7.2. Numba architecture

7.2.1. Introduction

Numba is a compiler for Python bytecode with optional type-specialization.

Suppose you enter a function like this into the standard Python interpreter (henceforward referred to as “CPython”):

def add(a, b):
    return a + b

The interpreter will immediately parse the function and convert it into a bytecode representation that describes how the CPython interpreter should execute the function at a low level. For the example above, it looks something like this:

>>> import dis
>>> dis.dis(add)
2           0 LOAD_FAST                0 (a)
            3 LOAD_FAST                1 (b)
            6 BINARY_ADD
            7 RETURN_VALUE

CPython uses a stack-based interpreter (much like an HP calculator), so the code first pushes two local variables onto the stack. The BINARY_ADD opcode pops the top two arguments off the stack and makes a Python C API function call that is equivalent to calling a.__add__(b). The result is then pushed onto the top of the interpreter stack. Finally, the RETURN_VALUE opcode returns value on the top of the stack as the result of the function call.

Numba can take this bytecode and compile it to machine code that performs the same operations as the CPython interpreter, treating a and b as generic Python objects. The full semantics of Python are preserved, and the compiled function can be used with any kind of objects that have the add operator defined. When a Numba function is compiled this way, we say that it has been compiled in object mode, because the code still manipulates Python objects.

Numba code compiled in object mode is not much faster than executing the original Python function in the CPython interpreter. However, if we specialize the function to only run with certain data types, Numba can generate much shorter and more efficient code that manipulates the data natively without any calls into the Python C API. When code has been compiled for specific data types so that the function body no longer relies on the Python runtime, we say the function has been compiled in nopython mode. Numeric code compiled in nopython mode can be hundreds of times faster than the original Python.

7.2.2. Compiler architecture

Like many compilers, Numba can be conceptually divided into a frontend and a backend.

The Numba frontend comprises the stages which analyze the Python bytecode, translate it to Numba IR and perform various transformations and analysis steps on the IR. One of the key steps is type inference. The frontend must succeed in typing all variables unambiguously in order for the backend to generate code in nopython mode, because the backend uses type information to match appropriate code generators with the values they operate on.

The Numba backend walks the Numba IR resulting from the frontend analyses and exploits the type information deduced by the type inference phase to produce the right LLVM code for each encountered operation. After LLVM code is produced, the LLVM library is asked to optimize it and generate native processor code for the final, native function.

There are other pieces besides the compiler frontend and backend, such as the caching machinery for JIT functions. Those pieces are not considered in this document.

7.2.3. Contexts

Numba is quite flexible, allowing it to generate code for different hardware architectures like CPUs and GPUs. In order to support these different applications, Numba uses a typing context and a target context.

A typing context is used in the compiler frontend to perform type inference on operations and values in the function. Similar typing contexts could be used for many architectures because for nearly all cases, typing inference is hardware-independent. However, Numba currently has a different typing context for each target.

A target context is used to generate the specific instruction sequence required to operate on the Numba types identified during type inference. Target contexts are architecture-specific and are flexible in defining the execution model and available Python APIs. For example, Numba has a “cpu” and a “cuda” context for those two kinds of architecture, and a “parallel” context which produces multithreaded CPU code.

7.2.4. Compiler stages

The jit() decorator in Numba ultimately calls numba.compiler.compile_extra() which compiles the Python function in a multi-stage process, described below. Stage 1: Analyze bytecode

At the start of compilation, the function bytecode is passed to an instance of the Numba interpreter (numba.interpreter). The interpreter object analyzes the bytecode to find the control flow graph (numba.controlflow). The control flow graph (CFG) describes the ways that execution can move from one block to the next inside the function as a result of loops and branches.

The data flow analysis (numba.dataflow) takes the control flow graph and traces how values get pushed and popped off the Python interpreter stack for different code paths. This is important to understand the lifetimes of variables on the stack, which are needed in Stage 2.

If you set the environment variable NUMBA_DUMP_CFG to 1, Numba will dump the results of the control flow graph analysis to the screen. Our add() example is pretty boring, since there is only one statement block:

CFG adjacency lists:
{0: []}
CFG dominators:
{0: set([0])}
CFG post-dominators:
{0: set([0])}
CFG back edges: []
CFG loops:
CFG node-to-loops:
{0: []}

A function with more complex flow control will have a more interesting control flow graph. This function:

def doloops(n):
    acc = 0
    for i in range(n):
        acc += 1
        if n == 10:
    return acc

compiles to this bytecode:

 9           0 LOAD_CONST               1 (0)
             3 STORE_FAST               1 (acc)

10           6 SETUP_LOOP              46 (to 55)
             9 LOAD_GLOBAL              0 (range)
            12 LOAD_FAST                0 (n)
            15 CALL_FUNCTION            1
            18 GET_ITER
       >>   19 FOR_ITER                32 (to 54)
            22 STORE_FAST               2 (i)

