# C++

## Basic Usage¶

In this section, we will illustrate how to use XAD to compute first order derivatives in both forward and adjoint mode.

As an example, we choose a simple function with 4 inputs and 1 output variable, defined as:

double f(double x0, double x1, double x2, double x3)
{
double a = sin(x0) * cos(x1);
double b = x2 * x3 - tan(x1 - x2);
double c = a + 2* c;
return c*c;
}


We will compute derivatives of this function at the point:

double x0 = 1.0;
double x1 = 1.5;
double x2 = 1.3;
double x3 = 1.2;


### Prerequisite: Replace Active Variables¶

In order to use XAD to differentiate this function, we first must replace all independent data types and all values that depend on them with an active data type provided by XAD. In the above function, all variables depend on the inputs and thus all occurrences of double must be replaced.

This can be done in one of two ways:

1. The variables can be replaced directly, given the desired mode of differentiation. For example, for forward mode double is replaced by the type FReal and for adjoint mode the type AReal.
2. The function is made a template, so that it can be called with any data type, including the original double.

We choose the second approach for this tutorial, thus the function becomes:

template <class T>
T f(T x0, T x1, T x2, T x3)
{
T a = sin(x0) * cos(x1);
T b = x2 * x3 - tan(x1 - x2);
T c = a + 2* b;
return c*c;
}


This means we can use the same definition with both forward and adjoint modes.

### Forward Mode¶

As illustrated in Algorithmic Differentiation Background: Forward Mode, when applied to a function with a single output, the forward mode of algorithmic differentiation can compute one derivative at a time. For illustration, we choose to derive the function with respect to the input variable x0.

To initiate the forward mode, we must first declare active variables with the appropriate type. XAD provides convenience typedefs to select the mode of differentiation, illustrated in detail in AD Mode Interface. For forward mode, we can declare the types needed as:

typedef xad::fwd<double> mode;


We can then use the AD typedef for our variables.

The next step is to initialize the dependent variables, which is simply done by assigning the input values to new variables of type AD:

AD x0_ad = x0;


For forward mode, we must now seed the initial derivative for the variable we are interested in with the value 1 (as described in Algorithmic Differentiation Background: Forward Mode), as:

derivative(x0_ad) = 1.0;


The global function derivative is a convenience function that works on any active data type. Alternatively, we could have used the member function FReal::setDerivative.

At this point we are ready to call our function and it will compute the function value as well as the derivative we are interested in:

AD y = f(x0_ad, x1_ad, x2_ad, x3_ad);


We can now access the results using the value and derivative functions on the output (or the member functions FReal::getDerivative and FReal::getValue). For example, the following code outputs them to the console:

std::cout << "y = " << value(y) << "\n"
<< "dy/dx0 = " << derivative(y) << "\n";


This example is included with XAD (fwd_1st).

The adjoint mode of automatic differentiation is the natural choice for the function at hand, as it has a single output and multiple inputs. We can get all four derivatives in one execution.

Adjoint mode needs a tape to record the operations and their values during the valuation. After setting the adjoints of the outputs, this tape can then be rolled back to compute the adjoints of the inputs.

Both the active data type and the tape type can be obtained from the interface structure adj:

typedef xad::adj<double> mode;
typedef mode::tape_type tape_type;


The first step for computing adjoints is to initialise the tape::

tape_type tape;


This calls the default constructor Tape::Tape, which creates the tape and activates it.

Next, we create the input variables and register them with the tape:

AD x0_ad = x0;
tape.registerInput(x0);
tape.registerInput(x1);
tape.registerInput(x2);
tape.registerInput(x3);


Note that only variables registered as inputs with the tape and all variables dependent on them are recorded. Also note that before registering active variables, the current threads needs to have an active tape. To ensure thread-safety, every thread of the application can have its own active tape.

Once the independent variables are set, we can start recording derivatives on tape and run the algorithm:

tape.newRecording();



At this stage, we have all operations recorded and have the value computed. We now need to register the outputs with the tape as well, before we can seed the initial adjoint of the output wit 1 as explained in Algorithmic Differentiation Background: Adjoint Mode:

tape.registerOutput(y);
derivative(y) = 1.0;


This uses the global function derivative, which returns a reference to the stored derivative (or adjoint) of the given parameter. Alternatively the member functions AReal::setAdjoint or AReal::setDerivative can be used for the same purpose.

