This change re-instates the fast rational approximation of the logistic function for float32 in Eigen (removed in 66f07efeae), but uses the more accurate approximation 1/(1+exp(-1)) ~= exp(x) below -9. The exponential is only calculated on the vectorized path if at least one element in the SIMD input vector is less than -9.
This change also contains a few improvements to speed up the original float specialization of logistic:
- Introduce EIGEN_PREDICT_{FALSE,TRUE} for __builtin_predict and use it to predict that the logistic-only path is most likely (~2-3% speedup for the common case).
- Carefully set the upper clipping point to the smallest x where the approximation evaluates to exactly 1. This saves the explicit clamping of the output (~7% speedup).
The increased accuracy for tanh comes at a cost of 10-20% depending on instruction set.
The benchmarks below repeated calls
u = v.logistic() (u = v.tanh(), respectively)
where u and v are of type Eigen::ArrayXf, have length 8k, and v contains random numbers in [-1,1].
Benchmark numbers for logistic:
Before:
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
SSE
BM_eigen_logistic_float 4467 4468 155835 model_time: 4827
AVX
BM_eigen_logistic_float 2347 2347 299135 model_time: 2926
AVX+FMA
BM_eigen_logistic_float 1467 1467 476143 model_time: 2926
AVX512
BM_eigen_logistic_float 805 805 858696 model_time: 1463
After:
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
SSE
BM_eigen_logistic_float 2589 2590 270264 model_time: 4827
AVX
BM_eigen_logistic_float 1428 1428 489265 model_time: 2926
AVX+FMA
BM_eigen_logistic_float 1059 1059 662255 model_time: 2926
AVX512
BM_eigen_logistic_float 673 673 1000000 model_time: 1463
Benchmark numbers for tanh:
Before:
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
SSE
BM_eigen_tanh_float 2391 2391 292624 model_time: 4242
AVX
BM_eigen_tanh_float 1256 1256 554662 model_time: 2633
AVX+FMA
BM_eigen_tanh_float 823 823 866267 model_time: 1609
AVX512
BM_eigen_tanh_float 443 443 1578999 model_time: 805
After:
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
SSE
BM_eigen_tanh_float 2588 2588 273531 model_time: 4242
AVX
BM_eigen_tanh_float 1536 1536 452321 model_time: 2633
AVX+FMA
BM_eigen_tanh_float 1007 1007 694681 model_time: 1609
AVX512
BM_eigen_tanh_float 471 471 1472178 model_time: 805
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