11          25 LOAD_FAST                1 (acc)
            28 LOAD_CONST               2 (1)
            31 INPLACE_ADD
            32 STORE_FAST               1 (acc)

12          35 LOAD_FAST                0 (n)
            38 LOAD_CONST               3 (10)
            41 COMPARE_OP               2 (==)
            44 POP_JUMP_IF_FALSE       19

13          47 BREAK_LOOP
            48 JUMP_ABSOLUTE           19
            51 JUMP_ABSOLUTE           19
       >>   54 POP_BLOCK

14     >>   55 LOAD_FAST                1 (acc)
            58 RETURN_VALUE

The corresponding CFG for this bytecode is:

CFG adjacency lists:
{0: [6], 6: [19], 19: [54, 22], 22: [19, 47], 47: [55], 54: [55], 55: []}
CFG dominators:
{0: set([0]),
 6: set([0, 6]),
 19: set([0, 6, 19]),
 22: set([0, 6, 19, 22]),
 47: set([0, 6, 19, 22, 47]),
 54: set([0, 6, 19, 54]),
 55: set([0, 6, 19, 55])}
CFG post-dominators:
{0: set([0, 6, 19, 55]),
 6: set([6, 19, 55]),
 19: set([19, 55]),
 22: set([22, 55]),
 47: set([47, 55]),
 54: set([54, 55]),
 55: set([55])}
CFG back edges: [(22, 19)]
CFG loops:
{19: Loop(entries=set([6]), exits=set([54, 47]), header=19, body=set([19, 22]))}
CFG node-to-loops:
{0: [], 6: [], 19: [19], 22: [19], 47: [], 54: [], 55: []}

The numbers in the CFG refer to the bytecode offsets shown just to the left of the opcode names above. Stage 2: Generate the Numba IR

Once the control flow and data analyses are complete, the Numba interpreter can step through the bytecode and translate it into an Numba-internal intermediate representation. This translation process changes the function from a stack machine representation (used by the Python interpreter) to a register machine representation (used by LLVM).

Although the IR is stored in memory as a tree of objects, it can be serialized to a string for debugging. If you set the environment variable NUMBA_DUMP_IR equal to 1, the Numba IR will be dumped to the screen. For the add() function described above, the Numba IR looks like:

label 0:
    a = arg(0, name=a)                       ['a']
    b = arg(1, name=b)                       ['b']
    $0.3 = a + b                             ['$0.3', 'a', 'b']
    del b                                    []
    del a                                    []
    $0.4 = cast(value=$0.3)                  ['$0.3', '$0.4']
    del $0.3                                 []
    return $0.4                              ['$0.4']

The del instructions are produced by Live Variable Analysis. Those instructions ensure references are not leaked. In nopython mode, some objects are tracked by the numba runtime and some are not. For tracked objects, a dereference operation is emitted; otherwise, the instruction is an no-op. In object mode each variable contains an owned reference to a PyObject. Stage 3: Macro expansion

Now that the function has been translated into the Numba IR, macro expansion can be performed. Macro expansion converts specific attributes that are known to Numba into IR nodes representing function calls. This is initiated in the numba.compiler.translate_stage function, and is implemented in numba.macro.

Examples of attributes that are macro-expanded include the CUDA intrinsics for grid, block and thread dimensions and indices. For example, the assignment to tx in the following function:

def f(a):
    tx = cuda.threadIdx.x

has the following representation after translation to Numba IR:

$0.1 = global(cuda: <module 'numba.cuda' from '...'>) ['$0.1']
$0.2 = getattr(value=$0.1, attr=threadIdx) ['$0.1', '$0.2']
del $0.1                                 []
$0.3 = getattr(value=$0.2, attr=x)       ['$0.2', '$0.3']
del $0.2                                 []
tx = $0.3                                ['$0.3', 'tx']

After macro expansion, the $0.3 = getattr(value=$0.2, attr=x) IR node is translated into:

$0.3 = call tid.x(, )                    ['$0.3']

which represents an instance of the Intrinsic IR node for calling the tid.x intrinsic function. Stage 4: Rewrite untyped IR

Before running type inference, it may be desired to run certain transformations on the Numba IR. One such example is to detect raise statements which have an implicitly constant argument, so as to support them in nopython mode. Let’s say you compile the following function with Numba:

def f(x):
   if x == 0:
      raise ValueError("x cannot be zero")

If you set the NUMBA_DUMP_IR environment variable to 1, you’ll see the IR being rewritten before the type inference phase:

    del $0.3                                 []
    $12.1 = global(ValueError: <class 'ValueError'>) ['$12.1']
    $const12.2 = const(str, x cannot be zero) ['$const12.2']
    $12.3 = call $12.1($const12.2)           ['$12.1', '$12.3', '$const12.2']
    del $const12.2                           []
    del $12.1                                []
    raise $12.3                              ['$12.3']
    del $0.3                                 []
    $12.1 = global(ValueError: <class 'ValueError'>) ['$12.1']
    $const12.2 = const(str, x cannot be zero) ['$const12.2']
    $12.3 = call $12.1($const12.2)           ['$12.1', '$12.3', '$const12.2']
    del $const12.2                           []
    del $12.1                                []
    raise <class 'ValueError'>('x cannot be zero') [] Stage 5: Infer types

Now that the Numba IR has been generated and macro-expanded, type analysis can be performed. The types of the function arguments can be taken either from the explicit function signature given in the @jit decorator (such as @jit('float64(float64, float64)')), or they can be taken from the types of the actual function arguments if compilation is happening when the function is first called.

