Add checked arithmetic for /

[kimikage]

This also changes the implementation of / for Fixed. The new / also checks for overflow.

/ (checked_fdiv) throws DivideError for division by zero and OverflowError for overflow.

julia> 0.5Q0f7 / 0Q0f7 # v0.8.4 and this PR
ERROR: DivideError: integer division error

julia> 0.5N0f8 / 0N0f8 # v0.8.4
ERROR: ArgumentError: Normed{UInt8,8} is an 8-bit type representing 256 values from 0.0 to 1.0; cannot represent Inf

julia> 0.5N0f8 / 0N0f8 # this PR
ERROR: DivideError: integer division error

julia> -1Q0f7 / -1Q0f7 # v0.8.4
-1.0Q0f7

julia> -1Q0f7 / -1Q0f7 # this PR
ERROR: OverflowError: -1.0Q0f7 / -1.0Q0f7 overflowed for type Q0f7

julia> 1N0f8 / 0.5N0f8 # v0.8.4
ERROR: ArgumentError: Normed{UInt8,8} is an 8-bit type representing 256 values from 0.0 to 1.0; cannot represent 1.9921874

julia> 1N0f8 / 0.5N0f8 # this PR
ERROR: OverflowError: 1.0N0f8 / 0.502N0f8 overflowed for type N0f8

[codecov]

Codecov Report

Merging #222 into master will increase coverage by 0.11%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #222      +/-   ##
==========================================
+ Coverage   91.46%   91.58%   +0.11%     
==========================================
  Files           6        6              
  Lines         574      582       +8     
==========================================
+ Hits          525      533       +8     
  Misses         49       49              

Impacted Files	Coverage Δ
src/fixed.jl	95.45% <ø> (-0.04%)	⬇️
src/normed.jl	92.22% <ø> (-0.05%)	⬇️
src/FixedPointNumbers.jl	88.34% <100.00%> (+0.59%)	⬆️

---

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 7f6575e...04e18b8. Read the comment docs.

[kimikage]

Benchmark

x_q0f7  = rand(Q0f7,  1000, 1000); y_q0f7  = rand(Q0f7,  1000, 1000);
x_q0f15 = rand(Q0f15, 1000, 1000); y_q0f15 = rand(Q0f15, 1000, 1000);
x_q3f4  = rand(Q3f4,  1000, 1000); y_q3f4  = rand(Q3f4,  1000, 1000);
x_q3f12 = rand(Q3f12, 1000, 1000); y_q3f12 = rand(Q3f12, 1000, 1000);

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

non_zero(x) = x === zero(x) ? eps(x) : x;
z(x::X) where X = clamp(x * 0.5 * rand(X), X);

w_q0f7  = non_zero.(y_q0f7);  z_q0f7  = z.(w_q0f7);
w_q0f15 = non_zero.(y_q0f15); z_q0f15 = z.(w_q0f15);
w_q3f4  = non_zero.(y_q3f4);  z_q3f4  = z.(w_q3f4);
w_q3f12 = non_zero.(y_q3f12); z_q3f12 = z.(w_q3f12);

w_n0f8  = non_zero.(y_n0f8);  z_n0f8  = z.(w_n0f8);
w_n0f16 = non_zero.(y_n0f16); z_n0f16 = z.(w_n0f16);
w_n4f4  = non_zero.(y_n4f4);  z_n4f4  = z.(w_n4f4);
w_n4f12 = non_zero.(y_n4f12); z_n4f12 = z.(w_n4f12);

@btime $z_q0f7  ./ $w_q0f7;
@btime $z_q0f15 ./ $w_q0f15;
@btime $z_q3f4  ./ $w_q3f4;
@btime $z_q3f12 ./ $w_q3f12;

@btime $z_n0f8  ./ $w_n0f8;
@btime $z_n0f16 ./ $w_n0f16;
@btime $z_n4f4  ./ $w_n4f4;
@btime $z_n4f12 ./ $w_n4f12;

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

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

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

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

julia> versioninfo()
Julia Version 1.5.1
Commit 697e782ab8 (2020-08-25 20:08 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-9.0.1 (ORCJIT, skylake)

Fixed	before	after		Normed	before	after
z_q0f7 ./ w_q0f7	2.016 ms	1.880 ms		z_n0f8 ./ w_n0f8	2.157 ms	1.656 ms
z_q0f15 ./ w_q0f15	2.435 ms	2.576 ms		z_n0f16 ./ w_n0f16	3.017 ms	2.393 ms
z_q3f4 ./ w_q3f4	2.016 ms	1.940 ms		z_n4f4 ./ w_n4f4	2.216 ms	1.691 ms
z_q3f12 ./ w_q3f12	2.458 ms	2.599 ms		z_n4f12 ./ w_n4f12	3.611 ms	2.362 ms

	Fixed			Normed
wrapping_fdiv.	(x_q0f7 , y_q0f7 )	290.3 μs		(x_n0f8 , y_n0f8 )	286.8 μs
wrapping_fdiv.	(x_q0f15, y_q0f15)	920.4 μs		(x_n0f16, y_n0f16)	896.5 μs
wrapping_fdiv.	(x_q3f4 , y_q3f4 )	287.0 μs		(x_n4f4 , y_n4f4 )	286.7 μs
wrapping_fdiv.	(x_q3f12, y_q3f12)	896.6 μs		(x_n4f12, y_n4f12)	881.2 μs
saturating_fdiv.	(x_q0f7 , y_q0f7 )	272.3 μs		(x_n0f8 , y_n0f8 )	217.4 μs
saturating_fdiv.	(x_q0f15, y_q0f15)	892.7 μs		(x_n0f16, y_n0f16)	826.2 μs
saturating_fdiv.	(x_q3f4 , y_q3f4 )	270.5 μs		(x_n4f4 , y_n4f4 )	218.0 μs
saturating_fdiv.	(x_q3f12, y_q3f12)	894.2 μs		(x_n4f12, y_n4f12)	834.9 μs

[kimikage]

JuliaLang/julia #37085 was merged. ("Normed{UInt128,100}"-->"Normed{UInt128, 100}")
So, shall we update the test first? (cf. #139 (comment))

[timholy]

Interesting that we've had checked here a long time and I no longer remembered that...I can't remember the last time I divided FixedPoints, but I suspect it has come up.

[kimikage]

FYI, the Normed multiplication has also been checked so far. (cf. #213)

julia> 10N4f12 * 10N4f12 # v0.8.4
ERROR: ArgumentError: Normed{UInt16,12} is a 16-bit type representing 65536 values from 0.0 to 16.0037; cannot represent 100.0

julia> 10N4f12 * 10N4f12 # PR #213 (checked arithmetic for `Normed`)
ERROR: OverflowError: 10.0N4f12 * 10.0N4f12 overflowed for type N4f12

julia> 10Q4f11 * 10Q4f11 # PR #220 (wrapping arithmetic for `Fixed`)
4.0Q4f11

I will soon be asking for feedback on the default arithmetic in v0.9 and the state of arithmetic in v0.10 and later in the Discourse.

[timholy]

Yeah, I think with multiplication I always convert to float first.

[kimikage]

I ran rebase and moved the 100% tests to "test/common.jl".