# Chapter 03: Structure of nmath Kernel Programs

#### Kjell Nygren

#### 2026-06-11

## Why “programs” rather than compiled objects

Normal C++ code in an R package is compiled ahead of time by the package build toolchain and linked into a DLL/`.so`. OpenCL device code works differently: the GPU source is compiled at runtime, on the device, by the GPU driver. This means there is no offline linker for device code — all GPU code that will run together must be collected into a single source string before the driver sees it.

In OpenCL terms, that string is a program. The C++ call sequence is:

    // kernel_source is one big string containing all device code
    program = clCreateProgramWithSource(context, 1, &src_ptr, &src_len, &status);
    clBuildProgram(program, 0, nullptr, nullptr, nullptr, nullptr);
    kernel  = clCreateKernel(program, "dnorm_kernel", &status);

`clBuildProgram` invokes the vendor GPU compiler on `kernel_source` and produces a device binary. A build failure here is equivalent to a C++ compilation error, except it happens at the moment of first use, not at `R CMD INSTALL` time.

## No `#include` on the device

OpenCL C is a restricted dialect of C99. There is no file system visible to device code: `#include` is simply not available inside an OpenCL program string. Every declaration or definition that a kernel function needs must appear earlier in the same string.

This creates the fundamental porting challenge for `nmathopencl`. R’s `nmath` library is written as standard C99 and relies on a chain of `#include` directives:

    #include "nmath.h"       /* includes R_ext/Arith.h, R_ext/Error.h, ... */
    #include "dpq.h"

Those headers pull in standard C types (`FILE*`, `size_t`, `int32_t`), R-specific runtime types (`SEXP`, `Rboolean`), and R callbacks (`error()`, `warning()`, `R_alloc()`). None of these concepts exist on a GPU.

The solution is to assemble the program string in layers, prepending OpenCL-compatible replacements for everything the nmath sources expect to find via `#include`.

## The four-layer program structure

A complete `nmathopencl` kernel program is a concatenation of four layers:

    ┌─────────────────────────────────────────────────────────────┐
    │  Layer 0 – OPENCL.cl                                         │
    │  Global configuration: fp64 pragma, HAVE_* macros, ML_NAN   │
    ├─────────────────────────────────────────────────────────────┤
    │  Layer 1 – Upstream shims  (inst/cl/R_ext_types/, R_shims/, │
    │                              R_ext_runtime/, System/, ...)   │
    │  OpenCL-C replacements for C headers the nmath sources need  │
    ├─────────────────────────────────────────────────────────────┤
    │  Layer 2 – nmath library  (inst/cl/nmath/)                   │
    │  The ported Mathlib functions, exactly the subset required   │
    │  by this kernel (determined by @all_depends_nmath)           │
    ├─────────────────────────────────────────────────────────────┤
    │  Layer 3 – Kernel function  (inst/cl/src/)                   │
    │  The __kernel entry point: GPU-parallel version of a CPU     │
    │  loop over the nmath function                                │
    └─────────────────────────────────────────────────────────────┘

Each layer is a plain text block that is physically concatenated (in order) to form the final string passed to `clCreateProgramWithSource`.

## Layer 0: `OPENCL.cl` — the global configuration header

`inst/cl/OPENCL.cl` is always placed at the very top. It sets the conditions under which the rest of the program will compile:

    #pragma OPENCL EXTENSION cl_khr_fp64 : enable   /* require 64-bit floats */

    /* Enable expm1/log1p built-ins on OpenCL >= 1.2 */
    #if defined(__OPENCL_C_VERSION__) && (__OPENCL_C_VERSION__ >= 120)
      #define HAVE_EXPM1 1
      #define HAVE_LOG1P 1
    #endif
    #undef HAVE_LONG_DOUBLE      /* no long double on GPU */

    #define ML_NAN   (0.0/0.0)
    #define ML_POSINF INFINITY
    #define ML_NEGINF -INFINITY
    #define INLINE static inline

This one file handles the biggest portability concern: double-precision support (`cl_khr_fp64`) must be declared before any `double` variable appears in device code, and the `HAVE_*` macros tell the nmath sources which C99 math built-ins are available.

