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

.. only:: html

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

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

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

.. _sphx_glr_examples_reciprocal_lookup.py:


Reciprocal lookup table
=======================

In this example a lookup table for floating-point reciprocals is designed,
assuming that each bit is implemented using an N-ínput LUT, suitable for FPGA, and
single-precision floating-point representation.

.. GENERATED FROM PYTHON SOURCE LINES 9-30

.. code-block:: Python


    import apytypes as apy
    import numpy as np
    import matplotlib.pyplot as plt

    N = 6
    WL = 9

    step = apy.APyFixed(1, 3, N)
    r = apy.arange(1, 2 + 2**-6, step=step)

    table = apy.zeros((2**6,), 1, WL)
    float_table = []
    for i in range(2**6):
        start = float(r[i])
        stop = float(r[i + 1])

        meanreciprocal = (1 / start + 1 / stop) / 2
        float_table.append(meanreciprocal)
        table[i] = apy.fx(meanreciprocal, 1, WL)








.. GENERATED FROM PYTHON SOURCE LINES 31-32

Print the resulting table

.. GENERATED FROM PYTHON SOURCE LINES 32-36

.. code-block:: Python

    fstring = f"{{:{WL}b}}"
    for b in table.to_bits():
        print(fstring.format(b))





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

 .. code-block:: none

    111111100
    111110100
    111101101
    111100101
    111011110
    111011000
    111010001
    111001010
    111000100
    110111110
    110111000
    110110010
    110101100
    110100111
    110100001
    110011100
    110010111
    110010010
    110001101
    110001000
    110000100
    101111111
    101111011
    101110111
    101110010
    101101110
    101101010
    101100110
    101100010
    101011110
    101011011
    101010111
    101010100
    101010000
    101001101
    101001001
    101000110
    101000011
    101000000
    100111101
    100111010
    100110111
    100110100
    100110001
    100101110
    100101011
    100101001
    100100110
    100100011
    100100001
    100011110
    100011100
    100011001
    100010111
    100010101
    100010010
    100010000
    100001110
    100001011
    100001001
    100000111
    100000101
    100000011
    100000001




.. GENERATED FROM PYTHON SOURCE LINES 37-38

Plot the approximation, with and without fixed-point values

.. GENERATED FROM PYTHON SOURCE LINES 38-50

.. code-block:: Python


    f = np.linspace(1, 2, 10000)
    fig, ax = plt.subplots()

    ax.plot(f, 1 / f)
    ax.step(r[:-1], table, where="post")
    ax.step(r[:-1], float_table, where="post")
    ax.stairs(table.to_numpy(), r.to_numpy())
    ax.stairs(float_table, r.to_numpy())
    ax.set_title("Function and approximation values")
    plt.show()




.. image-sg:: /examples/images/sphx_glr_reciprocal_lookup_001.png
   :alt: Function and approximation values
   :srcset: /examples/images/sphx_glr_reciprocal_lookup_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 51-52

Compute seed error and after one Newton-Raphson iteration

.. GENERATED FROM PYTHON SOURCE LINES 52-76

.. code-block:: Python


    points = 10000


    def eval_seed(x, table):
        table_row = x.man >> (23 - N)  # N most significant bits of x
        new_exp = 253 - x.exp
        new_man = (table[table_row].to_bits() & ((1 << (WL - 1)) - 1)) << (23 - WL + 1)
        res = apy.APyFloat(x.sign, new_exp, new_man, 8, 23)
        return res


    fval = np.linspace(1, 2, points)
    seed_table = apy.zeros((points,), exp_bits=8, man_bits=23)
    nr_table = apy.zeros((points,), exp_bits=8, man_bits=23)

    for i in range(points):
        x = apy.fp(fval[i], 8, 23)
        seed = eval_seed(x, table)
        seed_table[i] = seed
        # Perform one Newton-Raphson iteration
        refined = seed + seed * (1 - seed * x)
        nr_table[i] = refined








.. GENERATED FROM PYTHON SOURCE LINES 77-78

Plot resulting errors

.. GENERATED FROM PYTHON SOURCE LINES 78-86

.. code-block:: Python


    fig, ax = plt.subplots()
    ax.plot(fval, 1 / fval - seed_table, label="Seed error")
    ax.plot(fval, 1 / fval - nr_table, label="Error after one Newton-Raphson iteration")
    ax.set_yscale("symlog", linthresh=2**-23)
    ax.legend()
    ax.set_title("Approximation errors")
    plt.show()



.. image-sg:: /examples/images/sphx_glr_reciprocal_lookup_002.png
   :alt: Approximation errors
   :srcset: /examples/images/sphx_glr_reciprocal_lookup_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.310 seconds)


.. _sphx_glr_download_examples_reciprocal_lookup.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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