Clad enables automatic differentiation (AD) for C++. It is based on LLVM compiler infrastructure and is a plugin for Clang compiler. Clad is based on source code transformation. Given C++ source code of a mathematical function, it can automatically generate C++ code for computing derivatives of the function. It supports both forward-mode and reverse-mode AD.

How to use Clad

Since Clad is a Clang plugin, it must be properly attached when Clang compiler is invoked. First, the plugin must be built to get (or .dylib). To compile SourceFile.cpp with Clad enabled use:

clang -cc1 -x c++ -std=c++11 -load /full/path/to/lib/ -plugin clad SourceFile.cpp

Clad provides five API functions:

  • clad::differentiate to use forward-mode AD
  • clad::gradient to use reverse-mode AD
  • clad::hessian to compute Hessian matrix using a combination of forward-mode and reverse-mode AD
  • clad::jacobian to compute Jacobian matrix using reverse-mode AD
  • clad::estimate-error to compute the floating-point error of the given program using reverse-mode AD.

API functions are used to label an existing function for differentiation. Both functions return a functor object containing the generated derivative which can be called via .execute method, which forwards provided arguments to the generated derivative function. Example:

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

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

int main() {
  auto f_dx = clad::differentiate(f, "x");
  std::cout << f_dx.execute(3, 4) << std::endl; // prints: 4
  f_dx.dump(); // prints:
  /* double f_darg0(double x, double y) {
       double _d_x = 1; double _d_y = 0;
       return _d_x * y + x * _d_y;
     } */ 

Forward mode

For a function f of several inputs and single (scalar) output, forward mode AD can be used to compute (or, in case of Clad, create a function) computing a directional derivative of f with respect to a single specified input variable. Derivative function created by the forward-mode AD is guaranteed to have at most a constant factor (around 2-3) more arithmetical operations compared to the original function.

clad::differentiate(f, ARGS) takes 2 arguments:

  1. f is a pointer to a function or a method to be differentiated
  2. ARGS is either:
    • a single numerical literal indicating an index of independent variable (e.g. 0 for x, 1 for y)
    • a string literal with the name of independent variable (as stated in the definition of f, e.g. "x" or "y")

Generated derivative function has the same signature as the original function f, however its return value is the value of the derivative.

Reverse mode

When a function has many inputs and derivatives w.r.t. every input (i.e. gradient vector) are required, reverse-mode AD is a better alternative to the forward-mode. Reverse-mode AD allows to compute the gradient of f using at most a constant factor (around 4) more arithmetical operations compared to the original function. While its constant factor and memory overhead is higher than that of the forward-mode, it is independent of the number of inputs. E.g. for a function having N inputs and consisting of T arithmetical operations, computing its gradient takes a single execution of the reverse-mode AD and around 4*T operations, while it would take N executions of the forward-mode, this requiring up to N*3*T operations.

clad::gradient(f, /*optional*/ ARGS) takes 1 or 2 arguments:

  1. f is a pointer to a function or a method to be differentiated
  2. ARGS is either:
    • not provided, then f is differentiated w.r.t. its every argument
    • a string literal with comma-separated names of independent variables (e.g. "x" or "y" or "x, y" or "y, x")

Since a vector of derivatives must be returned from a function generated by the reverse mode, its signature is slightly different. The generated function has void return type and same input arguments. The function has additional, last input argument of the type T*, where T is the return type of f. This is the “result” argument which has to point to the beginning of the vector where the gradient will be stored. The caller is responsible for allocating and zeroing-out the gradient storage. Example:

auto f_grad = clad::gradient(f);
double result1[2] = {};
f_grad.execute(x, y, result1);
std::cout << "dx: " << result1[0] << ' ' << "dy: " << result1[1] << std::endl;

auto f_dx_dy = clad::gradient(f, "x, y"); // same effect as before

auto f_dy_dx = clad::gradient(f, "y, x");
double result2[2] = {};
f_dy_dx.execute(x, y, result2);
// note that the derivatives are mapped to the "result" indices in the same order as they were specified in the argument:
std::cout << "dy: " << result2[0] << ' ' << "dx: " << result2[1] << std::endl;

Note: we are working on improving the gradient interface.

What can be differentiated

Clad is based on compile-time analysis and transformation of C++ abstract syntax tree (Clang AST). This means that Clad must be able to see the body of a function to differentiate it (e.g. if a function is defined in an external library there is no way for Clad to get its AST).

We aim to support every piece of modern C++ syntax, however at the moment only the a subset of C++ is supported and there are some constraints on functions that can be differentiated with Clad:

  • Only builtin C++ scalar numeric types (e.g. double, float, int) are fully supported
  • Differentiated functions must return a single value of supported scalar numeric type
  • Differentiated functions can have arbitrary number or inputs of supported scalar numeric types
  • Clad can also differentiate struct/class methods, however at the moment there is no way to differentiate them w.r.t. member fields

Note: we are currently working on vector inputs

The following subset of C++ syntax is supported at the moment:

  • Numerical literals, builtin arithmetic operators +, -, *, /
  • Variable declarations of supported types (including local variables in {} blocks)
  • Inside functions, builtin arrays (e.g. double x[1][2][3];) of supported types and subscript operator x[i]
  • Direct assignments to variables via = and +=, -=, *=, /=, ++, --
  • Conditional operator ?: and boolean expressions
  • Comma operator ,
  • Control flow: if statements and for loops (work on loops in the reverse-mode is in progress)
  • Calls to other functions, including recursion

Specifying custom derivatives

Sometimes Clad may be unable to differentiate your function (e.g. if its definition is in a library and source code is not available). Alternatively, an efficient/more numerically stable expression for derivatives may be know. In such cases, it is useful to be able to specify a custom derivatives for your function.

