Structural Details of Code Generation with SymPy

Several submodules in SymPy allow one to generate directly compilable and executable code in a variety of different programming languages from SymPy expressions. In addition, there are functions that generate Python importable objects that can evaluate SymPy expressions very efficiently.

We will start with a brief introduction to the components that make up the code generation capabilities of SymPy.

Introduction

There are four main levels of abstractions:

expression
   |
code printers
   |
code generators
   |
autowrap

sympy.utilities.autowrap uses codegen, and codegen uses the code printers. sympy.utilities.autowrap does everything: it lets you go from SymPy expression to numerical function in the same Python process in one step. Codegen is actual code generation, i.e., to compile and use later, or to include in some larger project.

The code printers translate the SymPy objects into actual code, like abs(x) -> fabs(x) (for C).

The code printers don’t print optimal code in many cases. An example of this is powers in C. x**2 prints as pow(x, 2) instead of x*x. Other optimizations (like mathematical simplifications) should happen before the code printers.

Currently, sympy.simplify.cse_main.cse() is not applied automatically anywhere in this chain. It ideally happens at the codegen level, or somewhere above it.

We will iterate through the levels below.

The following three lines will be used to setup each example:

>>> from sympy import *
>>> init_printing(use_unicode=True)
>>> from sympy.abc import a, e, k, n, r, t, x, y, z, T, Z
>>> from sympy.abc import beta, omega, tau
>>> f, g = symbols('f, g', cls=Function)

Code printers (sympy.printing)

This is where the meat of code generation is; the translation of SymPy actually more like a lightweight version of codegen for Python, and Python (sympy.printing.pycode.pycode()), and sympy.printing.lambdarepr.lambdarepr(), which supports many libraries (like NumPy), and theano (sympy.printing.theanocode.theano_function()). The code printers are special cases of the other prints in SymPy (str printer, pretty printer, etc.).

An important distinction is that the code printer has to deal with assignments (using the sympy.codegen.ast.Assignment object). This serves as building blocks for the code printers and hence the codegen module. An example that shows the use of Assignment in C code:

>>> from sympy.codegen.ast import Assignment
>>> print(ccode(Assignment(x, y + 1)))
x = y + 1;

Here is another simple example of printing a C version of a SymPy expression:

>>> expr = (Rational(-1, 2) * Z * k * (e**2) / r)
>>> expr
    2
-Z⋅e ⋅k
────────
  2⋅r
>>> ccode(expr)
-1.0/2.0*Z*pow(e, 2)*k/r
>>> from sympy.codegen.ast import real, float80
>>> ccode(expr, assign_to="E", type_aliases={real: float80})
E = -1.0L/2.0L*Z*powl(e, 2)*k/r;

To generate code with some math functions provided by e.g. the C99 standard we need to import functions from sympy.codegen.cfunctions:

>>> from sympy.codegen.cfunctions import expm1
>>> ccode(expm1(x), standard='C99')
expm1(x)

Piecewise expressions are converted into conditionals. If an assign_to variable is provided an if statement is created, otherwise the ternary operator is used. Note that if the Piecewise lacks a default term, represented by (expr, True) then an error will be thrown. This is to prevent generating an expression that may not evaluate to anything. A use case for Piecewise:

>>> expr = Piecewise((x + 1, x > 0), (x, True))
>>> print(fcode(expr, tau))
      if (x > 0) then
         tau = x + 1
      else
         tau = x
      end if

The various printers also tend to support Indexed objects well. With contract=True these expressions will be turned into loops, whereas contract=False will just print the assignment expression that should be looped over:

>>> len_y = 5
>>> mat_1 = IndexedBase('mat_1', shape=(len_y,))
>>> mat_2 = IndexedBase('mat_2', shape=(len_y,))
>>> Dy = IndexedBase('Dy', shape=(len_y-1,))
>>> i = Idx('i', len_y-1)
>>> eq = Eq(Dy[i], (mat_1[i+1] - mat_1[i]) / (mat_2[i+1] - mat_2[i]))
>>> print(jscode(eq.rhs, assign_to=eq.lhs, contract=False))
Dy[i] = (mat_1[i + 1] - mat_1[i])/(mat_2[i + 1] - mat_2[i]);
>>> Res = IndexedBase('Res', shape=(len_y,))
>>> j = Idx('j', len_y)
>>> eq = Eq(Res[j], mat_1[j]*mat_2[j])
>>> print(jscode(eq.rhs, assign_to=eq.lhs, contract=True))
for (var j=0; j<5; j++){
   Res[j] = 0;
}
for (var j=0; j<5; j++){
   for (var j=0; j<5; j++){
      Res[j] = Res[j] + mat_1[j]*mat_2[j];
   }
}
>>> print(jscode(eq.rhs, assign_to=eq.lhs, contract=False))
Res[j] = mat_1[j]*mat_2[j];

Custom printing can be defined for certain types by passing a dictionary of “type” : “function” to the user_functions kwarg. Alternatively, the dictionary value can be a list of tuples i.e., [(argument_test, cfunction_string)]. This can be used to call a custom Octave function:

>>> custom_functions = {
...   "f": "existing_octave_fcn",
...   "g": [(lambda x: x.is_Matrix, "my_mat_fcn"),
...         (lambda x: not x.is_Matrix, "my_fcn")]
... }
>>> mat = Matrix([[1, x]])
>>> octave_code(f(x) + g(x) + g(mat), user_functions=custom_functions)
existing_octave_fcn(x) + my_fcn(x) + my_mat_fcn([1 x])

An example of Mathematica code printer:

>>> x_ = Function('x')
>>> expr = x_(n*T) * sin((t - n*T) / T)
>>> expr = expr / ((-T*n + t) / T)
>>> expr
            ⎛-T⋅n + t⎞
T⋅x(T⋅n)⋅sin⎜────────⎟
            ⎝   T    ⎠
──────────────────────
       -T⋅n + t

>>> expr = summation(expr, (n, -1, 1))
>>> mathematica_code(expr)
T*x[-T]*Sin[(T + t)/T]/(T + t) + T*x[T]*Sin[(-T + t)/T]/(-T + t) + T*x[0]*Sin[
t/T]/t

We can go through a common expression in different languages we support and see how it works:

>>> k, g1, g2, r, I, S = symbols("k, gamma_1, gamma_2, r, I, S")
>>> expr = k * g1 * g2 / (r**3)
>>> expr = expr * 2 * I * S * (3 * (cos(beta))**2 - 1) / 2
>>> expr
            ⎛     2       ⎞
I⋅S⋅γ₁⋅γ₂⋅k⋅⎝3⋅cos (β) - 1⎠
───────────────────────────
              3
             r
>>> print(jscode(expr, assign_to="H_is"))
H_is = I*S*gamma_1*gamma_2*k*(3*Math.pow(Math.cos(beta), 2) - 1)/Math.pow(r, 3);
>>> print(ccode(expr, assign_to="H_is", standard='C89'))
H_is = I*S*gamma_1*gamma_2*k*(3*pow(cos(beta), 2) - 1)/pow(r, 3);
>>> print(fcode(expr, assign_to="H_is"))
      H_is = I*S*gamma_1*gamma_2*k*(3*cos(beta)**2 - 1)/r**3
>>> print(julia_code(expr, assign_to="H_is"))
H_is = I.*S.*gamma_1.*gamma_2.*k.*(3*cos(beta).^2 - 1)./r.^3
>>> print(octave_code(expr, assign_to="H_is"))
H_is = I.*S.*gamma_1.*gamma_2.*k.*(3*cos(beta).^2 - 1)./r.^3;
>>> print(rust_code(expr, assign_to="H_is"))
H_is = I*S*gamma_1*gamma_2*k*(3*beta.cos().powi(2) - 1)/r.powi(3);
>>> print(mathematica_code(expr))
I*S*gamma_1*gamma_2*k*(3*Cos[beta]^2 - 1)/r^3

Codegen (sympy.utilities.codegen)

This module deals with creating compilable code from SymPy expressions. This is lower level than autowrap, as it doesn’t actually attempt to compile the code, but higher level than the printers, as it generates compilable files (including header files), rather than just code snippets.

The user friendly functions, here, are codegen and make_routine. codegen takes a list of (variable, expression) pairs and a language (C, F95, and Octave/Matlab are supported). It returns, as strings, a code file and a header file (for relevant languages). The variables are created as functions that return the value of the expression as output.

