```Issue #14383 has been updated by jzakiya (Jabari Zakiya).

Well, I did say "serious" math, didn't I.

```
2.5.0 :097 > 2**256
=> 115792089237316195423570985008687907853269984665640564039457584007913129639936

2.5.0 :099 > n = 2**256 + 1; tm{ pp n.factors }
[[1238926361552897, 1],
[93461639715357977769163558199606896584051237541638188580280321, 1]]
=> 11.187528889

2.5.0 :103 > n = 2**256 + 6; tm{ pp n.factors }
[[2, 1],
[9663703905367, 1],
[5991082217089035545953414273093775102416031327093273407023490613, 1]]
=> 181.896643599

2.5.0 :105 > n = 2**256 + 7; tm{ pp n.factors }
[[92243, 1],
[14633710594132193, 1],
[39071613028785859, 1],
[2195480924803008082289717129761953851423, 1]]
=> 86.285821283

2.5.0 :107 > n = 2**256 + 8; tm{ pp n.factors }
[[2, 3],
[3, 1],
[683, 1],
[4049, 1],
[85009, 1],
[2796203, 1],
[31797547, 1],
[81776791273, 1],
[2822551529460330847604262086149015242689, 1]]
=> 210.062944465

2.5.0 :109 > n = 2**256 + 9; tm{ pp n.factors }
[[5, 1],
[37181, 1],
[210150995838577, 1],
[2963851002430239530676411809410149856603062505748058817897, 1]]
=> 8.70362184
```

----------------------------------------
Feature #14383: Making prime_division in prime.rb Ruby 3 ready.
https://bugs.ruby-lang.org/issues/14383#change-69730

* Author: jzakiya (Jabari Zakiya)
* Status: Open
* Priority: Normal
* Assignee:
* Target version:
----------------------------------------
I have been running old code in Ruby 2.5.0 (released 2017.12.25) to check for
speed and compatibility. I still see the codebase in `prime.rb` hardly has
changed at all (except for replacing `Math.sqrt` with `Integer.sqrt`).

To achieve the Ruby 3 goal to make it at least three times faster than Ruby 2
there are three general areas where Ruby improvements can occur.

* increase the speed of its implementation at the machine level
* rewrite its existing codebase in a more efficient|faster manner
* use faster algorithms to implement routines and functions

I want to suggest how to address the later two ways to improve performance of
specifically the `prime_division` method in the `prime.rb` library.

I've raised and made suggestions to some of these issues here
[ruby-issues forum](https://bugs.ruby-lang.org/issues/12676) and now hope to invigorate additional discussion.

Hopefully with the release of 2.5.0, and Ruby 3 conceptually closer to reality,
more consideration will be given to coding and algorithmic improvements to
increase its performance too.

**Mathematical correctness**

First I'd like to raise what I consider *math bugs* in `prime_division`, in how
it handles `0` and `-1` inputs.

```
> -1.prime_division
=> [[-1,1]]

> 0.prime_division
Traceback (most recent call last):
4: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/bin/irb:11:in `<main>'
3: from (irb):85
2: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/lib/ruby/2.5.0/prime.rb:30:in `prime_division'
1: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/lib/ruby/2.5.0/prime.rb:203:in `prime_division'
ZeroDivisionError (ZeroDivisionError)
```
First, `0` is a perfectly respectable integer, and is non-prime, so its output should be `[]`,
an empty array to denote it has no prime factors. The existing behavior is solely a matter of
`prime_division`'s' implementation, and does not take this mathematical reality into account.

The output for `-1` is also mathematically wrong because `1` is also non-prime (and correctly
returns `[]`), well then mathematically so should `-1`.  Thus, `prime_division` treats `-1` as
a new prime number, and factorization, that has no mathematical basis.  Thus, for mathematical
correctness and consistency `-1` and `0` should both return `[]`, as none have prime factors.

