Add checked, wrapping and saturating arithmetic for div/cld/fld

[kimikage]

The default arithmetic for div is still checked arithmetic.
This changes the error type for overflow from DivideError to OverflowError.

julia> div(-1Q0f7, -eps(Q0f7))
ERROR: DivideError: integer division error # v0.8.4
ERROR: OverflowError: div(-1.0Q0f7, -0.008Q0f7) overflowed for type Int8 # this PR

This also adds the support for cld and 3-arg div.

[codecov]

Codecov Report

Merging #226 into master will increase coverage by 0.22%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #226      +/-   ##
==========================================
+ Coverage   95.19%   95.42%   +0.22%     
==========================================
  Files           6        6              
  Lines         645      677      +32     
==========================================
+ Hits          614      646      +32     
  Misses         31       31              

Impacted Files	Coverage Δ
src/FixedPointNumbers.jl	96.15% <100.00%> (+0.53%)	⬆️

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[kimikage]

Benchmark

See #222 (comment) for detail of variables.

for op in (div, fld)
    @btime $op.($z_q0f7 , $w_q0f7 );
    @btime $op.($z_q0f15, $w_q0f15);
    @btime $op.($z_q3f4 , $w_q3f4 );
    @btime $op.($z_q3f12, $w_q3f12);

    @btime $op.($z_n0f8 , $w_n0f8 );
    @btime $op.($z_n0f16, $w_n0f16);
    @btime $op.($z_n4f4 , $w_n4f4 );
    @btime $op.($z_n4f12, $w_n4f12);
end

@btime wrapping_div.($x_q0f7 , $y_q0f7 );
@btime wrapping_fld.($x_q0f7 , $y_q0f7 );
@btime wrapping_cld.($x_q0f7 , $y_q0f7 );
@btime wrapping_div.($x_q0f15, $y_q0f15);
@btime wrapping_div.($x_q3f4 , $y_q3f4 );
@btime wrapping_div.($x_q3f12, $y_q3f12);

@btime wrapping_div.($x_n0f8 , $y_n0f8 );
@btime wrapping_fld.($x_n0f8 , $y_n0f8 );
@btime wrapping_cld.($x_n0f8 , $y_n0f8 );
@btime wrapping_div.($x_n0f16, $y_n0f16);
@btime wrapping_div.($x_n4f4 , $y_n4f4 );
@btime wrapping_div.($x_n4f12, $y_n4f12);

@btime saturating_div.($x_q0f7 , $y_q0f7 );
@btime saturating_fld.($x_q0f7 , $y_q0f7 );
@btime saturating_cld.($x_q0f7 , $y_q0f7 );
@btime saturating_div.($x_q0f15, $y_q0f15);
@btime saturating_div.($x_q3f4 , $y_q3f4 );
@btime saturating_div.($x_q3f12, $y_q3f12);

@btime saturating_div.($x_n0f8 , $y_n0f8 );
@btime saturating_fld.($x_n0f8 , $y_n0f8 );
@btime saturating_cld.($x_n0f8 , $y_n0f8 );
@btime saturating_div.($x_n0f16, $y_n0f16);
@btime saturating_div.($x_n4f4 , $y_n4f4 );
@btime saturating_div.($x_n4f12, $y_n4f12);

div (checked_div)

Fixed	before	after		Normed	before	after
(z_q0f7 , w_q0f7 )	2.019 ms	1.541 ms		(z_n0f8 , w_n0f8 )	1.793 ms	1.321 ms
(z_q0f15, w_q0f15)	3.085 ms	2.173 ms		(z_n0f16, w_n0f16)	2.427 ms	1.932 ms
(z_q3f4 , w_q3f4 )	2.019 ms	1.544 ms		(z_n4f4 , w_n4f4 )	1.791 ms	1.291 ms
(z_q3f12, w_q3f12)	3.320 ms	2.428 ms		(z_n4f12, w_n4f12)	2.423 ms	2.228 ms

fld (checked_fld)

Fixed	before	after		Normed	before	after
(z_q0f7 , w_q0f7 )	3.363 ms	1.540 ms		(z_n0f8 , w_n0f8 )	1.793 ms	1.327 ms
(z_q0f15, w_q0f15)	3.989 ms	2.194 ms		(z_n0f16, w_n0f16)	2.431 ms	1.967 ms
(z_q3f4 , w_q3f4 )	3.356 ms	1.544 ms		(z_n4f4 , w_n4f4 )	1.795 ms	1.312 ms
(z_q3f12, w_q3f12)	4.030 ms	2.227 ms		(z_n4f12, w_n4f12)	2.434 ms	2.164 ms

wrapping_*

	Fixed			Normed
wrapping_div.	(x_q0f7 , y_q0f7 )	496.4 μs		(x_n0f8 , y_n0f8 )	323.1 μs
wrapping_fld.	(x_q0f7 , y_q0f7 )	496.1 μs		(x_n0f8 , y_n0f8 )	309.3 μs
wrapping_cld.	(x_q0f7 , y_q0f7 )	496.6 μs		(x_n0f8 , y_n0f8 )	309.0 μs
wrapping_div.	(x_q0f15, y_q0f15)	1.054 ms		(x_n0f16, y_n0f16)	909.5 μs
wrapping_div.	(x_q3f4 , y_q3f4 )	496.4 μs		(x_n4f4 , y_n4f4 )	309.9 μs
wrapping_div.	(x_q3f12, y_q3f12)	1.105 ms		(x_n4f12, y_n4f12)	869.3 μs

saturating_*

	Fixed			Normed
saturating_div.	(x_q0f7 , y_q0f7 )	319.8 μs		(x_n0f8 , y_n0f8 )	489.3 μs
saturating_fld.	(x_q0f7 , y_q0f7 )	319.7 μs		(x_n0f8 , y_n0f8 )	485.4 μs
saturating_cld.	(x_q0f7 , y_q0f7 )	320.2 μs		(x_n0f8 , y_n0f8 )	485.7 μs
saturating_div.	(x_q0f15, y_q0f15)	936.1 μs		(x_n0f16, y_n0f16)	1.014 ms
saturating_div.	(x_q3f4 , y_q3f4 )	319.8 μs		(x_n4f4 , y_n4f4 )	485.5 μs
saturating_div.	(x_q3f12, y_q3f12)	973.9 μs		(x_n4f12, y_n4f12)	967.2 μs

[kimikage]

I'm going to move on to the implementation of checked_rem and checked_mod for now.

Probably most people are not interested in the division as I'm not interested in it.:joy: