# float-cmp

float-cmp defines and implements traits for approximate comparison of floating point types which have fallen away from exact equality due to the limited precision available within floating point representations. Implementations of these traits are provided for `f32` and `f64` types.

When I was a kid in the '80s, the programming rule was "Never compare floating point numbers". If you can follow that rule and still get the outcome you desire, then more power to you. However, if you really do need to compare them, this crate provides a reasonable way to do so.

Another crate `efloat` offers another solution by providing a floating point type that tracks its error bounds as operations are performed on it, and thus can implement the `ApproxEq` trait in this crate more accurately, without specifying a `Margin`.

The recommended go-to solution (although it may not be appropriate in all cases) is the `approx_eq()` function in the `ApproxEq` trait (or better yet, the macros). For `f32` and `f64`, the `F32Margin` and `F64Margin` types are provided for specifying margins as both an epsilon value and an ULPs value, and defaults are provided via `Default` (although there is no perfect default value that is always appropriate, so beware).

Several other traits are provided including `Ulps`, `ApproxEqUlps`, `ApproxOrdUlps`, and `ApproxEqRatio`.

## The problem

Floating point operations must round answers to the nearest representable number. Multiple operations may result in an answer different from what you expect. In the following example, the assert will fail, even though the printed output says "0.45 == 0.45":

```  let a: f32 = 0.15 + 0.15 + 0.15;
let b: f32 = 0.1 + 0.1 + 0.25;
println!("{} == {}", a, b);
assert!(a==b)  // Fails, because they are not exactly equal```

This fails because the correct answer to most operations isn't exactly representable, and so your computer's processor chooses to represent the answer with the closest value it has available. This introduces error, and this error can accumulate as multiple operations are performed.

## The solution

With `ApproxEq`, we can get the answer we intend:

```  let a: f32 = 0.15 + 0.15 + 0.15;
let b: f32 = 0.1 + 0.1 + 0.25;
println!("{} == {}", a, b);
// They are equal, within 2 ulps
assert!( approx_eq!(f32, a, b, ulps = 2) );```

## Some explanation

We use the term ULP (units of least precision, or units in the last place) to mean the difference between two adjacent floating point representations (adjacent meaning that there is no floating point number between them). This term is borrowed from prior work (personally I would have chosen "quanta"). The size of an ULP (measured as a float) varies depending on the exponents of the floating point numbers in question. That is a good thing, because as numbers fall away from equality due to the imprecise nature of their representation, they fall away in ULPs terms, not in absolute terms. Pure epsilon-based comparisons are absolute and thus don't map well to the nature of the additive error issue. They work fine for many ranges of numbers, but not for others (consider comparing -0.0000000028 to +0.00000097).

## Using this crate

You can use the `ApproxEq` trait directly like so:

`    assert!( a.approx_eq(b, F32Margin { ulps: 2, epsilon: 0.0 }) );`

We have implemented `From<(f32,i32)>` for `F32Margin` (and similarly for `F64Margin`) so you can use this shorthand:

`    assert!( a.approx_eq(b, (0.0, 2)) );`

With macros, it is easier to be explicit about which type of margin you wish to set, without mentioning the other one (the other one will be zero). But the downside is that you have to specify the type you are dealing with:

```    assert!( approx_eq!(f32, a, b, ulps = 2) );
assert!( approx_eq!(f32, a, b, epsilon = 0.00000003) );
assert!( approx_eq!(f32, a, b, epsilon = 0.00000003, ulps = 2) );
assert!( approx_eq!(f32, a, b, (0.0, 2)) );
assert!( approx_eq!(f32, a, b, F32Margin { epsilon: 0.0, ulps: 2 }) );
assert!( approx_eq!(f32, a, b, F32Margin::default()) );
assert!( approx_eq!(f32, a, b) ); // uses the default```

For most cases, I recommend you use a smallish integer for the `ulps` parameter (1 to 5 or so), and a similar small multiple of the floating point's EPSILON constant (1.0 to 5.0 or so), but there are plenty of cases where this is insufficient.

## Implementing these traits

You can implement `ApproxEq` for your own complex types like shown below. The floating point type `F` must be `Copy`, but for large types you can implement it for references to your type as shown.

```use float_cmp::ApproxEq;

pub struct Vec2<F> {
pub x: F,
pub y: F,
}

impl<'a, M: Copy + Default, F: Copy + ApproxEq<Margin=M>> ApproxEq for &'a Vec2<F> {
type Margin = M;

fn approx_eq<T: Into<Self::Margin>>(self, other: Self, margin: T) -> bool {
let margin = margin.into();
self.x.approx_eq(other.x, margin)
&& self.y.approx_eq(other.y, margin)
}
}```

## Non floating-point types

`ApproxEq` can be implemented for non floating-point types as well, since `Margin` is an associated type.

The `efloat` crate implements (or soon will implement) `ApproxEq` for a compound type that tracks floating point error bounds by checking if the error bounds overlap. In that case `type Margin = ()`.

## Inspiration

This crate was inspired by this Random ASCII blog post:

https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/

## Macros

 approx_eq

## Structs

 F32Margin This type defines a margin within two f32s might be considered equal and is intended as the associated type for the `ApproxEq` trait. F64Margin This type defines a margin within two f32s might be considered equal and is intended as the associated type for the `ApproxEq` trait.

## Traits

 ApproxEq ApproxEq is a trait for approximate equality comparisons. ApproxEqRatio ApproxEqRatio is a trait for approximate equality comparisons bounding the ratio of the difference to the larger. ApproxEqUlps ApproxEqUlps is a trait for approximate equality comparisons. The associated type Flt is a floating point type which implements Ulps, and is required so that this trait can be implemented for compound types (e.g. vectors), not just for the floats themselves. Ulps

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