Clad is an open source clang plugin which supports automatic differentiation of mathematical functions in C++. Currently Clad supports four modes for automatic differentiation namely forward, reverse, Hessian, Jacobian.

The Forward mode

Clad supports forward mode automatic differentiation through the clad::differentiate API call.

#include <iostream>
#include "clad/Differentiator/Differentiator.h"

double func(int x) { return x * x; }

int main() {
  /*Calling clad::differentiate to get the forward mode derivative of
  the given mathematical function*/
  auto d_func = clad::differentiate(func, "x");
  // execute the generated derivative function.
  std::cout << d_func.execute(/*x =*/3) << std::endl;
  // Dump the generated derivative code to std output.

Here we are differentiating a function func which takes an input x and returns a scaler value x * x.`.dump()` method is used to get a dump of generated derivative function to the standard output.

The Reverse Mode

Clad also supports reverse mode automatic differentiation, through the clad::gradient API call.

#include <iostream>
#include "clad/Differentiator/Differentiator.h"

double f(double x, double y, double z) { return x * y * z; }

int main() {
  auto d_f = clad::gradient(f, "x, y");
  double dx = 0, dy = 0;
  d_f.execute(/*x=*/2, /*y=*/3, /*z=*/4, &dx, &dy);
  std::cout << "dx : " << dx << "dy :" << dy << std::endl;

In the above example we are differentiating w.r.t x and y we can also differentiate w.r.t to single argument i.e. either x or y as clad::gradient(f, “x”) not writing any argument i.e. clad::gradient(f) will result in differentiation of the function w.r.t to each input.

The Hessian Mode

Clad can also produce an hessian matrix through the clad::hessian API call. It returns the hessian matrix as a flattened vector in row major format.

#include <iostream>
#include "clad/Differentiator/Differentiator.h"

double f(double x, double y, double z) { return x * y * z; }

// Function with array input

double f_arr(double x, double y, double z[2]) { return x * y * z[0] * z[1]; }

int main() {
  // Workflow similar to clad::gradient for non-array input arguments.
  auto f_hess = clad::hessian(f, "x, y");
  double matrix_f[9] = {0};
  clad::array_ref<double> matrix_f_ref(matrix_f, 9);
  f_hess.execute(3, 4, 5, matrix_f_ref);
  std::cout << "[" << matrix_f_ref[0] << ", " << matrix_f_ref[1]
            << matrix_f_ref[2] << "\n"
            << matrix_f_ref[3] << ", " << matrix_f_ref[4] << matrix_f_ref[5]
            << "\n"
            << matrix_f_ref[6] << ", " << matrix_f_ref[7] << matrix_f_ref[8]
            << "]"
            << "\n";

When arrays are involved we need to specify the array index that needs to be differentiated. For example if we want to differentiate w.r.t to the first two elements of the array along with x and y we will write clad::hessian(f_arr, z[0:1]) for the above example rest of the steps for execution are similar to reverse mode. Here the clad::array_ref variable stores the hessian matrix.

The Jacobian Mode

Clad can produce Jacobian of a function using its reverse mode. It returns the jacobian matrix as a flattened vector with elements arranged in row-major format.

#include <iostream>
#include "clad/Differentiator/Differentiator.h"

void f(double x, double y, double z, double* output) {
  output[0] = x * y;
  output[1] = y * y * x;
  output[2] = 6 * x * y * z;

int main() {
  auto f_jac = clad::jacobian(f);

  double jac[9] = {0};
  double output[3] = {0};
  f_jac.execute(3, 4, 5, output, jac);
  std::cout << jac[0] << " " << jac[1] << std::endl
            << jac[2] << " " << jac[3] << std::endl
            << jac[4] << " " << jac[5] << std::endl
            << jac[6] << " " << jac[7] << std::endl
            << jac[8] << std::endl;

The jacobian matrix size should be equal to no. of independent variables times the number of outputs in the original function in the above example it would be an array of size 3x3 = 9.

Error Estimation API

Clad is capable of annotating a given function with floating point error estimation code using reverse mode AD.

#include <iostream>
#include "clad/Differentiator/Differentiator.h"

double func(double x, double y) { return x * y; }

int main() {

  auto dfunc_error = clad::estimate_error(func);
  // Used to print generated code to standard output.
  double x, y, d_x, d_y, final_error = 0;
  // Call execute
  dfunc_error.execute(x, y, &d_x, &d_y, final_error);

  std::cout << final_error;

The function signature is similar to clad::gradient except we need to add an extra argument of type double& which is used to store the total floating point error.