.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/dual_numbers/plot_01_forward_mode_autodiff.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_calculus_dual_numbers_plot_01_forward_mode_autodiff.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_calculus_dual_numbers_plot_01_forward_mode_autodiff.py:


Forward-mode automatic differentiation via dual numbers
============================================================

Dual numbers compute a function's value and its exact derivative
together, with no finite-difference step-size tradeoff between
truncation error (large ``h``) and floating-point cancellation (small
``h``).

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python

    import math

    import numpy as np

    from mathematicskit.calculus.systems.dual_numbers import derivative
    from mathematicskit.calculus.systems.finite_differences import central_difference








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Exact derivative, compared against central differences at various h
----------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-35

.. code-block:: Python


    f_dual = lambda x: (x * x).sin()  # d/dx sin(x^2) = 2x cos(x^2)
    f_plain = lambda x: math.sin(x * x)
    x0 = 1.3
    exact = 2.0 * x0 * math.cos(x0**2)

    dual_result = derivative(f_dual, x0)
    print(f"dual-number derivative: {dual_result:.15f} (exact: {exact:.15f}, error {abs(dual_result - exact):.2e})")

    for h in (1e-1, 1e-3, 1e-5, 1e-7, 1e-9, 1e-11):
        cd = central_difference(f_plain, x0, h=h)
        print(f"central diff h={h:.0e}: {cd:.15f}, error {abs(cd - exact):.2e}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    dual-number derivative: -0.309196080171192 (exact: -0.309196080171192, error 0.00e+00)
    central diff h=1e-01: -0.331234137017592, error 2.20e-02
    central diff h=1e-03: -0.309198313356795, error 2.23e-06
    central diff h=1e-05: -0.309196080394702, error 2.24e-10
    central diff h=1e-07: -0.309196080405805, error 2.35e-10
    central diff h=1e-09: -0.309196113157384, error 3.30e-08
    central diff h=1e-11: -0.309197112358106, error 1.03e-06




.. GENERATED FROM PYTHON SOURCE LINES 36-38

Composite functions and the chain rule fall out automatically
----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-44

.. code-block:: Python


    g = lambda x: (x.exp() / (x + 1.0)).log()
    x1 = 2.0
    grad = derivative(g, x1)
    print(f"d/dx log(exp(x)/(x+1)) at x={x1}: {grad:.10f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    d/dx log(exp(x)/(x+1)) at x=2.0: 0.6666666667




.. GENERATED FROM PYTHON SOURCE LINES 45-47

A NumPy sanity check against a nearby closed-form derivative
----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-50

.. code-block:: Python


    numpy_gradient_estimate = np.gradient([f_plain(x0 - 1e-6), f_plain(x0), f_plain(x0 + 1e-6)], 1e-6)[1]
    print("close to numpy finite-difference estimate:", bool(np.isclose(dual_result, numpy_gradient_estimate, atol=1e-3)))




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    close to numpy finite-difference estimate: True





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_calculus_dual_numbers_plot_01_forward_mode_autodiff.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_forward_mode_autodiff.ipynb <plot_01_forward_mode_autodiff.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_forward_mode_autodiff.py <plot_01_forward_mode_autodiff.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_forward_mode_autodiff.zip <plot_01_forward_mode_autodiff.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_