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

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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
This commit is contained in:
Rasmus Munk Larsen
2019-11-15 17:09:46 -08:00
parent 2cb2915f90
commit f1e8307308
4 changed files with 109 additions and 72 deletions
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6
test/packetmath.cpp
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@@ -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) {