How to efficiently normalize vector C++ - c++

Question

I want to know how to efficiently normalize a vector in C++. So far, this is what I have. Is there a way to make it more efficient and / or do it in a single pass.
std::array<float, MyClass::FEATURE_LENGTH> MyClass::normalize(const std::array<float, FEATURE_LENGTH>& arr) {
std::array<float, MyClass::FEATURE_LENGTH> output{};
double mod = 0.0;
for (size_t i = 0; i < arr.size(); ++i) {
mod += arr[i] * arr[i];
}
double mag = std::sqrt(mod);
if (mag == 0) {
throw std::logic_error("The input vector is a zero vector");
}
for (size_t i = 0; i < arr.size(); ++i) {
output[i] = arr[i] / mag;
}
return output;
}

Answer 1

There are many ways to optimize implementations of this algorithm, depending on the particulars of your problem.
For all of your loops, you can use SIMD vectorization to increase throughput.
If your vectors are very wide then you can use multiple threads to compute the magnitude. Each would compute a partial sum, then some serial code would collect the results.
You can work entirely in floats, rather than doubles, if your values are within range.
You can compute the inverse square root of the magnitude by using intrinsics (such as RSQRTSS on x86) or using Quake's method if such intrinsics are unavailable. Then you would scale by that value.
Additionally, you can get much faster code by fusing operations with the normalization. Say you want to add two vectors and normalize the result. You can compute their sum and their magnitude in a single pass and then scale in a second.

Answer 2

How can you do it in a single pass. It is obvious than you need to compute mag using all items and that you must have compute it before updating items?
As it might more take to do a division than a multiplication, one possible optimization would be to add:
double mag_inv = 1.0 / mag;
Then you could multiply items like that:
output[i] = arr[i] * mag_inv;
If there is a relatively high probability that a vector is already normalized, you might want to check if mag is equal to 1.0.

Answer 3

In case, if someone needs it here's an example of SIMD vectorization code:
#include <immintrin.h> //header for SIMD functions
void Normalize(const float lpInput[4], float lpOutput[4]) {
__m128 vInput = _mm_load_ps(lpInput); // load input vector (x, y, z, a)
__m128 vSquared = _mm_mul_ps(vInput, vInput); // square the input values
__m128 vHalfSum = _mm_hadd_ps(vSquared, vSquared);
__m128 vSum = _mm_hadd_ps(vHalfSum, vHalfSum); // compute the sum of values
float fInvSqrt; _mm_store_ss(&fInvSqrt, _mm_rsqrt_ss(vSum)); // compute the inverse sqrt
__m128 vNormalized = _mm_mul_ps(vInput, _mm_set1_ps(fInvSqrt)); // normalize the input vector
_mm_store_ps(lpOutput, vNormalized); // store normalized vector (x, y, z, a)
}
In order to compile it properly you'll need to enable SSE and AVX instructions in compiler options (-msse -mavx for gcc or clang || /arch:sse /arch:avx for msvc)