Fix definitions of `==` and `isless`

[devmotion]

trafficstars

Given the definitions of == and isequal in ForwardDiff#master, the PR reverts https://github.com/JuliaDiff/DualNumbers.jl/commit/337539f32622b8d713a7ea122aeaafb4f1e0de3b (see #10) and defines both ==(x, y) and isequal(x, y) in terms of both value(x) and epsilon(x) etc. This makes

• the definitions in ForwardDiff and DualNumbers consistent,
• the definitions of == and isequal in DualNumbers consistent,
• the definitions of == and isequal mathematically correct.

Additionally, the PR fixes isless: According to its docstring, isless(x, y) should

Test whether `x` is less than `y`, according to a fixed total order (defined together with `isequal`). `isless` is not defined for pairs `(x, y)` of all types. However, if it is defined, it is expected to satisfy the following:
- If `isless(x, y)` is defined, then so is `isless(y, x)` and `isequal(x, y)`, and exactly one of those three yields true.
- The relation defined by `isless` is transitive, i.e., `isless(x, y)` && `isless(y, z)` implies `isless(x, z)`.

On the master branch, however, the first requirement is not satisfied (e.g., choose x = dual(3, 0.5) and y = dual(3, 1.0): then neither isless(x, y), isless(y, x), nor isequal(x, y) is satisfied).

cc @andreasnoack who was involved in #10

[codecov[bot]]

Codecov Report

:x: Patch coverage is 83.33333% with 1 line in your changes missing coverage. Please review. :white_check_mark: Project coverage is 51.88%. Comparing base (d9251a7) to head (d52f6a3). :warning: Report is 2 commits behind head on master.

Files with missing lines	Patch %	Lines
src/dual.jl	83.33%	1 Missing :warning:

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #103      +/-   ##
==========================================
+ Coverage   50.94%   51.88%   +0.94%     
==========================================
  Files           2        2              
  Lines         212      212              
==========================================
+ Hits          108      110       +2     
+ Misses        104      102       -2     

:umbrella: View full report in Codecov by Sentry.
:loudspeaker: Have feedback on the report? Share it here.

:rocket: New features to boost your workflow:

• :snowflake: Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[devmotion]

@mcabbott I think you might be interested in this change.

[mcabbott]

This looks right to me, at first glance.

I see that https://github.com/JuliaDiff/DualNumbers.jl/pull/104 recently adopted the other convention.

But I have never used this package and am unsure who does, or the value of maintaining it in parallel to ForwardDiff.

[dlfivefifty]

This package predates ForwardDiff.jl and its value is mostly in its simplicity and accessibility.

The "right" thing to do is move ForwardDiff.Dual here. But that would require someone to take the initiative, and some functionality would need to be added to ForwardDiff.Dual to maintain feature parity.