Optimize multiplication for Normed

[kimikage]

This adds wrapping_mul, saturating_mul and checked_mul binary operations. However, this does not specialize them for Fixed and does not change * for Fixed.

This applies the wrapping arithmetic as the default arithmetic of * for the abstract FixedPoint, but as mentioned above, the *for Fixed is not affected by the default arithmetic. Furthermore, * for Normed overrides the default arithmetic with the "checked" arithmetic for backward compatibility. (Strictly speaking, the error type is changed from ArgumentError to OverflowError.)

This replaces most of Normed's implementation of multiplication with integer operations. This improves the speed in many cases and the accuracy in some cases.

Fixes #174

[codecov]

Codecov Report

Merging #213 into master will increase coverage by 0.61%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #213      +/-   ##
==========================================
+ Coverage   90.39%   91.01%   +0.61%     
==========================================
  Files           6        6              
  Lines         510      534      +24     
==========================================
+ Hits          461      486      +25     
+ Misses         49       48       -1     

Impacted Files	Coverage Δ
src/FixedPointNumbers.jl	87.75% <100.00%> (+0.32%)	⬆️
src/normed.jl	92.14% <100.00%> (+0.86%)	⬆️
src/utilities.jl	96.00% <0.00%> (+4.00%)	⬆️

---

Continue to review full report at Codecov.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update 5794adf...1dea1e9. Read the comment docs.

[kimikage]

Benchmark

x_n0f8  = collect(rand(N0f8,  1000, 1000)); y_n0f8  = collect(rand(N0f8,  1000, 1000));
x_n0f16 = collect(rand(N0f16, 1000, 1000)); y_n0f16 = collect(rand(N0f16, 1000, 1000));
x_n4f4  = collect(rand(N4f4,  1000, 1000)); y_n4f4  = collect(rand(N4f4,  1000, 1000));
x_n4f12 = collect(rand(N4f12, 1000, 1000)); y_n4f12 = collect(rand(N4f12, 1000, 1000));
z_n4f4  = N4f4.( rand(1000, 1000));
z_n4f12 = N4f12.(rand(1000, 1000));

@btime $x_n0f8 .* $y_n0f8;
@btime $x_n0f16 .* $y_n0f16;
@btime saturating_mul.($x_n4f4, $y_n4f4);
@btime saturating_mul.($x_n4f12, $y_n4f12);
@btime $x_n4f4 .* $z_n4f4;
@btime $x_n4f12 .* $z_n4f12;

	Julia v1.5.0 before	Julia v1.5.0 after	Julia v1.4.2 before	Julia v1.4.2 after
x_n0f8 .* y_n0f8	1.965 ms	77.600 μs	1.978 ms	85.100 μs
x_n0f16 .* y_n0f16	2.172 ms	165.900 μs	2.168 ms	166.200 μs
saturating_mul.(x_n4f4, y_n4f4)	-	105.900 μs	-	104.100 μs
saturating_mul.(x_n4f12, y_n4f12)	-	304.200 μs	-	302.700 μs
$x_n4f4 .* $z_n4f4 (checked_mul)	2.018 ms	794.400 μs	1.991 ms	794.700 μs
$x_n4f12 .* $z_n4f12(checked_mul)	2.636 ms	1.086 ms	2.688 ms	1.090 ms

[timholy]

Remarkable work as usual!

[kimikage]

Thank you for all your reviews.
As for the default arithmetic settings, I will ask for feedback again (in Discourse?) before the release of v0.9.