What is left is interpreting the tape to compute the adjoints of the independent variables:

tape.computeAdjoints();


We can now access the adjoints of the inputs, which are the derivatives we are interested in, via the global derivative function or the member function AReal::getDerivative:

std::cout << "y     = " << value(y) << "\n"
<< "dy/dx0 = " << derivative(x0_ad) << "\n"
<< "dy/dx1 = " << derivative(x1_ad) << "\n"
<< "dy/dx2 = " << derivative(x2_ad) << "\n"
<< "dy/dx3 = " << derivative(x3_ad) << "\n";


This example is included with XAD (adj_1st).

### Best Practices¶

When the algorithm to be evaluated has less outputs than inputs, adjoint mode should be preferred. However, when only a small number of derivatives are needed (e.g. less than 5), the memory for the tape can be avoided by using forward mode. Experimentation is advised to find the optimal mode for the given algorithm.

## External Functions¶

Often parts of the algorithm to be differentiated are not available in source code. For example, a routine from an external math library may be called. Reimplementing it may not be desirable (for performance or development effort reasons), in which case the derivatives of this function need to be implemented manually in some form.

This can be achieved by either:

• Applying finite differences to the library function (bumping),
• Implementing the adjoint code of the function by hand, or
• Computing the derivatives analytically, possibly using other library functions.

In these cases, the external function interface of XAD can be used to integrate the manual derivatives, which is described below. With the same technique, performance- or memory-critical parts of the application may be hand-tuned.

### Example Algorithm¶

We pick an example algorithm which computes the length of a multi dimensional vector. This is defined as:

$y = \sqrt{\sum_0^{N-1} x_i^2}$

The goal is to compute the derivatives of $$y$$ with respect to all input vector elements using adjoint mode.

The algorithm can be implemented in C++ code as:

std::vector<double> xsqr(n);
for (int i = 0; i < n; ++i)
xsqr[i] = x[i] * x[i];
double y = sqrt(sum_elements(x, n));


For this example, we assume that the sum_elements is an external function implemented in a library that we do not have source code of. It has the prototype:

double sum_elements(const double* x, int n);


### External Function For Adjoint Mode¶

To use the external function, we follow this procedure:

1. At the point of the call, convert the values of the input active variables to the underlying plain data type (double)
2. Call the external function passively
3. Assign the result values to active output variables so the tape recording can continue
4. Store the tape slots of the inputs and outputs with a checkpoint callback object and register it with the tape.
5. When computing adjoints, this callback needs to load the adjoint of the outputs, propagate to them to the inputs manually, and increment the input adjoints by these values.

We put all the functionality into a callback object. We derive from the CheckpointCallback base class and implement at least the virtual method CheckpointCallback::computeAdjoint. This method gets called during tape rollback. We also place the forward computation within the same object (this could also be done outside of the callback class). The declaration of our callback class looks like this:

template <class Tape>
{
public:
typedef typename Tape::slot_type   slot_type;   // type for slot in the tape
typedef typename Tape::value_type  value_type;  // double
typedef typename Tape::active_type active_type; // AReal<double>

active_type computeExternal(const active_type* x, int n); // forward compute

private:
std::vector<slot_type> inputSlots_;             // slots of inputs in tape
slot_type outputSlot_;                          // slot of output in tape
};


We declare it as a template for arbitrary tape types, which is good practice as it allows to reuse this implementation with higher order derivatives too.

#### computeExternal Method¶

Within the computeExternal method, we first store the slots in the tape for the input variables, as we will need them during adjoint computation to increment the corresponding adjoints. We use the inputSlots_ member vector to keep this information:

for (int i = 0; i < n; ++i)
inputSlots_.push_back(x[i].getSlot());


Then we create a copy of the active inputs and store them in a vector of passive values, with which we can call the external function:

std::vector<value_type> x_p(n);
for (int i = 0; i < n; ++i)
x_p[i] = value(x[i]);

value_type y = sum_elements(&x_p[0], n);


We now need to store this result in an active variable, register it as an output of the external function (to allow the tape to continue recording dependent variables), and keep its slot in the tape for the later adjoint computation:

active_type ret = y;
Tape::getActive()->registerOutput(ret);
outputSlot_ = ret.getSlot();


Finally we need to insert the callback into the tape, hence requesting it to be called during adjoint rollback of the tape, and return:

