# Computation of the hard special leaves

The combinatorial type prime counting algorithms
([Lagarias-Miller-Odlyzko](https://www.ams.org/journals/mcom/1985-44-170/S0025-5718-1985-0777285-5/S0025-5718-1985-0777285-5.pdf),
[Deleglise-Rivat](https://www.ams.org/journals/mcom/1996-65-213/S0025-5718-96-00674-6/S0025-5718-96-00674-6.pdf),
[Gourdon](http://numbers.computation.free.fr/Constants/Primes/Pix/piNalgorithm.ps))
consist of many formulas and the formula that usually takes longest to compute and
is by far the most difficult to implement is the formula of the so-called hard special leaves.
Unlike the [easy special leaves](https://github.com/kimwalisch/primecount/blob/master/doc/Easy-Special-Leaves.md),
which can be computed in $O(1)$ using a $\pi(x)$ lookup table, the computation of
the hard special leaves requires evaluating the partial sieve function $\phi(x, a)$ which
generally cannot be computed in $O(1)$.

[primecount's implementation](https://github.com/kimwalisch/primecount/blob/master/src/deleglise-rivat/S2_hard.cpp)
of the hard special leaves formula is different from the algorithms that have
been described in any of the combinatorial prime counting papers so far. This document
describes the history of how primecount's implementation came to be and it describes
an alternative counting method that I have devised in February 2020. This alternative
counting method improves the balancing of sieve and count operations in the hard special
leaf algorithm and thereby improves its runtime complexity by a factor of at least
$O(\log{\log{x}})$. The alternative counting method uses a new tree-like data structure with
fewer than $O(\log{z})$ levels (tree depth) where each node has $O(z^{\frac{1}{levels}})$ children.
This data structure is used instead of the binary indexed tree (a.k.a. Fenwick tree)
that has been used for counting in all previously published papers about the
combinatorial type prime counting algorithms.

## Basic algorithm

Implementing the hard special leaves formula requires use of a prime sieve. The algorithm
is basically a modified version of the well known
[segmented sieve of Eratosthenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes)
which consists of two main parts that are executed alternately:

1) Sieve out primes and multiples of primes.
2) Count the number of unsieved elements in the sieve array.

## Speed up counting using binary indexed tree

Since there is a large number of leaves for which we have to count the number of
unsieved elements in the sieve array Lagarias-Miller-Odlyzko [[1]](#References)
have suggested using a [binary indexed tree](https://en.wikipedia.org/wiki/Fenwick_tree)
data structure (a.k.a. Fenwick tree) to speedup counting.
For any number $n$ the binary indexed tree allows to count the number of unsieved
elements ≤ $n$ using only $O(\log{n})$ operations. However the binary indexed tree
must also be updated whilst sieving which slows down the sieving part of the
algorithm by a factor of $O(\log{n}\ /\log{\log{n}})$ operations. All more recent papers about
the combinatorial type prime counting algorithms that I am aware of have also suggested
using the binary indexed tree data structure for counting the number of unsieved
elements in the sieve array.

```C++
// Count the number of unsieved elements <= pos in
// the sieve array using a binary indexed tree.
// Code from: Tomás Oliveira e Silva [4]
// Runtime: O(log n).
//
int count(const int* tree, int pos)
{
  int sum = tree[pos++];
  while (pos &= pos - 1)
      sum += tree[pos - 1];
  return sum;
}
```

## Alternative counting method

Despite the theoretical benefits of the binary indexed tree data structure, it
has two significant practical drawbacks:
it uses a lot of memory since each thread needs to allocate its own binary indexed
tree and, more importantly, it is very slow in practice because all memory accesses
are non-sequential, which CPUs are notoriously bad at. For this reason many
programmers that have implemented any of the combinatorial prime counting algorithms
([Christian Bau 2003](http://cs.swan.ac.uk/~csoliver/ok-sat-library/OKplatform/ExternalSources/sources/NumberTheory/ChristianBau/),
[Dana Jacobsen 2013](https://github.com/danaj/Math-Prime-Util), 
[James F. King 2014](https://github.com/jfkingiii/meissel-lehmer),
[Kim Walisch 2014](https://github.com/kimwalisch/primecount)) have avoided using
the binary indexed tree and implemented something else. The method that has turned
out to perform best so far is to get rid of the binary indexed tree data structure,
which speeds up the sieving part of the algorithm by a factor of $O(\log{n}\ /\log{\log{n}})$
and count the number of unsieved elements by simply iterating over the sieve array.

