December | 2013 | BuddyPanda

Automatic differentiation is a great tool for various scientific and financial computing needs. The best approach, I think, is to use operator overloading so that the resulting automatic differentiation class can be a drop-in replacement for common variable types such as double. This approach does make it harder to create the class though, as one has to be careful to avoid temporaries that can drastically slow down calculations. The traditional way to fight temporaries is to use expression templates, but there are other ways around it. I remember at one place I used to work, the approach was to not allow binary operators and only use addition-assignment (and others) operators. Obviously, that approach makes the code a pain to read, but the resulting code was very fast.

Now that I have access to a c++11 development environment, there is a new alternative: to use move constructors and move assignment operators. So, below is my attempt to embrace it in this context. The class ardv is designed to be a drop-in replacement for the double type. A typical usage pattern might be:

     template <class T>
     T addNumbers(T const& a, T const& b) {
         T res = a + b;
         return res;
     }

     adrv a1(1.0, 2);
     adrv b1(2.0, 2);
     a1.d(0) = 1.0;
     b1.d(1) = 1.0;

     double a2(1.0);
     double b2(2.0);

     adrv c1;
     double c2;

     c1 = addNumbers(a1, b1);
     c2 = addNumbers(a2, b2);

     //c1.d(0) is the derivative of c1 with respect to a1
     //c1.d(1) is the derivative of c1 with respect to b1
     std::cout << c1.v() << " " << c1.d(0) << " " << c1.d(1) << std::endl;

template <class T>

T addNumbers(T const& a, T const& b) {

T res = a + b;

return res;

}

adrv a1(1.0, 2);

adrv b1(2.0, 2);

a1.d(0) = 1.0;

b1.d(1) = 1.0;

double a2(1.0);

double b2(2.0);

adrv c1;

double c2;

c1 = addNumbers(a1, b1);

c2 = addNumbers(a2, b2);

//c1.d(0) is the derivative of c1 with respect to a1

//c1.d(1) is the derivative of c1 with respect to b1

std::cout << c1.v() << " " << c1.d(0) << " " << c1.d(1) << std::endl;

And below is the actual class. Note that I make absolutely no guarantee as to its accuracy.

 static const double AutoDiff_pi = 4.0 * atan(1.0);
 static const double AutoDiff_2_over_sqrt_pi = 2.0 / std::sqrt(AutoDiff_pi);

 template <class T>
 class adrv {
     T v_;
     std::vector<T> deriv_;
 public:
     typedef T value_type;

 	/** Constructors **/
     adrv() : v_(0.0) {}
     explicit adrv(T const& x) : v_(x) {}
     explicit adrv(int const& x) : v_(static_cast<T>(x)) {}
     adrv(T const& x, std::size_t const& nDerivs) : v_(x), deriv_(std::vector<T>(nDerivs, 0)) {}

     /** Copy constructor **/
     adrv(adrv<T> const& orig);

     /** Move constructor **/
     adrv(adrv<T>&& orig);

 	/** Accessors **/
     inline T& v() { return v_; }
     inline const T v() const { return v_; }
     inline const T d(std::size_t const& indx) const { return deriv_[indx]; }
     inline T& d(std::size_t const& indx) { return deriv_[indx]; }
     inline std::vector<T>& d() { return deriv_; }

     inline std::size_t numDerivs() const { return deriv_.size(); }
     inline std::size_t size() const { return deriv_.size(); }

     adrv<T>& operator=(T const& rhs);
     adrv<T>& operator=(adrv<T> const& rhs); //Copy assignment operator
     adrv<T>& operator=(adrv<T>&& rhs); //Move assignment operator

     adrv<T>& operator+=(T const& x);
     adrv<T>& operator-=(T const& x);
     adrv<T>& operator*=(T const& x);
     adrv<T>& operator/=(T const& x);

     adrv<T>& operator+=(adrv<T> const& x);
     adrv<T>& operator-=(adrv<T> const& x);
     adrv<T>& operator*=(adrv<T> const& x);
     adrv<T>& operator/=(adrv<T> const& x);

     adrv<T>& operator-();
     adrv<T>& operator+();

     inline bool operator>(T const& x) const;
     inline bool operator>=(T const& x) const;
     inline bool operator<(T const& x) const;
     inline bool operator<=(T const& x) const;
     inline bool operator==(T const& x) const;
     inline bool operator!=(T const& x) const;

     inline bool operator>(adrv<T> const& x) const;
     inline bool operator>=(adrv<T> const& x) const;
     inline bool operator<(adrv<T> const& x) const;
     inline bool operator<=(adrv<T> const& x) const;
     inline bool operator==(adrv<T> const& x) const;
     inline bool operator!=(adrv<T> const& x) const;

