1. Fix a bug in psqrt and make it return 0 for +inf arguments.

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

1 files changed, 4 insertions, 2 deletions

diff --git a/test/packetmath.cpp b/test/packetmath.cpp
index d652082b0..64dd3dbf6 100644
--- a/test/packetmath.cpp
+++ b/test/packetmath.cpp
@@ -605,9 +605,8 @@ template<typename Scalar,typename Packet> void packetmath_real()
   }
 
   if(internal::random<float>(0,1)<0.1f)
-    data1[internal::random<int>(0, PacketSize)] = 0;
+     data1[internal::random<int>(0, PacketSize)] = 0;
   CHECK_CWISE1_IF(PacketTraits::HasSqrt, std::sqrt, internal::psqrt);
-  CHECK_CWISE1_IF(PacketTraits::HasSqrt, Scalar(1)/std::sqrt, internal::prsqrt);
   CHECK_CWISE1_IF(PacketTraits::HasLog, std::log, internal::plog);
   CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i0, internal::pbessel_i0);
   CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i0e, internal::pbessel_i0e);
@@ -616,6 +615,9 @@ template<typename Scalar,typename Packet> void packetmath_real()
   CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_j0, internal::pbessel_j0);
   CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_j1, internal::pbessel_j1);
 
+  data1[0] = std::numeric_limits<Scalar>::infinity();
+  CHECK_CWISE1_IF(PacketTraits::HasRsqrt, Scalar(1)/std::sqrt, internal::prsqrt);
+
   // Use a smaller data range for the positive bessel operations as these
   // can have much more error at very small and very large values.
   for (int i=0; i<size; ++i) {