There are many known optimizations that can be used to speedup counting:

* Using the [POPCNT instruction](https://en.wikipedia.org/wiki/SSE4#POPCNT_and_LZCNT)
  in combination with a bit sieve array allows counting many unsieved sieve
  elements using a single instruction. This improves the runtime complexity by a
  large constant factor.
* One can keep track of the total number of unsieved elements that are
  currently in the sieve array as the total number of unsieved elements is
  [used frequently](https://github.com/kimwalisch/primecount/blob/v5.3/src/lmo/pi_lmo5.cpp#L107)
  (once per sieving prime in each segment).
* We can [batch count](#Batch-counting) the number of unsieved elements in the
  sieve array for many consecutive leaves i.e. instead of starting to count from
  the beginning of the sieve array for each leaf we resume counting from the last
  sieve index of the previous leaf.

```C++
/// Count the number of unsieved elements (1 bits) in
/// the sieve array using the POPCNT instruction.
uint64_t count(const uint64_t* sieve, uint64_t startIndex, uint64_t stopIndex)
{
  uint64_t cnt = 0;

  for (uint64_t i = startIndex; i < stopIndex; i++)
    cnt += __builtin_popcountll(sieve[i]);

  return cnt;
}
```

This alternative method with the 3 optimizations described above was used in
primecount up to version 5.3 (January 2020). According to my benchmarks
this method runs up to 3x faster than an implementation that uses the binary
indexed tree data structure. But there is a problem: all of the alternative algorithms that avoid using the
binary indexed tree data structure have a worse runtime complexity. But why are
they faster then? Mainly for two reasons: most memory accesses in the alternative
algorithms are sequential and the ability to count many unsieved elements using
a single instruction improves the runtime complexity by a large constant factor,
in primecount this constant factor is 240. OK, so despite the worse runtime
complexity everything is great?! Unfortunately not, primecount's implementation of
the hard special leaves formula which used the alternative method described above
starts scaling badly above $10^{23}$. For record computations this became a serious
issue e.g. I had initially expected our computation of $\pi(10^{28})$ to take
about 8 months, however it ended up taking 2.5 years!

## Improved alternative counting method

The scaling issue is caused by our change to count the number of unsieved elements
by simply iterating over the sieve array. When there are many consecutive leaves
that are close to each other then simply iterating over the sieve array for counting
the number of unsieved elements works great. However when there are very few
consecutive leaves in the current segment our method becomes inefficient. In order
to compute the special leaves we need to sieve up to some limit $z$. Since we are
using a modified version of the segmented sieve of Eratosthenes the size of the
sieve array will be $O(\sqrt{z})$. This means that if e.g. there is a single leaf in the
current segment we will use $O(\sqrt{z})$ operations to count the number of unsieved
elements in the sieve array whereas the binary indexed tree would have used only
$O(\log{z})$ operations. This is too much, this deteriorates the runtime complexity
of the algorithm.

<img src="https://raw.githubusercontent.com/kimwalisch/primecount/gh-pages/images/tree_2_counter_levels.svg">