 	/** basic math functions **/
     template <class U> friend adrv<U> sqrt(adrv<U> x);
     template <class U> friend adrv<U> exp(adrv<U> x);
     template <class U> friend adrv<U> log(adrv<U> x);
     template <class U> friend adrv<U> erfc(adrv<U> x);
     template <class U> friend adrv<U> sin(adrv<U> x);
     template <class U> friend adrv<U> cos(adrv<U> x);
     template <class U> friend adrv<U> tan(adrv<U> x);s
     template <class U> friend adrv<U> sinh(adrv<U> x);
     template <class U> friend adrv<U> cosh(adrv<U> x);
     template <class U> friend adrv<U> tanh(adrv<U> x);
     template <class U> friend adrv<U> abs(adrv<U> x);

 };

 //-----------------------------------------------------------------------------------------------
 // Definitions:
 //-----------------------------------------------------------------------------------------------

 template <class T>
 adrv<T>::adrv(adrv<T> const& orig) {
     if (this != &orig) {
         v_ = orig.v_;
         deriv_ = orig.deriv_;
     }
 }

 template <class T>
 adrv<T>::adrv(adrv<T>&& orig)
 : v_( std::move(orig.v_) ), deriv_(std::move(orig.deriv_))
 {}

 template <class T> inline adrv<T>& adrv<T>::operator=(T const& rhs) {
     v_ = rhs;
     if (deriv_.size() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = 0.0;
         }
     }
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator=(adrv<T> const& rhs) {
     if (this != &rhs) {
         v_ = rhs.v_;
         if (deriv_.size() != rhs.size()) deriv_.resize(rhs.size());
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = rhs.d(indx);
         }
     }
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator=(adrv<T>&& rhs) {
     v_ = std::move(rhs.v_);
     deriv_ = std::move(rhs.deriv_);
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator+=(T const& x) {
     v_ += x;
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator-=(T const& x) {
     v_ -= x;
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator*=(T const& x) {
     v_ *= x;
     for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
         deriv_[indx] *= x;
     }
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator/=(T const& x) {
     v_ /= x;
     for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
         deriv_[indx] /= x;
     }
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator+=(adrv<T> const& x) {
     if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {
         throw "sizes of derivative vectors must match";
     }

     //allow 0 size... in this case, resize the current d_ vector to the size of x.d()
     if (deriv_.size() == 0 && x.numDerivs() > 0)
         deriv_.resize(x.numDerivs(), 0.0);

     if (x.numDerivs() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] += x.d(indx);
         }
     }
     v_ += x.v();
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator-=(adrv<T> const& x) {
     if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {
         throw "sizes of derivative vectors must match";
     }

     //allow 0 size... in this case, resize the current d_ vector to the size of x.d()
     if (deriv_.size() == 0 && x.size() > 0)
         deriv_.resize(x.size(), 0.0);

     if (x.numDerivs() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] -= x.d(indx);
         }
     }
     v_ -= x.v();
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator*=(adrv<T> const& x) {
     if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {
         throw "sizes of derivative vectors must match";
     }

     //allow 0 size... in this case, resize the current d_ vector to the size of x.d()
     if (deriv_.size() == 0 && x.numDerivs() > 0)
         deriv_.resize(x.numDerivs(), 0.0);

     const T xv = x.v();
     if (x.numDerivs() > 0 && deriv_.size() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = v_ * x.d(indx) + deriv_[indx] * xv;
         }
     } else if (x.numDerivs() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = v_ * x.d(indx);
         }
     } else {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = deriv_[indx] * xv;
         }
     }
     v_ *= x.v();
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator/=(adrv<T> const& x) {
     if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {
         throw "sizes of derivative vectors must match";
     }