## Layer 1: The upstream shim layer

R’s nmath sources transitively include a long chain of headers. The shim layer provides OpenCL C replacements, grouped by what they replace:

### Type stubs

Files | What they replace
--- | ---
`System/stdint.cl` | `<stdint.h>`: `int32_t`, `uint64_t`, … using OpenCL built-in integer types
`R_ext_types/Boolean.cl`, `Complex.cl`, `Constants.cl`, etc. | Type-only parts of `<R_ext/Boolean.h>`, `<R_ext/Complex.h>`, etc.
`R_shims/Rinternals.cl` | Skeletal `SEXP`, `SEXPREC`, `Rboolean` so transitively included headers compile

None of these types are used by the nmath math logic; the stubs exist so the preprocessor does not abort on an unrecognized identifier.

### Capability macros

`R_shims/Rconfig.cl` reproduces what R’s `configure` script writes into `config.h`:

    #define HAVE_EXPM1 1
    #define HAVE_LOG1P 1
    #define IEEE_754   1

Without this, conditional compilation branches inside nmath would take the wrong path (e.g., falling back to a Taylor series instead of calling the built-in `expm1`).

### Runtime no-ops

R’s `error()` and `warning()` are host-side callbacks that `longjmp` into the R runtime — they cannot exist on the GPU. `R_ext_runtime/Error.cl` replaces them with silent macros:

    #define error(...)   /* no-op */
    #define warning(...) /* no-op */

Similarly, `R_ext_runtime/Memory.cl` stubs `R_alloc()` and friends (nmath’s hot paths do not actually allocate, so these stubs satisfy the preprocessor without ever executing), and `R_ext_runtime/Print.cl` stubs `Rprintf`.

### Math declarations

`R_ext_runtime/Arith.cl` is the most actively used shim. It maps R’s numeric predicates to their OpenCL equivalents:

    #define R_FINITE(x)  isfinite(x)
    #define ISNAN(x)     isnan(x)

`nmath/Rmath.cl` and `nmath/nmath.cl` close out the shim layer by providing the forward declarations, mathematical constants, and error-code macros that the nmath function files expect to find in `Rmath.h` / `nmath.h`.

### The design principle

Every shim is minimally intrusive: it defines only what the nmath math logic actually needs, and stubs or silences everything else. The goal is that the nmath `.cl` files are as close to verbatim copies of the C sources as possible; the shims carry all the differences.

## Layer 2: The nmath library and its annotation scheme

`inst/cl/nmath/` contains approximately 180 `.cl` files, one per source file in R’s `src/nmath/`. They implement the full suite of Mathlib functions in OpenCL C.

### `@provides` and `@depends` annotations

Because the program is assembled at runtime, the loader must know the correct concatenation order. Each nmath file declares what it provides and what it depends on:

    // From inst/cl/nmath/dnorm.cl
    // @source_origin: dnorm.c
    // @depends:  nmath, dpq
    // @provides: dnorm, dnorm4
    // @all_depends: dpq, Rmath, nmath
    // @load_order: 31

`opencltools::load_kernel_library()` reads these tags, builds a dependency graph, and performs a topological sort — files with no unsatisfied dependencies are emitted first, then files whose dependencies have been emitted, and so on. The result is a load order in which every function is declared before it is called.

### Only the required subset is loaded

A kernel that calls `dnorm` needs only 4 nmath files. A kernel that calls `dbinom` needs 18. Loading all ~180 files for every kernel would be wasteful and slow to compile. Instead, each kernel file carries a pre-computed `@all_depends_nmath` annotation listing the exact transitive closure:

    // dnorm_kernel.cl  -- needs 4 files
    // @all_depends_nmath: dpq, Rmath, nmath, dnorm

    // dbinom_kernel.cl -- needs 18 files
    // @all_depends_nmath: dpq, refactored, Rmath, nmath, stirlerr_cycle_free,
    //   chebyshev, cospi, fmax2, gammalims, lgammacor, log1p, gamma, lgamma,
    //   pgamma_utils, stirlerr_cycle_dependent, bd0, stirlerr, dbinom

`load_library_for_kernel()` reads `@all_depends_nmath`, looks up each stem in a pre-built index (`kernel_dependency_index.tsv`) to get the global load order, and concatenates exactly those files in the correct sequence. No topological sort is computed at install time — the sort was done once when the index was built.