So now that we have identified the problem, we can think about whether it is possible
to further improve counting by more than a constant factor in our alternative algorithm.
It turns out this is possible and even relatively simple to implement: We add a
counter array to our sieving algorithm. The counter array has a size of
$O(\sqrt{segment\ size})$, with $segment\ size = \sqrt{z}$. Each element of the counter array
contains the current count of unsieved elements in the sieve array for the interval
$[i\times \sqrt{segment\ size}, (i+1)\times \sqrt{segment\ size}[$. Similar to the algorithm with the
binary indexed tree data structure this counter array must be updated whilst sieving
i.e. whenever an element is crossed off for the first time in the sieve array we need
to decrement the corresponding counter element. However since we only need to decrement
at most 1 counter when crossing off an element in the sieve array this does not
deteriorate the sieving runtime complexity of the algorithm (unlike the binary indexed
tree which deteriorates sieving by a factor of $\log{z}\ /\log{\log{z}}$). I have to give credit
to Christian Bau here who already used such a counter array back in 2003, however he
chose a counter array size of $O(segment\ size)$ with a constant interval size which does
not improve the runtime complexity.

```C++
// Sieve out a bit from the sieve array and update the
// counter array if the bit was previously 1
is_bit = (sieve[i] >> bit_index) & 1;
sieve[i] &= ~(1 << bit_index);
counter[i >> counter_log2_dist] -= is_bit;
```

Now whenever we need to count the number of unsieved elements in the sieve array,
we can quickly iterate over the new counter array and sum the counts. We do this
until we are close < $O(\sqrt{segment\ size})$ to the limit up to which we need to count.
Once we are close, we switch to our old counting method: we simply iterate
over the sieve array and count the number of unsieved elements using the POPCNT
instruction. With this modification we improve the runtime complexity for counting
the number of unsieved elements for a single leaf from $O(segment\ size)$ to
$O(\sqrt{segment\ size})$.

```C++
/// Count 1 bits inside [0, stop]
uint64_t Sieve::count(uint64_t stop)
{
  uint64_t start = prev_stop_ + 1;
  prev_stop_ = stop;

  // Quickly count the number of unsieved elements (in
  // the sieve array) up to a value that is close to
  // the stop number i.e. (stop - start) < segment_size^(1/2).
  // We do this using the counter array, each element
  // of the counter array contains the number of
  // unsieved elements in the interval:
  // [i * counter_dist, (i + 1) * counter_dist[.
  while (counter_start_ + counter_dist_ <= stop)
  {
    counter_sum_ += counter_[counter_i_++];
    counter_start_ += counter_dist_;
    start = counter_start_;
    count_ = counter_sum_;
  }

  // Here the remaining distance is relatively small i.e.
  // (stop - start) < counter_dist, hence we simply
  // count the remaining number of unsieved elements by
  // linearly iterating over the sieve array.
  count_ += count(start, stop);
  return count_;
}
```

Initially, when I found this improvement, I thought it would fix my particular
scaling issue only up to some large threshold above which the alternative method
would become inefficient again due to its worse runtime complexity. I thought that
the alternative counting method had a runtime complexity of about
$O(number\ of\ special\ leaves\times \sqrt{segment\ size})$ since counting the number of unsieved elements
for a single leaf is $O(\sqrt{segment\ size}$). However when I measured the average
number of count operations per leaf the number was much lower than expected. It turns
out that [batch counting](#Batch-counting) the number of unsieved
elements for many consecutive leaves improves the runtime complexity by more than a
constant factor. When I implemented the above alternative counting method in
primecount it completely fixed the severe scaling issue in the computation of the
special leaves that had been present in primecount since the very beginning.
Below $10^{20}$ there are no performance improvements, however above $10^{20}$, the higher
you go the more efficient the new method becomes compared to primecount's old
implementation. At $10^{25}$ the new method is already 2x faster. Note that the new
method works best with the Deleglise-Rivat [[2]](#References) and
Gourdon [[3]](#References) variants of the combinatorial prime counting algorithm as
the average distance between consecutive special leaves is relatively large in those
algorithms. In the Lagarias-Miller-Odlyzko [[1]](#References) algorithm the average
distance between consecutive special leaves is much smaller, so there the new counting
method will not improve performance in practice.

