SymPy Code Generation and Printing

SymPy Code Generation and Printing

This document covers SymPy's capabilities for generating executable code in various languages, converting expressions to different output formats, and customizing printing behavior.

Code Generation

Converting to NumPy Functions

from sympy import symbols, sin, cos, lambdify
import numpy as np

x, y = symbols('x y')
expr = sin(x) + cos(y)

# Create NumPy function
f = lambdify((x, y), expr, 'numpy')

# Use with NumPy arrays
x_vals = np.linspace(0, 2*np.pi, 100)
y_vals = np.linspace(0, 2*np.pi, 100)
result = f(x_vals, y_vals)

Lambdify Options

from sympy import lambdify, exp, sqrt

# Different backends
f_numpy = lambdify(x, expr, 'numpy')      # NumPy
f_scipy = lambdify(x, expr, 'scipy')      # SciPy
f_mpmath = lambdify(x, expr, 'mpmath')    # mpmath (arbitrary precision)
f_math = lambdify(x, expr, 'math')        # Python math module

# Custom function mapping
custom_funcs = {'sin': lambda x: x}  # Replace sin with identity
f = lambdify(x, sin(x), modules=[custom_funcs, 'numpy'])

# Multiple expressions
exprs = [x**2, x**3, x**4]
f = lambdify(x, exprs, 'numpy')
# Returns tuple of results

Generating C/C++ Code

from sympy.utilities.codegen import codegen
from sympy import symbols

x, y = symbols('x y')
expr = x**2 + y**2

# Generate C code
[(c_name, c_code), (h_name, h_header)] = codegen(
    ('distance_squared', expr),
    'C',
    header=False,
    empty=False
)

print(c_code)
# Outputs valid C function

Generating Fortran Code

from sympy.utilities.codegen import codegen

[(f_name, f_code), (h_name, h_interface)] = codegen(
    ('my_function', expr),
    'F95',  # Fortran 95
    header=False
)

print(f_code)

Advanced Code Generation

from sympy.utilities.codegen import CCodeGen, make_routine
from sympy import MatrixSymbol, Matrix

# Matrix operations
A = MatrixSymbol('A', 3, 3)
expr = A + A.T

# Create routine
routine = make_routine('matrix_sum', expr)

# Generate code
gen = CCodeGen()
code = gen.write([routine], prefix='my_module')

Code Printers

from sympy.printing.c import C99CodePrinter, C89CodePrinter
from sympy.printing.fortran import FCodePrinter
from sympy.printing.cxx import CXX11CodePrinter

# C code
c_printer = C99CodePrinter()
c_code = c_printer.doprint(expr)

# Fortran code
f_printer = FCodePrinter()
f_code = f_printer.doprint(expr)

# C++ code
cxx_printer = CXX11CodePrinter()
cxx_code = cxx_printer.doprint(expr)

Printing and Output Formats

Pretty Printing

from sympy import init_printing, pprint, pretty, symbols
from sympy import Integral, sqrt, pi

# Initialize pretty printing (for Jupyter notebooks and terminal)
init_printing()

x = symbols('x')
expr = Integral(sqrt(1/x), (x, 0, pi))

# Pretty print to terminal
pprint(expr)
#   π
#   ⌠
#   ⎮   1
#   ⎮  ───  dx
#   ⎮  √x
#   ⌡
#   0

# Get pretty string
s = pretty(expr)
print(s)

LaTeX Output

from sympy import latex, symbols, Integral, sin, sqrt

x, y = symbols('x y')
expr = Integral(sin(x)**2, (x, 0, pi))

# Convert to LaTeX
latex_str = latex(expr)
print(latex_str)
# \int\limits_{0}^{\pi} \sin^{2}{\left(x \right)}\, dx

# Custom LaTeX formatting
latex_str = latex(expr, mode='equation')  # Wrapped in equation environment
latex_str = latex(expr, mode='inline')    # Inline math

# For matrices
from sympy import Matrix
M = Matrix([[1, 2], [3, 4]])
latex(M)  # \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]

MathML Output

from sympy.printing.mathml import mathml, print_mathml
from sympy import sin, pi

expr = sin(pi/4)

# Content MathML
mathml_str = mathml(expr)

# Presentation MathML
mathml_str = mathml(expr, printer='presentation')

# Print to console
print_mathml(expr)

String Representations

from sympy import symbols, sin, pi, srepr, sstr

x = symbols('x')
expr = sin(x)**2

# Standard string (what you see in Python)
str(expr)  # 'sin(x)**2'

# String representation (prettier)
sstr(expr)  # 'sin(x)**2'

# Reproducible representation
srepr(expr)  # "Pow(sin(Symbol('x')), Integer(2))"
# This can be eval()'ed to recreate the expression

