After defining how your custom primitive behaves abstractly in terms of shapes and data types, the next significant step is to teach JAX how to actually execute it on specific hardware like CPUs, GPUs, or TPUs. This process is known as "lowering," where the high-level primitive is translated into low-level instructions that the hardware backend understands, typically via the XLA (Accelerated Linear Algebra) compiler.

Why Lowering Rules are Necessary

JAX achieves its performance by JIT-compiling Python functions into optimized executable code using XLA. XLA operates on its own intermediate representation, HLO (High-Level Optimizer IR). When JAX encounters a primitive during compilation (like jax.lax.add or your custom one), it needs a way to translate that primitive into a corresponding sequence of HLO instructions. Without a lowering rule, XLA wouldn't know what machine code to generate for your custom operation.

Each backend (CPU, GPU, TPU) might have different optimal ways to implement the same operation. Therefore, you need to provide specific lowering rules tailored to each backend you intend to support.

The Target: XLA HLO

The lowering process in JAX almost always targets XLA HLO. Your lowering rule's job is to take the inputs to your primitive (represented as XLA values) and use an XLA builder object to construct the HLO computation graph that performs the desired operation. XLA then takes this HLO graph and compiles it further into highly optimized machine code for the target accelerator.

Registering Lowering Functions

You register a lowering rule for a specific primitive and backend using JAX's internal APIs. A common way involves functions like xla_client.register_translation. You associate your primitive object with a Python function that performs the lowering.

# Conceptual Example: Registering a lowering rule for a hypothetical 'my_custom_op_p' primitive
from jax.lib import xla_client
from jax.interpreters import xla

# Assume 'my_custom_op_p' is your Primitive object

def my_custom_op_lowering_cpu(builder, *args, **params):
  """Lowering rule for my_custom_op on CPU."""
  # 'builder' is an XlaBuilder instance
  # 'args' are the XLA computation values for the inputs
  # 'params' are the static parameters of the primitive
  
  # Use the builder to construct HLO operations...
  # Example: If my_custom_op adds 1.0
  # operand = args[0]
  # one = builder.ConstantF32Scalar(1.0)
  # result = builder.Add(operand, one)
  # return [result] # Return a list of output XLA values
  
  # Replace with actual HLO construction for your primitive
  pass 

# Register for the CPU backend
xla_client.register_translation(my_custom_op_p, 
                                my_custom_op_lowering_cpu, 
                                platform='cpu')

# Similarly, define and register for 'gpu' and 'tpu' platforms
# def my_custom_op_lowering_gpu(builder, *args, **params): ...
# xla_client.register_translation(my_custom_op_p, my_custom_op_lowering_gpu, platform='gpu') 

# def my_custom_op_lowering_tpu(builder, *args, **params): ...
# xla_client.register_translation(my_custom_op_p, my_custom_op_lowering_tpu, platform='tpu')

(Note: The exact API details might evolve; consult the current JAX documentation for the latest registration methods.)

Using the XLA Builder

The builder object passed to your lowering function is the primary tool for constructing HLO. It provides methods for:

Creating constants (e.g., builder.ConstantF32Scalar(1.0)).
Performing elementwise arithmetic (e.g., builder.Add, builder.Mul, builder.Log1p).
Matrix operations (e.g., builder.DotGeneral).
Data manipulation (e.g., builder.Reshape, builder.Transpose, builder.Slice).
Control flow (though often handled at a higher level in JAX).
And many more specialized operations.

You use these methods, chaining them together, taking the input args (which are handles to XLA values) and producing the output XLA values that represent the result of your primitive. The abstract evaluation rule you defined earlier provides the shape and dtype information that the builder often needs to construct these operations correctly.

Backend-Specific Considerations

While some primitives might have identical lowering rules across backends, performance-critical operations often benefit from backend specialization:

CPU: Lowering might generate standard HLO operations that XLA compiles effectively for multi-core CPUs, potentially using vector instructions (like AVX).
GPU: Lowering rules might generate HLO that maps well to CUDA kernels. This could involve using specific HLO operations known to be efficient on GPUs or structuring the computation to maximize parallelism and minimize memory transfers. For operations not directly available in HLO, you might need to use gpu_kernel calls within the lowering rule, embedding CUDA code (which adds significant complexity).
TPU: Lowering needs to consider the TPU's architecture, particularly its powerful matrix multiplication units (MXUs) and vector processing units. Efficient TPU code often involves structuring computations as large matrix multiplies or specific HLO operations optimized for the TPU hardware.

Example: Lowering a Simple `relu`

Let's imagine we're defining a custom ReLU primitive, my_relu_p. The ReLU operation is $max(0, x)$ .

# Conceptual Lowering for my_relu_p
# Assume my_relu_p is the Primitive object
# Assume abstract evaluation rule is already defined

def my_relu_lowering(builder, x_operand, **params):
  """Lowers my_relu(x) = max(0, x) to HLO."""
  
  # Get the shape and dtype from the operand
  # Abstract evaluation ensures x_operand has the correct abstract value
  shape = builder.GetShape(x_operand) 
  dtype = shape.element_type() # e.g., F32
  
  # Create a constant zero of the same type and shape
  zero = builder.Broadcast(builder.ConstantElement(0, dtype), shape.dimensions())
  
  # Compute the maximum elementwise
  result = builder.Max(zero, x_operand)
  
  return [result] # Return list containing the single output

# Register this rule for relevant platforms
# xla_client.register_translation(my_relu_p, my_relu_lowering, platform='cpu')
# xla_client.register_translation(my_relu_p, my_relu_lowering, platform='gpu')
# xla_client.register_translation(my_relu_p, my_relu_lowering, platform='tpu')

In this case, the lowering is straightforward: create a zero tensor of the same shape and type as the input and use the Max HLO operation. This rule is likely sufficient for all backends, as XLA knows how to compile Max efficiently for each.

Complexity and Advanced Usage

Writing lowering rules can range from simple, like the ReLU example, to highly complex. If your operation doesn't map cleanly onto existing HLO instructions, you might need to:

Compose multiple HLO operations: Build your functionality from existing HLO primitives.
Implement backend-specific kernels: For GPUs, write custom CUDA code and invoke it via HLO's CustomCall. This requires expertise in both HLO and the target hardware programming model (e.g., CUDA).
Understand XLA Optimizations: Be aware of how XLA might fuse or optimize the HLO you generate.

Debugging lowering rules can also be challenging, often requiring inspection of the generated HLO itself using tools provided by JAX or XLA.

By implementing the lowering rule(s), you provide the final piece needed for JAX to compile and run your custom primitive efficiently on your desired hardware, making it a first-class citizen within the JAX ecosystem, ready to be used inside jit, vmap, pmap, and even differentiated (once you define its differentiation rules).

Implementing Lowering Rules for Backends (CPU/GPU/TPU)