Tape::getActive()->insertCallback(this);
return ret;


#### computeAdjoint Method¶

The computeAdjoint method gets called by XAD during tape rollback. We need to override this method and implement the manual adjoint code. For a simple sum operation, this is straightforward: all input adjoints are equal to the output adjoint since all partial derivatives are 1. Thus we need to obtain the output adjoint and increment all input adjoints by this value:

value_type output_adj = tape->getAndResetOutputAdjoint(outputSlot_);
for (int i = 0; i < inputSlots_.size(); ++i)


The function Tape::getAndResetOutputAdjoint obtains the adjoint value corresponding to the given slot and resets it to zero. This reset is necessary in general as the output variable may have been overwriting other values in the forward computation. The Tape::incrementAdjoint function simply increments the adjoint with the given slot by the given value.

#### Wrapper Function¶

With the checkpointing callback class in place, we can implement a sum_elements overload for AReal that wraps the creation of this callback::

template <class T>
{
tape_type* tape = tape_type::getActive();
ExternalSumElementsCallback<tape_type>* ckp =
new ExternalSumElementsCallback<tape_type>;
tape->pushCallback(ckp);

return ckp->computeExternal(x, n);
}


This function dynamically allocates the checkpoint callback object and lets the tape manage its destruction via the Tape::pushCallback function. This call simply ensures that the callback object is destroyed when the tape is destroyed, making sure that no memory is leaked. If the callback object was managed elsewhere, this call would not be necessary. It then redirects the computation to the computeExternal function of the checkpoint callback class. Using this wrapper class, the sum_elements function can be used for active types in the same fashion as the original external function sum_elements for double. Defining it as a template allows us to re-use this function for higher-order derivatives, should we need them in future.

#### Call-Site¶

The call site then can be implemented as (assuming that x_ad is the vector holding the independent variables, already registered on tape):

tape.newRecording();

for (int i = 0; i < n; ++i)
AD y = sqrt(sum_elements(xsqr.data(), n)); // calls external function wrapper

tape.registerOutput(y);
derivative(y) = 1.0;

std::cout << "y = " << value(y) << "\n";
for (int i = 0; i < n; ++i)
std::cout << "dy/dx" << i << " = " << derivative(x[i]) << "\n";


This follows exactly the same procedure as given in Basic Usage.

This example is included with XAD (external_function).

### External Function For Forward Mode¶

Since forward mode involves no tape, a manual implementation of the derivative computation needs to be implemented together with computing the value. The manual derivatives can be updated directly in the output values using the derivative function.

In our example, we can implement the external function in forward mode as:

template <class T>
{

std::vector<T> x_p(n);
for (int i = 0; i < n; ++i)
x_p[i] = value(x[i]);

T y_p = sum_elements(&x_p[0], n);

active_type y = y_p;

for (int i = 0; i < n; ++i)
derivative(y) += derivative(x[i]);

return y;
}


We first extract the passive values from the x vector and call the external library function to get the passive output value y_p. This value is then assigned to the active output variable y, which also initializes its derivative to 0.

As we have a simple sum in this example, the derivative of the output is a sum of the derivatives of the inputs, which is computed by the loop in the end.

This example is included with XAD (external_function).

## Checkpointing¶

Checkpointing is a technique to reduce the memory footprint of the tape in adjoint mode algorithmic differentiation. Instead of recording the full algorithm on tape, which can quickly result in gigabytes of memory in a large computation, the tape is recorded for specific stages of the algorithm, one at a time. This is illustrated in the following figure:

The algorithm is divided into stages, where the input data of each stage is stored in a checkpoint and the outputs are computed passively (without recording on tape). Once the final output of the algorithm is computed, the adjoint of the output is initialized and at each checkpoint during tape rollback:

1. The inputs to the checkpoint are loaded,
2. The operations of this stage only are recorded on tape,
3. The output adjoints of this stage are initialized,
4. The tape is rolled back for this stage, computing the adjoints of the stage inputs,
5. The input adjoints are incremented by these values, and
6. The tape is wiped before proceeding with the previous stage.

Using this method, the tape memory is limited by the amount needed to record one algorithm stage instead of the full algorithm. However, each forward computation is computed twice, hence checkpointing trades computation for memory.