     //allow 0 size... in this case, resize the current d_ vector to the size of x.d()
     if (deriv_.size() == 0 && x.size() > 0)
         deriv_.resize(x.size(), 0.0);

     T xv = x.v();
     if (x.numDerivs() > 0 && deriv_.size() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = (deriv_[indx] * xv - v_ * x.d(indx)) / (xv * xv);
         }
     } else if (x.numDerivs() > 0) {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = -v_ * x.d(indx) / (xv * xv);
         }
     } else {
         for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {
             deriv_[indx] = deriv_[indx] / xv;
         }
     }
     v_ /= x.v();
     return *this;
 }

 //--------------------------------------------------------------------------------------
 //--------------------------------------------------------------------------------------
 // Unary operators
 //--------------------------------------------------------------------------------------
 //--------------------------------------------------------------------------------------

 template <class T> inline adrv<T>& adrv<T>::operator-() {
     for(std::size_t indx = 0; indx < deriv_.size(); ++indx) {
         deriv_[indx] = -deriv_[indx];
     }
     v_ = -v_;
     return *this;
 }

 template <class T> inline adrv<T>& adrv<T>::operator+() {
     return *this;
 }

 //--------------------------------------------------------------------------------------
 //--------------------------------------------------------------------------------------
 // Basic binary arithmetics
 //--------------------------------------------------------------------------------------
 //--------------------------------------------------------------------------------------

 template <class T> adrv<T> operator+(adrv<T> lhs, adrv<T> const& rhs) {
     lhs += rhs; return lhs;
 }

 template <class T> adrv<T> operator+(adrv<T> lhs, T const& rhs) {
     lhs += rhs; return lhs;
 }

 template <class T> adrv<T> operator+(T const& lhs, adrv<T> rhs) {
     rhs += lhs; return rhs;
 }

 template <class T> adrv<T> operator-(adrv<T> lhs, adrv<T> const& rhs) {
     lhs -= rhs; return lhs;
 }

 template <class T> adrv<T> operator-(adrv<T> lhs, T const& rhs) {
     lhs -= rhs; return lhs;
 }

 template <class T> adrv<T> operator-(T const& lhs, adrv<T> rhs) {
     rhs = -rhs;
     rhs += lhs;
     return rhs;
 }

 template <class T> adrv<T> operator*(adrv<T> lhs, adrv<T> const& rhs) {
     lhs *= rhs; return lhs;
 }

 template <class T> adrv<T> operator*(adrv<T> lhs, T const& rhs) {
     lhs *= rhs; return lhs;
 }

 template <class T> adrv<T> operator*(T const& lhs, adrv<T> rhs) {
     rhs *= lhs; return rhs;
 }

 template <class T> adrv<T> operator/(adrv<T> lhs, adrv<T> const& rhs) {
     lhs /= rhs; return lhs;
 }

 template <class T> adrv<T> operator/(adrv<T> lhs, T const& rhs) {
     lhs /= rhs; return lhs;
 }

 template <class T> adrv<T> operator/(T const& lhs, adrv<T> rhs) {
     for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
         rhs.d()[indx] *= -lhs / (rhs.v() * rhs.v());
     }
     rhs.v() = lhs / rhs.v();
     return rhs;
 }

 template <class T> adrv<T> pow(adrv<T> lhs, adrv<T> const& rhs) {
     if (lhs.numDerivs() != rhs.numDerivs() && (lhs.numDerivs() != 0 && rhs.numDerivs() != 0)) {
         throw "sizes of derivative vectors must match";
     }

     if (lhs.numDerivs() > 0 && rhs.numDerivs() > 0) {
         const T log_lhs = std::log(lhs.v());
         const T pow_term = std::pow(lhs.v(), rhs.v() - 1.0);
         for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
             lhs.d()[indx] = rhs.v() * pow_term * lhs.d(indx) + log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);
         }
     } else if (lhs.numDerivs() > 0) {
         const T pow_term = std::pow(lhs.v(), rhs.v() - 1.0);
         for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
             lhs.d()[indx] = rhs.v() * pow_term * lhs.d(indx);
         }
     } else {
         const T log_lhs = std::log(lhs.v());
         for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
             lhs.d()[indx] = log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);
         }
     }
     lhs.v() = std::pow(lhs.v(), rhs.v());
     return lhs;
 }

 template <class T> adrv<T> pow(adrv<T> lhs, T const& rhs) {
     const T pow_term = std::pow(lhs.v(), rhs - 1.0);
     for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
         lhs.d()[indx] = rhs * pow_term * lhs.d(indx);
     }
     lhs.v() = std::pow(lhs.v(), rhs.v());
     return lhs;
 }

 template <class T> adrv<T> pow(T const& lhs, adrv<T> rhs) {
     const T log_lhs = std::log(lhs);
     for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {
         rhs.d()[indx] = log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);
     }
     rhs.v() = std::pow(lhs, rhs.v());
     return rhs;
 }

