ext/bcmath: Performance improvement bcsqrt()

[SakiTakamachi]

Test

Approximately 80 million calculations were performed using values of various magnitudes and scales, and all results matched those from the previous implementation.

Benchmark

Performance improvements are particularly noticeable in the following cases:

• When dealing with small values (due to the use of a fast path)
• When handling decimal numbers less than 1 with many leading zeros (as leading zeros are removed before computation)
• When the scale of the input number (num) is much larger than the required result scale (since unnecessary fractional digits are now ignored during computation)

For large values that do not fall into the above categories, most of the execution time is spent on the iterative process of the Newton-Raphson method, especially on division operations. As a result, while there may be some minor gains from reducing memory allocations and lowering the cost of converting BCD to BC_VECTOR, there is no significant improvement in overall performance.

Small size value (fast path)

Code:

for ($i = 0; $i < 1000000; $i++) {
    bcsqrt(20, 6);
}

Result:

Benchmark 1: /php-dev2/sapi/cli/php /mount/bc/sqrt/0.php
  Time (mean ± σ):      65.9 ms ±   0.8 ms    [User: 60.4 ms, System: 3.2 ms]
  Range (min … max):    64.1 ms …  67.3 ms    46 runs
 
Benchmark 2: /master/sapi/cli/php /mount/bc/sqrt/0.php
  Time (mean ± σ):     746.1 ms ±   4.4 ms    [User: 740.2 ms, System: 3.5 ms]
  Range (min … max):   741.5 ms … 756.3 ms    10 runs
 
Summary
  '/php-dev2/sapi/cli/php /mount/bc/sqrt/0.php' ran
   11.32 ± 0.15 times faster than '/master/sapi/cli/php /mount/bc/sqrt/0.php'

Middle size value (< 1) (standard path)

Code:

for ($i = 0; $i < 500000; $i++) {
    bcsqrt('0.000045689101', 20);
}

Result:

Benchmark 1: /php-dev2/sapi/cli/php /mount/bc/sqrt/1.php
  Time (mean ± σ):     108.0 ms ±   0.3 ms    [User: 102.5 ms, System: 3.2 ms]
  Range (min … max):   107.5 ms … 108.8 ms    27 runs
 
Benchmark 2: /master/sapi/cli/php /mount/bc/sqrt/1.php
  Time (mean ± σ):      1.082 s ±  0.002 s    [User: 1.076 s, System: 0.003 s]
  Range (min … max):    1.078 s …  1.085 s    10 runs
 
Summary
  '/php-dev2/sapi/cli/php /mount/bc/sqrt/1.php' ran
   10.01 ± 0.04 times faster than '/master/sapi/cli/php /mount/bc/sqrt/1.php'

Middle size value 1 (fast path)

The new logic ignores unnecessary scales in the calculation, so this is a fast path.

Code:

for ($i = 0; $i < 700000; $i++) {
    bcsqrt('15151324141414.412312232141241', 0);
}

Result:

Benchmark 1: /php-dev2/sapi/cli/php /mount/bc/sqrt/2.php
  Time (mean ± σ):      55.0 ms ±   2.4 ms    [User: 49.6 ms, System: 3.1 ms]
  Range (min … max):    52.6 ms …  71.1 ms    53 runs

Benchmark 2: /master/sapi/cli/php /mount/bc/sqrt/2.php
  Time (mean ± σ):      1.099 s ±  0.003 s    [User: 1.092 s, System: 0.004 s]
  Range (min … max):    1.093 s …  1.102 s    10 runs
 
Summary
  '/php-dev2/sapi/cli/php /mount/bc/sqrt/2.php' ran
   19.98 ± 0.89 times faster than '/master/sapi/cli/php /mount/bc/sqrt/2.php'

Middle size value 2 (standard path)

Code:

for ($i = 0; $i < 500000; $i++) {
    bcsqrt('15151324141414.412312232141241', 30);
}

Result:

Benchmark 1: /php-dev2/sapi/cli/php /mount/bc/sqrt/3.php
  Time (mean ± σ):     204.1 ms ±   0.8 ms    [User: 198.5 ms, System: 3.3 ms]
  Range (min … max):   202.7 ms … 205.9 ms    14 runs
 
Benchmark 2: /master/sapi/cli/php /mount/bc/sqrt/3.php
  Time (mean ± σ):      1.315 s ±  0.034 s    [User: 1.308 s, System: 0.003 s]
  Range (min … max):    1.297 s …  1.411 s    10 runs

Summary
  '/php-dev2/sapi/cli/php /mount/bc/sqrt/3.php' ran
    6.44 ± 0.17 times faster than '/master/sapi/cli/php /mount/bc/sqrt/3.php'

Big size value (standard path)

Code:

for ($i = 0; $i < 100; $i++) {
    bcsqrt('15151324141414.412312232141241', 3000);
}

Result:

Benchmark 1: /php-dev2/sapi/cli/php /mount/bc/sqrt/4.php
  Time (mean ± σ):     195.7 ms ±   1.8 ms    [User: 189.7 ms, System: 3.4 ms]
  Range (min … max):   194.2 ms … 201.8 ms    15 runs

Benchmark 2: /master/sapi/cli/php /mount/bc/sqrt/4.php
  Time (mean ± σ):     280.0 ms ±   4.5 ms    [User: 274.3 ms, System: 3.2 ms]
  Range (min … max):   276.2 ms … 289.2 ms    10 runs
 
Summary
  '/php-dev2/sapi/cli/php /mount/bc/sqrt/4.php' ran
    1.43 ± 0.03 times faster than '/master/sapi/cli/php /mount/bc/sqrt/4.php'

[SakiTakamachi]

Sorry, it was still incomplete.

[SakiTakamachi]

The code is ready for review.

[SakiTakamachi]

It looks like nearzero.c is no longer in use, so it can probably be removed.

[SakiTakamachi]

done

[SakiTakamachi]

No longer using bc_add_ex, bc_int2num and bc_raise_bc_exponent, so I'll remove them.

[SakiTakamachi]

done