<a href="https://autodiff.github.io" target="_blank">
    <img src='docs/img/header.svg' width='100%'>
</a>

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# Overview

**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.

## Demonstration

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

```c++
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)*:

```c++
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**.

### Forward 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:

```c++
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:

```c++
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.

### Reverse mode

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.

<img
    src='art/expression-tree-diagram.svg'
    style='max-width:100%;'
    title='Expression tree diagram.'>

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:

```c++
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:

```c++
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:

<a href="https://autodiff.github.io" target="_blank">
    <img src='docs/img/autodiff.github.io.svg' width='100%'>
</a>

# License

MIT License

Copyright (c) 2018–2019 Allan Leal

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