In practice, as using less memory leads to higher cache-efficiency, checkpointing may be faster overall than recording the full algorithm even though more computations are performed.

### Example Algorithm¶

To demonstrate the checkpointing method, we choose a simple repeated application of the sine function to a single input:

template <class T>
void repeated_sin(int n, T& x)
{
for (int i = 0; i < n; ++i)
x = sin(x);
}


We divide the for loop into equidistant stages and insert a checkpoint at each of these.

### Checkpoint Callback¶

To create a checkpoint, we need to store the inputs of the stage and the slots in the tape for the inputs and outputs in a callback object inheriting from CheckpointCallback. The virtual method CheckpointCallback::computeAdjoint needs to be overridden to perform the per-stage adjoint computation. As all stages are identical, we choose to implement the functionality of all checkpoints within a single callback object and store the required inputs in a stack data structure. Alternatively we could have created a new checkpoint callback object at every checkpoint. The prototype for our callback is:

template <class Tape>
{
public:
typedef typename Tape::slot_type   slot_type;   // type for slot in the tape
typedef typename Tape::value_type  value_type;  // double
typedef typename Tape::active_type active_type; // AReal<double>

active_type computeStage(int n, active_type& x); // forward computation

private:
std::stack<int> n_;                    // number of iterations in this stage
std::stack<value_type> x_;             // input values for this stage
std::stack<slot_type> slots_;          // tape slots for input and output
};


For convenience of implementation, we added the forward computation for one stage within the same class in the computeStage method, which could also be performed outside of the object.

#### computeStage Method¶

Within the computeStage method, we first store the input value, the number of iterations, and the slots of the input in the checkpoint object:

n_.push(n);
slots_.push(x.getSlot());
value_type x_p = value(x);
x_.push(x_p);


We then compute the stage with the passive variable (not recording on the tape):

repeated_sin(n, x_p);


The value of the output active variable needs to be updated with the result and we need to store the slot of the output variable in the checkpoint also:

value(x) = x_p;
slots_.push(x.getSlot());


Note that we did not need to register x as an output with the tape here, as we had to do with the external functions example before, since the variable is already registered on tape (it's both input and output).

What is left is to register this callback object with the tape so that its computeAdjoint method is called at this point when the tape is rolled back:

Tape::getActive()->insertCallback(this);


#### computeAdjoint Method¶

The computeAdjoint method is called automatically by XAD at the checkpoints in the tape. We first need to load the inputs to this computation stage and obtain the adjoint of the output:

slot_type outputidx = slots_.top();  slots_.pop();
slot_type inputidx = slots_.top();   slots_.pop();
int n = n_.top();                    n_.pop();


The function Tape::getAndResetOutputAdjoint reads the adjoint corresponding to the slot given and resets it to 0. This reset is generally required as the variable corresponding to the slot may be re-used (overwritten) in the algorithm, as is the case in the repeated_sin function.

We now want to use XAD to compute the adjoints just for this computation stage. This is performed by creating a nested recording within the global tape, than can be rolled back individually:

active_type x = x_.top();               // local independent variable
x_.pop();
tape->registerInput(x);                 // need to register to record

tape->registerOutput(x);                // register x as an output



In a similar fashion to simple adjoint mode (see Basic Usage), we first initialize the local independent variables as active data types and start a nested recording. This is performed by creating a local object nested of type ScopedNestedRecording, which wraps calls to Tape::newNestedRecording in its constructor and Tape::endNestedRecording in its destructor. It is recommended to use the ScopedNestedRecording for this purpose to make sure the nested recording is always finished when the scope is left.

Next we record the operations for this stage by running the algorithm actively. We then set the adjoint of the output and compute the adjoints of the inputs. The adjoints of the inputs to this stage can then be incremented.

Note that when the nested object goes out of scope, i.e. when its destructor is called, the nested tape for this computation stage is wiped and the memory can be reused for the previous stage. This saves overall memory.

### Call-Site¶

The full algorithm with checkpointing can then be initiated as follows:

tape_type tape;

tape.registerInput(x_ad);                // register with the tape
tape.newRecording();                     // start recording derivatives

SinCheckpointCallback<tape_type> chkpt;  // setup checkpointing object

int checkpt_distance = 4;                // we checkpoint every 4 iterations
for (int i = 0; i < n; i += checkpt_distance)
{
int m = min(checkpt_distance, n-i);
chkpt.computeStage(m, x_ad);             // one computation stage
}

std::cout << "xout       = " << value(x_ad) << "\n"
<< "dxout/dxin = " << derivative(x_ad) << "\n";


This follows largely the same procedure as given in Basic Usage, but setting up the checkpoint object and calling its computeStage member for every stage of the algorithm (4 iterations in this example).

