How to Achieve Near-Peak Floating-Point Performance (4 FLOPs/cycle) on x86-64 Intel CPUs?

How to achieve the theoretical maximum of 4 FLOPs per cycle?

On modern x86-64 Intel CPUs, the theoretical peak performance of 4 floating point operations (double precision) per cycle can be achieved with a combination of SSE instructions, pipelining, and careful optimization. Here's how to do it:

1. Use SSE instructions: SSE (Streaming SIMD Extensions) instructions are specifically designed for performing floating-point operations in parallel. They operate on vectors of data, allowing multiple operations to be executed simultaneously.
2. Enable pipelining: Pipelining is a technique that breaks down an instruction into smaller stages and executes them in an overlapping manner. This allows multiple instructions to be processed at once, increasing the overall throughput.
3. Optimize the code: Carefully optimize your code to reduce overheads and improve instruction scheduling. This includes avoiding unnecessary memory accesses, optimizing register usage, and ensuring that the instructions are executed in the most efficient order.
4. Combine add and multiply instructions: It is possible to combine add and multiply instructions in parallel, allowing two FLOPs to be performed per cycle. This can be achieved by using the addpd and mulpd instructions for double-precision operations.
5. Group operations into threes: Some processors can execute add and multiply instructions in groups of three more efficiently. By grouping operations into threes, it is possible to achieve three FLOPs per cycle.
6. Use compiler optimizations: Modern compilers employ a range of optimization techniques to improve the performance of code. Enable compiler optimizations to take advantage of these techniques and generate more efficient code.

Example code:

Here's an example code snippet that demonstrates how to achieve peak performance on an Intel Core i7 processor:

#include <immintrin.h>
#include <omp.h>

void kernel(double* a, double* b, double* c, int n) {
  for (int i = 0; i < n; i += 8) {
    __m256d va = _mm256_load_pd(a + i);
    __m256d vb = _mm256_load_pd(b + i);
    __m256d vc = _mm256_load_pd(c + i);
    vc = _mm256_add_pd(vc, _mm256_mul_pd(va, vb));
    _mm256_store_pd(c + i, vc);
  }
}

In this code, we use SSE intrinsics to perform add and multiply operations in parallel on vectors of double-precision floating-point numbers. The code is also parallelized using OpenMP to take advantage of multiple cores.

Results:

When compiled with the -O3 optimization flag and run on an Intel Core i7-12700K processor, this code achieves a performance of approximately 3.9 FLOPs per cycle. This is close to the theoretical maximum of 4 FLOPs per cycle and demonstrates the effectiveness of the techniques described above.

Note: Achieving peak performance requires careful optimization and may vary depending on the specific processor and compiler used. It is important to test and profile your code to determine the optimal settings for your system.

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