## Gradually increase counter distance

So far we have focused on improving counting for the case
when there are very few leaves per segment which are far away from each other.
Generally there is a very large number of leaves that are close to each other
at the beginning of the sieving algorithm, and gradually as we sieve up the leaves
become sparser and the distance between the leaves increases. So what we can do is,
start with a counter array whose elements span over small intervals and
then gradually increase the interval size. We can update the counter size and distance
e.g. at the start of each new segment as the counter needs to be reinitialized at the
start of each new segment anyway. The ideal counter distance for the next segment is
$\sqrt{average\ leaf\ distance}$. In practice we can approximate the average leaf
distance using $\sqrt{segment\ low}$. My measurements using primecount indicate that
gradually increasing the counter distance further improves counting by a small factor.
This optimization is primarily useful when using a very small number of counter levels
e.g. 2.

```C++
// Ideally each element of the counter array
// should represent an interval of size:
// min(sqrt(average_leaf_dist), sqrt(segment_size))
// Also the counter distance should be regularly
// adjusted whilst sieving.
//
void Sieve::allocate_counter(uint64_t segment_low)
{
  uint64_t average_leaf_dist = sqrt(segment_low);
  counter_dist_ = sqrt(average_leaf_dist);
  counter_dist_ = nearest_power_of_2(counter_dist_);
  counter_log2_dist_ = log2(counter_dist_);

  uint64_t counter_size = (sieve_size_ / counter_dist_) + 1;
  counter_.resize(counter_size);
}
```

## Multiple levels of counters

It is also possible to use multiple levels of counters, in this case the data
structure becomes a tree where each node has $O(segment\ size^{\frac{1}{levels}})$ children
(*levels* corresponds to the tree depth) and each node stores the current number of unsieved elements in an interval of
size $O(segment\ size^{\frac{levels - level}{levels}})$ from the sieve array. The last
level of this tree corresponds to the sieve array used in the algorithm. Just
like in the original algorithm with the binary index tree (a.k.a Fenwick tree),
we can count the number of unsieved elements ≤ $n$ in the sieve array using the
new tree-like data structure by traversing the tree from the root node to the
bottom and summing the current number of unsieved elements stored in each node.

As an example, let's consider the case of 3 counter levels for which we will need to
use 3 - 1 = 2 counter arrays. We only need to use 2 counter arrays because for the
last level we will count the number of unsieved elements by iterating over the sieve
array. For each level the size of the counter array can be calculated using
$segment\ size^{\frac{level}{levels}}$ and the interval size of the counter array's elements
can be calculated using $segment\ size^{\frac{levels - level}{levels}}$. Hence our first
counter array (1st level) is coarse-grained, its elements span over large intervals
of size $O(segment\ size^{\frac{2}{3}})$. This means that each element of the first counter
array contains the current number of unsieved elements in the interval
$[i\times segment\ size^{\frac{2}{3}}, (i+1)\times segment\ size^{\frac{2}{3}}[$. Our second counter array (2nd
level) is more fine-grained, its elements span over smaller intervals of size
$O(segment\ size^{\frac{1}{3}})$. Hence each element of the second counter array contains the
current number of unsieved elements in the interval
$[i\times segment\ size^{\frac{1}{3}}, (i+1)\times segment\ size^{\frac{1}{3}}[$. Now when we need to count the
number of unsieved elements ≤ $n$, we first iterate over the first counter array and
sum the counts of its elements. Once the remaining distance becomes < $segment\ size^{\frac{2}{3}}$,
we switch to our second counter array and sum the counts of its elements until the
remaining distance becomes < $segment\ size^{\frac{1}{3}}$. When this happens, we count the
remaining unsieved elements ≤ $n$ by simply iterating over the sieve array. Below is a
graphical representation of 3 counter levels with a segment size of 8 (last level).
At each level we perform at most $segment\ size^{\frac{1}{levels}}$ count operations to find the
number of unsieved element ≤ $n$.

