# Objective
Almost all of the `*InOut` easing functions are not actually smooth
(`SineInOut` is the one exception).
Because they're defined piecewise, they jump from accelerating upwards
to accelerating downwards, causing infinite jerk at t=½.
## Solution
This PR adds the well-known
[smoothstep](https://registry.khronos.org/OpenGL-Refpages/gl4/html/smoothstep.xhtml),
as well as its higher-degree version
[smootherstep](https://en.wikipedia.org/wiki/Smoothstep#Variations), as
easing functions.
Mathematically, these are the classic [Hermite
interpolation](https://en.wikipedia.org/wiki/Hermite_interpolation)
results:
- for smoothstep, the cubic with velocity zero at both ends
- for smootherstep, the quintic with velocity zero *and acceleration
zero* at both ends
And because they're simple polynomials, there's no branching and thus
they don't have the acceleration jump in the middle.
I also added some more information and cross-linking to the
documentation for these and some of the other easing functions, to help
clarify why one might want to use these over other existing ones. In
particular, I suspect that if people are willing to pay for a quintic
they might prefer `SmootherStep` to `QuinticInOut`.
For consistency with how everything else has triples, I added
`Smooth(er)Step{In,Out}` as well, in case people want to run the `In`
and `Out` versions separately for some reason. Qualitatively they're not
hugely different from `Quadratic{In,Out}` or `Cubic{In,Out}`, though, so
could be removed if you'd rather. They're low cost to keep, though, and
convenient for testing.
## Testing
These are simple polynomials, so their coefficients can be read directly
from the Horner's method implementation and compared to the reference
materials. The tests from #16910 were updated to also test these 6 new
easing functions, ensuring basic behaviour, plus one was updated to
better check that the InOut versions of things match their rescaled In
and Out versions.
Even small changes like
```diff
- (((2.5 + (-1.875 + 0.375*t) * t) * t) * t) * t
+ (((2.5 + (-1.85 + 0.375*t) * t) * t) * t) * t
```
are caught by multiple tests this way.
If you want to confirm them visually, here are the 6 new ones graphed:
<https://www.desmos.com/calculator/2d3ofujhry>
---
## Migration Guide
This version of bevy marks `EaseFunction` as `#[non_exhaustive]` to that
future changes to add more easing functions will be non-breaking. If you
were exhaustively matching that enum -- which you probably weren't --
you'll need to add a catch-all (`_ =>`) arm to cover unknown easing
functions.
f96653498b