Fix errors found in the benchmarking and fix several issues with documentation.

 * @f{equation}{\frac{\partial\vec{v}}{\partial t} = dt\vec{v}\times\vec{B}@f}

 * @f{equation}{\frac{\partial\vec{x}}{\partial t} = dt\vec{v}@f}

 *

 * We compared the graph framework against the MLX framework since it supports

 * Apple GPUs and JAX due to it's popularity. Source codes for this benchmark

 * case is available in the appendix. Figure \ref{fig:compare} shows the through put of

 * pushing $10^{8}$ particles for $10^{3}$ time steps. The graph framework

 * consistently shows the best throughput on both CPUs and GPUs. Note MLX CPU

 * throughput could by improved by splitting the problem to multiple threads.

 * We compared the graph framework against the

 * <a href="https://ml-explore.github.io/mlx/build/html/index.html">MLX</a>

 * framework since it supports Apple GPUs,

 * <a href="https://docs.jax.dev/en/latest/">JAX</a> due to it's popularity,

 * and <a href="https://kokkos.org">Kokkos</a> for its performance

 * portability. Source codes for this benchmark case is available in the

 * appendix. Figure \ref{fig:compare} shows the through put of pushing $10^{8}$

 * particles for $10^{3}$ time steps. The graph framework consistently shows the

 * best throughput on both CPUs and GPUs. Note MLX CPU throughput could by

 * improved by splitting the problem to multiple threads.

 *

 * @subsection code_performance_comparison_codes Source codes for throughput benchmark comparison

 * @subsubsection code_performance_comparison_graph Graph Framework

        auto v_next = v + dt*lorentz;

        auto pos_next = pos + dt*v_next;

        workflow::manager<float> work(0);

        workflow::manager<float> work(thread_number);

        work.add_item({

            graph::variable_cast(x),

            graph::variable_cast(y),

def push(x, y, z, vx, vy, vz):

    dt = 0.000001

    vx_next = vx + dt*(vy*1 - vz*0)

    vy_next = vy + dt*(vz*0 - vy*1)

    vy_next = vy + dt*(vz*0 - vx*1)

    vz_next = vz + dt*(vx*0 - vy*0)

    return vx_next, vy_next, vz_next,

           x + dt*vx_next, y + dt*vy_next, z + dt*vz_next

end = time.time()

print(end - start)

 @endcode

 *

 * @subsubsection code_performance_comparison_kokkos Kokkos

 * @code

const size_t size = 100000000;

const size_t steps = 1000;

using ViewVectorType = Kokkos::View<float *, Kokkos::SharedSpace>;

ViewVectorType x("x", size);

ViewVectorType y("y", size);

ViewVectorType z("z", size);

ViewVectorType vx("vx", size);

ViewVectorType vy("vy", size);

ViewVectorType vz("vz", size);

Kokkos::parallel_for(size, KOKKOS_LAMBDA(const int64_t index) {

    vx[index] = 1;

    vz[index] = 1;

});

const std::chrono::high_resolution_clock::time_point start = std::chrono::high_resolution_clock::now();

for (size_t i = 0; i < steps; i++) {

    Kokkos::parallel_for(size, KOKKOS_LAMBDA(const int64_t index) {

        const float dt = 0.000001;

        const float vx_next = vx[index] + dt*(vy[index]*1 - vz[index]*0);

        const float vy_next = vy[index] + dt*(vz[index]*0 - vx[index]*1);

        const float vz_next = vz[index] + dt*(vx[index]*0 - vy[index]*0);

        x[index] += dt*vx_next;

        y[index] += dt*vy_next;

        z[index] += dt*vz_next;

        vx[index] = vx_next;

        vy[index] = vy_next;

        vz[index] = vz_next;

    });

}

Kokkos::fence();

std::chrono::high_resolution_clock::time_point end = std::chrono::high_resolution_clock::now();

const auto total_time = end - start;

 @endcode

 */

 * @section discription_introduction Introduction

 * The basic functionality of this framework is to build expression graphs

 * representing mathematical equations. Reduce those graphs to simpler forms.

 * Transform those graph to take derivatives. Just-In-Time (JIT) compile them to

 * available compute device kernels. Then run those kernels in workflow. The

 * Transform those graphs to take derivatives. Just-In-Time (JIT) compile them

 * to available compute device kernels. Then run those kernels in workflows. The

 * code is written in using C++23 features. To simplify embedding into legacy

 * codes, there are additional language bindings for C and Fortran.

 *

 * be reduced to a single constant by calling the evaluate method. Sub-graph

 * expressions are combined, factored out, or moved to enable better reductions

 * on subsequent passes. As new ways of reducing the graph are implemented,

 * current and existing code built using this framework benefit from improved

 * speed. The figure above shows a visualization of the tree data structure for

 * the equation of a line, the derivative, and the subsequent reductions.

 * current and existing code built using this framework will benefit from

 * improved speed. The figure above shows a visualization of the tree data

 * structure for the equation of a line, the derivative, and the subsequent

 * reductions.

