2. Simplify handling of special cases by taking advantage of the fact that the
builtin vrsqrt approximation handles negative, zero and +inf arguments correctly.
This speeds up the SSE and AVX implementations by ~20%.
3. Make the Newton-Raphson formula used for rsqrt more numerically robust:
Before: y = y * (1.5 - x/2 * y^2)
After: y = y * (1.5 - y * (x/2) * y)
Forming y^2 can overflow for very large or very small (denormalized) values of x, while x*y ~= 1. For AVX512, this makes it possible to compute accurate results for denormal inputs down to ~1e-42 in single precision.
4. Add a faster double precision implementation for Knights Landing using the vrsqrt28 instruction and a single Newton-Raphson iteration.
Benchmark results: https://bitbucket.org/snippets/rmlarsen/5LBq9o
Add a new EIGEN_HAS_INTRINSIC_INT128 macro, and use this instead of __SIZEOF_INT128__. This fixes related issues with TensorIntDiv.h when building with Clang for Windows, where support for 128-bit integer arithmetic is advertised but broken in practice.
The errors were introduced by this commit : d38e6fbc27
After the above mentioned commit, some of the tests started failing with the following error
```
Building HIPCC object unsupported/test/CMakeFiles/cxx11_tensor_reduction_gpu_5.dir/cxx11_tensor_reduction_gpu_5_generated_cxx11_tensor_reduction_gpu.cu.o
In file included from /home/rocm-user/eigen/unsupported/test/cxx11_tensor_reduction_gpu.cu:16:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/Tensor:29:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/../SpecialFunctions:70:
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/SpecialFunctionsHalf.h:28:22: error: call to 'erf' is ambiguous
return Eigen::half(Eigen::numext::erf(static_cast<float>(a)));
^~~~~~~~~~~~~~~~~~
/home/rocm-user/eigen/unsupported/test/../../Eigen/src/Core/MathFunctions.h:1600:7: note: candidate function [with T = float]
float erf(const float &x) { return ::erff(x); }
^
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/SpecialFunctionsImpl.h:1897:5: note: candidate function [with Scalar = float]
erf(const Scalar& x) {
^
In file included from /home/rocm-user/eigen/unsupported/test/cxx11_tensor_reduction_gpu.cu:16:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/Tensor:29:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/../SpecialFunctions:75:
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/arch/GPU/GpuSpecialFunctions.h:87:23: error: call to 'erf' is ambiguous
return make_double2(erf(a.x), erf(a.y));
^~~
/home/rocm-user/eigen/unsupported/test/../../Eigen/src/Core/MathFunctions.h:1603:8: note: candidate function [with T = double]
double erf(const double &x) { return ::erf(x); }
^
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/SpecialFunctionsImpl.h:1897:5: note: candidate function [with Scalar = double]
erf(const Scalar& x) {
^
In file included from /home/rocm-user/eigen/unsupported/test/cxx11_tensor_reduction_gpu.cu:16:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/Tensor:29:
In file included from /home/rocm-user/eigen/unsupported/Eigen/CXX11/../SpecialFunctions:75:
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/arch/GPU/GpuSpecialFunctions.h:87:33: error: call to 'erf' is ambiguous
return make_double2(erf(a.x), erf(a.y));
^~~
/home/rocm-user/eigen/unsupported/test/../../Eigen/src/Core/MathFunctions.h:1603:8: note: candidate function [with T = double]
double erf(const double &x) { return ::erf(x); }
^
/home/rocm-user/eigen/unsupported/Eigen/CXX11/../src/SpecialFunctions/SpecialFunctionsImpl.h:1897:5: note: candidate function [with Scalar = double]
erf(const Scalar& x) {
^
3 errors generated.
```
This PR fixes the compile error by removing the "old" implementation for "erf" (assuming that the "new" implementation is what we want going forward. from a GPU point-of-view both implementations are the same).
This PR also fixes what seems like a cut-n-paste error in the aforementioned commit
- Split SpecialFunctions files in to a separate BesselFunctions file.
In particular add:
- Modified bessel functions of the second kind k0, k1, k0e, k1e
- Bessel functions of the first kind j0, j1
- Bessel functions of the second kind y0, y1
The fixes needed are
* adding EIGEN_DEVICE_FUNC attribute to a couple of funcs (else HIPCC will error out when non-device funcs are called from global/device funcs)
* switching to using ::<math_func> instead std::<math_func> (only for HIPCC) in cases where the std::<math_func> is not recognized as a device func by HIPCC
* removing an errant "j" from a testcase (don't know how that made it in to begin with!)
