Ruby

Background

This proposal emanates from issues I raised with regard to speeding up Ruby's implementation of it's gcd.

https://bugs.ruby-lang.org/issues/15166

The use case for these proposed methods exists for many mathematical|numerical
problems, but other too. Using these methods within the Ruby codebase alone
will also help facilitate Ruby's 3x3 goals, in addition to other improvements.

The compiler directive __builtin_ctz (count [of] trailing zeroes)
speeds up all the algorithms because it translates into 1 (a minimal) number
of machine instructions to perform this function.

https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html

https://stackoverflow.com/questions/13517232/gcc-builtin-functions

It's possible to mimic it to achieve better comparable performance. The above referenced algorithms use __builtin_ctz for two distinct purposes.

1. To find the specific count of trailing bits of a value.

vz = __builtin_ctz(v);

2. To reduce a value by its count of trailing bits.

v >>= __builtin_ctz(v);

The equivalent Ruby code used is: while ((v & 1) == 0) v >>= 1;

This is inefficient as it processes only 1 bit per iteration, but can be improved by doing 2 bits per iteration.

while ((v & 0b11) == 0) v >>= 2; if ((v & 1) == 0) v >>= 1; 

Using more bits improves performance more. For this I created an array ctz_bits[] which provides the count of trailing zeroes for a given value.

First I tried 3 bits per iteration.

int ctz_bits[8] = {0, 0, 1, 0, 2, 0, 1, 0}; while ((u & 0b111) == 0) u >>= 3; u >>= ctz_bits[u & 0b111]; 

Then 4 bits, with better performance,

int ctz_bits[16] = {0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0}; while ((u & 0xf) == 0) u >>= 4; u >>= ctz_bits[u & 0xf]; 

Then 5 bits, with even better performance (notice the pattern here).

int ctz_bits[32] = {0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0}; while ((u & 0x1f) == 0) u >>= 5; u >>= ctz_bits[u & 0x1f]; 

I settled on using 8 bits which is a 256 element array.

int ctz_bits[256] = {0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0}; while ((u & 0xff) == 0) u >>= 8; u >>= ctz_bits[u & 0xff]; 

This can be standardized to accommodate any bit size as follows.

int ctz_shift_bits = 8; int ctz_mask = 255; // 2**ctz_shift_bits - 1 int ctz_bits[256] = {0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0}; while ((u & ctz_mask) == 0) u >>= ctz_shift_bits; u >>= ctz_bits[u & ctz_mask]; 

By sacrificing a little bit of speed (maybe?) we can compress the
ctz_bits[] array. First we can eliminate odd index values, which
all return '0' shift bits (lsb is '1'), into a single 8 element array.
(This is conceptual code, that could accommodate any bit size).

int ctz_shift_bits = x; int ctz_mask = 2**ctz_shift_bits - 1 int ctz_bits[8] = {0, 1, 2, 1, 3, 1, 2, 1}; while ((u & ctz_mask) == 0) u >>= ctz_shift_bits; if ((u & 1) == 0) { u >>= (u % 16) == 0 ? correct_value : ctz_bits[correct_mask]; } 

Proposal

Create a method that mimic the function of __builtin_ctz.
These names are just descriptive to show functionality.

1) x.ctz_val -- 12.ctz_val => 3; 13.ctz_val => 13`` 

2) x.ctz_bits -- x = 12, x.ctz_bits => 2, x = 12 x = 13, x.ctz_bits => 0, x = 13 

The ideal case is to use -__builtin_ctz, but I understand the concern about
it's availability on all compilers. Creating these methods can provide the best
of both worlds, by aliasing ctz_bits to __builtin_ctz and choosing which
to use at compile time based on availability. Providing either (or both) will
definitely increase Ruby's internal performance, and help it reach its 3x3 goal,
while providing users with fast and standard methods for these functions.

You can see implementation comparisons with gcd from below.

https://gist.github.com/jzakiya/44eae4feeda8f6b048e19ff41a0c6566

The xx_a versions mimic those using __builtin_ctz.

Below shows the differences in ruby gcd implementations.
The standard lib implementation rubygcd is slowest, ruby_a is 1/3 faster,
and ruby_b (using, builtin_ctz) is almost fully 2x faster. They clearly
display specific, and possible, performance benefits of this proposals.

[jzakiya@jabari-pc ~]$ ./gcd2 gcd between numbers in [1 and 2000] gcdwikipedia7fast32 : time = 73 gcdwikipedia4fast : time = 113 gcdFranke : time = 133 gcdwikipedia3fast : time = 139 gcdwikipedia2fastswap : time = 162 gcdwikipedia5fast : time = 140 gcdwikipedia7fast : time = 129 gcdwikipedia2fast : time = 161 gcdwikipedia6fastxchg : time = 145 gcdwikipedia2fastxchg : time = 168 gcd_iterative_mod : time = 230 gcd_recursive : time = 232 basicgcd : time = 234 rubygcd : time = 305 gcdwikipedia2 : time = 312 gcdwikipedia7fast32_a : time = 129 gcdwikipedia4fast_a : time = 149 rubygcd_a : time = 193 rubygcd_b : time = 169 gcd between numbers in [1000000001 and 1000002000] gcdwikipedia7fast32 : time = 76 gcdwikipedia4fast : time = 106 gcdFranke : time = 121 gcdwikipedia3fast : time = 127 gcdwikipedia2fastswap : time = 153 gcdwikipedia5fast : time = 126 gcdwikipedia7fast : time = 118 gcdwikipedia2fast : time = 148 gcdwikipedia6fastxchg : time = 134 gcdwikipedia2fastxchg : time = 154 gcd_iterative_mod : time = 215 gcd_recursive : time = 214 basicgcd : time = 220 rubygcd : time = 287 gcdwikipedia2 : time = 289 gcdwikipedia7fast32_a : time = 116 gcdwikipedia4fast_a : time = 142 rubygcd_a : time = 180 rubygcd_b : time = 155