How Can We Efficiently Implement a Vectorized Logarithm Function Using AVX2?

Efficient Vectorized Logarithm Implementation in AVX2

The objective is to implement an efficient vectorized version of the log2 function for 4 double-precision numbers using AVX2, comparable to the performance of SVML's __m256d _mm256_log2_pd (__m256d a) but available on other compilers.

Implementation Approach

The common strategy for log2(a) involves computing the sum of the exponent and the log2 of the mantissa, which has a limited range of 1.0 to 2.0. This enables us to use a polynomial approximation for the log2 of the mantissa.

1. Extract Exponent: Extract the exponent part of the input vector and convert it back to a double-precision value, adjusting for the bias.
2. Extract and Adjust Mantissa: Extract the mantissa and adjust it to a range of [0.5, 1.0). This ensures that the polynomial approximation we use will be more accurate.
3. Polynomial Approximation: Use a polynomial approximation to compute the log2 of the adjusted mantissa. We can fit a polynomial using a series expansion or minimax techniques.
4. Combination: Add the computed exponent and the polynomial approximation of the log2 of the mantissa to obtain the final log2 result.

Optimizations

To improve accuracy, we can use a ratio of two polynomials instead of a single high-order polynomial. This technique reduces rounding errors and maintains high precision.

Additionally, we can skip checks for underflow, overflow, or denormal values if it's known that the input values are positive and finite. This optimization can significantly speed up the implementation.

Performance Considerations

• Instruction Latency: Modern hardware has long instruction latencies. To optimize performance, we can use faster polynomial evaluation schemes, such as Estrin's scheme, which allows for parallel execution of polynomial terms.
• Exploiting FMA: The fused-multiply-add (FMA) instruction is highly efficient. By employing FMA in our implementation, we can accelerate the polynomial evaluation process.

Accuracy and Range

The accuracy and range of the implementation depend on the specific polynomial approximation used. It is possible to achieve very high accuracy over a specific range of mantissa values.

Comparison to Existing Implementations

The proposed implementation aims to provide a fast and efficient vectorized log2 function that can be used on any platform with AVX2 support. It targets high performance comparable to Intel compilers' SVML implementation while being available for other compilers as well.

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