.. _numeric_computation:

===================
Numeric Computation
===================

Symbolic computer algebra systems like SymPy facilitate the construction and
manipulation of mathematical expressions.  Unfortunately when it comes time
to evaluate these expressions on numerical data, symbolic systems often have
poor performance.

Fortunately SymPy offers a number of easy-to-use hooks into other numeric
systems, allowing you to create mathematical expressions in SymPy and then
ship them off to the numeric system of your choice.  This page documents many
of the options available including the ``math`` library, the popular array
computing package ``numpy``, code generation in ``Fortran`` or ``C``, and the
use of the array compiler ``Aesara``.

Subs/evalf
----------

Subs is the slowest but simplest option.  It runs at SymPy speeds.
The ``.subs(...).evalf()`` method can substitute a numeric value
for a symbolic one and then evaluate the result within SymPy.


    >>> from sympy import *
    >>> from sympy.abc import x
    >>> expr = sin(x)/x
    >>> expr.evalf(subs={x: 3.14})
    0.000507214304613640

This method is slow.  You should use this method production only if performance
is not an issue.  You can expect ``.subs`` to take tens of microseconds. It
can be useful while prototyping or if you just want to see a value once.

Lambdify
--------

The ``lambdify`` function translates SymPy expressions into Python functions,
leveraging a variety of numerical libraries.  It is used as follows:

    >>> from sympy import *
    >>> from sympy.abc import x
    >>> expr = sin(x)/x
    >>> f = lambdify(x, expr)
    >>> f(3.14)
    0.000507214304614

Here lambdify makes a function that computes ``f(x) = sin(x)/x``.  By default
lambdify relies on implementations in the ``math`` standard library. This
numerical evaluation takes on the order of hundreds of nanoseconds, roughly two
orders of magnitude faster than the ``.subs`` method.  This is the speed
difference between SymPy and raw Python.

Lambdify can leverage a variety of numerical backends.  By default it uses the
``math`` library.  However it also supports ``mpmath`` and most notably,
``numpy``.  Using the ``numpy`` library gives the generated function access to
powerful vectorized ufuncs that are backed by compiled C code.

    >>> from sympy import *
    >>> from sympy.abc import x
    >>> expr = sin(x)/x
    >>> f = lambdify(x, expr, "numpy")

    >>> import numpy
    >>> data = numpy.linspace(1, 10, 10000)
    >>> f(data)
    [ 0.84147098  0.84119981  0.84092844 ... -0.05426074 -0.05433146
     -0.05440211]

If you have array-based data this can confer a considerable speedup, on the
order of 10 nano-seconds per element. Unfortunately numpy incurs some start-up
time and introduces an overhead of a few microseconds.

CuPy is a NumPy-compatible array library that mainly runs on CUDA, but has 
increasing support for other GPU manufacturers. It can in many cases be used as 
a drop-in replacement for numpy.

    >>> f = lambdify(x, expr, "cupy")
    >>> import cupy as cp
    >>> data = cp.linspace(1, 10, 10000)
    >>> y = f(data) # perform the computation
    >>> cp.asnumpy(y) # explicitly copy from GPU to CPU / numpy array
    [ 0.84147098  0.84119981  0.84092844 ... -0.05426074 -0.05433146
     -0.05440211]

uFuncify
--------

The ``autowrap`` module contains methods that help in efficient computation.

* :ref:`autowrap<autowrap>` method for compiling code generated by the
  :ref:`codegen<codegen_prose>` module, and wrap the binary for use in python.

* :ref:`binary_function<binary_function>` method automates the steps needed to autowrap
  the SymPy expression and attaching it to a ``Function`` object with ``implemented_function()``.

* :ref:`ufuncify<ufuncify_method>` generates a binary function that supports broadcasting
  on numpy arrays using different backends that are faster as compared to ``subs/evalf``
  and ``lambdify``.

The API reference of all the above is listed here: :py:func:`sympy.utilities.autowrap`.

Aesara
------

SymPy has a strong connection with
`Aesara <https://aesara.readthedocs.io/en/latest/>`_, a mathematical array
compiler.  SymPy expressions can be easily translated to Aesara graphs and then
compiled using the Aesara compiler chain.

    >>> from sympy import *
    >>> from sympy.abc import x
    >>> expr = sin(x)/x

    >>> from sympy.printing.aesaracode import aesara_function
    >>> f = aesara_function([x], [expr])

If array broadcasting or types are desired then Aesara requires this extra
information

    >>> f = aesara_function([x], [expr], dims={x: 1}, dtypes={x: 'float64'})

Aesara has a more sophisticated code generation system than SymPy's C/Fortran
code printers.  Among other things it handles common sub-expressions and
compilation onto the GPU.  Aesara also supports SymPy Matrix and Matrix
Expression objects.

So Which Should I Use?
----------------------

The options here were listed in order from slowest and least dependencies to
fastest and most dependencies.  For example, if you have Aesara installed then
that will often be the best choice.  If you don't have Aesara but do have
``f2py`` then you should use ``ufuncify``. If you have been comfortable using 
lambdify with the numpy module, but have a GPU, CuPy can provide substantial 
speedups with little effort.

+-----------------+-------+-----------------------------+---------------+
| Tool            | Speed | Qualities                   | Dependencies  |
+=================+=======+=============================+===============+
| subs/evalf      | 50us  | Simple                      | None          |
+-----------------+-------+-----------------------------+---------------+
| lambdify        | 1us   | Scalar functions            | math          |
+-----------------+-------+-----------------------------+---------------+
| lambdify-numpy  | 10ns  | Vector functions            | numpy         |
+-----------------+-------+-----------------------------+---------------+
| ufuncify        | 10ns  | Complex vector expressions  | f2py, Cython  |
+-----------------+-------+-----------------------------+---------------+
| lambdify-cupy   | 10ns  | Vector functions on GPUs    | cupy          |
+-----------------+-------+-----------------------------+---------------+
| Aesara          | 10ns  | Many outputs, CSE, GPUs     | Aesara        |
+-----------------+-------+-----------------------------+---------------+