Speed up lambdifying

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import logging

import ampform
import graphviz
import qrules
import sympy as sp
from ampform.dynamics.builder import create_relativistic_breit_wigner_with_ff
from IPython.display import HTML, SVG

from tensorwaves.function.sympy import (
    create_parametrized_function,
    fast_lambdify,
    split_expression,
)

logging.getLogger("tensorwaves.data").setLevel(logging.ERROR)  # hide progress bars

Note

Since #374, expressions are lambdified with common sub-expressions. This should already reduce lambdification time significantly and also results in faster computational functions.

Split expression

Lambdifying a SymPy expression can take rather long when an expression is complicated. Fortunately, TensorWaves offers a way to speed up the lambdify process. The idea is to split up an an expression into sub-expressions, separate those separately, and then recombining them. Let’s illustrate that idea with the following simplified example:

x, y, z = sp.symbols("x:z")
expr = x**z + 2 * y + sp.log(y * z)
expr

\[\displaystyle x^{z} + 2 y + \log{\left(y z \right)}\]

This expression can be represented in a tree of mathematical operations.

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dot = sp.dotprint(expr, bgcolor="none")
graphviz.Source(dot)

The function split_expression() can now be used to split up this expression tree into a ‘top expression’ plus definitions for each of the sub-expressions into which it was split:

top_expr, sub_expressions = split_expression(expr, max_complexity=3)
top_expr

\[\displaystyle f_{0} + f_{1} + f_{2}\]

sub_expressions

{f0: x**z, f1: 2*y, f2: log(y*z)}

The original expression can easily be reconstructed with subs() or xreplace():

top_expr.xreplace(sub_expressions)

\[\displaystyle x^{z} + 2 y + \log{\left(y z \right)}\]

Each of the expression trees are now smaller than the original:

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dot = sp.dotprint(top_expr, bgcolor="none")
graphviz.Source(dot)

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for symbol, definition in sub_expressions.items():
    dot = sp.dotprint(definition, bgcolor="none")
    graph = graphviz.Source(dot)
    graph.render(filename=f"sub_expr_{symbol}", format="svg")

html = "<table>\n"
html += "  <tr>\n"
html += "".join(
    f'    <th style="text-align:center; background-color:white">{symbol}</th>\n'
    for symbol in sub_expressions
)
html += "  </tr>\n"
html += "  <tr>\n"
for symbol in sub_expressions:
    svg = SVG(f"sub_expr_{symbol}.svg").data
    html += f'    <td style="background-color:white">{svg}</td>\n'
html += "  </tr>\n"
html += "</table>"
HTML(html)

f0	f1	f2

Fast lambdify

Generally, the lambdify time scales exponentially with the size of an expression tree. With larger expression trees, it’s therefore much faster to lambdify these sub-expressions separately and to recombine them. TensorWaves offers a function that does this for you: fast_lambdify(). We’ll use an HelicityModel to illustrate this:

reaction = qrules.generate_transitions(
    initial_state=("J/psi(1S)", [+1]),
    final_state=["gamma", "pi0", "pi0"],
    allowed_intermediate_particles=["f(0)"],
)
model_builder = ampform.get_builder(reaction)
for name in reaction.get_intermediate_particles().names:
    model_builder.dynamics.assign(name, create_relativistic_breit_wigner_with_ff)
model = model_builder.formulate()

expression = model.expression.doit()
sorted_symbols = sorted(expression.free_symbols, key=str)

%%time

split_function = fast_lambdify(
    expression,
    sorted_symbols,  # ty:ignore[invalid-argument-type]
    backend="numpy",
    max_complexity=100,
)

CPU times: user 330 ms, sys: 3.22 ms, total: 333 ms
Wall time: 332 ms

%%time

normal_function = sp.lambdify(sorted_symbols, expression)

CPU times: user 2.41 s, sys: 8.84 ms, total: 2.42 s
Wall time: 2.32 s

Specifying complexity

When creating a parametrized function, we use the create_parametrized_function() function. By default, this internally calls SymPy’s own lambdify() function. But if you specify its max_complexity argument, create_parametrized_function() uses TensorWaves’s fast_lambdify().

%%time

function = create_parametrized_function(
    expression=model.expression.doit(),
    parameters=model.parameter_defaults,
    backend="numpy",
    max_complexity=100,
)

CPU times: user 736 ms, sys: 6.02 ms, total: 742 ms
Wall time: 739 ms