Clad supports that functionality by allowing to specify your own derivatives in namespace custom_derivatives. For a function named FNAME you can specify:

  • a custom derivative w.r.t I-th argument by defining a function FNAME_dargI inside namespace custom_derivatives
  • a custom gradient w.r.t every argument by defining a function FNAME_grad inside namespace custom_derivatives

When Clad will encounter a function FNAME, it will first do a lookup inside the custom_derivatives namespace to try to find a suitable custom function, and only if none is found will proceed to automatically derive it.


  • Suppose that you have a function my_pow(x, y) which computes x to the power of y. However, Clad is not able to differentiate my_pow’s body (e.g. it calls an external library or uses some non-differentiable approximation):
    double my_pow(double x, double y) { // something non-differentiable here... }

    However, you know analytical formulas of its derivatives, and you can easily specify custom derivatives:

    namespace custom_derivatives {
    double my_pow_darg0(double x, double y) { return y * my_pow(x, y - 1); }
    double my_pow_darg1(dobule x, double y) { return my_pow(x, y) * std::log(x); }

    You can also specify a custom gradient:

    namespace custom_derivatives {
    void my_pow_grad(double x, double y, double* result) { 
       double t = my_pow(x, y - 1);
       result[0] = y * t;
       result[1] = x * t * std::log(x);

    Whenever Clad will encounter my_pow inside differentiated function, it will find and use provided custom funtions instead of attempting to differentiate it.

Note: Clad provides custom derivatives for some mathematical functions from <cmath> inside clad/Differentiator/BuiltinDerivatives.h.

Note: the concept of custom_derivatives will be reviewed soon, we intend to provide a different interface and avoid function name-based specifications and by-name lookups.

How Clad works

Clad is a plugin for the Clang compiler. It relies on the Clang to build the AST (Clang AST) of user’s source code. Then, CladPlugin, implemented as clang::ASTConsumer analyzes the AST to find differentiation requests for clad and process those requests by building Clang AST for derivative functions. The whole clad’s operation sequence is the following:

  • Clang parses user’s source code and builds the AST.
  • CladPlugin analyzes the built AST and starts the traversal via HandleTopLevelDecl.
  • DiffCollector does the traversal and looks at every call expression to find all the differentiation requests (through calls to clad::differentiate(f, ...) or clad::gradient(f, ...)).
  • CladPlugin processes every found request.
  • Processing each request involves calling to DerivativeBuilder::Derive which then switches do either ForwardModeVisitor::Derive or ReverseModeVisitor::Derive, depending on which AD mode was requested.
  • ForwardModeVisitor and ReverseModeVisitor are derived from clang::StmtVisitor. In the Derive method they analyze the AST of the declaration of the original function and create the AST for the declaration of derivative function. Then they proceed to recursively Visit every Stmt in original function’s body and build the body for the derivative function. Forward/Reverse mode AD algorithm is implemented in Visit... methods, which are executed depending on the kind of AST node visited.
  • The AST of the newly built derivative function’s declaration is returned to CladPlugin, where the call to clad::differentiate(f, ...)/clad::gradient(f, ...) is updated and f is replaced by a reference to the newly created derivative function f_.... This effectively results in the execution of clad::differentiate(f_...)/clad::gradient(f_...), which constructs CladFunction with a pointer to the newly created derivative. Therefore, user’s calls to .execute method will invoke the newly generated derivative.
  • Finally, derivative’s AST is passed for further processing by Clang compiler (LLVM IR generation, optimizations, machine code generation, etc.).

Citing Clad

% Peer-Reviewed Publication
% 16th International workshop on Advanced Computing and Analysis Techniques
% in physics research (ACAT), 1-5 September, 2014, Prague, The Czech Republic
  author = {Vassilev,V. and Vassilev,M. and Penev,A. and Moneta,L. and Ilieva,V.},
  title = {Clad -- Automatic Differentiation Using Clang and LLVM},
  journal = {Journal of Physics: Conference Series},
  year = 2015,
  month = {may},
  volume = {608},
  number = {1},
  pages = {012055},
  doi = {10.1088/1742-6596/608/1/012055},
  url = {},
  publisher = {IOP Publishing}

Some references

ACAT 2014 Slides
Martin’s GSoC2014 Final Report
LLVM Poster
Violeta’s GSoC2013 Final Report
DIANA-HEP Meeting 2018
Demo at CERN


Founder of the project is Vassil Vassilev as part of his research interests and vision. We have quite a few contributors. If we missed you, please contact us!

How to Contribute

Have a look at our open positions and open projects pages

How to install

Find instructions at our README