The type inference engine is found in numba.typeinfer. Its job is to assign a type to every intermediate variable in the Numba IR. The result of this pass can be seen by setting the NUMBA_DUMP_ANNOTATION environment variable to 1:

# File: archex.py
# --- LINE 4 ---


# --- LINE 5 ---

def add(a, b):

    # --- LINE 6 ---
    # label 0
    #   a = arg(0, name=a)  :: int64
    #   b = arg(1, name=b)  :: int64
    #   $0.3 = a + b  :: int64
    #   del b
    #   del a
    #   $0.4 = cast(value=$0.3)  :: int64
    #   del $0.3
    #   return $0.4

    return a + b

If type inference fails to find a consistent type assignment for all the intermediate variables, it will label every variable as type pyobject and fall back to object mode. Type inference can fail when unsupported Python types, language features, or functions are used in the function body. Stage 6a: Rewrite typed IR

This pass’s purpose is to perform any high-level optimizations that still require, or could at least benefit from, Numba IR type information.

One example of a problem domain that isn’t as easily optimized once lowered is the domain of multidimensional array operations. When Numba lowers an array operation, Numba treats the operation like a full ufunc kernel. During lowering a single array operation, Numba generates an inline broadcasting loop that creates a new result array. Then Numba generates an application loop that applies the operator over the array inputs. Recognizing and rewriting these loops once they are lowered into LLVM is hard, if not impossible.

An example pair of optimizations in the domain of array operators is loop fusion and shortcut deforestation. When the optimizer recognizes that the output of one array operator is being fed into another array operator, and only to that array operator, it can fuse the two loops into a single loop. The optimizer can further eliminate the temporary array allocated for the initial operation by directly feeding the result of the first operation into the second, skipping the store and load to the intermediate array. This elimination is known as shortcut deforestation. Numba currently uses the rewrite pass to implement these array optimizations. For more information, please consult the “Case study: Array Expressions” subsection, later in this document.

One can see the result of rewriting by setting the NUMBA_DUMP_IR environment variable to a non-zero value (such as 1). The following example shows the output of the rewrite pass as it recognizes an array expression consisting of a multiply and add, and outputs a fused kernel as a special operator, arrayexpr():

a0 = arg(0, name=a0)                     ['a0']
a1 = arg(1, name=a1)                     ['a1']
a2 = arg(2, name=a2)                     ['a2']
$0.3 = a0 * a1                           ['$0.3', 'a0', 'a1']
del a1                                   []
del a0                                   []
$0.5 = $0.3 + a2                         ['$0.3', '$0.5', 'a2']
del a2                                   []
del $0.3                                 []
$0.6 = cast(value=$0.5)                  ['$0.5', '$0.6']
del $0.5                                 []
return $0.6                              ['$0.6']
a0 = arg(0, name=a0)                     ['a0']
a1 = arg(1, name=a1)                     ['a1']
a2 = arg(2, name=a2)                     ['a2']
$0.5 = arrayexpr(ty=array(float64, 1d, C), expr=('+', [('*', [Var(a0, test.py (14)), Var(a1, test.py (14))]), Var(a2, test.py (14))])) ['$0.5', 'a0', 'a1', 'a2']
del a0                                   []
del a1                                   []
del a2                                   []
$0.6 = cast(value=$0.5)                  ['$0.5', '$0.6']
del $0.5                                 []
return $0.6                              ['$0.6']

Following this rewrite, Numba lowers the array expression into a new ufunc-like function that is inlined into a single loop that only allocates a single result array. Stage 6b: Perform Automatic Parallelization

This pass is only performed if the parallel option in the jit() decorator is set to True. This pass find parallelism implicit in the semantics of operations in the Numba IR and replaces those operations with explicitly parallel representations of those operations using a special parfor operator. Then, optimizations are performed to maximize the number of parfors that are adjacent to each other such that they can then be fused together into one parfor that takes only one pass over the data and will thus typically have better cache performance. Finally, during lowering, these parfor operators are converted to a form similar to guvectorize to implement the actual parallelism.