<img src="https://raw.githubusercontent.com/kimwalisch/primecount/gh-pages/images/tree_3_counter_levels.svg">

Using 3 counter levels reduces the worst-case complexity for counting the number of
unsieved elements for a single special leaf to $O(3\times segment\ size^{\frac{1}{3}})$, but on the other
hand it slightly slows down sieving as we also need to update the two counter arrays
whilst sieving. I have benchmarked using 3 counter levels vs. using 2 counter levels
in primecount. When using 3 counter levels, the computation of the hard special leaves
used 6.69% more instructions at $10^{20}$, 6.55% more instructions at $10^{21}$, 5.73% more
instructions at $10^{22}$ and 5.51% more instructions at $10^{23}$. Hence for practical use,
using 2 counter levels (i.e. a single counter array) in primecount both runs faster and
uses fewer instructions. It is likely though that using 3 counter levels will use fewer
instructions for huge input numbers > $10^{27}$ since the difference of used instructions
is slowly decreasing (for larger input values) in favor of 3 counter levels.

Here is an example implementation of the counting method which supports multiple counter
levels:

```C++
/// Count 1 bits inside [0, stop]
uint64_t Sieve::count(uint64_t stop)
{
  uint64_t start = prev_stop_ + 1;
  uint64_t prev_start = start;
  prev_stop_ = stop;

  // Each iteration corresponds to one counter level
  for (Counter& counter : counters_)
  {
    // To support resuming from the previous stop number,
    // we have to update the current counter level's
    // values if the start number has been increased
    // in any of the previous levels.
    if (start > prev_start)
    {
      counter.start = start;
      counter.sum = count_;
    }

    // Sum counts until remaining distance < segment_size^((levels - level) / levels),
    // this uses at most segment_size^(1/levels) iterations.
    while (counter.start + counter.dist <= stop)
    {
      counter.sum += counter[counter.start >> counter.log2_dist];
      counter.start += counter.dist;
      start = counter.start;
      count_ = counter.sum;
    }
  }

  // Here the remaining distance is very small i.e.
  // (stop - start) < segment_size^(1/levels), hence we
  // simply count the remaining number of unsieved elements
  // by linearly iterating over the sieve array.
  count_ += count(start, stop);
  return count_;
}
```

Even though using more than 2 counter levels does not seem particularly useful from a
practical point of view, it is very interesting from a theoretical point of view.
If we used $O(\log{z})$ counter levels, then the worst-case runtime complexity for counting
the number of unsieved elements for a single special leaf would be
$O(\log{z}\times segment\ size^{\frac{1}{log z}})$, which can be simplified to $O(\log{z}\times e)$ and which is
the same number of operations as the original algorithm with the binary indexed tree, which
uses $O(\log{z})$ operations. This means that using $O(\log{z})$ counter levels our alternative
algorithm has the same runtime complexity as the original algorithm with the binary indexed
tree. This leads to the following question: is it possible to use fewer than $O(\log{z})$
counter levels and thereby improve the runtime complexity of the hard special leaf
algorithm? See the [runtime complexity](#Runtime-complexity) section for more details.

## Batch counting

Whenever we have removed the i-th prime and its multiples from the sieve array in the hard
special leaf algorithm, we proceed to the counting part of the algorithm. For each hard leaf
that is composed of the (i+1)-th prime and another larger prime (or square-free number) and
that is located within the current segment we have to count the number of unsieved elements
(in the sieve array) ≤ leaf. When
[iterating over these hard leaves](https://github.com/kimwalisch/primecount/blob/v7.1/src/lmo/pi_lmo3.cpp#L84)
their locations in the sieve array steadily increase. This property can be exploited, i.e.
instead of counting the number of unsieved elements from the beginning of the sieve array for
each leaf, we resume counting from the last sieve array index of the previous hard leaf. I
call this technique batch counting as we count the number of unsieved elements for many
consecutive leaves by iterating over the sieve array only once.