Custom Printing

from sympy.printing.str import StrPrinter

class MyPrinter(StrPrinter):
    def _print_Symbol(self, expr):
        return f"<{expr.name}>"

    def _print_Add(self, expr):
        return " PLUS ".join(self._print(arg) for arg in expr.args)

printer = MyPrinter()
x, y = symbols('x y')
print(printer.doprint(x + y))  # "<x> PLUS <y>"

Python Code Generation

autowrap - Compile and Import

from sympy.utilities.autowrap import autowrap
from sympy import symbols

x, y = symbols('x y')
expr = x**2 + y**2

# Automatically compile C code and create Python wrapper
f = autowrap(expr, backend='cython')
# or backend='f2py' for Fortran

# Use like a regular function
result = f(3, 4)  # 25

ufuncify - Create NumPy ufuncs

from sympy.utilities.autowrap import ufuncify
import numpy as np

x, y = symbols('x y')
expr = x**2 + y**2

# Create universal function
f = ufuncify((x, y), expr)

# Works with NumPy broadcasting
x_arr = np.array([1, 2, 3])
y_arr = np.array([4, 5, 6])
result = f(x_arr, y_arr)  # [17, 29, 45]

Expression Tree Manipulation

Walking Expression Trees

from sympy import symbols, sin, cos, preorder_traversal, postorder_traversal

x, y = symbols('x y')
expr = sin(x) + cos(y)

# Preorder traversal (parent before children)
for arg in preorder_traversal(expr):
    print(arg)

# Postorder traversal (children before parent)
for arg in postorder_traversal(expr):
    print(arg)

# Get all subexpressions
subexprs = list(preorder_traversal(expr))

Expression Substitution in Trees

from sympy import Wild, symbols, sin, cos

x, y = symbols('x y')
a = Wild('a')

expr = sin(x) + cos(y)

# Pattern matching and replacement
new_expr = expr.replace(sin(a), a**2)  # sin(x) -> x**2

Jupyter Notebook Integration

Display Math

from sympy import init_printing, display
from IPython.display import display as ipy_display

# Initialize printing for Jupyter
init_printing(use_latex='mathjax')  # or 'png', 'svg'

# Display expressions beautifully
expr = Integral(sin(x)**2, x)
display(expr)  # Renders as LaTeX in notebook

# Multiple outputs
ipy_display(expr1, expr2, expr3)

Interactive Widgets

from sympy import symbols, sin
from IPython.display import display
from ipywidgets import interact, FloatSlider
import matplotlib.pyplot as plt
import numpy as np

x = symbols('x')
expr = sin(x)

@interact(a=FloatSlider(min=0, max=10, step=0.1, value=1))
def plot_expr(a):
    f = lambdify(x, a * expr, 'numpy')
    x_vals = np.linspace(-np.pi, np.pi, 100)
    plt.plot(x_vals, f(x_vals))
    plt.show()

Converting Between Representations

String to SymPy

from sympy.parsing.sympy_parser import parse_expr
from sympy import symbols

x, y = symbols('x y')

# Parse string to expression
expr = parse_expr('x**2 + 2*x + 1')
expr = parse_expr('sin(x) + cos(y)')

# With transformations
from sympy.parsing.sympy_parser import (
    standard_transformations,
    implicit_multiplication_application
)

transformations = standard_transformations + (implicit_multiplication_application,)
expr = parse_expr('2x', transformations=transformations)  # Treats '2x' as 2*x

LaTeX to SymPy

from sympy.parsing.latex import parse_latex

# Parse LaTeX
expr = parse_latex(r'\frac{x^2}{y}')
# Returns: x**2/y

expr = parse_latex(r'\int_0^\pi \sin(x) dx')

Mathematica to SymPy

from sympy.parsing.mathematica import parse_mathematica

# Parse Mathematica code
expr = parse_mathematica('Sin[x]^2 + Cos[y]^2')
# Returns SymPy expression

Exporting Results

Export to File

from sympy import symbols, sin
import json

x = symbols('x')
expr = sin(x)**2

# Export as LaTeX to file
with open('output.tex', 'w') as f:
    f.write(latex(expr))

# Export as string
with open('output.txt', 'w') as f:
    f.write(str(expr))

# Export as Python code
with open('output.py', 'w') as f:
    f.write(f"from numpy import sin\n")
    f.write(f"def f(x):\n")
    f.write(f"    return {lambdify(x, expr, 'numpy')}\n")

