Summary of our design decisions and some pointers to the literature.

SymEngine is used in different languages

The C++ SymEngine library doesn’t care about SymPy objects at all. We are just trying to implement things in some maintainable way, currently we settled on using Basic, Mul, Pow, … hierarchy and implement most functionality using the visitor pattern or single dispatch, so Basic doesn’t need many methods. We are keeping an option to perhaps do things differently if they turn out to be faster. Either way though, this shouldn’t matter at all for how it is actually used from Python, Ruby or Julia.

Let’s talk about just Python: the wrappers are in the project. They are implemented using Cython, and they are free to introduce any kind of classes (including SymPy’s Expr or Sage’s Expression if needed), and the point of the wrappers is to make sure that things work out of the box from SymPy and Sage. The only job of the C++ SymEngine library is to ensure that the library’s C++ API is implemented in such a way so that the wrappers can be written to do what they need. For example, we could easily introduce SymPy’s Expr into the wrappers, by simply introducing the Expr class and make all the other classes subclass from it instead of from Basic.

That was the reason we split the wrappers, so now in the (pure) C++ symengine/symengine repository, we only have to worry about speed, correctness, maintainability and a usable API, and we can concentrate on these things without worrying or even testing any kind of wrappers. In the wrappers (symengine/, or .jl, .rb), we simply just use the C++ (or C) API and the only thing we care is so that the (Python) wrapper can be used from sympy/Sage (and we test that in the test suite), and that it doesn’t introduce unnecessary overhead in terms of speed. Ruby or Julia wrappers then care about interoperability with other libraries in those languages.

Reference Counted Pointers

Teuchos RCP

Memory management is handled by RCP (reference counted pointers) from Trilinos (module Teuchos). We have copied the relevant files into src/teuchos, so no external dependency is needed. Brief code snippets of the most frequent operations are given in our C++ Style Guide, this is useful to consult if you are unsure about the syntax. In order to understand how it works under the hood, read Teuchos::RCP Beginner’s Guide (pdf). Finally, more thorough exposition is given in Teuchos C++ Memory Management Classes, Idioms, and Related Topics — The Complete Reference (pdf).

Teuchos’ RCP implements reference counting of objects, exactly like Python works. When an object runs out of scope, its reference count is decreased. When it is copied, its reference count is increased. When reference count goes to zero, it is deallocated. This all happens automatically, so as long as our C++ Style Guide is followed, things just work.

When CMAKE_BUILD_TYPE=Debug is set in our CMake (the default), then Teuchos is compiled with debugging support, which means that as long as you follow our C++ Style Guide, the C++ code should never segfault (since you never access raw pointers that could segfault and Teuchos raises a nice exception with full stack trace if there is any problem, very similar to Python). Use this mode when developing.

When CMAKE_BUILD_TYPE=Release, then Teuchos is compiled without debugging support, which means that all pointer operations become either as fast as raw pointers, or very close. As such, there is pretty much zero overhead. However, in this mode, the program can segfault if you access memory incorrectly. This segfault however would be prevented if CMAKE_BUILD_TYPE=Debug, so always use the Debug build to test your code, only when all tests pass you can enable Release mode.

The Trilinos RCP pointers as described above are only used when WITH_SYMENGINE_RCP=OFF is set in CMake. The default option is WITH_SYMENGINE_RCP=ON, which uses our own implementation of RCP (see src/symengine_rcp.h). Our implementation is faster, but it only implements a subset of all the functionality and it requires all our objects to have a refcount_ variable. Otherwise the usage of our RCP is identical to Teuchos::RCP, and Travis-CI tests both implementations of RCP to make sure the code works with both.