The automatic parallelization pass has a number of sub-passes, many of which are controllable using a dictionary of options passed via the parallel keyword argument to jit():

{ 'comprehension': True/False,  # parallel comprehension
  'prange':        True/False,  # parallel for-loop
  'numpy':         True/False,  # parallel numpy calls
  'reduction':     True/False,  # parallel reduce calls
  'setitem':       True/False,  # parallel setitem
  'stencil':       True/False,  # parallel stencils
  'fusion':        True/False,  # enable fusion or not

The default is set to True for all of them. The sub-passes are described in more detail in the following paragraphs.

  1. CFG Simplification

    Sometimes Numba IR will contain chains of blocks containing no loops which are merged in this sub-pass into single blocks. This sub-pass simplifies subsequent analysis of the IR.

  2. Numpy canonicalization

    Some Numpy operations can be written as operations on Numpy objects (e.g. arr.sum()), or as calls to Numpy taking those objects (e.g. numpy.sum(arr)). This sub-pass converts all such operations to the latter form for cleaner subsequent analysis.

  3. Array analysis

    A critical requirement for later parfor fusion is that parfors have identical iteration spaces and these iteration spaces typically correspond to the sizes of the dimensions of Numpy arrays. In this sub-pass, the IR is analyzed to determine equivalence classes for the dimensions of Numpy arrays. Consider the example, a = b + 1, where a and b are both Numpy arrays. Here, we know that each dimension of a must have the same equivalence class as the corresponding dimension of b. Typically, routines rich in Numpy operations will enable equivalence classes to be fully known for all arrays created within a function.

    Array analysis will also reason about size equivalvence for slice selection, and boolean array masking (one dimensional only). For example, it is able to infer that a[1 : n-1] is of the same size as b[0 : n-2].

    Array analysis may also insert safety assumptions to ensure pre-conditions related to array sizes are met before an operation can be parallelized. For example, np.dot(X, w) between a 2-D matrix X and a 1-D vector w requires that the second dimension of X is of the same size as w. Usually this kind of runtime check is automatically inserted, but if array analysis can infer such equivalence, it will skip them.

    Users can even help array analysis by turning implicit knowledge about array sizes into explicit assertions. For example, in the code below:

    def logistic_regression(Y, X, w, iterations):
        assert(X.shape == (Y.shape[0], w.shape[0]))
        for i in range(iterations):
            w -= np.dot(((1.0 / (1.0 + np.exp(-Y * np.dot(X, w))) - 1.0) * Y), X)
        return w

    Making the explicit assertion helps eliminate all bounds checks in the rest of the function.

  4. prange() to parfor

    The use of prange (Explicit Parallel Loops) in a for loop is an explicit indication from the programmer that all iterations of the for loop can execute in parallel. In this sub-pass, we analyze the CFG to locate loops and to convert those loops controlled by a prange object to the explicit parfor operator. Each explicit parfor operator consists of:

    1. A list of loop nest information that describes the iteration space of the parfor. Each entry in the loop nest list contains an indexing variable, the start of the range, the end of the range, and the step value for each iteration.
    2. An initialization (init) block which contains instructions to be executed one time before the parfor begins executing.
    3. A loop body comprising a set of basic blocks that correspond to the body of the loop and compute one point in the iteration space.
    4. The index variables used for each dimension of the iteration space.

    For parfor pranges, the loop nest is a single entry where the start, stop, and step fields come from the specified prange. The init block is empty for prange parfors and the loop body is the set of blocks in the loop minus the loop header.

    With parallelization on, array comprehensions (List comprehension) will also be translated to prange so as to run in parallel. This behavior be disabled by setting parallel={'comprehension': False}.

    Likewise, the overall prange to parfor translation can be disabled by setting parallel={'prange': False}, in which case prange is treated the same as range.

  5. Numpy to parfor

    In this sub-pass, Numpy functions such as ones, zeros, dot, most of the random number generating functions, arrayexprs (from Section Stage 6a: Rewrite typed IR), and Numpy reductions are converted to parfors. Generally, this conversion creates the loop nest list, whose length is equal to the number of dimensions of the left-hand side of the assignment instruction in the IR. The number and size of the dimensions of the left-hand-side array is taken from the array analysis information generated in sub-pass 3 above. An instruction to create the result Numpy array is generated and stored in the new parfor’s init block. A basic block is created for the loop body and an instruction is generated and added to the end of that block to store the result of the computation into the array at the current point in the iteration space. The result stored into the array depends on the operation that is being converted. For example, for ones, the value stored is a constant 1. For calls to generate a random array, the value comes from a call to the same random number function but with the size parameter dropped and therefore returning a scalar. For arrayexpr operators, the arrayexpr tree is converted to Numba IR and the value at the root of that expression tree is used to write into the output array. The translation from Numpy functions and arrayexpr operators to parfor can be disabled by setting parallel={'numpy': False}.

