typst/vendor/float-cmp

float-cmp

Documentation is available at https://docs.rs/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);
  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

By default this crate enables the ratio module providing the ApproxEqRatio trait. This feature pulls in num-traits. If you disable this feature, you'll need to either enable num-traits directly or else enable the std feature; otherwise it won't compile. This crate is #![no_std] unless you enable the std feature.

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, 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/