt a = eval1(A, B);
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_fpt b = eval1(A + 1, B + 1);
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if ((!is_neg(a) && !is_neg(b)) ||
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(!is_pos(a) && !is_pos(b)))
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return a + b;
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return convert(A[0] * A[0] * B[0] - A[1] * A[1] * B[1]) / (a - b);
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}
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// Evaluates expression (re = 16 EPS):
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// A[0] * sqrt(B[0]) + A[1] * sqrt(B[1]) + A[2] * sqrt(B[2]).
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_fpt eval3(_int* A, _int* B) {
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_fpt a = eval2(A, B);
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_fpt b = eval1(A + 2, B + 2);
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if ((!is_neg(a) && !is_neg(b)) ||
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(!is_pos(a) && !is_pos(b)))
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return a + b;
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tA[3] = A[0] * A[0] * B[0] + A[1] * A[1] * B[1] - A[2] * A[2] * B[2];
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tB[3] = 1;
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tA[4] = A[0] * A[1] * 2;
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tB[4] = B[0] * B[1];
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return eval2(tA + 3, tB + 3) / (a - b);
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}
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// Evaluates expression (re = 25 EPS):
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// A[0] * sqrt(B[0]) + A[1] * sqrt(B[1]) +
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// A[2] * sqrt(B[2]) + A[3] * sqrt(B[3]).
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_fpt eval4(_int* A, _int* B) {
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_fpt a = eval2(A, B);
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_fpt b = eval2(A + 2, B + 2);
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if ((!is_neg(a) && !is_neg(b)) ||
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(!is_pos(a) && !is_pos(b)))
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return a + b;
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tA[0] = A[0] * A[0] * B[0] + A[1] * A[1] * B[1] -
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A[2] * A[2] * B[2] - A[3] * A[3] * B[3];
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tB[0] = 1;
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tA[1] = A[0] * A[1] * 2;
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tB[1] = B[0] * B[1];
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tA[2] = A[2] * A[3] * -2;
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tB[2] = B[2] * B[3];
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return eval3(tA, tB) / (a - b);
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}
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private:
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_int tA[5];
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_int tB[5];
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_converter convert;
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};
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} // detail
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} // polygon
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} // boost
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#endif // BOOST_POLYGON_DETAIL_VORONOI_ROBUST_FPT
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