Numeric Computation

Symbolic computer algebra systems like Diofant 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 Diofant offers a number of easy-to-use hooks into other numeric systems, allowing you to create mathematical expressions in Diofant 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 Diofant speeds. The .subs(...).evalf() method can substitute a numeric value for a symbolic one and then evaluate the result within Diofant.

>>> expr = sin(x)/x
>>> expr.evalf(subs={x: 3.14}, strict=False)
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 Diofant expressions into Python functions, leveraging a variety of numerical libraries. It is used as follows:

>>> 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 Diofant 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.

>>> expr = sin(x)/x
>>> f = lambdify(x, expr, "numpy")

>>> import numpy
>>> data = numpy.linspace(1, 10, 10000)
>>> pprint(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

While NumPy operations are very efficient for vectorized data they sometimes incur unnecessary costs when chained together. Consider the following operation

x = get_numpy_array(...)
y = sin(x)/x

The operators sin and / call routines that execute tight for loops in C. The resulting computation looks something like this

for(int i = 0; i < n; i++)
{
    temp[i] = sin(x[i]);
}
for(int i = i; i < n; i++)
{
    y[i] = temp[i] / x[i];
}

This is slightly sub-optimal because

1. We allocate an extra temp array
2. We walk over x memory twice when once would have been sufficient

A better solution would fuse both element-wise operations into a single for loop

for(int i = i; i < n; i++)
{
    y[i] = sin(x[i]) / x[i];
}

Statically compiled projects like NumPy are unable to take advantage of such optimizations. Fortunately, Diofant is able to generate efficient low-level C or Fortran code. It can then depend on projects like Cython or f2py to compile and reconnect that code back up to Python. Fortunately this process is well automated and a Diofant user wishing to make use of this code generation should call the ufuncify function

>>> expr = sin(x)/x

>>> from diofant.utilities.autowrap import ufuncify
>>> f = ufuncify((x,), expr)

This function f consumes and returns a NumPy array. Generally ufuncify performs at least as well as lambdify. If the expression is complicated then ufuncify often significantly outperforms the NumPy backed solution. Jensen has a good blog post on this topic.

Theano

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

>>> expr = sin(x)/x

>>> from diofant.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 Diofant’s C/Fortran code printers. Among other things it handles common sub-expressions and compilation onto the GPU. Theano also supports Diofant 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