Note

The codegen callable is not in the sympy namespace automatically, to use it you must first import codegen from sympy.utilities.codegen

For instance:

>>> from sympy.utilities.codegen import codegen
>>> length, breadth, height = symbols('length, breadth, height')
>>> [(c_name, c_code), (h_name, c_header)] = \
... codegen(('volume', length*breadth*height), "C99", "test",
...         header=False, empty=False)
>>> print(c_name)
test.c
>>> print(c_code)
#include "test.h"
#include <math.h>
double volume(double breadth, double height, double length) {
   double volume_result;
   volume_result = breadth*height*length;
   return volume_result;
}
>>> print(h_name)
test.h
>>> print(c_header)
#ifndef PROJECT__TEST__H
#define PROJECT__TEST__H
double volume(double breadth, double height, double length);
#endif

Various flags to codegen let you modify things. The project name for preprocessor instructions can be varied using project. Variables listed as global variables in arg global_vars will not show up as function arguments.

language is a case-insensitive string that indicates the source code language. Currently, C, F95 and Octave are supported. Octave generates code compatible with both Octave and Matlab.

header when True, a header is written on top of each source file. empty when True, empty lines are used to structure the code. With argument_sequence a sequence of arguments for the routine can be defined in a preferred order.

prefix defines a prefix for the names of the files that contain the source code. If omitted, the name of the first name_expr tuple is used.

to_files when True, the code will be written to one or more files with the given prefix.

Here is an example:

>>> [(f_name, f_code), header] = codegen(("volume", length*breadth*height),
...     "F95", header=False, empty=False, argument_sequence=(breadth, length),
...     global_vars=(height,))
>>> print(f_code)
REAL*8 function volume(breadth, length)
implicit none
REAL*8, intent(in) :: breadth
REAL*8, intent(in) :: length
volume = breadth*height*length
end function

The method make_routine creates a Routine object, which represents an evaluation routine for a set of expressions. This is only good for internal use by the CodeGen objects, as an intermediate representation from SymPy expression to generated code. It is not recommended to make a Routine object yourself. You should instead use make_routine method. make_routine in turn calls the routine method of the CodeGen object depending upon the language of choice. This creates the internal objects representing assignments and so on, and creates the Routine class with them.

The various codegen objects such as Routine and Variable aren’t SymPy objects (they don’t subclass from Basic).

For example:

>>> from sympy.utilities.codegen import make_routine
>>> from sympy.physics.hydrogen import R_nl
>>> expr = R_nl(3, y, x, 6)
>>> routine = make_routine('my_routine', expr)
>>> [arg.result_var for arg in routine.results]   
[result₅₁₄₂₃₄₁₆₈₁₃₉₇₇₁₉₄₂₈]
>>> [arg.expr for arg in routine.results]
⎡                __________                                          ⎤
⎢          y    ╱ (2 - y)!   -2⋅x                                    ⎥
⎢4⋅√6⋅(4⋅x) ⋅  ╱  ──────── ⋅ℯ    ⋅assoc_laguerre(2 - y, 2⋅y + 1, 4⋅x)⎥
⎢            ╲╱   (y + 3)!                                           ⎥
⎢────────────────────────────────────────────────────────────────────⎥
⎣                                 3                                  ⎦
>>> [arg.name for arg in routine.arguments]
[x, y]

Another more complicated example with a mixture of specified and automatically-assigned names. Also has Matrix output:

>>> routine = make_routine('fcn', [x*y, Eq(a, 1), Eq(r, x + r), Matrix([[x, 2]])])
>>> [arg.result_var for arg in routine.results]   
[result_5397460570204848505]
>>> [arg.expr for arg in routine.results]
[x⋅y]
>>> [arg.name for arg in routine.arguments]   
[x, y, a, r, out_8598435338387848786]

We can examine the various arguments more closely:

>>> from sympy.utilities.codegen import (InputArgument, OutputArgument,
...                                      InOutArgument)
>>> [a.name for a in routine.arguments if isinstance(a, InputArgument)]
[x, y]

>>> [a.name for a in routine.arguments if isinstance(a, OutputArgument)]  
[a, out_8598435338387848786]
>>> [a.expr for a in routine.arguments if isinstance(a, OutputArgument)]
[1, [x  2]]

>>> [a.name for a in routine.arguments if isinstance(a, InOutArgument)]
[r]
>>> [a.expr for a in routine.arguments if isinstance(a, InOutArgument)]
[r + x]

The full API reference can be viewed here.

Autowrap

Autowrap automatically generates code, writes it to disk, compiles it, and imports it into the current session. Main functions of this module are autowrap, binary_function, and ufuncify.

It also automatically converts expressions containing Indexed objects into summations. The classes IndexedBase, Indexed and Idx represent a matrix element M[i, j]. See Tensor Module for more on this.

autowrap creates a wrapper using f2py or Cython and creates a numerical function.

Note

The autowrap callable is not in the sympy namespace automatically, to use it you must first import autowrap from sympy.utilities.autowrap

The callable returned from autowrap() is a binary Python function, not a SymPy object. For example:

>>> from sympy.utilities.autowrap import autowrap
>>> expr = ((x - y + z)**(13)).expand()
>>> binary_func = autowrap(expr)    
>>> binary_func(1, 4, 2)    
-1.0

The various flags available with autowrap() help to modify the services provided by the method. The argument tempdir tells autowrap to compile the code in a specific directory, and leave the files intact when finished. For instance:

>>> from sympy.utilities.autowrap import autowrap
>>> from sympy.physics.qho_1d import psi_n
>>> x_ = IndexedBase('x')
>>> y_ = IndexedBase('y')
>>> m = symbols('m', integer=True)
>>> i = Idx('i', m)
>>> qho = autowrap(Eq(y_[i], psi_n(0, x_[i], m, omega)), tempdir='/tmp')  

Checking the Fortran source code in the directory specified reveals this:

subroutine autofunc(m, omega, x, y)
implicit none
INTEGER*4, intent(in) :: m
REAL*8, intent(in) :: omega
REAL*8, intent(in), dimension(1:m) :: x
REAL*8, intent(out), dimension(1:m) :: y
INTEGER*4 :: i

REAL*8, parameter :: hbar = 1.05457162d-34
REAL*8, parameter :: pi = 3.14159265358979d0
do i = 1, m
   y(i) = (m*omega)**(1.0d0/4.0d0)*exp(-4.74126166983329d+33*m*omega*x(i &
         )**2)/(hbar**(1.0d0/4.0d0)*pi**(1.0d0/4.0d0))
end do

end subroutine

Using the argument args along with it changes argument sequence:

>>> eq = Eq(y_[i], psi_n(0, x_[i], m, omega))
>>> qho = autowrap(eq, tempdir='/tmp', args=[y, x, m, omega])  

yields:

subroutine autofunc(y, x, m, omega)
implicit none
INTEGER*4, intent(in) :: m
REAL*8, intent(in) :: omega
REAL*8, intent(out), dimension(1:m) :: y
REAL*8, intent(in), dimension(1:m) :: x
INTEGER*4 :: i

REAL*8, parameter :: hbar = 1.05457162d-34
REAL*8, parameter :: pi = 3.14159265358979d0
do i = 1, m
   y(i) = (m*omega)**(1.0d0/4.0d0)*exp(-4.74126166983329d+33*m*omega*x(i &
         )**2)/(hbar**(1.0d0/4.0d0)*pi**(1.0d0/4.0d0))
end do

end subroutine

The argument verbose is boolean, optional and if True, autowrap will not mute the command line backends. This can be helpful for debugging.

The argument language and backend are used to change defaults: Fortran and f2py to C and Cython. The argument helpers is used to define auxiliary expressions needed for the main expression. If the main expression needs to call a specialized function it should be put in the helpers iterable. Autowrap will then make sure that the compiled main expression can link to the helper routine. Items should be tuples with (<function_name>, <sympy_expression>, <arguments>). It is mandatory to supply an argument sequence to helper routines.

Another method available at the autowrap level is binary_function. It returns a sympy function. The advantage is that we can have very fast functions as compared to SymPy speeds. This is because we will be using compiled functions with Sympy attributes and methods. An illustration:

>>> from sympy.utilities.autowrap import binary_function
>>> from sympy.physics.hydrogen import R_nl
>>> psi_nl = R_nl(1, 0, a, r)
>>> f = binary_function('f', psi_nl)    
>>> f(a, r).evalf(3, subs={a: 1, r: 2})  
0.766

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

>>> x = get_numpy_array(...) # doctest: +SKIP
>>> 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, SymPy 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 SymPy user wishing to make use of this code generation should call the ufuncify function.

ufuncify is the third method available with Autowrap module. It basically implies ‘Universal functions’ and follows an ideology set by NumPy. The main point of ufuncify as compared to autowrap is that it allows arrays as arguments and can operate in an element-by-element fashion. The core operation done element-wise is in accordance to Numpy’s array broadcasting rules. See this for more.

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

>>> from sympy.utilities.autowrap import ufuncify
>>> f = ufuncify([x], expr) # doctest: +SKIP

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.