    For reductions, the loop nest list is similarly created using the array analysis information for the array being reduced. In the init block, the initial value is assigned to the reduction variable. The loop body consists of a single block in which the next value in the iteration space is fetched and the reduction operation is applied to that value and the current reduction value and the result stored back into the reduction value. The translation of reduction functions to parfor can be disabled by setting parallel={'reduction': False}.

    Setting the NUMBA_DEBUG_ARRAY_OPT_STATS environment variable to 1 will show some statistics about parfor conversions in general.

  6. Setitem to parfor

    Setting a range of array elements using a slice or boolean array selection can also run in parallel. Statement such as A[P] = B[Q] (or a simpler case A[P] = c, where c is a scalar) is translated to parfor if one of the following conditions is met:

    1. P and Q are slices or multi-dimensional selector involving scalar and slices, and A[P] and B[Q] are considered size equivalent by array analysis. Only 2-value slice/range is supported, 3-value with a step will not be translated to parfor.
    2. P and Q are the same boolean array.

    This translation can be disabled by setting parallel={'setitem': False}.

  7. Simplification

    Performs a copy propagation and dead code elimination pass.

  8. Fusion

    This sub-pass first processes each basic block and does a reordering of the instructions within the block with the goal of pushing parfors lower in the block and lifting non-parfors towards the start of the block. In practice, this approach does a good job of getting parfors adjacent to each other in the IR, which enables more parfors to then be fused. During parfor fusion, each basic block is repeatedly scanned until no further fusion is possible. During this scan, each set of adjacent instructions are considered. Adjacent instructions are fused together if:

    1. they are both parfors
    2. the parfors’ loop nests are the same size and the array equivalence classes for each dimension of the loop nests are the same, and
    3. the first parfor does not create a reduction variable used by the second parfor.

    The two parfors are fused together by adding the second parfor’s init block to the first’s, merging the two parfors’ loop bodies together and replacing the instances of the second parfor’s loop index variables in the second parfor’s body with the loop index variables for the first parfor. Fusion can be disabled by setting parallel={'fusion': False}.

    Setting the NUMBA_DEBUG_ARRAY_OPT_STATS environment variable to 1 will show some statistics about parfor fusions.

  9. Push call objects and compute parfor parameters

    In the lowering phase described in Section Stage 7a: Generate nopython LLVM IR, each parfor becomes a separate function executed in parallel in guvectorize (The @guvectorize decorator) style. Since parfors may use variables defined previously in a function, when those parfors become separate functions, those variables must be passed to the parfor function as parameters. In this sub-pass, a use-def scan is made over each parfor body and liveness information is used to determine which variables are used but not defined by the parfor. That list of variables is stored here in the parfor for use during lowering. Function variables are a special case in this process since function variables cannot be passed to functions compiled in nopython mode. Instead, for function variables, this sub-pass pushes the assignment instruction to the function variable into the parfor body so that those do not need to be passed as parameters.

    To see the intermediate IR between the above sub-passes and other debugging information, set the NUMBA_DEBUG_ARRAY_OPT environment variable to 1. For the example in Section Stage 6a: Rewrite typed IR, the following IR with a parfor is generated during this stage:

    label 0:
        a0 = arg(0, name=a0)                     ['a0']
        a0_sh_attr0.0 = getattr(attr=shape, value=a0) ['a0', 'a0_sh_attr0.0']
        $consta00.1 = const(int, 0)              ['$consta00.1']
        a0size0.2 = static_getitem(value=a0_sh_attr0.0, index_var=$consta00.1, index=0) ['$consta00.1', 'a0_sh_attr0.0', 'a0size0.2']
        a1 = arg(1, name=a1)                     ['a1']
        a1_sh_attr0.3 = getattr(attr=shape, value=a1) ['a1', 'a1_sh_attr0.3']
        $consta10.4 = const(int, 0)              ['$consta10.4']
        a1size0.5 = static_getitem(value=a1_sh_attr0.3, index_var=$consta10.4, index=0) ['$consta10.4', 'a1_sh_attr0.3', 'a1size0.5']
        a2 = arg(2, name=a2)                     ['a2']
        a2_sh_attr0.6 = getattr(attr=shape, value=a2) ['a2', 'a2_sh_attr0.6']
        $consta20.7 = const(int, 0)              ['$consta20.7']
        a2size0.8 = static_getitem(value=a2_sh_attr0.6, index_var=$consta20.7, index=0) ['$consta20.7', 'a2_sh_attr0.6', 'a2size0.8']
    ---begin parfor 0---
    index_var =  parfor_index.9
    LoopNest(index_variable=parfor_index.9, range=0,a0size0.2,1 correlation=5)
    init block:
        $np_g_var.10 = global(np: <module 'numpy' from '/usr/local/lib/python3.5/dist-packages/numpy/__init__.py'>) ['$np_g_var.10']
        $empty_attr_attr.11 = getattr(attr=empty, value=$np_g_var.10) ['$empty_attr_attr.11', '$np_g_var.10']
        $np_typ_var.12 = getattr(attr=float64, value=$np_g_var.10) ['$np_g_var.10', '$np_typ_var.12']
        $0.5 = call $empty_attr_attr.11(a0size0.2, $np_typ_var.12, kws=(), func=$empty_attr_attr.11, vararg=None, args=[Var(a0size0.2, test2.py (7)), Var($np_typ_var.12, test2.py (7))]) ['$0.5', '$empty_attr_attr.11', '$np_typ_var.12', 'a0size0.2']
    label 1:
        $arg_out_var.15 = getitem(value=a0, index=parfor_index.9) ['$arg_out_var.15', 'a0', 'parfor_index.9']
        $arg_out_var.16 = getitem(value=a1, index=parfor_index.9) ['$arg_out_var.16', 'a1', 'parfor_index.9']
        $arg_out_var.14 = $arg_out_var.15 * $arg_out_var.16 ['$arg_out_var.14', '$arg_out_var.15', '$arg_out_var.16']
        $arg_out_var.17 = getitem(value=a2, index=parfor_index.9) ['$arg_out_var.17', 'a2', 'parfor_index.9']
        $expr_out_var.13 = $arg_out_var.14 + $arg_out_var.17 ['$arg_out_var.14', '$arg_out_var.17', '$expr_out_var.13']
        $0.5[parfor_index.9] = $expr_out_var.13  ['$0.5', '$expr_out_var.13', 'parfor_index.9']
    ----end parfor 0----
        $0.6 = cast(value=$0.5)                  ['$0.5', '$0.6']
        return $0.6                              ['$0.6']
    ______________________________________________________________________ Stage 7a: Generate nopython LLVM IR

