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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).

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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
This commit is contained in:
Rasmus Munk Larsen
2019-12-16 21:33:42 +00:00
parent 8e5da71466
commit a566074480
9 changed files with 191 additions and 23 deletions
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34
Eigen/src/Core/arch/GPU/PacketMath.h
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@@ -156,6 +156,15 @@ EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double eq_mask(const double& a,
return __longlong_as_double(a == b ? 0xffffffffffffffffull : 0ull);
}
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float lt_mask(const float& a,
const float& b) {
return __int_as_float(a < b ? 0xffffffffu : 0u);
}
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double lt_mask(const double& a,
const double& b) {
return __longlong_as_double(a < b ? 0xffffffffffffffffull : 0ull);
}
} // namespace
template <>
@@ -213,10 +222,21 @@ EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float4 pcmp_eq<float4>(const float4& a,
eq_mask(a.w, b.w));
}
template <>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float4 pcmp_lt<float4>(const float4& a,
const float4& b) {
return make_float4(lt_mask(a.x, b.x), lt_mask(a.y, b.y), lt_mask(a.z, b.z),
lt_mask(a.w, b.w));
}
template <>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double2
pcmp_eq<double2>(const double2& a, const double2& b) {
return make_double2(eq_mask(a.x, b.x), eq_mask(a.y, b.y));
}
template <>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE double2
pcmp_lt<double2>(const double2& a, const double2& b) {
return make_double2(lt_mask(a.x, b.x), lt_mask(a.y, b.y));
}
#endif // EIGEN_CUDA_ARCH || defined(EIGEN_HIP_DEVICE_COMPILE)
template<> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float4 plset<float4>(const float& a) {
@@ -646,6 +666,20 @@ EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE half2 pcmp_eq<half2>(const half2& a,
return __halves2half2(eq1, eq2);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE half2 pcmp_lt<half2>(const half2& a,
const half2& b) {
half true_half = half_impl::raw_uint16_to_half(0xffffu);
half false_half = half_impl::raw_uint16_to_half(0x0000u);
half a1 = __low2half(a);
half a2 = __high2half(a);
half b1 = __low2half(b);
half b2 = __high2half(b);
half eq1 = __half2float(a1) < __half2float(b1) ? true_half : false_half;
half eq2 = __half2float(a2) < __half2float(b2) ? true_half : false_half;
return __halves2half2(eq1, eq2);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE half2 pand<half2>(const half2& a,
const half2& b) {