Let us see an example for some quantitative analysis:

>>> from sympy.physics.hydrogen import R_nl
>>> expr = R_nl(3, 1, x, 6)
>>> expr
               -2⋅x
8⋅x⋅(4 - 4⋅x)⋅ℯ
───────────────────
         3

The lambdify function translates SymPy expressions into Python functions, leveraging a variety of numerical libraries. By default lambdify relies on implementations in the math standard library. Naturally, Raw Python is faster than SymPy. 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.

Let us compare the speeds:

>>> from sympy.utilities.autowrap import ufuncify
>>> from sympy.utilities.lambdify import lambdify
>>> fn_numpy = lambdify(x, expr, 'numpy')   
>>> fn_fortran = ufuncify([x], expr, backend='f2py')    
>>> from numpy import linspace  
>>> xx = linspace(0, 1, 5)  
>>> fn_numpy(xx)    
[ 0.          1.21306132  0.98101184  0.44626032  0.        ]
>>> fn_fortran(xx)  
[ 0.          1.21306132  0.98101184  0.44626032  0.        ]
>>> import timeit
>>> timeit.timeit('fn_numpy(xx)', 'from __main__ import fn_numpy, xx', number=10000)    
0.18891601900395472
>>> timeit.timeit('fn_fortran(xx)', 'from __main__ import fn_fortran, xx', number=10000)    
0.004707066000264604

The options available with ufuncify are more or less the same as those available with autowrap.

There are other facilities available with SymPy to do efficient numeric computation. See this page for a comparison among them.

Classes and functions for rewriting expressions (sympy.codegen.rewriting)

Classes and functions useful for rewriting expressions for optimized code generation. Some languages (or standards thereof), e.g. C99, offer specialized math functions for better performance and/or precision.

Using the optimize function in this module, together with a collection of rules (represented as instances of Optimization), one can rewrite the expressions for this purpose:

>>> from sympy import Symbol, exp, log
>>> from sympy.codegen.rewriting import optimize, optims_c99
>>> x = Symbol('x')
>>> optimize(3*exp(2*x) - 3, optims_c99)
3*expm1(2*x)
>>> optimize(exp(2*x) - 3, optims_c99)
exp(2*x) - 3
>>> optimize(log(3*x + 3), optims_c99)
log1p(x) + log(3)
>>> optimize(log(2*x + 3), optims_c99)
log(2*x + 3)

The optims_c99 imported above is tuple containing the following instances (which may be imported from sympy.codegen.rewriting):

• expm1_opt

• log1p_opt

• exp2_opt

• log2_opt

• log2const_opt

class sympy.codegen.rewriting.Optimization(cost_function=None, priority=1)

Abstract base class for rewriting optimization.

Subclasses should implement __call__ taking an expression as argument.

Parameters

cost_function : callable returning number

priority : number

class sympy.codegen.rewriting.ReplaceOptim(query, value, **kwargs)

Rewriting optimization calling replace on expressions.

The instance can be used as a function on expressions for which it will apply the replace method (see sympy.core.basic.Basic.replace()).

Parameters

query : first argument passed to replace

value : second argument passed to replace

Examples

>>> from sympy import Symbol, Pow
>>> from sympy.codegen.rewriting import ReplaceOptim
>>> from sympy.codegen.cfunctions import exp2
>>> x = Symbol('x')
>>> exp2_opt = ReplaceOptim(lambda p: p.is_Pow and p.base == 2,
...     lambda p: exp2(p.exp))
>>> exp2_opt(2**x)
exp2(x)

sympy.codegen.rewriting.create_expand_pow_optimization(limit)

Creates an instance of ReplaceOptim for expanding Pow.

The requirements for expansions are that the base needs to be a symbol and the exponent needs to be an integer (and be less than or equal to limit).

Parameters

limit : int

The highest power which is expanded into multiplication.

Examples

>>> from sympy import Symbol, sin
>>> from sympy.codegen.rewriting import create_expand_pow_optimization
>>> x = Symbol('x')
>>> expand_opt = create_expand_pow_optimization(3)
>>> expand_opt(x**5 + x**3)
x**5 + x*x*x
>>> expand_opt(x**5 + x**3 + sin(x)**3)
x**5 + x*x*x + sin(x)**3

sympy.codegen.rewriting.optimize(expr, optimizations)

Apply optimizations to an expression.

Parameters

expr : expression

optimizations : iterable of Optimization instances

The optimizations will be sorted with respect to priority (highest first).

Examples

>>> from sympy import log, Symbol
>>> from sympy.codegen.rewriting import optims_c99, optimize
>>> x = Symbol('x')
>>> optimize(log(x+3)/log(2) + log(x**2 + 1), optims_c99)
log1p(x**2) + log2(x + 3)

Tools for simplifying expressions using approximations (sympy.codegen.approximations)

class sympy.codegen.approximations.SeriesApprox(bounds, reltol, max_order=4, n_point_checks=4, **kwargs)

Approximates functions by expanding them as a series

Parameters

bounds : dict

Mapping expressions to length 2 tuple of bounds (low, high).

reltol : number

Threshold for when to ignore a term. Taken relative to the largest lower bound among bounds.

max_order : int

Largest order to include in series expansion

n_point_checks : int (even)

The validity of an expansion (with respect to reltol) is checked at discrete points (linearly spaced over the bounds of the variable). The number of points used in this numerical check is given by this number.

Examples

>>> from sympy import sin, pi
>>> from sympy.abc import x, y
>>> from sympy.codegen.rewriting import optimize
>>> from sympy.codegen.approximations import SeriesApprox
>>> bounds = {x: (-.1, .1), y: (pi-1, pi+1)}
>>> series_approx2 = SeriesApprox(bounds, reltol=1e-2)
>>> series_approx3 = SeriesApprox(bounds, reltol=1e-3)
>>> series_approx8 = SeriesApprox(bounds, reltol=1e-8)
>>> expr = sin(x)*sin(y)
>>> optimize(expr, [series_approx2])
x*(-y + (y - pi)**3/6 + pi)
>>> optimize(expr, [series_approx3])
(-x**3/6 + x)*sin(y)
>>> optimize(expr, [series_approx8])
sin(x)*sin(y)

class sympy.codegen.approximations.SumApprox(bounds, reltol, **kwargs)

Approximates sum by neglecting small terms

If terms are expressions which can be determined to be monotonic, then bounds for those expressions are added.

Parameters

bounds : dict

Mapping expressions to length 2 tuple of bounds (low, high).

reltol : number

Threshold for when to ignore a term. Taken relative to the largest lower bound among bounds.

Examples

>>> from sympy import exp
>>> from sympy.abc import x, y, z
>>> from sympy.codegen.rewriting import optimize
>>> from sympy.codegen.approximations import SumApprox
>>> bounds = {x: (-1, 1), y: (1000, 2000), z: (-10, 3)}
>>> sum_approx3 = SumApprox(bounds, reltol=1e-3)
>>> sum_approx2 = SumApprox(bounds, reltol=1e-2)
>>> sum_approx1 = SumApprox(bounds, reltol=1e-1)
>>> expr = 3*(x + y + exp(z))
>>> optimize(expr, [sum_approx3])
3*(x + y + exp(z))
>>> optimize(expr, [sum_approx2])
3*y + 3*exp(z)
>>> optimize(expr, [sum_approx1])
3*y

Classes for abstract syntax trees (sympy.codegen.ast)

Types used to represent a full function/module as an Abstract Syntax Tree.

Most types are small, and are merely used as tokens in the AST. A tree diagram has been included below to illustrate the relationships between the AST types.

AST Type Tree

*Basic*
     |--->AssignmentBase
     |             |--->Assignment
     |             |--->AugmentedAssignment
     |                                    |--->AddAugmentedAssignment
     |                                    |--->SubAugmentedAssignment
     |                                    |--->MulAugmentedAssignment
     |                                    |--->DivAugmentedAssignment
     |                                    |--->ModAugmentedAssignment
     |
     |--->CodeBlock
     |
     |
     |--->Token
     |        |--->Attribute
     |        |--->For
     |        |--->String
     |        |       |--->QuotedString
     |        |       |--->Comment
     |        |--->Type
     |        |       |--->IntBaseType
     |        |       |              |--->_SizedIntType
     |        |       |                               |--->SignedIntType
     |        |       |                               |--->UnsignedIntType
     |        |       |--->FloatBaseType
     |        |                        |--->FloatType
     |        |                        |--->ComplexBaseType
     |        |                                           |--->ComplexType
     |        |--->Node
     |        |       |--->Variable
     |        |       |           |---> Pointer
     |        |       |--->FunctionPrototype
     |        |                            |--->FunctionDefinition
     |        |--->Element
     |        |--->Declaration
     |        |--->While
     |        |--->Scope
     |        |--->Stream
     |        |--->Print
     |        |--->FunctionCall
     |        |--->BreakToken
     |        |--->ContinueToken
     |        |--->NoneToken
     |
     |--->Statement
     |--->Return

Predefined types

A number of Type instances are provided in the sympy.codegen.ast module for convenience. Perhaps the two most common ones for code-generation (of numeric codes) are float32 and float64 (known as single and double precision respectively). There are also precision generic versions of Types (for which the codeprinters selects the underlying data type at time of printing): real, integer, complex_, bool_.