## Layer 3: The kernel function

The `__kernel` function is the entry point that the GPU driver exposes. Conceptually it is a GPU-parallel version of a CPU `for` loop:

    /* CPU equivalent:
       for (int i = 0; i < n; i++) out[i] = dnorm(x[i], mu[i], sigma[i], gl); */

    __kernel void dnorm_kernel(
        __global const double* x,
        __global const double* mu,
        __global const double* sigma,
        __global const int*    give_log,
        __global double*       out,
        const int              len
    ) {
        int i = get_global_id(0);    /* this work-item handles element i */
        if (i >= len) return;
        int gl = (give_log[i] != 0) ? 1 : 0;
        out[i] = dnorm(x[i], mu[i], sigma[i], gl);   /* same nmath call */
    }

`get_global_id(0)` returns the index of the current work-item in the first dimension of the work grid. The host launches exactly `n` work-items so each element of the input vector is handled by one work-item in parallel.

The kernel calls `dnorm` — the same function name as in the C source. All the layers below it (shims + nmath files) ensure that `dnorm` is defined and reachable in the same program string.

## Assembling the program in C++

`build_rmath_opencl_program()` in `src/kernel_loader.cpp` shows precisely how the four layers are concatenated:

    std::string build_rmath_opencl_program(
        const std::string& kernel_relative_path,
        const std::string& package,
        const std::string& nmath_depends_annotation)
    {
      // Load exactly the nmath subset this kernel needs (via @all_depends_nmath)
      const std::string nmath_src = load_library_for_kernel(
          kernel_relative_path, "nmath", package, nmath_depends_annotation);

      return
        load_kernel_source("OPENCL.cl",      package)        + "\n"  // Layer 0
        + load_kernel_library("libR_shims",  package, false) + "\n"  // Layer 1
        + load_kernel_library("R_ext_types", package, false) + "\n"
        + load_kernel_library("R_shims",     package, false) + "\n"
        + load_kernel_library("R_ext_runtime", package, false) + "\n"
        + load_kernel_library("R_ext_internals", package, false) + "\n"
        + load_kernel_library("System",      package, false) + "\n"
        + nmath_src                                          + "\n"  // Layer 2
        + load_kernel_source(kernel_relative_path, package);         // Layer 3
    }

The returned string is passed directly to `clCreateProgramWithSource` and then compiled by the device driver. At that point the GPU can execute the kernel function.

Note the dual guard: each loader function itself is compiled to a no-op when `USE_OPENCL` is absent (see Chapter 02), so `build_rmath_opencl_program` is always callable but returns an empty string on CPU-only builds. Callers check `nmathopencl_has_opencl()` before dispatching to the GPU path.

## What this means for downstream developers

A downstream package that builds on `nmathopencl` does not need to replicate this assembly machinery. Kernel text loading uses opencltools (`load_kernel_source`, `load_kernel_library` with `package = "nmathopencl"`). `load_library_for_kernel` is re-exported from nmathopencl (and available via `LinkingTo: nmathopencl`) and handles all of layers 0, 1, and 2 automatically.

The downstream developer’s responsibilities are:

1. Write the `__kernel` function (Layer 3): call the nmath functions your computation needs.
2. Annotate it with `@depends_nmath` and `@all_depends_nmath` so the loader knows which nmath subset to pull in.
3. Write the C++ runner that calls `build_rmath_opencl_program()` (or the equivalent chain), dispatches work-items, and transfers results.

Chapter 07 covers how to write and annotate kernel functions. Chapter 10 walks through all three steps end-to-end for the `ex_glmbayes` example.