How Can AVX2 Be Used to Efficiently Implement log2(__m256d)?

Efficient Implementation of log2(__m256d) in AVX2

Introduction

Logarithmic calculations are essential in various scientific and engineering applications. This article explores the implementation of an efficient log2() function for 4-element double-precision floating-point vectors using Advanced Vector Extensions 2 (AVX2).

__m256d log2_pd Intrinsic in SVML

Intel's Scalable Vector Math Library (SVML) provides an intrinsic function __m256d _mm256_log2_pd (__m256d a) for performing log2 operations on 4-bit vectors. However, this intrinsic is only available in Intel compilers and is reported to have performance drawbacks on AMD processors.

Polynomial Approximation

To implement log2() without relying on compiler-specific intrinsics, we can leverage polynomial approximations. We can express log2(x) as a Taylor series expanded around x = 1, or more specifically, we can use multiple polynomial terms to approximate log2(mantissa) in the range of [1.0, 2.0].

Implementation Details

The following C implementation provides a highly efficient log2() function for 4-bit double-precision vectors using AVX2 and a custom polynomial approximation:

__m256d __vectorcall Log2(__m256d x) {
  // Extract exponent and normalize it

  // Calculate t=(y-1)/(y+1) and t**2
  // Calculate log2(y) and add exponent

  return log2_x;
}

The approximation formula used can be visualized as:

The polynomial coefficients were fitted to minimize the maximum absolute error over the range [1.0, 2.0].

Performance Analysis

Benchmarks show that this implementation outperforms both std::log2() and std::log() by a significant margin, achieving around 4 times the performance of std::log2().

Limitations and Considerations

The accuracy of the implementation can be tailored by adding more polynomial terms. However, increasing the polynomial order will increase the number of floating-point operations and potentially reduce performance.

Conclusion

The provided AVX2 implementation of log2() offers high efficiency and performance for vectorized logarithmic calculations. By leveraging custom polynomial approximations, this function provides a portable and efficient solution for log2 operations on 4-bit double-precision floating-point vectors.

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