# 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 Theano.

## 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.

## uFuncify¶

The autowrap module contains methods that help in efficient computation.

• autowrap method for compiling code generated by the codegen module, and wrap the binary for use in python.

• binary_function method automates the steps needed to autowrap the SymPy expression and attaching it to a Function object with implemented_function().

• ufuncify 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: sympy.utilities.autowrap().

## Theano¶

SymPy has a strong connection with Theano, a mathematical array compiler. SymPy expressions can be easily translated to Theano graphs and then compiled using the Theano compiler chain.

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

>>> from sympy.printing.theanocode import theano_function
>>> f = theano_function([x], [expr])


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

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


Theano 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. Theano 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 Theano installed then that will often be the best choice. If you don’t have Theano but do have f2py then you should use ufuncify.

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

Theano

10ns

Many outputs, CSE, GPUs

Theano