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. Letâs 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
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 âtop expressionâ 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
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)
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.name}", format="svg")
html = "<table>\n"
html += " <tr>\n"
html += "".join(
f' <th style="text-align:center; background-color:white">{symbol.name}</th>\n'
for symbol in sub_expressions
)
html += " </tr>\n"
html += " <tr>\n"
for symbol in sub_expressions:
svg = SVG(f"sub_expr_{symbol.name}.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, itâs therefore much faster to lambdify these sub-expressions separately and to recombine them. TensorWaves offers a function that does this for you: fast_lambdify()
. Weâll 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=lambda s: s.name)
%%time
split_function = fast_lambdify(
expression,
sorted_symbols,
max_complexity=100,
backend="numpy",
)
CPU times: user 322 ms, sys: 183 Îźs, total: 322 ms
Wall time: 320 ms
%%time
normal_function = sp.lambdify(sorted_symbols, expression)
CPU times: user 2.2 s, sys: 16.1 ms, total: 2.22 s
Wall time: 2.22 s
Specifying complexity#
When creating a parametrized function, we use the create_parametrized_function()
function. By default, this internally calls SymPyâs own lambdify()
function. But if you specify its max_complexity
argument, create_parametrized_function()
uses TensorWavesâs fast_lambdify()
.
%%time
function = create_parametrized_function(
expression=model.expression.doit(),
parameters=model.parameter_defaults,
max_complexity=100,
backend="numpy",
)
CPU times: user 655 ms, sys: 0 ns, total: 655 ms
Wall time: 653 ms