 //---------------------------------------------------------------------------------------------
 //---------------------------------------------------------------------------------------------
 // Functions
 //---------------------------------------------------------------------------------------------
 //---------------------------------------------------------------------------------------------
 template <class U> adrv<U> sqrt(adrv<U> x) {
     U tmp = 0.5 / std::sqrt(x.v());
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= tmp;
     }
     x.v() = std::sqrt(x.v());
     return x;
 }

 template <class U> adrv<U> exp(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= std::exp(x.v());
     }
     x.v() = std::exp(x.v());
     return x;
 }

 template <class U> adrv<U> log(adrv<U> x) {
     U tmp = 1.0 / x.v();
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= tmp;
     }
     x.v() = std::log(x.v());
     return x;
 }

 template <class U> adrv<U> erfc(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= -AutoDiff_2_over_sqrt_pi*std::exp(-x.v() * x.v());
     }
     x.v() = std::erfc(x.v());
     return x;
 }

 template <class U> adrv<U> sin(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= std::cos(x.v());
     }
     x.v() = std::sin(x.v());
     return x;
 }

 template <class U> adrv<U> cos(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= -std::sin(x.v());
     }
     x.v() = std::cos(x.v());
     return x;
 }

 template <class U> adrv<U> tan(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] /= std::cos(x.v()) * std::cos(x.v());
     }
     x.v() = std::tan(x.v());
     return x;
 }

 template <class U> adrv<U> sinh(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= std::cosh(x.v());
     }
     x.v() = std::sinh(x.v());
     return x;
 }

 template <class U> adrv<U> cosh(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] *= std::sinh(x.v());
     }
     x.v() = std::cosh(x.v());
     return x;
 }

 template <class U> adrv<U> tanh(adrv<U> x) {
     for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
         x.deriv_[indx] /= std::cosh(x.v()) * std::cosh(x.v());
     }
     x.v() = std::tanh(x.v());
     return x;
 }

 template <class U> adrv<U> abs(adrv<U> x) {
     if (x.v() < 0) {
         for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {
             x.deriv_[indx] = -x.deriv_[indx];
         }
     }
     x.v() = std::abs(x.v());
     return x;
 }

 //------------------------------------------------------------------------------------------------
 //------------------------------------------------------------------------------------------------
 // Various comparison operators
 //------------------------------------------------------------------------------------------------
 //------------------------------------------------------------------------------------------------
 template <class T> inline bool adrv<T>::operator> (T const& x) const { return v_ >  x; }
 template <class T> inline bool adrv<T>::operator>=(T const& x) const { return v_ >= x; }
 template <class T> inline bool adrv<T>::operator< (T const& x) const { return v_ <  x; }
 template <class T> inline bool adrv<T>::operator<=(T const& x) const { return v_ <= x; }
 template <class T> inline bool adrv<T>::operator==(T const& x) const { return v_ == x; }
 template <class T> inline bool adrv<T>::operator!=(T const& x) const { return v_ != x; }

 template <class T> inline bool adrv<T>::operator> (adrv<T> const& x) const { return v_ >  x.v_; }
 template <class T> inline bool adrv<T>::operator>=(adrv<T> const& x) const { return v_ >= x.v_; }
 template <class T> inline bool adrv<T>::operator< (adrv<T> const& x) const { return v_ <  x.v_; }
 template <class T> inline bool adrv<T>::operator<=(adrv<T> const& x) const { return v_ <= x.v_; }
 template <class T> inline bool adrv<T>::operator==(adrv<T> const& x) const { return v_ == x.v_; }
 template <class T> inline bool adrv<T>::operator!=(adrv<T> const& x) const { return v_ != x.v_; }