Note

It is important that the checkpoint callback object is valid when Tape::computeAdjoints is called. It should not be destroyed before.

See Checkpoint Callback Memory Management for how to use tape-based destruction with dynamically allocated checkpoint callbacks.

This example is included with XAD (checkpointing).

### Other Usage Patterns¶

Alternative methods may be used to update the adjoints within a checkpoint's CheckpointCallback::computeAdjoint method, such as:

• Forward mode algorithmic differentiation within an outer adjoint mode
• Finite differences (bumping)
• Analytic derivatives
• External library functions (see External Functions)

Checkpointing can also be used recursively, i.e., new checkpoints are created within a nested tape in a checkpoint.

The benefits of each of these approaches are highly application-dependent.

## Higher-Order Derivatives¶

As explained in Algorithmic Differentiation Background: Higher Orders, higher order derivatives can be computed by nesting first order algorithmic differentiation techniques. For example, one can obtain second order by computing forward mode over adjoint mode. With XAD, this technique can be used directly to compute higher order derivatives.

XAD's automatic differentiation interface structures (see AD Mode Interface) define second order mode data types for easy access. Types for third or higher orders need to defined manually from the basic first-order types.

We will demonstrate second-order derivatives using forward-over-adjoint mode in the following.

### Example Algorithm¶

For demonstration purposes, we use the same algorithm from Basic Usage:

template <class T>
T f(T x0, T x1, T x2, T x3)
{
T a = sin(x0) * cos(x1);
T b = x2 * x3 - tan(x1 - x2);
T c = a + 2* b;
return c*c;
}


We are interested in derivatives at the point:

double x0 = 1.0;
double x1 = 1.5;
double x2 = 1.3;
double x3 = 1.2;


In this mode, we can compute all first-order derivatives (as a single output function derived with adjoints gives all first order derivatives), and the first row of the Hessian matrix of second order derivatives. The full Hessian is defined as:

$\pmb{H} = \left[ \begin{array}{cccc} \frac{\partial^2 f}{\partial x_0^2} & \frac{\partial^2 f}{\partial x_0 \partial x_1} & \frac{\partial^2 f}{\partial x_0 \partial x_2} & \frac{\partial^2 f}{\partial x_0 \partial x_3} \\[6pt] \frac{\partial^2 f}{\partial x_1 \partial x_0} & \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_1 \partial x_3} \\[6pt] \frac{\partial^2 f}{\partial x_2 \partial x_0} & \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \frac{\partial^2 f}{\partial x_2 \partial x_3} \\[6pt] \frac{\partial^2 f}{\partial x_3 \partial x_0} & \frac{\partial^2 f}{\partial x_3 \partial x_1} & \frac{\partial^2 f}{\partial x_3 \partial x_2} & \frac{\partial^2 f}{\partial x_3^2} \end{array}\right]$

Note that the Hessian matrix is typically symmetric, which can be used to reduce the amount of computation needed for the full Hessian.

The first step is to set up the tape and active data types needed for this computation:

typedef xad::fwd_adj<double> mode;
typedef mode::tape_type tape_type;

tape_type tape;


Note that the active type for this mode is actually AReal<FReal<double>>.

Now we need to setup the independent variables and register them:

AD x0_ad = x0;



As we compute the second order using forward mode, we need to seed the initial derivative for the second order before running the algorithm:

derivative(value(x0_ad)) = 1.0;


The inner call to value takes the value of the outer type, i.e. it returns the value as the type FReal<double>, of which we set the derivative to 1.

Now we can start recording derivatives on the tape and run the algorithm:

tape.newRecording();



For the inner adjoint mode, we need to register the output and seed the initial adjoint with 1:

tape.registerOutput(y);
value(derivative(y)) = 1.0;


Here, the inner call to derivative gives the derivative of the outer type, i.e. the derivative of the adjoint-mode active type. This is of type FReal<double>, for which we set the value to 1.