```
> -1.prime_division
=> []

> 0.prime_division
=> []

> 1.prime_division
=> []
```
There's a very simple one-line fix to `prime_division` to do this:

```
# prime.rb

class Prime

def prime_division(value, generator = Prime::Generator23.new)
-- raise ZeroDivisionError if value == 0
++ return [] if (value.abs | 1) == 1
```

**Simple Code and Algorithmic Improvements**

As stated above, besides the machine implementation improvements, the other
areas of performance improvements will come from coding rewrites and better
algorithms. Below is the coding of `prime_division`. This coding has existed at
least since Ruby 2.0 (the farthest I've gone back).

```
# prime.rb

class Integer

# Returns the factorization of +self+.
#
# See Prime#prime_division for more details.
def prime_division(generator = Prime::Generator23.new)
Prime.prime_division(self, generator)
end

end

class Prime

def prime_division(value, generator = Prime::Generator23.new)
raise ZeroDivisionError if value == 0
if value < 0
value = -value
pv = [[-1, 1]]
else
pv = []
end
generator.each do |prime|
count = 0
while (value1, mod = value.divmod(prime)
mod) == 0
value = value1
count += 1
end
if count != 0
pv.push [prime, count]
end
break if value1 <= prime
end
if value > 1
pv.push [value, 1]
end
pv
end

end
```

This can be rewritten in more modern and idiomatic Ruby, to become much shorter
and easier to understand.

```
require 'prime.rb'

class Integer
def prime_division1(generator = Prime::Generator23.new)
Prime.prime_division1(self, generator)
end
end

class Prime

def prime_division1(value, generator = Prime::Generator23.new)
# raise ZeroDivisionError if value == 0
return [] if (value.abs | 1) == 1
pv = value < 0 ? [[-1, 1]] : []
value = value.abs
generator.each do |prime|
count = 0
while (value1, mod = value.divmod(prime); mod) == 0
value = value1
count += 1
end
pv.push [prime, count] unless count == 0
break if prime > value1
end
pv.push [value, 1] if value > 1
pv
end

end
```
By merely rewriting it we get smaller|concise code, that's easier to understand,
which is slightly faster. A *triple win!* Just paste the above code into a 2.5.0
terminal session, and run the benchmarks below.

```
def tm; s=Time.now; yield; Time.now-s end

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division }
[[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]]
=> 27.02951016

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division1 }
[[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]]
=> 25.959149721
```
Again, we get a *triple win* to this old codebase by merely rewriting it. It can
be made 3x faster by leveraging the `prime?` method from the `OpenSSL` library to
perform a more efficient|faster factoring algorithm, and implementation.

```
require 'prime.rb'
require 'openssl'

class Integer

def prime_division2(generator = Prime::Generator23.new)
return [] if (self.abs | 1) == 1
pv = self < 0 ? [-1] : []
value = self.abs
prime = generator.next
until value.to_bn.prime? or value == 1
while prime
(pv << prime; value /= prime; break) if value % prime == 0
prime = generator.next
end
end
pv << value if value > 1
pv.group_by {|prm| prm }.map{|prm, exp| [prm, exp.size] }
end

end
```
Here we're making much better use of Ruby idioms and libraries (`enumerable` and
`openssl`), leading to a much greater performance increase. A bigger *triple win*.
Pasting this code into a 2.5.0 terminal session gives the following results.

```
# Hardware: System76 laptop; I7 cpu @ 3.5GHz, 64-bit Linux

def tm; s=Time.now; yield; Time.now-s end

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division }
[[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]]
=> 27.02951016

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division1 }
[[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]]
=> 25.959149721

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division2 }
[[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]]
=> 9.39650374
```
`prime_division2` is much more usable for significantly larger numbers and use
cases than `prime_division`. I can even do multiple times better than this, if
you review the above cited forum thread.

My emphasis here is to show there are a lot of possible *low hanging fruit*
performance gains ripe for the picking to achieve Ruby 3 performance goals, if we
look (at minimum) for simpler|better code rewrites, and then algorithmic upgrades.

So the question is, are the devs willing to upgrade the codebase to provide the
demonstrated performance increases shown here for `prime_division`?

--
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```