improve impl of __sqrth, sqrtf, sqrt, __sqrtx and sqrtq

[Zhenming-Lin]

No description provided.

[Zhenming-Lin]

Some u80 or u128 integer operations are not yet supported under self-hosted aarch64 architecture.

[andrewrk]

Can you share a short summary of what improvements you made, and how you made them?

[Zhenming-Lin]

The impls(prev & cur) are all referenced from the libc, and the current impl is a newer version. The previous version used bit-by-bit calculations and was slow, but the current version uses lookup tables combined with Goldschmidt iterations to significantly improve the speed.

• The zig impl of sqrtf, sqrt and sqrtq is almost identical to the C impl.
• The Zig impl of __sqrth is based on the impl of sqrt, with the number of iterations appropriately reduced according to the precision requirements of f16 to improve speed.
• The Zig impl of __sqrtx is based on the impl of sqrtq, and the number of fractional bits in the last iteration is changed to 80-bit according to the precision requirements of f80 to improve speed.

Benchmark 1 (7605 runs): prev sqrtf
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           579us ±  140us     505us … 7.40ms        645 ( 8%)        0%
Benchmark 2 (10000 runs): sqrtf
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           102us ± 64.2us    77.3us … 4.44ms        879 ( 9%)        ⚡- 82.3% ±  0.5%

Benchmark 1 (2040 runs): prev sqrt
  measurement          mean ± σ            min … max           outliers         delta
  wall_time          2.35ms ±  318us    2.05ms … 4.88ms        124 ( 6%)        0%
Benchmark 2 (10000 runs): sqrt
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           143us ± 53.7us     117us … 1.34ms        839 ( 8%)        ⚡- 93.9% ±  0.3%

Benchmark 1 (10000 runs): __sqrth
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           101us ± 49.6us    79.5us … 2.04ms        451 ( 5%)        0%
Benchmark 2 (10000 runs): sqrtf
  measurement          mean ± σ            min … max           outliers         delta
  wall_time          94.8us ± 55.1us    76.6us … 2.41ms        955 (10%)        ⚡-  5.8% ±  1.4%
Benchmark 3 (10000 runs): sqrt
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           145us ± 62.7us     117us … 2.34ms        882 ( 9%)        💩+ 43.8% ±  1.6%
Benchmark 4 (9413 runs): __sqrtx
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           458us ±  142us     359us … 5.09ms        575 ( 6%)        💩+355.2% ±  2.9%
Benchmark 5 (6763 runs): sqrtq
  measurement          mean ± σ            min … max           outliers         delta
  wall_time           664us ±  136us     568us … 3.70ms        476 ( 7%)        💩+560.0% ±  2.9%