Ptr and RCP

The RCP has an overhead with every assignment (refcount increase/decrease). You can get access to the inner pointer as Ptr (just call .ptr()), which has the same performance as raw pointer in Release mode, yet it is 100% safe in Debug mode (i.e. if the original RCP gets out of scope, the object deallocated, then the Ptr becomes dangling, but you get a nice exception in Debug mode if you try to access it —- in Release mode it will segfault or have undefined behavior, just like raw pointers). The idea is that for non-owning access — i.e. typically you just want to read some term in Add, we should be passing around Ptr, not RCP (which has the extra refcount increase/decrease, which is a waste since we do not plan to own it). This we can implement already in SymEngine.


std::unique_ptr has a performance just like raw pointers + manual new/delete, so there is no reason to use manual new/delete and raw pointers, one should use std::unique_ptr. One issue with std::unique_ptr is that if you get access to the raw pointer using .get(), then it will segfault if it becomes dangling (i.e. there are no Debug time checks implemented in the standard library for this because it returns a raw pointer, not a Ptr). This can however be fixed by writing a new class UniquePtr that returns Ptr instead of raw pointer if you want to pass it around and is 100% Debug time checked, so it can’t segfault in Debug mode. I am implementing it in, it’s a bit of work since it needs to work with custom deallocators and be a drop-in replacement for std::unique_ptr. But it will get done. The beauty of this new UniquePtr is that together with Ptr, there is no need to ever use manual new/delete and raw pointers. UniquePtr has the same performance in Release mode, yet it is 100% safe in Debug mode. It’s great.

However, the issue is that even manual new/delete (or the equivalent UniquePtr) is slow, so we want to avoid it, or only do it as little as possible. I am still thinking if we could perhaps use UniquePtr instead of RCP. It would mean that for example, the Add container would deallocate its contents (i.e. instead of having a hashtable full of RCP, it would have a hashtable full of UniquePtr), and if you access a given term, you either get just a Ptr (thus very fast, but could become dangling if Add goes out of scope — which would be Debug time checked, so no segfault, but in Release it would segfault, and for example if one does it at runtime in Python in Release mode, it would segfault, so one would have to make sure this is not exposed to the user), or you need to make a copy. For example, if you create some symbols and then use them in expressions, then currently we just pass around RCP, i.e. reference counted pointers to the original symbol. With the new approach, we would need to make a copy. Since we do not want to copy the std::string from inside Symbol, we want to store the symbols in some kind of table, and only pass a simple reference to the table (and also we need to deallocate things from the table if they are not used anymore). In other words, we just reinvented RCP again. So for Symbols, it seems it wouldn’t have many benefits. It might have some benefit for classes like Add. If they internally use UniquePtr, we can do an optimization in Release mode and store the contents directly in the hashtable (i.e. no pointers at all), and still pass around the Ptr to other code (i.e in Debug mode it would use UniquePtr, thus we would make sure that things are not dangling, and in Release mode we just pass around Ptr, with the performance of a raw pointer, to the internal array), that way we avoid new/delete. Also with this one can do custom allocator, i.e. allocate a chunk of memory for the hashtable, and just do placement new. I played with this, and surprisingly, the performance wasn’t much different to UniquePtr (for smaller objects it was a bit faster, but for larger objects — remember they are stored by value now — it was even slower). Also the creation of RCP vs UniquePtr was almost the same fast a well. The reason is that a simple refcount initialization is negligible in terms of time compared to the new call. What is slow is if you pass around RCP instead of Ptr, because raw pointer (which is what Ptr is in Release mode) is much faster than a refcount increase/decrease. We should still investigate if we can get rid of new/delete using the approaches from this paragraph.


  • We should pass around Ptr instead of RCP whenever possible, and we can do this right away.

  • Use UniquePtr (after it is implemented) whenever possible instead of RCP — though most places in SymEngine seem to require RCP. But we should keep this in mind, there might still be one or two places where UniquePtr is the way to go.

  • Never use raw new/delete and never use raw pointers (use UniquePtr + Ptr, and if it is not sufficient, use the slower RCP + Ptr).