The other Type instances defined are:

• intc: Integer type used by C’s “int”.

• intp: Integer type used by C’s “unsigned”.

• int8, int16, int32, int64: n-bit integers.

• uint8, uint16, uint32, uint64: n-bit unsigned integers.

• float80: known as “extended precision” on modern x86/amd64 hardware.

• complex64: Complex number represented by two float32 numbers

• complex128: Complex number represented by two float64 numbers

Using the nodes

It is possible to construct simple algorithms using the AST nodes. Let’s construct a loop applying Newton’s method:

>>> from sympy import symbols, cos
>>> from sympy.codegen.ast import While, Assignment, aug_assign, Print
>>> t, dx, x = symbols('tol delta val')
>>> expr = cos(x) - x**3
>>> whl = While(abs(dx) > t, [
...     Assignment(dx, -expr/expr.diff(x)),
...     aug_assign(x, '+', dx),
...     Print([x])
... ])
>>> from sympy.printing import pycode
>>> py_str = pycode(whl)
>>> print(py_str)
while (abs(delta) > tol):
    delta = (val**3 - math.cos(val))/(-3*val**2 - math.sin(val))
    val += delta
    print(val)
>>> import math
>>> tol, val, delta = 1e-5, 0.5, float('inf')
>>> exec(py_str)
1.1121416371
0.909672693737
0.867263818209
0.865477135298
0.865474033111
>>> print('%3.1g' % (math.cos(val) - val**3))
-3e-11

If we want to generate Fortran code for the same while loop we simple call fcode:

>>> from sympy.printing.fcode import fcode
>>> print(fcode(whl, standard=2003, source_format='free'))
do while (abs(delta) > tol)
   delta = (val**3 - cos(val))/(-3*val**2 - sin(val))
   val = val + delta
   print *, val
end do

There is a function constructing a loop (or a complete function) like this in sympy.codegen.algorithms.

class sympy.codegen.ast.Assignment

Represents variable assignment for code generation.

Parameters

lhs : Expr

Sympy object representing the lhs of the expression. These should be singular objects, such as one would use in writing code. Notable types include Symbol, MatrixSymbol, MatrixElement, and Indexed. Types that subclass these types are also supported.

rhs : Expr

Sympy object representing the rhs of the expression. This can be any type, provided its shape corresponds to that of the lhs. For example, a Matrix type can be assigned to MatrixSymbol, but not to Symbol, as the dimensions will not align.

Examples

>>> from sympy import symbols, MatrixSymbol, Matrix
>>> from sympy.codegen.ast import Assignment
>>> x, y, z = symbols('x, y, z')
>>> Assignment(x, y)
Assignment(x, y)
>>> Assignment(x, 0)
Assignment(x, 0)
>>> A = MatrixSymbol('A', 1, 3)
>>> mat = Matrix([x, y, z]).T
>>> Assignment(A, mat)
Assignment(A, Matrix([[x, y, z]]))
>>> Assignment(A[0, 1], x)
Assignment(A[0, 1], x)

class sympy.codegen.ast.AssignmentBase

Abstract base class for Assignment and AugmentedAssignment.

Attributes:

opstr

Symbol for assignment operator, e.g. “=”, “+=”, etc.

class sympy.codegen.ast.Attribute(possibly parametrized)

For use with sympy.codegen.ast.Node (which takes instances of Attribute as attrs).

Parameters

name : str

parameters : Tuple

Examples

>>> from sympy.codegen.ast import Attribute
>>> volatile = Attribute('volatile')
>>> volatile
volatile
>>> print(repr(volatile))
Attribute(String('volatile'))
>>> a = Attribute('foo', [1, 2, 3])
>>> a
foo(1, 2, 3)
>>> a.parameters == (1, 2, 3)
True

class sympy.codegen.ast.AugmentedAssignment

Base class for augmented assignments.

Attributes:

binopstr

Symbol for binary operation being applied in the assignment, such as “+”, “*”, etc.

class sympy.codegen.ast.BreakToken

Represents ‘break’ in C/Python (‘exit’ in Fortran).

Use the premade instance break_ or instantiate manually.

Examples

>>> from sympy.printing import ccode, fcode
>>> from sympy.codegen.ast import break_
>>> ccode(break_)
'break'
>>> fcode(break_, source_format='free')
'exit'

class sympy.codegen.ast.CodeBlock

Represents a block of code

For now only assignments are supported. This restriction will be lifted in the future.

Useful attributes on this object are:

left_hand_sides:

Tuple of left-hand sides of assignments, in order.

left_hand_sides:

Tuple of right-hand sides of assignments, in order.

free_symbols: Free symbols of the expressions in the right-hand sides

which do not appear in the left-hand side of an assignment.

Useful methods on this object are:

topological_sort:

Class method. Return a CodeBlock with assignments sorted so that variables are assigned before they are used.

cse:

Return a new CodeBlock with common subexpressions eliminated and pulled out as assignments.

Examples

>>> from sympy import symbols, ccode
>>> from sympy.codegen.ast import CodeBlock, Assignment
>>> x, y = symbols('x y')
>>> c = CodeBlock(Assignment(x, 1), Assignment(y, x + 1))
>>> print(ccode(c))
x = 1;
y = x + 1;

cse(symbols=None, optimizations=None, postprocess=None, order='canonical')

Return a new code block with common subexpressions eliminated

See the docstring of sympy.simplify.cse_main.cse() for more information.

Examples

>>> from sympy import symbols, sin
>>> from sympy.codegen.ast import CodeBlock, Assignment
>>> x, y, z = symbols('x y z')

>>> c = CodeBlock(
...     Assignment(x, 1),
...     Assignment(y, sin(x) + 1),
...     Assignment(z, sin(x) - 1),
... )
...
>>> c.cse()
CodeBlock(
    Assignment(x, 1),
    Assignment(x0, sin(x)),
    Assignment(y, x0 + 1),
    Assignment(z, x0 - 1)
)

classmethod topological_sort(assignments)

Return a CodeBlock with topologically sorted assignments so that variables are assigned before they are used.

The existing order of assignments is preserved as much as possible.

This function assumes that variables are assigned to only once.

This is a class constructor so that the default constructor for CodeBlock can error when variables are used before they are assigned.

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import CodeBlock, Assignment
>>> x, y, z = symbols('x y z')

>>> assignments = [
...     Assignment(x, y + z),
...     Assignment(y, z + 1),
...     Assignment(z, 2),
... ]
>>> CodeBlock.topological_sort(assignments)
CodeBlock(
    Assignment(z, 2),
    Assignment(y, z + 1),
    Assignment(x, y + z)
)

class sympy.codegen.ast.Comment

Represents a comment.

class sympy.codegen.ast.ComplexType

Represents a complex floating point number.

class sympy.codegen.ast.ContinueToken

Represents ‘continue’ in C/Python (‘cycle’ in Fortran)

Use the premade instance continue_ or instantiate manually.

Examples

>>> from sympy.printing import ccode, fcode
>>> from sympy.codegen.ast import continue_
>>> ccode(continue_)
'continue'
>>> fcode(continue_, source_format='free')
'cycle'

class sympy.codegen.ast.Declaration

Represents a variable declaration

Parameters

variable : Variable

Examples

>>> from sympy import Symbol
>>> from sympy.codegen.ast import Declaration, Type, Variable, integer, untyped
>>> z = Declaration('z')
>>> z.variable.type == untyped
True
>>> z.variable.value == None
True

class sympy.codegen.ast.Element

Element in (a possibly N-dimensional) array.

Examples

>>> from sympy.codegen.ast import Element
>>> elem = Element('x', 'ijk')
>>> elem.symbol.name == 'x'
True
>>> elem.indices
(i, j, k)
>>> from sympy import ccode
>>> ccode(elem)
'x[i][j][k]'
>>> ccode(Element('x', 'ijk', strides='lmn', offset='o'))
'x[i*l + j*m + k*n + o]'

class sympy.codegen.ast.FloatBaseType

Represents a floating point number type.

cast_nocheck

alias of sympy.core.numbers.Float

class sympy.codegen.ast.FloatType

Represents a floating point type with fixed bit width.

Base 2 & one sign bit is assumed.

Parameters

name : str

Name of the type.

nbits : integer

Number of bits used (storage).

nmant : integer

Number of bits used to represent the mantissa.

nexp : integer

Number of bits used to represent the mantissa.