Next we compute the adjoints, which computes both the first and second order derivatives:

tape.computeAdjoints();


We can now output the result:

std::cout << "y = " << value(value(y)) << "\n";


And the first order derivatives:

std::cout << "dy/dx0 = " << value(derivative(x0_ad)) << "\n"
<< "dy/dx1 = " << value(derivative(x1_ad)) << "\n"
<< "dy/dx2 = " << value(derivative(x2_ad)) << "\n"
<< "dy/dx3 = " << value(derivative(x3_ad)) << "\n";


Note again that the inner call to derivative obtains the derivative of the outer active data type, hence it gives a FReal<double> reference that represents the first order adjoint value. We can get this value as a double using the value call.

The second order derivatives w.r.t. x0 can be obtained as:

std::cout << "d2y/dx0dx0 = " << derivative(derivative(x0_ad)) << "\n"
<< "d2y/dx0dx1 = " << derivative(derivative(x1_ad)) << "\n"
<< "d2y/dx0dx2 = " << derivative(derivative(x2_ad)) << "\n"
<< "d2y/dx0dx3 = " << derivative(derivative(x3_ad)) << "\n";


which 'unwraps' the derivatives of the first and second order active types.

The result of the running the application for the given inputs is:

y      = 7.69565
dy/dx0 = 0.21205
dy/dx1 = -16.2093
dy/dx2 = 24.8681
dy/dx3 = 14.4253
d2y/dx0dx0 = -0.327326
d2y/dx0dx1 = -3.21352
d2y/dx0dx2 = 0.342613
d2y/dx0dx3 = 0.198741


Forward over adjoint is the recommended mode for second-order derivatives.

This example is included with XAD (fwd_adj_2nd).

### Other Second-Order Modes¶

Other second-order modes work in a similar fashion. They are briefly described in the following.

#### Forward Over Forward¶

With forward-over-forward mode, there is no tape needed and the derivatives of both orders need to be seeded before running the algorithm. One element of the Hessian and one first-order derivative can be computed with this method, if the function has one output. The derivative initialization sequence in this mode is typically:

value(derivative(x)) = 1.0;   // initialize the first-order derivative
derivative(value(x)) = 1.0;   // initialize the second-order derivative


After the computation, the first order derivative can be retrieved as:

std::cout << "dy/dx = " << derivative(value(y)) << "\n";


And the second order derivative as:

std::cout << "d2y/dxdx = " << derivative(derivative(y)) << "\n";


With different initial seeding, different elements of the Hessian can be obtained.

Here the inner mode is forward, computing one derivative in a tape-less fashion, and the outer mode is adjoint, requiring a tape. With this mode, we need to initialize the forward-mode derivative with:

value(derivative(x)) = 1.0;   // initialize the first-order derivative


As the derivative of the output corresponds to the first order result, we need to seed its derivative (i.e. the adjoint) after running the algorithm:

derivative(derivative(y)) = 1.0;


After tape interpretation, we can now obtain the first-order derivative as:

std::cout << "dy/dx = " << value(derivative(y)) << "\n";


Due to the symmetries in this mode of operation, the same first-order derivatives can also be obtained as:

std::cout << "dy/dx = " << derivative(derivative(x)) << "\n";


Which allows to get all first-order derivatives w.r.t. to all inputs in this mode, similar to the forward-over-adjoint mode.

The second-order derivatives can be obtained as:

std::cout << "d2y/dxdx = " << derivative(value(x))


As both nested modes are adjoint, this mode needs to two tapes for both orders. Hence the types defined in the interface structure adj_adj need an inner and an outer tape type:

typedef xad::adj_adj<double> mode;
typedef mode::inner_tape_type inner_tape_type;
typedef mode::outer_tape_type outer_tape_type;


In this mode, no initial derivatives need to be set, but it is important that both tapes are initialized and a new recording is started on both before running the algorithm.

After the execution, the outer derivative needs to be seeded as:

value(derivative(y)) = 1.0;


And then the outer tape needs to compute the adjoints. This computes the value(derivative(x)) as an output, and the derivative of this needs to be set before interpreting the inner tape:

derivative(derivative(x)) = 1.0;


After calling computeAdjoints() on the inner tape, we can read the first-order derivatives as:

std::cout << "dy/dx = " << value(derivative(x)) << "\n;


And the second-order derivatives as:

std::cout << "d2y/dxdx" << derivative(value(x)) << "\n";