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CTA allows differentiation of basic functions such as (x^2-1)^.5 * atan((x^2-1)^.5) at compile time.

Prerequisites

You'll need a compiler with support for C++14, cmake, and boost for the boost preprocessor library.

Installing

  • mkdir build in the CTA folder
  • cmake ..
  • make all test
  • make install

Usage

CTA is a header-only library:

#include <cta/cta.hpp>
using namespace cta;

Let's declare some variables:

auto x = make_var<double, 0>();
auto y = make_var<double, 1>();

The first template argument defines the type of the variable, the second defines the index. Here's a couple of functions:

auto f = x*y + cos(x);
auto g = f + x;

All CTA expressions can be printed to ostreams:

std::cout << f << std::endl; // prints ((x0*x1)+cos(x0))

Use operator() to evaluate functions:

f(10, 20); // ~199.16

Differentiation:

auto fdx = differentiate(f, x);

Exponentiation currently is only supported with constant exponents. When the exponent is an integer, there is a form that allows slightly better simplification:

auto h = pow(x, 2);
auto i = pow<2>(x);

std::cout << differentiate(h, x) << std::endl; // prints (2*pow(x0, 1))
std::cout << differentiate(i, x) << std::endl; // prints (2*x0)

CTA can also handle the case where you know a value and its derivative(s) (think taylor series):

auto t = make_var<double, 0>();
auto g = make_dvar<double, 1>();
auto f = t + pow<2>(g);
auto fdt = differentiate(g, t); // 1 + 2*g'*g

std::cout << fdt(1, { 5, 10 }) << std::endl; // prints 101

Not supplying a derivative required by an expression (for instance, calling fdt(1, { 5 })) results in a compile error.

Supported operations

  • unary -
  • binary +, -, *, /,
  • pow, sin, cos, atan

Limitations

CTA can't differentiate everything. For instance:

d/dx((x^2 - 1)^.5 - atan((x^2 - 1)^.5)) = (x^2 - 1)^.5 / x

As you can see, this is defined at x = 1. CTA's simplification logic isn't powerful to find this derivative, instead getting:

x/(x^2-1)^.5 - x/((x^2-1)^.5 * x^2)

Both of these terms have a pole at x = 1, so the result at x = 1 will be undefined and numerical stability in the vicinity will be poor.

Files

  • cta.hpp: includes all of the following files
  • terms.hpp: defines templates for the operations (e.g. cta::detail::sum for binary +).
  • operators.hpp: defines operator+-*/
  • functions.hpp: defines pow, sin, ...
  • evaluator.hpp: defines operator()
  • differentiator.hpp: defines differentiate()
  • is_constant.hpp: used for checking if an expression is a constant.
  • simplifier.hpp: defines simplify()
  • output.hpp: defines operator<<
  • types.hpp: used by most of the other files to generate code.