If type inference succeeds in finding a Numba type for every intermediate variable, then Numba can (potentially) generate specialized native code. This process is called lowering. The Numba IR tree is translated into LLVM IR by using helper classes from llvmlite. The machine-generated LLVM IR can seem unnecessarily verbose, but the LLVM toolchain is able to optimize it quite easily into compact, efficient code.

The basic lowering algorithm is generic, but the specifics of how particular Numba IR nodes are translated to LLVM instructions is handled by the target context selected for compilation. The default target context is the “cpu” context, defined in numba.targets.cpu.

The LLVM IR can be displayed by setting the NUMBA_DUMP_LLVM environment variable to 1. For the “cpu” context, our add() example would look like:

define i32 @"__main__.add$1.int64.int64"(i64* %"retptr",
                                         {i8*, i32}** %"excinfo",
                                         i8* %"env",
                                         i64 %"arg.a", i64 %"arg.b")
     %"a" = alloca i64
     %"b" = alloca i64
     %"$0.3" = alloca i64
     %"$0.4" = alloca i64
     br label %"B0"
     store i64 %"arg.a", i64* %"a"
     store i64 %"arg.b", i64* %"b"
     %".8" = load i64* %"a"
     %".9" = load i64* %"b"
     %".10" = add i64 %".8", %".9"
     store i64 %".10", i64* %"$0.3"
     %".12" = load i64* %"$0.3"
     store i64 %".12", i64* %"$0.4"
     %".14" = load i64* %"$0.4"
     store i64 %".14", i64* %"retptr"
     ret i32 0

The post-optimization LLVM IR can be output by setting NUMBA_DUMP_OPTIMIZED to 1. The optimizer shortens the code generated above quite significantly:

define i32 @"__main__.add$1.int64.int64"(i64* nocapture %retptr,
                                         { i8*, i32 }** nocapture readnone %excinfo,
                                         i8* nocapture readnone %env,
                                         i64 %arg.a, i64 %arg.b)
     %.10 = add i64 %arg.b, %arg.a
     store i64 %.10, i64* %retptr, align 8
     ret i32 0

If created during Stage 6b: Perform Automatic Parallelization, parfor operations are lowered in the following manner. First, instructions in the parfor’s init block are lowered into the existing function using the normal lowering code. Second, the loop body of the parfor is turned into a separate GUFunc. Third, code is emitted for the current function to call the parallel GUFunc.

To create a GUFunc from the parfor body, the signature of the GUFunc is created by taking the parfor parameters as identified in step 9 of Stage Stage 6b: Perform Automatic Parallelization and adding to that a special schedule parameter, across which the GUFunc will be parallelized. The schedule parameter is in effect a static schedule mapping portions of the parfor iteration space to Numba threads and so the length of the schedule array is the same as the number of configured Numba threads. To make this process easier and somewhat less dependent on changes to Numba IR, this stage creates a Python function as text that contains the parameters to the GUFunc and iteration code that takes the current schedule entry and loops through the specified portion of the iteration space. In the body of that loop, a special sentinel is inserted for subsequent easy location. This code that handles the processing of the iteration space is then eval’ed into existence and the Numba compiler’s run_frontend function is called to generate IR. That IR is scanned to locate the sentinel and the sentinel is replaced with the loop body of the parfor. Then, the process of creating the parallel GUFunc is completed by compiling this merged IR with the Numba compiler’s compile_ir function.