 template <class T, class U> bool operator> (adrv<T> const& expr, U const& val) { return expr.v() >  static_cast<T>(val); }
 template <class T, class U> bool operator>=(adrv<T> const& expr, U const& val) { return expr.v() >= static_cast<T>(val); }
 template <class T, class U> bool operator< (adrv<T> const& expr, U const& val) { return expr.v() <  static_cast<T>(val); }
 template <class T, class U> bool operator<=(adrv<T> const& expr, U const& val) { return expr.v() <= static_cast<T>(val); }
 template <class T, class U> bool operator==(adrv<T> const& expr, U const& val) { return expr.v() == static_cast<T>(val); }
 template <class T, class U> bool operator!=(adrv<T> const& expr, U const& val) { return expr.v() != static_cast<T>(val); }

 template <class T, class U> bool operator> (U const& val, adrv<T> const& expr) { return static_cast<T>(val) >  expr.v(); }
 template <class T, class U> bool operator>=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) >= expr.v(); }
 template <class T, class U> bool operator< (U const& val, adrv<T> const& expr) { return static_cast<T>(val) <  expr.v(); }
 template <class T, class U> bool operator<=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) <= expr.v(); }
 template <class T, class U> bool operator==(U const& val, adrv<T> const& expr) { return static_cast<T>(val) == expr.v(); }
 template <class T, class U> bool operator!=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) != expr.v(); }

 /** IO operators **/
 template <class T> std::ostream& operator<<(std::ostream& os, adrv<T> const& val) {
     os <<  val.v();
     return os;
 }

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static const double AutoDiff_pi = 4.0 * atan(1.0);

static const double AutoDiff_2_over_sqrt_pi = 2.0 / std::sqrt(AutoDiff_pi);

template <class T>

class adrv {

T v_;

std::vector<T> deriv_;

public:

typedef T value_type;

/** Constructors **/

adrv() : v_(0.0) {}

explicit adrv(T const& x) : v_(x) {}

explicit adrv(int const& x) : v_(static_cast<T>(x)) {}

adrv(T const& x, std::size_t const& nDerivs) : v_(x), deriv_(std::vector<T>(nDerivs, 0)) {}

/** Copy constructor **/

adrv(adrv<T> const& orig);

/** Move constructor **/

adrv(adrv<T>&& orig);

/** Accessors **/

inline T& v() { return v_; }

inline const T v() const { return v_; }

inline const T d(std::size_t const& indx) const { return deriv_[indx]; }

inline T& d(std::size_t const& indx) { return deriv_[indx]; }

inline std::vector<T>& d() { return deriv_; }

inline std::size_t numDerivs() const { return deriv_.size(); }

inline std::size_t size() const { return deriv_.size(); }

adrv<T>& operator=(T const& rhs);

adrv<T>& operator=(adrv<T> const& rhs); //Copy assignment operator

adrv<T>& operator=(adrv<T>&& rhs); //Move assignment operator

adrv<T>& operator+=(T const& x);

adrv<T>& operator-=(T const& x);

adrv<T>& operator*=(T const& x);

adrv<T>& operator/=(T const& x);

adrv<T>& operator+=(adrv<T> const& x);

adrv<T>& operator-=(adrv<T> const& x);

adrv<T>& operator*=(adrv<T> const& x);

adrv<T>& operator/=(adrv<T> const& x);

adrv<T>& operator-();

adrv<T>& operator+();

inline bool operator>(T const& x) const;

inline bool operator>=(T const& x) const;

inline bool operator<(T const& x) const;

inline bool operator<=(T const& x) const;

inline bool operator==(T const& x) const;

inline bool operator!=(T const& x) const;

inline bool operator>(adrv<T> const& x) const;

inline bool operator>=(adrv<T> const& x) const;

inline bool operator<(adrv<T> const& x) const;

inline bool operator<=(adrv<T> const& x) const;

inline bool operator==(adrv<T> const& x) const;

inline bool operator!=(adrv<T> const& x) const;

