autodiff is a C 17 library that uses modern and advanced programming techniques to enable automatic computation of derivatives in an efficient and easy way.

Consider the following function f(x, y, z):

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

which we use use to evaluate variable u = f(x, y, z):

double x = 1.0;
double y = 2.0;
double z = 3.0;
double u = f(x, y, z);

How can we minimally transform this code so that not only u, but also its derivatives ∂u/∂x, ∂u/∂y, and ∂u/∂z, can be computed?

The next two sections present how this can be achieved using two automatic differentiation algorithms implemented in autodiff: forward mode and reverse mode.

In a forward mode automatic differentiation algorithm, both output variables and one or more of their derivatives are computed together. For example, the function evaluation f(x, y, z) can be transformed in a way that it will not only produce the value of u, the output variable, but also one or more of its derivatives (∂u/∂x, ∂u/∂y, ∂u/∂z) with respect to the input variables (x, y, z).

Enabling forward automatic differentiation for the calculation of derivatives using {{autodiff}} is relatively simple. For our previous function f, we only need to replace the floating-point type double to autodiff::dual for both input and output variables:

dual f(const dual& x, const dual& y, const dual& z)
{
    return (x   y   z) * exp(x * y * z);
}

We can now compute the derivatives ∂u/∂x, ∂u/∂y, and ∂u/∂z as follows:

dual x = 1.0;
dual y = 2.0;
dual z = 3.0;
dual u = f(x, y, z);

double dudx = derivative(f, wrt(x), at(x, y, z));
double dudy = derivative(f, wrt(y), at(x, y, z));
double dudz = derivative(f, wrt(z), at(x, y, z));

The auxiliary function autodiff::wrt, an acronym for with respect to, is used to indicate which input variable (x, y, z) is the selected one to compute the partial derivative of f. The auxiliary function autodiff::at is used to indicate where (at which values of its parameters) the derivative of f is evaluated.

In a reverse mode automatic differentiation algorithm, the output variable of a function is evaluated first. During this function evaluation, all mathematical operations between the input variables are "recorded" in an expression tree. By traversing this tree from top-level (output variable as the root node) to bottom-level (input variables as the leaf nodes), it is possible to compute the contribution of each branch on the derivatives of the output variable with respect to input variables.

Thus, a single pass in a reverse mode calculation computes all derivatives, in contrast with forward mode, which requires one pass for each input variable. Note, however, that it is possible to change the behavior of a forward pass so that many (even all) derivatives of an output variable are computed simultaneously (e.g., in a single forward pass, ∂u/∂x, ∂u/∂y, and ∂u/∂z are evaluated together with u, in contrast with three forward passes, each one computing the individual derivatives).

Similar as before, we can use {{autodiff}} to enable reverse automatic differentiation for our function f by simply replacing type double by autodiff::var as follows:

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

The code below demonstrates how the derivatives ∂u/∂x, ∂u/∂y, and ∂u/∂z can be calculated:

var x = 1.0;
var y = 2.0;
var z = 3.0;
var u = f(x, y, z);

Derivatives dud = derivatives(u);

double dudx = dud(x);
double dudy = dud(y);
double dudz = dud(z);

The function autodiff::derivatives will traverse the expression tree stored in variable u and compute all its derivatives with respect to the input variables (x, y, z), which are then stored in the object dud. The derivative of u with respect to input variable x (i.e., ∂u/∂x) can then be extracted from dud using dud(x). The operations dud(x), dud(y), dud(z) involve no computations! Just extraction of derivatives previously computed with a call to function autodiff::derivatives.

Check the documentation website for more details:

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Copyright (c) 2018–2020 Allan Leal

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