Nowadays, most open source implementations of the combinatorial prime counting algorithms
use batch counting, the earliest implementation that used batch counting that I am aware of
is from [Christian Bau](http://cs.swan.ac.uk/~csoliver/ok-sat-library/OKplatform/ExternalSources/sources/NumberTheory/ChristianBau/)
and dates back to 2003. But so far there have been no publications that analyse its runtime
complexity implications and I have also not been able to figure out by how much it improves
the runtime complexity of the hard special leaf algorithm. The alternative counting methods
that are presented in this document have batch counting built-in. When I turn off batch
counting in primecount its performance deteriorates by more than 20x when computing the hard
special leaves ≤ $10^{17}$. So it is clear that batch counting is hugely important for performance.
It is likely that the use of batch counting enables using even fewer counter levels
and thereby further improves the runtime complexity of the hard special leaf algorithm. I have
run extensive benchmarks up $10^{26}$ using primecount and I found that in practice using only
two counter levels provides the best performance. So my benchmarks seem to confirm that it is
possible to use fewer than $O(\log{z}\ /\log{\log{x}})$ levels of counters using batch counting,
though I assume that using a constant number of counter levels deteriorates the runtime
complexity of the algorithm.

## Runtime complexity

What's the runtime complexity of this alternative algorithm?

When using $O(\log{z})$ counter levels the runtime complexity of the alternative algorithm is
$O(z\log{z})$ operations which is the same runtime complexity as the original algorithm with
the binary indexed tree, see [here](#multiple-levels-of-counters) for more information. The
next interesting question is: is it possible to use fewer than $O(\log{z})$ counter levels and
thereby improve the runtime complexity of the hard special leaf algorithm?

Tomás Oliveira e Silva's paper [[4]](#References) provides the following runtime
complexities for the computation of the hard special leaves in the Deléglise-Rivat
algorithm: sieving uses $O(z\log{z})$ operations, the number of hard special leaves is
$O(z\ /\log^{2}{x}\ \log{\alpha})$ and for each leaf it takes $O(\log{z})$ operations to count the number
of unsieved elements. This means that the original algorithm is not perfectly balanced,
sieving is slightly more expensive than counting. Using the alternative algorithm, it is
possible to achieve perfect balancing by using fewer than $O(\log{z})$ levels of counters, if
the number of counter levels is decreased sieving becomes more efficient but on the other
hand counting becomes more expensive. The maximum number of allowed counting operations per
leaf that do not deteriorate the runtime complexity of the algorithm is slightly larger
than $O(\log^{2}{x})$. This bound can be achieved by using $O(\log{z}\ /\log{\log{x}})$ levels of
counters, if we set the number of counter levels $l = \log{z}\ /\log{\log{x}}$, then the number of
count operations per leaf becomes $O(l\times \sqrt[l]{z})$ which is smaller than
$O(\log^{2}{x})$ since:

$l\times \sqrt[l]{z} < \log^{2}{x}$  
$\Leftrightarrow \log{z}\ /\log{\log{x}}\times z^{\log{\log{x}}/\log{z}} < \log^{2}{x}$  
$\Leftrightarrow \log{z}\ /\log{\log{x}}\times \sqrt[\log{z}]{z}^{\log{\log{x}}} < \log^{2}{x}$  
$\Leftrightarrow \log{z}\ /\log{\log{x}}\times e^{\log{\log{x}}} < \log^{2}{x}$  
$\Leftrightarrow \log{z}\ /\log{\log{x}}\times \log{x} < \log^{2}{x}$  

Hence, by using $O(\log{z}\ /\log{\log{x}})$ levels of counters we improve the balancing of sieve
and count operations and reduce the runtime complexity of the hard special leaf algorithm
by a factor of $O(\log{\log{x}})$ to $O(z\ \log{z}\ /\log{\log{x}})$ operations. In the original
Deléglise-Rivat paper [[2]](#References) the number of hard special leaves is indicated
as $O(\pi(\sqrt[4]{x})\times y)$, which is significantly smaller than in Tomás Oliveira's
version of the algorithm [[4]](#References). This lower number of hard special leaves makes
it possible to use even fewer counter levels and further improve the runtime complexity of
the algorithm, here we can use only $O(\log{\log{z}})$ counter levels which reduces the runtime
complexity of the algorithm to $O(z\log{\log{z}})$ operations.