To call the parallel GUFunc, the static schedule must be created. Code is inserted to call a function named do_scheduling. This function is called with the size of each of the parfor’s dimensions and the number N of configured Numba threads (NUMBA_NUM_THREADS). The do_scheduling function will divide the iteration space into N approximately equal sized regions (linear for 1D, rectangular for 2D, or hyperrectangles for 3+D) and the resulting schedule is passed to the parallel GUFunc. The number of threads dedicated to a given dimension of the full iteration space is roughly proportional to the ratio of the size of the given dimension to the sum of the sizes of all the dimensions of the iteration space.

Parallel reductions are not natively provided by GUFuncs but the parfor lowering strategy allows us to use GUFuncs in a way that reductions can be performed in parallel. To accomplish this, for each reduction variable computed by a parfor, the parallel GUFunc and the code that calls it are modified to make the scalar reduction variable into an array of reduction variables whose length is equal to the number of Numba threads. In addition, the GUFunc still contains a scalar version of the reduction variable that is updated by the parfor body during each iteration. One time at the end of the GUFunc this local reduction variable is copied into the reduction array. In this way, false sharing of the reduction array is prevented. Code is also inserted into the main function after the parallel GUFunc has returned that does a reduction across this smaller reduction array and this final reduction value is then stored into the original scalar reduction variable.

The GUFunc corresponding to the example from Section Stage 6b: Perform Automatic Parallelization can be seen below:

label 0:
    sched.29 = arg(0, name=sched)            ['sched.29']
    a0 = arg(1, name=a0)                     ['a0']
    a1 = arg(2, name=a1)                     ['a1']
    a2 = arg(3, name=a2)                     ['a2']
    _0_5 = arg(4, name=_0_5)                 ['_0_5']
    $3.1.24 = global(range: <class 'range'>) ['$3.1.24']
    $const3.3.21 = const(int, 0)             ['$const3.3.21']
    $3.4.23 = getitem(value=sched.29, index=$const3.3.21) ['$3.4.23', '$const3.3.21', 'sched.29']
    $const3.6.28 = const(int, 1)             ['$const3.6.28']
    $3.7.27 = getitem(value=sched.29, index=$const3.6.28) ['$3.7.27', '$const3.6.28', 'sched.29']
    $const3.8.32 = const(int, 1)             ['$const3.8.32']
    $3.9.31 = $3.7.27 + $const3.8.32         ['$3.7.27', '$3.9.31', '$const3.8.32']
    $3.10.36 = call $3.1.24($3.4.23, $3.9.31, kws=[], func=$3.1.24, vararg=None, args=[Var($3.4.23, <string> (2)), Var($3.9.31, <string> (2))]) ['$3.1.24', '$3.10.36', '$3.4.23', '$3.9.31']
    $3.11.30 = getiter(value=$3.10.36)       ['$3.10.36', '$3.11.30']
    jump 1                                   []
label 1:
    $28.2.35 = iternext(value=$3.11.30)      ['$28.2.35', '$3.11.30']
    $28.3.25 = pair_first(value=$28.2.35)    ['$28.2.35', '$28.3.25']
    $28.4.40 = pair_second(value=$28.2.35)   ['$28.2.35', '$28.4.40']
    branch $28.4.40, 2, 3                    ['$28.4.40']
label 2:
    $arg_out_var.15 = getitem(value=a0, index=$28.3.25) ['$28.3.25', '$arg_out_var.15', 'a0']
    $arg_out_var.16 = getitem(value=a1, index=$28.3.25) ['$28.3.25', '$arg_out_var.16', 'a1']
    $arg_out_var.14 = $arg_out_var.15 * $arg_out_var.16 ['$arg_out_var.14', '$arg_out_var.15', '$arg_out_var.16']
    $arg_out_var.17 = getitem(value=a2, index=$28.3.25) ['$28.3.25', '$arg_out_var.17', 'a2']
    $expr_out_var.13 = $arg_out_var.14 + $arg_out_var.17 ['$arg_out_var.14', '$arg_out_var.17', '$expr_out_var.13']
    _0_5[$28.3.25] = $expr_out_var.13        ['$28.3.25', '$expr_out_var.13', '_0_5']
    jump 1                                   []
label 3:
    $const44.1.33 = const(NoneType, None)    ['$const44.1.33']
    $44.2.39 = cast(value=$const44.1.33)     ['$44.2.39', '$const44.1.33']
    return $44.2.39                          ['$44.2.39']
______________________________________________________________________ Stage 7b: Generate object mode LLVM IR

If type inference fails to find Numba types for all values inside a function, the function will be compiled in object mode. The generated LLVM will be significantly longer, as the compiled code will need to make calls to the Python C API to perform basically all operations. The optimized LLVM for our example add() function is:

