I didn't want to be limited to a fixed size integer and making lists of similar commands with hardcoded constants, so I developed a C++11 solution which makes use of template metaprogramming to generate the functions and the constants. The assembly code generated with -O3 seems as tight as it can get without using BMI:
andl $0x55555555, %eax
movl %eax, %ecx
shrl %ecx
orl %eax, %ecx
andl $0x33333333, %ecx
movl %ecx, %eax
shrl $2, %eax
orl %ecx, %eax
andl $0xF0F0F0F, %eax
movl %eax, %ecx
shrl $4, %ecx
orl %eax, %ecx
movzbl %cl, %esi
shrl $8, %ecx
andl $0xFF00, %ecx
orl %ecx, %esi
TL;DR source repo and live demo.
Implementation
Basically every step in the morton1 function works by shifting and adding to a sequence of constants which look like this:
0b0101010101010101 (alternate 1 and 0)0b0011001100110011 (alternate 2x 1 and 0)0b0000111100001111 (alternate 4x 1 and 0)0b0000000011111111 (alternate 8x 1 and 0)
If we were to use D dimensions, we would have a pattern with D-1 zeros and 1 one. So to generate these it's enough to generate consecutive ones and apply some bitwise or:
/// @brief Generates 0b1...1 with @tparam n ones
template <class T, unsigned n>
using n_ones = std::integral_constant<T, (~static_cast<T>(0) >> (sizeof(T) * 8 - n))>;
/// @brief Performs `@tparam input | (@tparam input << @tparam width` @tparam repeat times.
template <class T, T input, unsigned width, unsigned repeat>
struct lshift_add :
public lshift_add<T, lshift_add<T, input, width, 1>::value, width, repeat - 1> {
};
/// @brief Specialization for 1 repetition, just does the shift-and-add operation.
template <class T, T input, unsigned width>
struct lshift_add<T, input, width, 1> : public std::integral_constant<T,
(input & n_ones<T, width>::value) | (input << (width < sizeof(T) * 8 ? width : 0))> {
};
Now that we can generate the constants at compile time for arbitrary dimensions with the following:
template <class T, unsigned step, unsigned dimensions = 2u>
using mask = lshift_add<T, n_ones<T, 1 << step>::value, dimensions * (1 << step), sizeof(T) * 8 / (2 << step)>;
With the same type of recursion, we can generate functions for each of the steps of the algorithm x = (x | (x >> K)) & M:
template <class T, unsigned step, unsigned dimensions>
struct deinterleave {
static T work(T input) {
input = deinterleave<T, step - 1, dimensions>::work(input);
return (input | (input >> ((dimensions - 1) * (1 << (step - 1))))) & mask<T, step, dimensions>::value;
}
};
// Omitted specialization for step 0, where there is just a bitwise and
It remains to answer the question "how many steps do we need?". This depends also on the number of dimensions. In general, k steps compute 2^k - 1 output bits; the maximum number of meaningful bits for each dimension is given by z = sizeof(T) * 8 / dimensions, therefore it is enough to take 1 + log_2 z steps. The problem is now that we need this as constexpr in order to use it as a template parameter. The best way I found to work around this is to define log2 via metaprogramming:
template <unsigned arg>
struct log2 : public std::integral_constant<unsigned, log2<(arg >> 1)>::value + 1> {};
template <>
struct log2<1u> : public std::integral_constant<unsigned, 0u> {};
/// @brief Helper constexpr which returns the number of steps needed to fully interleave a type @tparam T.
template <class T, unsigned dimensions>
using num_steps = std::integral_constant<unsigned, log2<sizeof(T) * 8 / dimensions>::value + 1>;
And finally, we can perform one single call:
/// @brief Helper function which combines @see deinterleave and @see num_steps into a single call.
template <class T, unsigned dimensions>
T deinterleave_first(T n) {
return deinterleave<T, num_steps<T, dimensions>::value - 1, dimensions>::work(n);
}