  • Never pass pointers to some internal data — pass Ptr and have it Debug time checked by using UniquePtr in Debug mode, and use the data directly in Release mode

As an example of the last point, e.g. to give access to an internal std::map (as a pointer, so that the map is not copied), here is how to do it:

class A {
    UniquePtr<std::map<int, int> > m;
    std::map<int, int> m;
    Ptr<std::map<int, int>> get_access() {
        return m.ptr();
        return ptrFromRef(m);

That way, in debug mode, you can catch dangling references but in the optimized mode it is optimally fast.

Object creation and is_canonical()

Classes like Add, Mul, Pow are initialized through their constructor using their internal representation. Add, Mul have a coeff and dict, while Pow has base and exp. There are restrictions on what coeff and dict can be (for example coeff cannot be zero in Mul, and if Mul is used inside Add, then Mul’s coeff must be one, etc.). All these restrictions are checked when SYMENGINE_ASSERT is enabled inside the constructors using the is_canonical() method. That way, you don’t have to worry about creating Add/Mul/Pow with wrong arguments, as it will be caught by the tests. In the Release mode no checks are done, so you can construct Add/Mul/Pow very quickly. The idea is that depending on the algorithm, you sometimes know that things are already canonical, so you simply pass it directly to Add/Mul/Pow and you avoid expensive type checking and canonicalization. At the same time, you need to make sure that tests are still running with SYMENGINE_ASSERT enabled, so that Add/Mul/Pow is never in an inconsistent state.

The philosophy of symengine is that you impose as many restrictions in is_canonical() for each class as you can (and only check that in Debug mode), so that inside the class you can assume all those things and call faster algorithms (e.g. in Rational you know it’s not an integer, so you don’t need to worry about that special case, at the same time if you have an integer, you are forced to use the Integer class, thus automatically using faster algorithms for just integers). Then the idea is to use the information about the algorithm to construct arguments of the symengine classes in canonical form and then call the constructor without any checks.

For cases where you can’t or don’t want to bother constructing in canonical form, we provide high-level functions like add, mul, pow, rational, where you just provide arguments that are not necessarily in canonical form, and these functions will check and simplify. E.g. add(x, x) will check and simplify to Mul(2, x), e.g. you never have the instance Add(x, x). In the same spirit, rational(2, 1) will check and convert to Integer(2), e.g. you never have Rational(2, 1).

Summary: always try to construct objects directly using their constructors and all the knowledge that you have for the given algorithm, that way things will be very fast. If you want slower but simpler code, you can use the add(), mul(), pow(), rational() functions that perform general and possibly slow canonicalization first.

Operator Overloading

Ideally, we would like to be able to do:

RCP<Basic> x  = make_rcp<Symbol>("x");
RCP<Basic> y  = make_rcp<Symbol>("y");
RCP<Basic> r = (x + y) + (y + x);
std::cout << r << std::endl;

But the problem is, that the +, - and << operations must be defined on the RCP objects. However, just as you should not redefine what double + double is, you should not try to redefine operator overloading for an existing type (RCP). We can override operators for Basic objects, like so:

((*x) + (*y)) + ((*y) + (*x))

But here the problem is that the + operator only gets access to Basic, but it needs access to RCP<Basic> for memory management. In order to allow for operator overloaded types that use dynamic memory allocation, we will need to create our own “handle” classes. It is hard to write a handle class in C++ that are const-correct and clean and simple to use for most C++ developers. It can be done, but it is very hard, especially since we care about performance. In our opinion, we are better off writing such a layer in Python. An example of handle classes is [2] — it is non-const correct, but should give the ok performance.

Solution: using non-member non-friend functions is much more clear and much cleaner:

add(add(x, y), add(y, x))

The function signature of add is:

inline RCP<Basic> add(const RCP<Basic> &a, const RCP<Basic> &b);

For more complicated expressions, instead of add, we might also consider using the naming scheme proposed in [1]. Another advantage of this approach is that compiler errors are much easier to understand since it either finds our function or it does not, while when overloading operators of our templated classes (and RCP), any mistake typically results in pages of compiler errors in gcc.

The Python wrappers then just call this add function and provide natural mathematical syntax (x + y) + (y + x) at the Python level.