Pickle SymPy Objects

import pickle
from sympy import symbols, sin

x = symbols('x')
expr = sin(x)**2 + x

# Save
with open('expr.pkl', 'wb') as f:
    pickle.dump(expr, f)

# Load
with open('expr.pkl', 'rb') as f:
    loaded_expr = pickle.load(f)

Numerical Evaluation and Precision

High-Precision Evaluation

from sympy import symbols, pi, sqrt, E, exp, sin
from mpmath import mp

x = symbols('x')

# Standard precision
pi.evalf()  # 3.14159265358979

# High precision (1000 digits)
pi.evalf(1000)

# Set global precision with mpmath
mp.dps = 50  # 50 decimal places
expr = exp(pi * sqrt(163))
float(expr.evalf())

# For expressions
result = (sqrt(2) + sqrt(3)).evalf(100)

Numerical Substitution

from sympy import symbols, sin, cos

x, y = symbols('x y')
expr = sin(x) + cos(y)

# Numerical evaluation
result = expr.evalf(subs={x: 1.5, y: 2.3})

# With units
from sympy.physics.units import meter, second
distance = 100 * meter
time = 10 * second
speed = distance / time
speed.evalf()

Common Patterns

Pattern 1: Generate and Execute Code

from sympy import symbols, lambdify
import numpy as np

# 1. Define symbolic expression
x, y = symbols('x y')
expr = x**2 + y**2

# 2. Generate function
f = lambdify((x, y), expr, 'numpy')

# 3. Execute with numerical data
data_x = np.random.rand(1000)
data_y = np.random.rand(1000)
results = f(data_x, data_y)

Pattern 2: Create LaTeX Documentation

from sympy import symbols, Integral, latex
from sympy.abc import x

# Define mathematical content
expr = Integral(x**2, (x, 0, 1))
result = expr.doit()

# Generate LaTeX document
latex_doc = f"""
\\documentclass{{article}}
\\usepackage{{amsmath}}
\\begin{{document}}

We compute the integral:
\\begin{{equation}}
{latex(expr)} = {latex(result)}
\\end{{equation}}

\\end{{document}}
"""

with open('document.tex', 'w') as f:
    f.write(latex_doc)

Pattern 3: Interactive Computation

from sympy import symbols, simplify, expand
from sympy.parsing.sympy_parser import parse_expr

x, y = symbols('x y')

# Interactive input
user_input = input("Enter expression: ")
expr = parse_expr(user_input)

# Process
simplified = simplify(expr)
expanded = expand(expr)

# Display
print(f"Simplified: {simplified}")
print(f"Expanded: {expanded}")
print(f"LaTeX: {latex(expr)}")

Pattern 4: Batch Code Generation

from sympy import symbols, lambdify
from sympy.utilities.codegen import codegen

# Multiple functions
x = symbols('x')
functions = {
    'f1': x**2,
    'f2': x**3,
    'f3': x**4
}

# Generate C code for all
for name, expr in functions.items():
    [(c_name, c_code), _] = codegen((name, expr), 'C')
    with open(f'{name}.c', 'w') as f:
        f.write(c_code)

Pattern 5: Performance Optimization

from sympy import symbols, sin, cos, cse
import numpy as np

x, y = symbols('x y')

# Complex expression with repeated subexpressions
expr = sin(x + y)**2 + cos(x + y)**2 + sin(x + y)

# Common subexpression elimination
replacements, reduced = cse(expr)
# replacements: [(x0, sin(x + y)), (x1, cos(x + y))]
# reduced: [x0**2 + x1**2 + x0]

# Generate optimized code
for var, subexpr in replacements:
    print(f"{var} = {subexpr}")
print(f"result = {reduced[0]}")

Important Notes

1. NumPy compatibility: When using lambdify with NumPy, ensure your expression uses functions available in NumPy.

2. Performance: For numerical work, always use lambdify or code generation rather than subs() + evalf() in loops.

3. Precision: Use mpmath for arbitrary precision arithmetic when needed.

4. Code generation caveats: Generated code may not handle all edge cases. Test thoroughly.

5. Compilation: autowrap and ufuncify require a C/Fortran compiler and may need configuration on your system.

6. Parsing: When parsing user input, validate and sanitize to avoid code injection vulnerabilities.

7. Jupyter: For best results in Jupyter notebooks, call init_printing() at the start of your session.