Examples

>>> from sympy import S, Float
>>> from sympy.codegen.ast import FloatType
>>> half_precision = FloatType('f16', nbits=16, nmant=10, nexp=5)
>>> half_precision.max
65504
>>> half_precision.tiny == S(2)**-14
True
>>> half_precision.eps == S(2)**-10
True
>>> half_precision.dig == 3
True
>>> half_precision.decimal_dig == 5
True
>>> half_precision.cast_check(1.0)
1.0
>>> half_precision.cast_check(1e5)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Maximum value for data type smaller than new value.

cast_nocheck(value)

Casts without checking if out of bounds or subnormal.

decimal_dig

Number of digits needed to store & load without loss.

Number of decimal digits needed to guarantee that two consecutive conversions (float -> text -> float) to be idempotent. This is useful when one do not want to loose precision due to rounding errors when storing a floating point value as text.

dig

Number of decimal digits that are guaranteed to be preserved in text.

When converting text -> float -> text, you are guaranteed that at least dig number of digits are preserved with respect to rounding or overflow.

eps

Difference between 1.0 and the next representable value.

max

Maximum value representable.

max_exponent

The largest positive number n, such that 2**(n - 1) is a representable finite value.

min_exponent

The lowest negative number n, such that 2**(n - 1) is a valid normalized number.

tiny

The minimum positive normalized value.

class sympy.codegen.ast.For

Represents a ‘for-loop’ in the code.

Expressions are of the form:

“for target in iter:

body…”

Parameters

target : symbol

iter : iterable body : CodeBlock or iterable

! When passed an iterable it is used to instantiate a CodeBlock.

Examples

>>> from sympy import symbols, Range
>>> from sympy.codegen.ast import aug_assign, For
>>> x, i, j, k = symbols('x i j k')
>>> for_i = For(i, Range(10), [aug_assign(x, '+', i*j*k)])
>>> for_i  # doctest: -NORMALIZE_WHITESPACE
For(i, iterable=Range(0, 10, 1), body=CodeBlock(
    AddAugmentedAssignment(x, i*j*k)
))
>>> for_ji = For(j, Range(7), [for_i])
>>> for_ji  # doctest: -NORMALIZE_WHITESPACE
For(j, iterable=Range(0, 7, 1), body=CodeBlock(
    For(i, iterable=Range(0, 10, 1), body=CodeBlock(
        AddAugmentedAssignment(x, i*j*k)
    ))
))
>>> for_kji =For(k, Range(5), [for_ji])
>>> for_kji  # doctest: -NORMALIZE_WHITESPACE
For(k, iterable=Range(0, 5, 1), body=CodeBlock(
    For(j, iterable=Range(0, 7, 1), body=CodeBlock(
        For(i, iterable=Range(0, 10, 1), body=CodeBlock(
            AddAugmentedAssignment(x, i*j*k)
        ))
    ))
))

class sympy.codegen.ast.FunctionCall

Represents a call to a function in the code.

Parameters

name : str

function_args : Tuple

Examples

>>> from sympy.codegen.ast import FunctionCall
>>> from sympy.printing.pycode import pycode
>>> fcall = FunctionCall('foo', 'bar baz'.split())
>>> print(pycode(fcall))
foo(bar, baz)

class sympy.codegen.ast.FunctionDefinition

Represents a function definition in the code.

Parameters

return_type : Type

name : str

parameters: iterable of Variable instances

body : CodeBlock or iterable

attrs : iterable of Attribute instances

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import real, FunctionPrototype
>>> from sympy.printing.ccode import ccode
>>> x, y = symbols('x y', real=True)
>>> fp = FunctionPrototype(real, 'foo', [x, y])
>>> ccode(fp)
'double foo(double x, double y)'
>>> from sympy.codegen.ast import FunctionDefinition, Return
>>> body = [Return(x*y)]
>>> fd = FunctionDefinition.from_FunctionPrototype(fp, body)
>>> print(ccode(fd))
double foo(double x, double y){
    return x*y;
}

class sympy.codegen.ast.FunctionPrototype

Represents a function prototype

Allows the user to generate forward declaration in e.g. C/C++.

Parameters

return_type : Type

name : str

parameters: iterable of Variable instances

attrs : iterable of Attribute instances

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import real, FunctionPrototype
>>> from sympy.printing.ccode import ccode
>>> x, y = symbols('x y', real=True)
>>> fp = FunctionPrototype(real, 'foo', [x, y])
>>> ccode(fp)
'double foo(double x, double y)'

class sympy.codegen.ast.IntBaseType

Integer base type, contains no size information.

class sympy.codegen.ast.Node

Subclass of Token, carrying the attribute ‘attrs’ (Tuple)

Examples

>>> from sympy.codegen.ast import Node, value_const, pointer_const
>>> n1 = Node([value_const])
>>> n1.attr_params('value_const')  # get the parameters of attribute (by name)
()
>>> from sympy.codegen.fnodes import dimension
>>> n2 = Node([value_const, dimension(5, 3)])
>>> n2.attr_params(value_const)  # get the parameters of attribute (by Attribute instance)
()
>>> n2.attr_params('dimension')  # get the parameters of attribute (by name)
(5, 3)
>>> n2.attr_params(pointer_const) is None
True

attr_params(looking_for)

Returns the parameters of the Attribute with name looking_for in self.attrs

class sympy.codegen.ast.NoneToken

The AST equivalence of Python’s NoneType

The corresponding instance of Python’s None is none.

Examples

>>> from sympy.codegen.ast import none, Variable
>>> from sympy.printing.pycode import pycode
>>> print(pycode(Variable('x').as_Declaration(value=none)))
x = None

class sympy.codegen.ast.Pointer

Represents a pointer. See Variable.

Examples

Can create instances of Element:

>>> from sympy import Symbol
>>> from sympy.codegen.ast import Pointer
>>> i = Symbol('i', integer=True)
>>> p = Pointer('x')
>>> p[i+1]
Element(x, indices=((i + 1,),))

class sympy.codegen.ast.Print

Represents print command in the code.

Parameters

formatstring : str

*args : Basic instances (or convertible to such through sympify)

Examples

>>> from sympy.codegen.ast import Print
>>> from sympy.printing.pycode import pycode
>>> print(pycode(Print('x y'.split(), "coordinate: %12.5g %12.5g")))
print("coordinate: %12.5g %12.5g" % (x, y))

class sympy.codegen.ast.QuotedString

Represents a string which should be printed with quotes.

class sympy.codegen.ast.Return

Represents a return command in the code.

class sympy.codegen.ast.Scope

Represents a scope in the code.

Parameters

body : CodeBlock or iterable

When passed an iterable it is used to instantiate a CodeBlock.

class sympy.codegen.ast.SignedIntType

Represents a signed integer type.

class sympy.codegen.ast.Stream

Represents a stream.

There are two predefined Stream instances stdout & stderr.

Parameters

name : str

Examples

>>> from sympy import Symbol
>>> from sympy.printing.pycode import pycode
>>> from sympy.codegen.ast import Print, stderr, QuotedString
>>> print(pycode(Print(['x'], file=stderr)))
print(x, file=sys.stderr)
>>> x = Symbol('x')
>>> print(pycode(Print([QuotedString('x')], file=stderr)))  # print literally "x"
print("x", file=sys.stderr)

class sympy.codegen.ast.String

SymPy object representing a string.

Atomic object which is not an expression (as opposed to Symbol).

Parameters

text : str

Examples

>>> from sympy.codegen.ast import String
>>> f = String('foo')
>>> f
foo
>>> str(f)
'foo'
>>> f.text
'foo'
>>> print(repr(f))
String('foo')

class sympy.codegen.ast.Token

Base class for the AST types.

Defining fields are set in __slots__. Attributes (defined in __slots__) are only allowed to contain instances of Basic (unless atomic, see String). The arguments to __new__() correspond to the attributes in the order defined in __slots__`. The ``defaults class attribute is a dictionary mapping attribute names to their default values.

Subclasses should not need to override the __new__() method. They may define a class or static method named _construct_<attr> for each attribute to process the value passed to __new__(). Attributes listed in the class attribute not_in_args are not passed to sympy.Basic.

kwargs(exclude=(), apply=None)

Get instance’s attributes as dict of keyword arguments.

Parameters

exclude : collection of str

Collection of keywords to exclude.

apply : callable, optional

Function to apply to all values.

class sympy.codegen.ast.Type

Represents a type.

The naming is a super-set of NumPy naming. Type has a classmethod from_expr which offer type deduction. It also has a method cast_check which casts the argument to its type, possibly raising an exception if rounding error is not within tolerances, or if the value is not representable by the underlying data type (e.g. unsigned integers).

Parameters

name : str

Name of the type, e.g. object, int16, float16 (where the latter two would use the Type sub-classes IntType and FloatType respectively). If a Type instance is given, the said instance is returned.