/** basic math functions **/

template <class U> friend adrv sqrt(adrv x);

template <class U> friend adrv exp(adrv x);

template <class U> friend adrv log(adrv x);

template <class U> friend adrv erfc(adrv x);

template <class U> friend adrv sin(adrv x);

template <class U> friend adrv cos(adrv x);

template <class U> friend adrv tan(adrv x);s

template <class U> friend adrv sinh(adrv x);

template <class U> friend adrv cosh(adrv x);

template <class U> friend adrv tanh(adrv x);

template <class U> friend adrv abs(adrv x);

};

//-----------------------------------------------------------------------------------------------

// Definitions:

//-----------------------------------------------------------------------------------------------

template <class T>

adrv<T>::adrv(adrv<T> const& orig) {

if (this != &orig) {

v_ = orig.v_;

deriv_ = orig.deriv_;

}

template <class T>

adrv<T>::adrv(adrv<T>&& orig)

: v_( std::move(orig.v_) ), deriv_(std::move(orig.deriv_))

{}

template <class T> inline adrv<T>& adrv<T>::operator=(T const& rhs) {

v_ = rhs;

if (deriv_.size() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = 0.0;

}

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator=(adrv<T> const& rhs) {

if (this != &rhs) {

v_ = rhs.v_;

if (deriv_.size() != rhs.size()) deriv_.resize(rhs.size());

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = rhs.d(indx);

}

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator=(adrv<T>&& rhs) {

v_ = std::move(rhs.v_);

deriv_ = std::move(rhs.deriv_);

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator+=(T const& x) {

v_ += x;

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator-=(T const& x) {

v_ -= x;

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator*=(T const& x) {

v_ *= x;

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] *= x;

}

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator/=(T const& x) {

v_ /= x;

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] /= x;

}

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator+=(adrv<T> const& x) {

if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {

throw "sizes of derivative vectors must match";

}

//allow 0 size... in this case, resize the current d_ vector to the size of x.d()

if (deriv_.size() == 0 && x.numDerivs() > 0)

deriv_.resize(x.numDerivs(), 0.0);

if (x.numDerivs() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] += x.d(indx);

}

v_ += x.v();

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator-=(adrv<T> const& x) {

if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {

throw "sizes of derivative vectors must match";

}

//allow 0 size... in this case, resize the current d_ vector to the size of x.d()

if (deriv_.size() == 0 && x.size() > 0)

deriv_.resize(x.size(), 0.0);

if (x.numDerivs() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] -= x.d(indx);

}

v_ -= x.v();

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator*=(adrv<T> const& x) {

if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {

throw "sizes of derivative vectors must match";

}

//allow 0 size... in this case, resize the current d_ vector to the size of x.d()

if (deriv_.size() == 0 && x.numDerivs() > 0)

deriv_.resize(x.numDerivs(), 0.0);

const T xv = x.v();

if (x.numDerivs() > 0 && deriv_.size() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = v_ * x.d(indx) + deriv_[indx] * xv;

}

} else if (x.numDerivs() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = v_ * x.d(indx);

}

} else {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = deriv_[indx] * xv;

}

v_ *= x.v();

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator/=(adrv<T> const& x) {

if (x.numDerivs() != deriv_.size() && (deriv_.size() != 0 && x.numDerivs() != 0)) {

throw "sizes of derivative vectors must match";

}

//allow 0 size... in this case, resize the current d_ vector to the size of x.d()

if (deriv_.size() == 0 && x.size() > 0)

deriv_.resize(x.size(), 0.0);

T xv = x.v();

if (x.numDerivs() > 0 && deriv_.size() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = (deriv_[indx] * xv - v_ * x.d(indx)) / (xv * xv);

}

} else if (x.numDerivs() > 0) {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = -v_ * x.d(indx) / (xv * xv);

}

} else {

for(std::size_t indx = 0, endIndx = deriv_.size(); indx < endIndx; ++indx) {

deriv_[indx] = deriv_[indx] / xv;

}

v_ /= x.v();

return *this;

}

//--------------------------------------------------------------------------------------

// Unary operators

//--------------------------------------------------------------------------------------

template <class T> inline adrv<T>& adrv<T>::operator-() {

for(std::size_t indx = 0; indx < deriv_.size(); ++indx) {

deriv_[indx] = -deriv_[indx];

}

v_ = -v_;

return *this;

}

template <class T> inline adrv<T>& adrv<T>::operator+() {

return *this;

}

//--------------------------------------------------------------------------------------