@PyExc_SystemError = external global i8
@".const.Numba_internal_error:_object_mode_function_called_without_an_environment" = internal constant [73 x i8] c"Numba internal error: object mode function called without an environment\00"
@".const.name_'a'_is_not_defined" = internal constant [24 x i8] c"name 'a' is not defined\00"
@PyExc_NameError = external global i8
@".const.name_'b'_is_not_defined" = internal constant [24 x i8] c"name 'b' is not defined\00"

define i32 @"__main__.add$1.pyobject.pyobject"(i8** nocapture %retptr, { i8*, i32 }** nocapture readnone %excinfo, i8* readnone %env, i8* %arg.a, i8* %arg.b) {
  %.6 = icmp eq i8* %env, null
  br i1 %.6, label %entry.if, label %entry.endif, !prof !0

entry.if:                                         ; preds = %entry
  tail call void @PyErr_SetString(i8* @PyExc_SystemError, i8* getelementptr inbounds ([73 x i8]* @".const.Numba_internal_error:_object_mode_function_called_without_an_environment", i64 0, i64 0))
  ret i32 -1

entry.endif:                                      ; preds = %entry
  tail call void @Py_IncRef(i8* %arg.a)
  tail call void @Py_IncRef(i8* %arg.b)
  %.21 = icmp eq i8* %arg.a, null
  br i1 %.21, label %B0.if, label %B0.endif, !prof !0

B0.if:                                            ; preds = %entry.endif
  tail call void @PyErr_SetString(i8* @PyExc_NameError, i8* getelementptr inbounds ([24 x i8]* @".const.name_'a'_is_not_defined", i64 0, i64 0))
  tail call void @Py_DecRef(i8* null)
  tail call void @Py_DecRef(i8* %arg.b)
  ret i32 -1

B0.endif:                                         ; preds = %entry.endif
  %.30 = icmp eq i8* %arg.b, null
  br i1 %.30, label %B0.endif1, label %B0.endif1.1, !prof !0

B0.endif1:                                        ; preds = %B0.endif
  tail call void @PyErr_SetString(i8* @PyExc_NameError, i8* getelementptr inbounds ([24 x i8]* @".const.name_'b'_is_not_defined", i64 0, i64 0))
  tail call void @Py_DecRef(i8* %arg.a)
  tail call void @Py_DecRef(i8* null)
  ret i32 -1

B0.endif1.1:                                      ; preds = %B0.endif
  %.38 = tail call i8* @PyNumber_Add(i8* %arg.a, i8* %arg.b)
  %.39 = icmp eq i8* %.38, null
  br i1 %.39, label %B0.endif1.1.if, label %B0.endif1.1.endif, !prof !0

B0.endif1.1.if:                                   ; preds = %B0.endif1.1
  tail call void @Py_DecRef(i8* %arg.a)
  tail call void @Py_DecRef(i8* %arg.b)
  ret i32 -1

B0.endif1.1.endif:                                ; preds = %B0.endif1.1
  tail call void @Py_DecRef(i8* %arg.b)
  tail call void @Py_DecRef(i8* %arg.a)
  tail call void @Py_IncRef(i8* %.38)
  tail call void @Py_DecRef(i8* %.38)
  store i8* %.38, i8** %retptr, align 8
  ret i32 0

declare void @PyErr_SetString(i8*, i8*)

declare void @Py_IncRef(i8*)

declare void @Py_DecRef(i8*)

declare i8* @PyNumber_Add(i8*, i8*)

The careful reader might notice several unnecessary calls to Py_IncRef and Py_DecRef in the generated code. Currently Numba isn’t able to optimize those away.

Object mode compilation will also attempt to identify loops which can be extracted and statically-typed for “nopython” compilation. This process is called loop-lifting, and results in the creation of a hidden nopython mode function just containing the loop which is then called from the original function. Loop-lifting helps improve the performance of functions that need to access uncompilable code (such as I/O or plotting code) but still contain a time-intensive section of compilable code. Stage 8: Compile LLVM IR to machine code

In both object mode and nopython mode, the generated LLVM IR is compiled by the LLVM JIT compiler and the machine code is loaded into memory. A Python wrapper is also created (defined in numba.dispatcher.Dispatcher) which can do the dynamic dispatch to the correct version of the compiled function if multiple type specializations were generated (for example, for both float32 and float64 versions of the same function).

The machine assembly code generated by LLVM can be dumped to the screen by setting the NUMBA_DUMP_ASSEMBLY environment variable to 1:

        .globl  __main__.add$1.int64.int64
        .align  16, 0x90
        .type   __main__.add$1.int64.int64,@function
        addq    %r8, %rcx
        movq    %rcx, (%rdi)
        xorl    %eax, %eax

The assembly output will also include the generated wrapper function that translates the Python arguments to native data types.