Examples

>>> from sympy.codegen.ast import Type
>>> t = Type.from_expr(42)
>>> t
integer
>>> print(repr(t))
IntBaseType(String('integer'))
>>> from sympy.codegen.ast import uint8
>>> uint8.cast_check(-1)   # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Minimum value for data type bigger than new value.
>>> from sympy.codegen.ast import float32
>>> v6 = 0.123456
>>> float32.cast_check(v6)
0.123456
>>> v10 = 12345.67894
>>> float32.cast_check(v10)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Casting gives a significantly different value.
>>> boost_mp50 = Type('boost::multiprecision::cpp_dec_float_50')
>>> from sympy import Symbol
>>> from sympy.printing.cxxcode import cxxcode
>>> from sympy.codegen.ast import Declaration, Variable
>>> cxxcode(Declaration(Variable('x', type=boost_mp50)))
'boost::multiprecision::cpp_dec_float_50 x'

References

R25

https://docs.scipy.org/doc/numpy/user/basics.types.html

cast_check(value, rtol=None, atol=0, limits=None, precision_targets=None)

Casts a value to the data type of the instance.

Parameters

value : number

rtol : floating point number

Relative tolerance. (will be deduced if not given).

atol : floating point number

Absolute tolerance (in addition to rtol).

limits : dict

Values given by limits.h, x86/IEEE754 defaults if not given. Default: default_limits.

type_aliases : dict

Maps substitutions for Type, e.g. {integer: int64, real: float32}

Examples

>>> from sympy.codegen.ast import Type, integer, float32, int8
>>> integer.cast_check(3.0) == 3
True
>>> float32.cast_check(1e-40)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Minimum value for data type bigger than new value.
>>> int8.cast_check(256)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Maximum value for data type smaller than new value.
>>> v10 = 12345.67894
>>> float32.cast_check(v10)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Casting gives a significantly different value.
>>> from sympy.codegen.ast import float64
>>> float64.cast_check(v10)
12345.67894
>>> from sympy import Float
>>> v18 = Float('0.123456789012345646')
>>> float64.cast_check(v18)
Traceback (most recent call last):
  ...
ValueError: Casting gives a significantly different value.
>>> from sympy.codegen.ast import float80
>>> float80.cast_check(v18)
0.123456789012345649

classmethod from_expr(expr)

Deduces type from an expression or a Symbol.

Parameters

expr : number or SymPy object

The type will be deduced from type or properties.

Raises

ValueError when type deduction fails.

Examples

>>> from sympy.codegen.ast import Type, integer, complex_
>>> Type.from_expr(2) == integer
True
>>> from sympy import Symbol
>>> Type.from_expr(Symbol('z', complex=True)) == complex_
True
>>> Type.from_expr(sum)  # doctest: +ELLIPSIS
Traceback (most recent call last):
  ...
ValueError: Could not deduce type from expr.

class sympy.codegen.ast.UnsignedIntType

Represents an unsigned integer type.

class sympy.codegen.ast.Variable

Represents a variable

Parameters

symbol : Symbol

type : Type (optional)

Type of the variable.

attrs : iterable of Attribute instances

Will be stored as a Tuple.

Examples

>>> from sympy import Symbol
>>> from sympy.codegen.ast import Variable, float32, integer
>>> x = Symbol('x')
>>> v = Variable(x, type=float32)
>>> v.attrs
()
>>> v == Variable('x')
False
>>> v == Variable('x', type=float32)
True
>>> v
Variable(x, type=float32)

One may also construct a Variable instance with the type deduced from assumptions about the symbol using the deduced classmethod:

>>> i = Symbol('i', integer=True)
>>> v = Variable.deduced(i)
>>> v.type == integer
True
>>> v == Variable('i')
False
>>> from sympy.codegen.ast import value_const
>>> value_const in v.attrs
False
>>> w = Variable('w', attrs=[value_const])
>>> w
Variable(w, attrs=(value_const,))
>>> value_const in w.attrs
True
>>> w.as_Declaration(value=42)
Declaration(Variable(w, value=42, attrs=(value_const,)))

as_Declaration(**kwargs)

Convenience method for creating a Declaration instance.

If the variable of the Declaration need to wrap a modified variable keyword arguments may be passed (overriding e.g. the value of the Variable instance).

Examples

>>> from sympy.codegen.ast import Variable
>>> x = Variable('x')
>>> decl1 = x.as_Declaration()
>>> decl1.variable.value == None
True
>>> decl2 = x.as_Declaration(value=42.0)
>>> decl2.variable.value == 42
True

classmethod deduced(symbol, value=None, attrs=(), cast_check=True)

Alt. constructor with type deduction from Type.from_expr.

Deduces type primarily from symbol, secondarily from value.

Parameters

symbol : Symbol

value : expr

(optional) value of the variable.

attrs : iterable of Attribute instances

cast_check : bool

Whether to apply Type.cast_check on value.

Examples

>>> from sympy import Symbol
>>> from sympy.codegen.ast import Variable, complex_
>>> n = Symbol('n', integer=True)
>>> str(Variable.deduced(n).type)
'integer'
>>> x = Symbol('x', real=True)
>>> v = Variable.deduced(x)
>>> v.type
real
>>> z = Symbol('z', complex=True)
>>> Variable.deduced(z).type == complex_
True

class sympy.codegen.ast.While

Represents a ‘for-loop’ in the code.

Expressions are of the form:

“while condition:

body…”

Parameters

condition : expression convertable to Boolean

body : CodeBlock or iterable

When passed an iterable it is used to instantiate a CodeBlock.

Examples

>>> from sympy import symbols, Gt, Abs
>>> from sympy.codegen import aug_assign, Assignment, While
>>> x, dx = symbols('x dx')
>>> expr = 1 - x**2
>>> whl = While(Gt(Abs(dx), 1e-9), [
...     Assignment(dx, -expr/expr.diff(x)),
...     aug_assign(x, '+', dx)
... ])

sympy.codegen.ast.aug_assign(lhs, op, rhs)

Create ‘lhs op= rhs’.

Represents augmented variable assignment for code generation. This is a convenience function. You can also use the AugmentedAssignment classes directly, like AddAugmentedAssignment(x, y).

Parameters

lhs : Expr

Sympy object representing the lhs of the expression. These should be singular objects, such as one would use in writing code. Notable types include Symbol, MatrixSymbol, MatrixElement, and Indexed. Types that subclass these types are also supported.

op : str

Operator (+, -, /, *, %).

rhs : Expr

Sympy object representing the rhs of the expression. This can be any type, provided its shape corresponds to that of the lhs. For example, a Matrix type can be assigned to MatrixSymbol, but not to Symbol, as the dimensions will not align.

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import aug_assign
>>> x, y = symbols('x, y')
>>> aug_assign(x, '+', y)
AddAugmentedAssignment(x, y)

Special C math functions (sympy.codegen.cfunctions)

This module contains SymPy functions mathcin corresponding to special math functions in the C standard library (since C99, also available in C++11).

The functions defined in this module allows the user to express functions such as expm1 as a SymPy function for symbolic manipulation.

class sympy.codegen.cfunctions.Cbrt

Represents the cube root function.

The reason why one would use Cbrt(x) over cbrt(x) is that the latter is internally represented as Pow(x, Rational(1, 3)) which may not be what one wants when doing code-generation.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import Cbrt
>>> Cbrt(x)
Cbrt(x)
>>> Cbrt(x).diff(x)
1/(3*x**(2/3))

See also

Sqrt

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.Sqrt

Represents the square root function.

The reason why one would use Sqrt(x) over sqrt(x) is that the latter is internally represented as Pow(x, S.Half) which may not be what one wants when doing code-generation.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import Sqrt
>>> Sqrt(x)
Sqrt(x)
>>> Sqrt(x).diff(x)
1/(2*sqrt(x))

See also

Cbrt

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.exp2

Represents the exponential function with base two.

The benefit of using exp2(x) over 2**x is that the latter is not as efficient under finite precision arithmetic.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import exp2
>>> exp2(2).evalf() == 4
True
>>> exp2(x).diff(x)
log(2)*exp2(x)

See also

log2

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.expm1

Represents the exponential function minus one.

The benefit of using expm1(x) over exp(x) - 1 is that the latter is prone to cancellation under finite precision arithmetic when x is close to zero.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import expm1
>>> '%.0e' % expm1(1e-99).evalf()
'1e-99'
>>> from math import exp
>>> exp(1e-99) - 1
0.0
>>> expm1(x).diff(x)
exp(x)

See also

log1p

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.fma

Represents “fused multiply add”.

The benefit of using fma(x, y, z) over x*y + z is that, under finite precision arithmetic, the former is supported by special instructions on some CPUs.

Examples

>>> from sympy.abc import x, y, z
>>> from sympy.codegen.cfunctions import fma
>>> fma(x, y, z).diff(x)
y

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.hypot

Represents the hypotenuse function.