// Basic binary arithmetics

//--------------------------------------------------------------------------------------

template <class T> adrv<T> operator+(adrv<T> lhs, adrv<T> const& rhs) {

lhs += rhs; return lhs;

}

template <class T> adrv<T> operator+(adrv<T> lhs, T const& rhs) {

lhs += rhs; return lhs;

}

template <class T> adrv<T> operator+(T const& lhs, adrv<T> rhs) {

rhs += lhs; return rhs;

}

template <class T> adrv<T> operator-(adrv<T> lhs, adrv<T> const& rhs) {

lhs -= rhs; return lhs;

}

template <class T> adrv<T> operator-(adrv<T> lhs, T const& rhs) {

lhs -= rhs; return lhs;

}

template <class T> adrv<T> operator-(T const& lhs, adrv<T> rhs) {

rhs = -rhs;

rhs += lhs;

return rhs;

}

template <class T> adrv<T> operator*(adrv<T> lhs, adrv<T> const& rhs) {

lhs *= rhs; return lhs;

}

template <class T> adrv<T> operator*(adrv<T> lhs, T const& rhs) {

lhs *= rhs; return lhs;

}

template <class T> adrv<T> operator*(T const& lhs, adrv<T> rhs) {

rhs *= lhs; return rhs;

}

template <class T> adrv<T> operator/(adrv<T> lhs, adrv<T> const& rhs) {

lhs /= rhs; return lhs;

}

template <class T> adrv<T> operator/(adrv<T> lhs, T const& rhs) {

lhs /= rhs; return lhs;

}

template <class T> adrv<T> operator/(T const& lhs, adrv<T> rhs) {

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

rhs.d()[indx] *= -lhs / (rhs.v() * rhs.v());

}

rhs.v() = lhs / rhs.v();

return rhs;

}

template <class T> adrv<T> pow(adrv<T> lhs, adrv<T> const& rhs) {

if (lhs.numDerivs() != rhs.numDerivs() && (lhs.numDerivs() != 0 && rhs.numDerivs() != 0)) {

throw "sizes of derivative vectors must match";

}

if (lhs.numDerivs() > 0 && rhs.numDerivs() > 0) {

const T log_lhs = std::log(lhs.v());

const T pow_term = std::pow(lhs.v(), rhs.v() - 1.0);

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

lhs.d()[indx] = rhs.v() * pow_term * lhs.d(indx) + log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);

}

} else if (lhs.numDerivs() > 0) {

const T pow_term = std::pow(lhs.v(), rhs.v() - 1.0);

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

lhs.d()[indx] = rhs.v() * pow_term * lhs.d(indx);

}

} else {

const T log_lhs = std::log(lhs.v());

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

lhs.d()[indx] = log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);

}

lhs.v() = std::pow(lhs.v(), rhs.v());

return lhs;

}

template <class T> adrv<T> pow(adrv<T> lhs, T const& rhs) {

const T pow_term = std::pow(lhs.v(), rhs - 1.0);

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

lhs.d()[indx] = rhs * pow_term * lhs.d(indx);

}

lhs.v() = std::pow(lhs.v(), rhs.v());

return lhs;

}

template <class T> adrv<T> pow(T const& lhs, adrv<T> rhs) {

const T log_lhs = std::log(lhs);

for(std::size_t indx = 0; indx < rhs.numDerivs(); ++indx) {

rhs.d()[indx] = log_lhs * std::exp(rhs.v() * log_lhs) * rhs.d(indx);

}

rhs.v() = std::pow(lhs, rhs.v());

return rhs;

}

//---------------------------------------------------------------------------------------------

// Functions

//---------------------------------------------------------------------------------------------

template <class U> adrv sqrt(adrv x) {

U tmp = 0.5 / std::sqrt(x.v());

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= tmp;

}

x.v() = std::sqrt(x.v());

return x;

}

template <class U> adrv exp(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= std::exp(x.v());

}

x.v() = std::exp(x.v());

return x;

}

template <class U> adrv log(adrv x) {

U tmp = 1.0 / x.v();

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= tmp;

}

x.v() = std::log(x.v());

return x;

}

template <class U> adrv erfc(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= -AutoDiff_2_over_sqrt_pi*std::exp(-x.v() * x.v());