 *

 * @subsubsection discription_graphs_builds Building Graphs

 * As an example building an expression of line @f$y=mx+b@f$ accomplished by

 * running them in order. One @ref workflow::manager is created for each device

 * or thread. The user is responsible for creating threads. Each kernel is

 * generated through a @ref workflow::work_item. A work item is defined by

 * kernel @ref graph::input_nodes, @ref graph::output_nodes and

 * @ref graph::map_nodes. Map items are used to take the results of kernel and

 * kernel @ref graph::input_nodes, @ref graph::output_nodes, and

 * @ref graph::map_nodes. Map items are used to take the results of a kernel and

 * update an input buffer. Using our example of line equation, we can create a

 * workflow to compute @f$y@f$ and @f$\frac{\partial y}{\partial x}@f$.

 * @code

 * elements in the inputs. Multiple work items can be created and will be

 * executed in order of creation.

 *

 * Once the work items are defined that can be JIT compiled to a backend device.

 * Once the work items are defined they can be JIT compiled to a backend device.

 * The graph framework supports back ends for generic CPUs, Apple Metal GPUs,

 * Nvidia Cuda GPUs, and initial HIP support of AMD GPUs. Each back end supplies

 * relevant driver code to build the kernel source, compile the kernel, build

 * as either variables @f$x@f$ or constants @f$m,b@f$. These nodes are connected

 * by nodes for multiply and addition operations. The output @f$y@f$ represents

 * the entire graph of operations.

 * @image{} html line_graph.png "The graph structure for @f$y=mx+b@f$."

 * @image{} html line_graph.png "The graph structure for y = mx + b."

 * Evaluation of graphs start from the top most node in this case the @f$+@f$

 * operation. Evaluation of a node is not performed until all sub-nodes are

 * evaluated starting with the left operand. Evaluation starts by recursively

 * graphs of a function derivative. For an example of taking derivatives see the

 * @ref tutorial_derivatives "auto differentiation tutorial". Lets say that we

 * want to take the derivative of @f$\frac{\partial y}{\partial x}@f$. This is

 * achieved by evaluating the until bottom left most node is reached. Then a new

 * graph is build starting with @f$\frac{\partial m}{\partial x}=0@f$. Applying

 * the first half of the chain rule we build a new graph for @f$0x@f$

 * achieved by evaluating the graph until the bottom left most node is reached.

 * Then a new graph is constucted starting with

 * @f$\frac{\partial m}{\partial x}=0@f$. Applying the first half of the chain

 * rule we build a new graph for @f$0x@f$

 * @image{} html line_graph_dydf1.png ""

 * Then we take the derivative of the right operand and apply the second half

 * of the chain rule to build a new graph for @f$0x=0@f$.

 * The final expression for @f$\frac{\partial y}{\partial x}@f$ contains many

 * unnecessary nodes in the graph. Instead of building full graphs, we can

 * simplify and eliminate nodes as we build them. For instance, when the

 * expression @f$0x@f$ this created can be immediately reduce it  to a single

 * node.

 * expression @f$0\times x@f$ is created, this can be immediately reduced to a

 * single node @f$0@f$.

 * @image{} html line_graph_reduce1.png ""

 * Applying all possible reductions reduces the final expression to

 * @f$\frac{\partial y}{\partial x}=m@f$.

 * @subsection general_concepts_compile_maps Maps

 * Maps enable the results of an output node to be stored in an input node. This

 * is used for a wide varity of cases. For instance take a gradient decent step.

 * @f{equation}{y = y + \frac{\partial f}{\partial x}@f}

 * @f{equation}{y_{i+1} = y_{i} + \frac{\partial f}{\partial x}@f}

 * In this case the output of the expression

 * @f$y + \frac{\partial f}{\partial x}@f$

 * can be mapped to update @f$y@f$.

 * <hr>

 * @section general_concepts_safe_math Safe Math

 * There are some conditions where mathematically, a graph should evaluate to a

 * normal number. However, when evaluated suing floating point precision, can

 * normal number. However, when evaluated using floating point precision, can

 * lead to <tt>Inf</tt> or <tt>NaN</tt>. An example of this the

 * @f$\exp\left(x\right)@f$ function. For large argument values,

 * @f$\exp\left(x\right)@f$ overflows the maximum floating point precision and

 * executable target which can be used to test out the API's of this framework.

 * The playground starts with a blank main function.

 * @code

#include "../graph_framework/jit.hpp"

#include "graph_framework.hpp"

int main(int argc, const char * argv[]) {

    START_GPU

 * main. This will allow us to play with different floating point types. For now

 * we will start with a simple float type.

 * @code

#include "../graph_framework/jit.hpp"

#include "graph_framework.hpp"

template<jit::float_scalar T>

void run_tutorial() {

 * so all method are called using the <tt>-></tt> operator.

 *

 * @subsection tutorial_constant Constant Nodes

 * Next we want to define a constant. There are two method to define constants

 * Next we want to define a constant. There are two methods to define constants

 * explicitly or implicitly.

 * @code

template<jit::float_scalar T>