Implementation below is 4x faster than std::pow with relatively higher tolerance (0.000001) on an AVX-512 CPU. It is made of vertically auto-vectorizable loops for every basic operation like multiplication and division so that it computes 8,16,32 elements at once instead of horizontally vectorizing the Newton-Raphson loop.
#include <cmath>
/*
Newton-Raphson iterative solution
f_err(x) = x*x*x - N
f'_err(x) = 3*x*x
x = x - (x*x*x - N)/(3*x*x)
x = x - (x - N/(x*x))/3 <--- repeat until error < tolerance
but with vertical-parallelization
*/
template < typename Type, int Simd, int inverseTolerance>
inline
void cubeRootFast(Type *const __restrict__ data,
Type *const __restrict__ result) noexcept
{
// alignment 64 required for AVX512 vectorization
alignas(64)
Type xd[Simd];
alignas(64)
Type resultData[Simd];
alignas(64)
Type xSqr[Simd];
alignas(64)
Type nDivXsqr[Simd];
alignas(64)
Type diff[Simd];
// cube root checking mask
for (int i = 0; i < Simd; i++)
{
xd[i] = data[i] <= Type(0.000001);
}
// skips division by zero if input is zero or close to zero
for (int i = 0; i < Simd; i++)
{
resultData[i] = xd[i] ? Type(1.0) : data[i];
}
// Newton-Raphson Iterations in parallel
bool work = true;
while (work)
{
// compute x*x
for (int i = 0; i < Simd; i++)
{
xSqr[i] = resultData[i] *resultData[i];
}
// compute N/(x*x)
for (int i = 0; i < Simd; i++)
{
nDivXsqr[i] = data[i] / xSqr[i];
}
// compute x - N/(x*x)
for (int i = 0; i < Simd; i++)
{
nDivXsqr[i] = resultData[i] - nDivXsqr[i];
}
// compute (x-N/(x*x))/3
for (int i = 0; i < Simd; i++)
{
nDivXsqr[i] = nDivXsqr[i] / Type(3.0);
}
// compute x - (x-N/(x*x))/3
for (int i = 0; i < Simd; i++)
{
diff[i] = resultData[i] - nDivXsqr[i];
}
// compute error
for (int i = 0; i < Simd; i++)
{
diff[i] = resultData[i] - diff[i];
}
// compute absolute error
for (int i = 0; i < Simd; i++)
{
diff[i] = std::abs(diff[i]);
}
// compute condition to stop looping (error < tolerance)?
for (int i = 0; i < Simd; i++)
{
diff[i] = diff[i] > Type(1.0/inverseTolerance);
}
// all SIMD lanes have to have zero work left to end
Type check = 0;
for (int i = 0; i < Simd; i++)
{
check += diff[i];
}
work = (check > Type(0.0));
// compute the next x guess
for (int i = 0; i < Simd; i++)
{
resultData[i] = resultData[i] - nDivXsqr[i];
}
}
// if input was close to zero, output zero
// output result otherwise
for (int i = 0; i < Simd; i++)
{
result[i] = xd[i] ? Type(0.0) : resultData[i];
}
}
#include <iostream>
int main()
{
constexpr int n = 8192;
constexpr int simd = 16;
constexpr int inverseTolerance = 1000;
float data[n];
for (int i = 0; i < n; i++)
{
data[i] = i;
}
for (int i = 0; i < n; i += simd)
{
cubeRootFast<float, simd, inverseTolerance> (data + i, data + i);
}
for (int i = 0; i < 10; i++)
std::cout << data[i *i *i] << std::endl;
return 0;
}
It is tested only with GCC so it may require extra MSVC-pragmas on each loop to force auto-vectorization. If you have OpenMP, then you can also use #pragma omp simd safelen(Simd)
to achieve same thing.
The performance only holds within [0,1] range. To use bigger values, you should use range reduction like this:
// example: max value is 1000
for(auto & input:inputs)
input = input/1000.0f // normalize
for(..)
cubeRootFast<float, simd, inverseTolerance> (input + i, input + i)
for(auto & input:inputs)
input = 10.0f*input // de-normalize (1000 = 10 x 10 x 10)
If you need only 0.005 error on low-range like [0,1000] with 16x speedup, you can try below implementation that uses polynomial approximation (Horner-Scheme is applied to compute with FMA instructions and no explicit auto-vectorization required since it doesn't include any branching/loop inside):
// optimized for [0,1] range: ~1 cycles on AVX512, 0.003 average error
// polynomial approximation with Horner Scheme for FMA optimization
template<typename T>
T cubeRootFast(T x)
{
T xd = x-T(1.0);
T result = T(-55913.0/4782969.0);
result *= xd;
result += T(21505.0/1594323.0);
result *= xd;
result += T(-935.0/59049.0);
result *= xd;
result += T(374.0/19683.0);
result *= xd;
result += T(-154.0/6561.0);
result *= xd;
result += T(22.0/729.0);
result *= xd;
result += T(-10.0/243.0);
result *= xd;
result += T(5.0/81.0);
result *= xd;
result += T(-1.0/9.0);
result *= xd;
result += T(1.0/3.0);
result *= xd;
result += T(1.0);
return result;
}
// range reduction + dereduction: ~ 1 cycles on AVX512
for(int i=0;i<8192;i++)
{
float inp = input[i];
// scaling + descaling for range [1,999]
float scaling = (inp>333.0f)?(1000.0f):(333.0f);
scaling = (inp>103.0f)?scaling:(103.0f);
scaling = (inp>29.0f)?scaling:(29.0f);
scaling = (inp>7.0f)?scaling:(7.0f);
scaling = (inp>3.0f)?scaling:(3.0f);
output[i] = powf(scaling,0.33333333333f)*cubeRootFast<float>(inp/scaling);
}