}

x.v() = std::erfc(x.v());

return x;

}

template <class U> adrv sin(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= std::cos(x.v());

}

x.v() = std::sin(x.v());

return x;

}

template <class U> adrv cos(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= -std::sin(x.v());

}

x.v() = std::cos(x.v());

return x;

}

template <class U> adrv tan(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] /= std::cos(x.v()) * std::cos(x.v());

}

x.v() = std::tan(x.v());

return x;

}

template <class U> adrv sinh(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= std::cosh(x.v());

}

x.v() = std::sinh(x.v());

return x;

}

template <class U> adrv cosh(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] *= std::sinh(x.v());

}

x.v() = std::cosh(x.v());

return x;

}

template <class U> adrv tanh(adrv x) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] /= std::cosh(x.v()) * std::cosh(x.v());

}

x.v() = std::tanh(x.v());

return x;

}

template <class U> adrv abs(adrv x) {

if (x.v() < 0) {

for(std::size_t indx = 0; indx < x.numDerivs(); ++indx) {

x.deriv_[indx] = -x.deriv_[indx];

}

x.v() = std::abs(x.v());

return x;

}

//------------------------------------------------------------------------------------------------

// Various comparison operators

//------------------------------------------------------------------------------------------------

template <class T> inline bool adrv<T>::operator> (T const& x) const { return v_ > x; }

template <class T> inline bool adrv<T>::operator>=(T const& x) const { return v_ >= x; }

template <class T> inline bool adrv<T>::operator< (T const& x) const { return v_ < x; }

template <class T> inline bool adrv<T>::operator<=(T const& x) const { return v_ <= x; }

template <class T> inline bool adrv<T>::operator==(T const& x) const { return v_ == x; }

template <class T> inline bool adrv<T>::operator!=(T const& x) const { return v_ != x; }

template <class T> inline bool adrv<T>::operator> (adrv<T> const& x) const { return v_ > x.v_; }

template <class T> inline bool adrv<T>::operator>=(adrv<T> const& x) const { return v_ >= x.v_; }

template <class T> inline bool adrv<T>::operator< (adrv<T> const& x) const { return v_ < x.v_; }

template <class T> inline bool adrv<T>::operator<=(adrv<T> const& x) const { return v_ <= x.v_; }

template <class T> inline bool adrv<T>::operator==(adrv<T> const& x) const { return v_ == x.v_; }

template <class T> inline bool adrv<T>::operator!=(adrv<T> const& x) const { return v_ != x.v_; }

template <class T, class U> bool operator> (adrv<T> const& expr, U const& val) { return expr.v() > static_cast<T>(val); }

template <class T, class U> bool operator>=(adrv<T> const& expr, U const& val) { return expr.v() >= static_cast<T>(val); }

template <class T, class U> bool operator< (adrv<T> const& expr, U const& val) { return expr.v() < static_cast<T>(val); }

template <class T, class U> bool operator<=(adrv<T> const& expr, U const& val) { return expr.v() <= static_cast<T>(val); }

template <class T, class U> bool operator==(adrv<T> const& expr, U const& val) { return expr.v() == static_cast<T>(val); }

template <class T, class U> bool operator!=(adrv<T> const& expr, U const& val) { return expr.v() != static_cast<T>(val); }

template <class T, class U> bool operator> (U const& val, adrv<T> const& expr) { return static_cast<T>(val) > expr.v(); }

template <class T, class U> bool operator>=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) >= expr.v(); }

template <class T, class U> bool operator< (U const& val, adrv<T> const& expr) { return static_cast<T>(val) < expr.v(); }

template <class T, class U> bool operator<=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) <= expr.v(); }

template <class T, class U> bool operator==(U const& val, adrv<T> const& expr) { return static_cast<T>(val) == expr.v(); }

template <class T, class U> bool operator!=(U const& val, adrv<T> const& expr) { return static_cast<T>(val) != expr.v(); }

/** IO operators **/

template <class T> std::ostream& operator<<(std::ostream& os, adrv<T> const& val) {

os << val.v();

return os;

}

In a future post, I’ll write about the efficiency of this approach compared to a version using expression templates. In brief though, it is pretty competitive under most circumstances.

Monthly Archives: December 2013

Automatic Differentiation using Move Constructors