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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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34
Eigen/src/Core/arch/SSE/MathFunctions.h
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@@ -98,30 +98,34 @@ Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); }
template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000u);
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000u);
_EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f);
_EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f);
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000u);
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000u);
Packet4f neg_half = pmul(_x, p4f_minus_half);
// select only the inverse sqrt of positive normal inputs (denormals are
// flushed to zero and cause infs as well).
Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min);
Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x));
// Identity infinite, zero, negative and denormal arguments.
Packet4f lt_min_mask = _mm_cmplt_ps(_x, p4f_flt_min);
Packet4f inf_mask = _mm_cmpeq_ps(_x, p4f_inf);
Packet4f not_normal_finite_mask = _mm_or_ps(lt_min_mask, inf_mask);
// Fill in NaNs and Infs for the negative/zero entries.
Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps());
Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask);
Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan),
_mm_and_ps(zero_mask, p4f_inf));
// Compute an approximate result using the rsqrt intrinsic.
Packet4f y_approx = _mm_rsqrt_ps(_x);
// Do a single step of Newton's iteration.
x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));
// Do a single step of Newton-Raphson iteration to improve the approximation.
// This uses the formula y_{n+1} = y_n * (1.5 - y_n * (0.5 * x) * y_n).
// It is essential to evaluate the inner term like this because forming
// y_n^2 may over- or underflow.
Packet4f y_newton = pmul(
y_approx, pmadd(y_approx, pmul(neg_half, y_approx), p4f_one_point_five));
// Insert NaNs and Infs in all the right places.
return _mm_or_ps(x, infs_and_nans);
// Select the result of the Newton-Raphson step for positive normal arguments.
// For other arguments, choose the output of the intrinsic. This will
// return rsqrt(+inf) = 0, rsqrt(x) = NaN if x < 0, and rsqrt(x) = +inf if
// x is zero or a positive denormalized float (equivalent to flushing positive
// denormalized inputs to zero).
return pselect<Packet4f>(not_normal_finite_mask, y_approx, y_newton);
}
#else