The hypotenuse function is provided by e.g. the math library in the C99 standard, hence one may want to represent the function symbolically when doing code-generation.

Examples

>>> from sympy.abc import x, y
>>> from sympy.codegen.cfunctions import hypot
>>> hypot(3, 4).evalf() == 5
True
>>> hypot(x, y)
hypot(x, y)
>>> hypot(x, y).diff(x)
x/hypot(x, y)

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.log10

Represents the logarithm function with base ten.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log10
>>> log10(100).evalf() == 2
True
>>> log10(x).diff(x)
1/(x*log(10))

See also

log2

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.log1p

Represents the natural logarithm of a number plus one.

The benefit of using log1p(x) over log(x + 1) is that the latter is prone to cancellation under finite precision arithmetic when x is close to zero.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log1p
>>> from sympy.core.function import expand_log
>>> '%.0e' % expand_log(log1p(1e-99)).evalf()
'1e-99'
>>> from math import log
>>> log(1 + 1e-99)
0.0
>>> log1p(x).diff(x)
1/(x + 1)

See also

expm1

fdiff(argindex=1)

Returns the first derivative of this function.

class sympy.codegen.cfunctions.log2

Represents the logarithm function with base two.

The benefit of using log2(x) over log(x)/log(2) is that the latter is not as efficient under finite precision arithmetic.

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log2
>>> log2(4).evalf() == 2
True
>>> log2(x).diff(x)
1/(x*log(2))

See also

exp2, log10

fdiff(argindex=1)

Returns the first derivative of this function.

C specific AST nodes (sympy.codegen.cnodes)

AST nodes specific to the C family of languages

class sympy.codegen.cnodes.CommaOperator

Represents the comma operator in C

class sympy.codegen.cnodes.Label

Label for use with e.g. goto statement.

Examples

>>> from sympy.codegen.cnodes import Label
>>> from sympy.printing.ccode import ccode
>>> print(ccode(Label('foo')))
foo:

class sympy.codegen.cnodes.PostDecrement

Represents the post-decrement operator

class sympy.codegen.cnodes.PostIncrement

Represents the post-increment operator

class sympy.codegen.cnodes.PreDecrement

Represents the pre-decrement operator

Examples

>>> from sympy.abc import x
>>> from sympy.codegen.cnodes import PreDecrement
>>> from sympy.printing.ccode import ccode
>>> ccode(PreDecrement(x))
'--(x)'

class sympy.codegen.cnodes.PreIncrement

Represents the pre-increment operator

sympy.codegen.cnodes.alignof(arg)

Generate of FunctionCall instance for calling ‘alignof’

class sympy.codegen.cnodes.goto

Represents goto in C

sympy.codegen.cnodes.sizeof(arg)

Generate of FunctionCall instance for calling ‘sizeof’

Examples

>>> from sympy.codegen.ast import real
>>> from sympy.codegen.cnodes import sizeof
>>> from sympy.printing.ccode import ccode
>>> ccode(sizeof(real))
'sizeof(double)'

class sympy.codegen.cnodes.struct

Represents a struct in C

class sympy.codegen.cnodes.union

Represents a union in C

C++ specific AST nodes (sympy.codegen.cxxnodes)

AST nodes specific to C++.

class sympy.codegen.cxxnodes.using

Represents a ‘using’ statement in C++

Fortran specific AST nodes (sympy.codegen.fnodes)

AST nodes specific to Fortran.

The functions defined in this module allows the user to express functions such as dsign as a SymPy function for symbolic manipulation.

class sympy.codegen.fnodes.ArrayConstructor

Represents an array constructor

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.fnodes import ArrayConstructor
>>> ac = ArrayConstructor([1, 2, 3])
>>> fcode(ac, standard=95, source_format='free')
'(/1, 2, 3/)'
>>> fcode(ac, standard=2003, source_format='free')
'[1, 2, 3]'

class sympy.codegen.fnodes.Do

Represents a Do loop in in Fortran

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import aug_assign, Print
>>> from sympy.codegen.fnodes import Do
>>> from sympy.printing import fcode
>>> i, n = symbols('i n', integer=True)
>>> r = symbols('r', real=True)
>>> body = [aug_assign(r, '+', 1/i), Print([i, r])]
>>> do1 = Do(body, i, 1, n)
>>> print(fcode(do1, source_format='free'))
do i = 1, n
    r = r + 1d0/i
    print *, i, r
end do
>>> do2 = Do(body, i, 1, n, 2)
>>> print(fcode(do2, source_format='free'))
do i = 1, n, 2
    r = r + 1d0/i
    print *, i, r
end do

class sympy.codegen.fnodes.Extent

Represents a dimension extent.

Examples

>>> from sympy.codegen.fnodes import Extent
>>> e = Extent(-3, 3)  # -3, -2, -1, 0, 1, 2, 3
>>> from sympy.printing import fcode
>>> fcode(e, source_format='free')
'-3:3'
>>> from sympy.codegen.ast import Variable, real
>>> from sympy.codegen.fnodes import dimension, intent_out
>>> dim = dimension(e, e)
>>> arr = Variable('x', real, attrs=[dim, intent_out])
>>> fcode(arr.as_Declaration(), source_format='free', standard=2003)
'real*8, dimension(-3:3, -3:3), intent(out) :: x'

class sympy.codegen.fnodes.FortranReturn

AST node explicitly mapped to a fortran “return”.

Because a return statement in fortran is different from C, and in order to aid reuse of our codegen ASTs the ordinary .codegen.ast.Return is interpreted as assignment to the result variable of the function. If one for some reason needs to generate a fortran RETURN statement, this node should be used.

Examples

>>> from sympy.codegen.fnodes import FortranReturn
>>> from sympy.printing import fcode
>>> fcode(FortranReturn('x'))
'       return x'

class sympy.codegen.fnodes.GoTo

Represents a goto statement in Fortran

Examples

>>> from sympy.codegen.fnodes import GoTo
>>> go = GoTo([10, 20, 30], 'i')
>>> from sympy.printing import fcode
>>> fcode(go, source_format='free')
'go to (10, 20, 30), i'

class sympy.codegen.fnodes.ImpliedDoLoop

Represents an implied do loop in Fortran

Examples

>>> from sympy import Symbol, fcode
>>> from sympy.codegen.fnodes import ImpliedDoLoop, ArrayConstructor
>>> i = Symbol('i', integer=True)
>>> idl = ImpliedDoLoop(i**3, i, -3, 3, 2)  # -27, -1, 1, 27
>>> ac = ArrayConstructor([-28, idl, 28]) # -28, -27, -1, 1, 27, 28
>>> fcode(ac, standard=2003, source_format='free')
'[-28, (i**3, i = -3, 3, 2), 28]'

class sympy.codegen.fnodes.Module

Represents a module in Fortran

Examples

>>> from sympy.codegen.fnodes import Module
>>> from sympy.printing import fcode
>>> print(fcode(Module('signallib', ['implicit none'], []), source_format='free'))
module signallib
implicit none
<BLANKLINE>
contains
<BLANKLINE>
<BLANKLINE>
end module

class sympy.codegen.fnodes.Program

Represents a ‘program’ block in Fortran

Examples

>>> from sympy.codegen.ast import Print
>>> from sympy.codegen.fnodes import Program
>>> prog = Program('myprogram', [Print([42])])
>>> from sympy.printing import fcode
>>> print(fcode(prog, source_format='free'))
program myprogram
    print *, 42
end program

class sympy.codegen.fnodes.Subroutine

Represents a subroutine in Fortran

Examples

>>> from sympy import symbols
>>> from sympy.codegen.ast import Print
>>> from sympy.codegen.fnodes import Subroutine
>>> from sympy.printing import fcode
>>> x, y = symbols('x y', real=True)
>>> sub = Subroutine('mysub', [x, y], [Print([x**2 + y**2, x*y])])
>>> print(fcode(sub, source_format='free', standard=2003))
subroutine mysub(x, y)
real*8 :: x
real*8 :: y
print *, x**2 + y**2, x*y
end subroutine

class sympy.codegen.fnodes.SubroutineCall

Represents a call to a subroutine in Fortran

Examples

>>> from sympy.codegen.fnodes import SubroutineCall
>>> from sympy.printing import fcode
>>> fcode(SubroutineCall('mysub', 'x y'.split()))
'       call mysub(x, y)'

sympy.codegen.fnodes.allocated(array)

Creates an AST node for a function call to Fortran’s “allocated(…)”

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.fnodes import allocated
>>> alloc = allocated('x')
>>> fcode(alloc, source_format='free')
'allocated(x)'

sympy.codegen.fnodes.array(symbol, dim, intent=None, **kwargs)

