| commit | a5660744801116ad2f9ab4e9e389f194ba307a35 | [log] [tgz] |
|---|---|---|
| author | Rasmus Munk Larsen <rmlarsen@google.com> | Mon Dec 16 21:33:42 2019 +0000 |
| committer | Rasmus Munk Larsen <rmlarsen@google.com> | Mon Dec 16 21:33:42 2019 +0000 |
| tree | a7150bf1b016baab78b82ab8d2195b5e91916126 | |
| parent | 8e5da71466591cc24352782b08dc78ddb94f0717 [diff] |
Improve accuracy of fast approximate tanh and the logistic functions in Eigen, such that they preserve relative accuracy to within a few ULPs where their function values tend to zero (around x=0 for tanh, and for large negative x for the logistic function). This change re-instates the fast rational approximation of the logistic function for float32 in Eigen (removed in https://gitlab.com/libeigen/eigen/commit/66f07efeaed39d6a67005343d7e0caf7d9eeacdb), 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
Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
For more information go to http://eigen.tuxfamily.org/.
For pull request, bug reports, and feature requests, go to https://gitlab.com/libeigen/eigen.