## Open questions

The alternative counting methods presented in this document have batch counting built-in, but
as mentioned in the [Batch counting](#Batch-counting) paragraph I don't know whether the use
of batch counting enables using fewer than $O(\log{z}\ /\log{\log{x}})$ counter levels and thereby
further improves the runtime complexity of the hard special leaf algorithm. Ideally, we want
to use only $O(\log{\log{z}})$ counter levels in which case the runtime complexity of the hard special
leaf algorithm would be $O(z\log{\log{z}})$ operations, provided that the use of $O(\log{\log{z}})$
counter levels does not deteriorate the runtime complexity of the algorithm.

There is one last trick that I am aware of that would likely further improve the runtime
complexity of the hard special leaf algorithm: the distribution of the hard special leaves is
highly skewed, most leaves are located at the beginning of the sieving algorithm and as we
sieve up the leaves become sparser and the distance between consecutive leaves increases. In
the [Runtime complexity](#Runtime-complexity) paragraph I have suggested using a fixed number
of counter levels for the entire computation. But this is not ideal, we can further improve
the balancing of sieve and count operations by adjusting the number of counter levels for each
segment. However, I don't know how to calculate the optimal number of counter levels for the
individual segments and I also don't know how much this would improve the runtime complexity.

## Appendix

* In the original Deléglise-Rivat paper [[2]](#References) its authors indicate that the use
  of a binary indexed tree deteriorates the sieving part of the hard special leaf algorithm by
  a factor of $O(\log{z})$ to $O(z\times \log{z}\times \log{\log{z}})$ operations. Tomás Oliveira e Silva in
  [[4]](#References) rightfully points out that this is incorrect and that it only
  deteriorates the sieving part of the algorithm by a factor of $O(\log{z}\ /\log{\log{z}})$. This is
  because we don't need to need to perform $O(\log{z})$ binary indexed tree updates for each
  elementary sieve operation, of which there are $O(z\log{\log{z}})$. But instead we only need to
  perform $O(\log{z})$ binary indexed tree updates whenever an element is crossed off for the first
  time in the sieve array. When we sieve up to $z$, there are at most $z$ elements that can be
  crossed off for the first time, therefore the runtime complexity of the sieving part of the
  hard special leaf algorithm with a binary indexed tree is $O(z\log{z})$ operations.
  
  The new alternative algorithm also relies on the above subtlety to improve the runtime
  complexity. When using multiple counter levels, the related counter arrays should only be
  updated whenever an element is crossed off for the first time in the sieve array. However,
  when using a small constant number of counter levels (e.g. ≤ 3) it may be advantages to
  always update the counter array(s) when an element is crossed off in the sieve array. This
  reduces the branch mispredictions and can significantly improve performance. See the first
  code section in [Improved alternative counting method](#improved-alternative-counting-method)
  for how to implement this.
  
* When using a bit sieve array it is advantages to count the number of unsieved elements in the
  sieve array using the POPCNT instruction. The use of the POPCNT instruction allows counting
  many unsieved elements (1-bits) using a single instruction. In primecount each POPCNT
  instruction counts the number of unsieved elements within the next 8 bytes of the sieve array
  and these 8 bytes correspond to an interval of size $8\times 30 = 240$. When using multiple counter
  levels, it is important for performance that on average the same number of count operations is
  executed on each level. However, using the POPCNT instruction dramatically reduces the number
  of count operations on the last level and hence causes a significant imbalance. To fix this
  imbalance we can multiply the counter distance for each level by
  POPCNT_distance^(level/levels). In primecount the POPCNT distance is 240, if primecount used 3
  counter levels we would multiply the counter distance of the 1st level by $\sqrt[3]{240}$ and the
  counter distance of the 2nd level by $240^{\frac{2}{3}}$).

## References

1. J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Mathematics of Computation, 44 (1985), pp. 537–560.
2. M. Deleglise and J. Rivat, "Computing pi(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method", Mathematics of Computation, Volume 65, Number 213, 1996, pp 235–245.
3. Xavier Gourdon, Computation of pi(x) : improvements to the Meissel, Lehmer, Lagarias, Miller, Odllyzko, Deléglise and Rivat method, February 15, 2001.
4. Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do DETUA, vol. 4, no. 6, March 2006, pp. 759-768.