Convenience function for creating a Variable instance for a Fortran array

Parameters

symbol : symbol

dim : Attribute or iterable

If dim is an Attribute it need to have the name ‘dimension’. If it is not an Attribute, then it is passsed to dimension() as *dim

intent : str

One of: ‘in’, ‘out’, ‘inout’ or None

**kwargs:

Keyword arguments for Variable (‘type’ & ‘value’)

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.ast import integer, real
>>> from sympy.codegen.fnodes import array
>>> arr = array('a', '*', 'in', type=integer)
>>> print(fcode(arr.as_Declaration(), source_format='free', standard=2003))
integer*4, dimension(*), intent(in) :: a
>>> x = array('x', [3, ':', ':'], intent='out', type=real)
>>> print(fcode(x.as_Declaration(value=1), source_format='free', standard=2003))
real*8, dimension(3, :, :), intent(out) :: x = 1

sympy.codegen.fnodes.bind_C(name=None)

Creates an Attribute bind_C with a name

Parameters

name : str

Examples

>>> from sympy import Symbol
>>> from sympy.printing import fcode
>>> from sympy.codegen.ast import FunctionDefinition, real, Return, Variable
>>> from sympy.codegen.fnodes import array, sum_, size, bind_C
>>> a = Symbol('a', real=True)
>>> s = Symbol('s', integer=True)
>>> arr = array(a, dim=[s], intent='in')
>>> body = [Return((sum_(a**2)/s)**.5)]
>>> fd = FunctionDefinition(real, 'rms', [arr, s], body, attrs=[bind_C('rms')])
>>> print(fcode(fd, source_format='free', standard=2003))
real*8 function rms(a, s) bind(C, name="rms")
real*8, dimension(s), intent(in) :: a
integer*4 :: s
rms = sqrt(sum(a**2)/s)
end function

class sympy.codegen.fnodes.cmplx

Fortran complex conversion function.

sympy.codegen.fnodes.dimension(*args)

Creates a ‘dimension’ Attribute with (up to 7) extents.

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.fnodes import dimension, intent_in
>>> dim = dimension('2', ':')  # 2 rows, runtime determined number of columns
>>> from sympy.codegen.ast import Variable, integer
>>> arr = Variable('a', integer, attrs=[dim, intent_in])
>>> fcode(arr.as_Declaration(), source_format='free', standard=2003)
'integer*4, dimension(2, :), intent(in) :: a'

class sympy.codegen.fnodes.dsign

Fortran sign intrinsic for double precision arguments.

class sympy.codegen.fnodes.isign

Fortran sign intrinsic for integer arguments.

class sympy.codegen.fnodes.kind

Fortran kind function.

sympy.codegen.fnodes.lbound(array, dim=None, kind=None)

Creates an AST node for a function call to Fortran’s “lbound(…)”

Parameters

array : Symbol or String

dim : expr

kind : expr

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.fnodes import lbound
>>> lb = lbound('arr', dim=2)
>>> fcode(lb, source_format='free')
'lbound(arr, 2)'

class sympy.codegen.fnodes.literal_dp

Fortran double precision real literal

class sympy.codegen.fnodes.literal_sp

Fortran single precision real literal

class sympy.codegen.fnodes.merge

Fortran merge function

sympy.codegen.fnodes.reshape(source, shape, pad=None, order=None)

Creates an AST node for a function call to Fortran’s “reshape(…)”

Parameters

source : Symbol or String

shape : ArrayExpr

sympy.codegen.fnodes.shape(source, kind=None)

Creates an AST node for a function call to Fortran’s “shape(…)”

Parameters

source : Symbol or String

kind : expr

Examples

>>> from sympy.printing import fcode
>>> from sympy.codegen.fnodes import shape
>>> shp = shape('x')
>>> fcode(shp, source_format='free')
'shape(x)'

sympy.codegen.fnodes.size(array, dim=None, kind=None)

Creates an AST node for a function call to Fortran’s “size(…)”

Examples

>>> from sympy import Symbol
>>> from sympy.printing import fcode
>>> from sympy.codegen.ast import FunctionDefinition, real, Return, Variable
>>> from sympy.codegen.fnodes import array, sum_, size
>>> a = Symbol('a', real=True)
>>> body = [Return((sum_(a**2)/size(a))**.5)]
>>> arr = array(a, dim=[':'], intent='in')
>>> fd = FunctionDefinition(real, 'rms', [arr], body)
>>> print(fcode(fd, source_format='free', standard=2003))
real*8 function rms(a)
real*8, dimension(:), intent(in) :: a
rms = sqrt(sum(a**2)*1d0/size(a))
end function

class sympy.codegen.fnodes.use

Represents a use statement in Fortran

Examples

>>> from sympy.codegen.fnodes import use
>>> from sympy.printing import fcode
>>> fcode(use('signallib'), source_format='free')
'use signallib'
>>> fcode(use('signallib', [('metric', 'snr')]), source_format='free')
'use signallib, metric => snr'
>>> fcode(use('signallib', only=['snr', 'convolution2d']), source_format='free')
'use signallib, only: snr, convolution2d'

class sympy.codegen.fnodes.use_rename

Represents a renaming in a use statement in Fortran

Examples

>>> from sympy.codegen.fnodes import use_rename, use
>>> from sympy.printing import fcode
>>> ren = use_rename("thingy", "convolution2d")
>>> print(fcode(ren, source_format='free'))
thingy => convolution2d
>>> full = use('signallib', only=['snr', ren])
>>> print(fcode(full, source_format='free'))
use signallib, only: snr, thingy => convolution2d

Algorithms (sympy.codegen.algorithms)

sympy.codegen.algorithms.newtons_method(expr, wrt, atol=1e-12, delta=None, debug=False, itermax=None, counter=None)

Generates an AST for Newton-Raphson method (a root-finding algorithm).

Returns an abstract syntax tree (AST) based on sympy.codegen.ast for Netwon’s method of root-finding.

Parameters

expr : expression

wrt : Symbol

With respect to, i.e. what is the variable.

atol : number or expr

Absolute tolerance (stopping criterion)

delta : Symbol

Will be a Dummy if None.

debug : bool

Whether to print convergence information during iterations

itermax : number or expr

Maximum number of iterations.

counter : Symbol

Will be a Dummy if None.

Examples

>>> from sympy import symbols, cos
>>> from sympy.codegen.ast import Assignment
>>> from sympy.codegen.algorithms import newtons_method
>>> x, dx, atol = symbols('x dx atol')
>>> expr = cos(x) - x**3
>>> algo = newtons_method(expr, x, atol, dx)
>>> algo.has(Assignment(dx, -expr/expr.diff(x)))
True

References

R26

https://en.wikipedia.org/wiki/Newton%27s_method

sympy.codegen.algorithms.newtons_method_function(expr, wrt, params=None, func_name='newton', attrs=(), **kwargs)

Generates an AST for a function implementing the Newton-Raphson method.

Parameters

expr : expression

wrt : Symbol

With respect to, i.e. what is the variable

params : iterable of symbols

Symbols appearing in expr that are taken as constants during the iterations (these will be accepted as parameters to the generated function).

func_name : str

Name of the generated function.

attrs : Tuple

Attribute instances passed as attrs to FunctionDefinition.

**kwargs :

Keyword arguments passed to sympy.codegen.algorithms.newtons_method().

Examples

>>> from sympy import symbols, cos
>>> from sympy.codegen.algorithms import newtons_method_function
>>> from sympy.codegen.pyutils import render_as_module
>>> from sympy.core.compatibility import exec_
>>> x = symbols('x')
>>> expr = cos(x) - x**3
>>> func = newtons_method_function(expr, x)
>>> py_mod = render_as_module(func)  # source code as string
>>> namespace = {}
>>> exec_(py_mod, namespace, namespace)
>>> res = eval('newton(0.5)', namespace)
>>> abs(res - 0.865474033102) < 1e-12
True

See also

sympy.codegen.ast.newtons_method

Python utilities (sympy.codegen.pyutils)

sympy.codegen.pyutils.render_as_module(content)

Renders python code as a module (with the required imports)

C utilities (sympy.codegen.cutils)

sympy.codegen.cutils.render_as_source_file(content, Printer=<class 'sympy.printing.ccode.C99CodePrinter'>, settings=None)

Renders a C source file (with required #include statements)

Fortran utilities (sympy.codegen.futils)

sympy.codegen.futils.render_as_module(definitions, name, declarations=(), printer_settings=None)

Creates a Module instance and renders it as a string.

This generates Fortran source code for a module with the correct use statements.

Parameters

definitions : iterable

Passed to sympy.codegen.fnodes.Module.

name : str

Passed to sympy.codegen.fnodes.Module.

declarations : iterable

Passed to sympy.codegen.fnodes.Module. It will be extended with use statements, ‘implicit none’ and public list generated from definitions.

printer_settings : dict

Passed to FCodePrinter (default: {'standard': 2003, 'source_format': 'free'}).