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// Copyright 2014-2018 Optimal Computing (NZ) Ltd.
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// Licensed under the MIT license. See LICENSE for details.
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//! # float-cmp
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//!
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//! float-cmp defines and implements traits for approximate comparison of floating point types
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//! which have fallen away from exact equality due to the limited precision available within
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//! floating point representations. Implementations of these traits are provided for `f32`
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//! and `f64` types.
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//!
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//! When I was a kid in the '80s, the programming rule was "Never compare floating point
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//! numbers". If you can follow that rule and still get the outcome you desire, then more
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//! power to you. However, if you really do need to compare them, this crate provides a
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//! reasonable way to do so.
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//!
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//! Another crate `efloat` offers another solution by providing a floating point type that
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//! tracks its error bounds as operations are performed on it, and thus can implement the
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//! `ApproxEq` trait in this crate more accurately, without specifying a `Margin`.
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//!
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//! The recommended go-to solution (although it may not be appropriate in all cases) is the
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//! `approx_eq()` function in the `ApproxEq` trait (or better yet, the macros). For `f32`
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//! and `f64`, the `F32Margin` and `F64Margin` types are provided for specifying margins as
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//! both an epsilon value and an ULPs value, and defaults are provided via `Default`
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//! (although there is no perfect default value that is always appropriate, so beware).
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//!
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//! Several other traits are provided including `Ulps`, `ApproxEqUlps`, `ApproxOrdUlps`, and
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//! `ApproxEqRatio`.
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//!
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//! ## The problem
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//!
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//! Floating point operations must round answers to the nearest representable number. Multiple
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//! operations may result in an answer different from what you expect. In the following example,
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//! the assert will fail, even though the printed output says "0.45 == 0.45":
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//!
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//! ```should_panic
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//! # extern crate float_cmp;
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//! # use float_cmp::ApproxEq;
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//! # fn main() {
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//! let a: f32 = 0.15 + 0.15 + 0.15;
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//! let b: f32 = 0.1 + 0.1 + 0.25;
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//! println!("{} == {}", a, b);
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//! assert!(a==b) // Fails, because they are not exactly equal
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//! # }
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//! ```
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//!
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//! This fails because the correct answer to most operations isn't exactly representable, and so
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//! your computer's processor chooses to represent the answer with the closest value it has
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//! available. This introduces error, and this error can accumulate as multiple operations are
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//! performed.
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//!
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//! ## The solution
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//!
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//! With `ApproxEq`, we can get the answer we intend:
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//!
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//! ```
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//! # #[macro_use]
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//! # extern crate float_cmp;
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//! # use float_cmp::{ApproxEq, F32Margin};
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//! # fn main() {
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//! let a: f32 = 0.15 + 0.15 + 0.15;
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//! let b: f32 = 0.1 + 0.1 + 0.25;
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//! println!("{} == {}", a, b);
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//! // They are equal, within 2 ulps
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//! assert!( approx_eq!(f32, a, b, ulps = 2) );
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//! # }
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//! ```
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//!
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//! ## Some explanation
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//!
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//! We use the term ULP (units of least precision, or units in the last place) to mean the
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//! difference between two adjacent floating point representations (adjacent meaning that there is
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//! no floating point number between them). This term is borrowed from prior work (personally I
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//! would have chosen "quanta"). The size of an ULP (measured as a float) varies
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//! depending on the exponents of the floating point numbers in question. That is a good thing,
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//! because as numbers fall away from equality due to the imprecise nature of their representation,
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//! they fall away in ULPs terms, not in absolute terms. Pure epsilon-based comparisons are
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//! absolute and thus don't map well to the nature of the additive error issue. They work fine
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//! for many ranges of numbers, but not for others (consider comparing -0.0000000028
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//! to +0.00000097).
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//!
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//! ## Using this crate
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//!
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//! You can use the `ApproxEq` trait directly like so:
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//!
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//! ```
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//! # extern crate float_cmp;
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//! # use float_cmp::{ApproxEq, F32Margin};
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//! # fn main() {
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//! # let a: f32 = 0.15 + 0.15 + 0.15;
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//! # let b: f32 = 0.1 + 0.1 + 0.25;
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//! assert!( a.approx_eq(b, F32Margin { ulps: 2, epsilon: 0.0 }) );
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//! # }
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//! ```
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//!
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//! We have implemented `From<(f32,i32)>` for `F32Margin` (and similarly for `F64Margin`)
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//! so you can use this shorthand:
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//!
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//! ```
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//! # extern crate float_cmp;
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//! # use float_cmp::{ApproxEq, F32Margin};
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//! # fn main() {
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//! # let a: f32 = 0.15 + 0.15 + 0.15;
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//! # let b: f32 = 0.1 + 0.1 + 0.25;
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//! assert!( a.approx_eq(b, (0.0, 2)) );
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//! # }
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//! ```
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//!
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//! With macros, it is easier to be explicit about which type of margin you wish to set,
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//! without mentioning the other one (the other one will be zero). But the downside is
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//! that you have to specify the type you are dealing with:
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//!
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//! ```
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//! # #[macro_use]
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//! # extern crate float_cmp;
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//! # use float_cmp::{ApproxEq, F32Margin};
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//! # fn main() {
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//! # let a: f32 = 0.15 + 0.15 + 0.15;
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//! # let b: f32 = 0.1 + 0.1 + 0.25;
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//! assert!( approx_eq!(f32, a, b, ulps = 2) );
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//! assert!( approx_eq!(f32, a, b, epsilon = 0.00000003) );
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//! assert!( approx_eq!(f32, a, b, epsilon = 0.00000003, ulps = 2) );
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//! assert!( approx_eq!(f32, a, b, (0.0, 2)) );
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//! assert!( approx_eq!(f32, a, b, F32Margin { epsilon: 0.0, ulps: 2 }) );
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//! assert!( approx_eq!(f32, a, b, F32Margin::default()) );
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//! assert!( approx_eq!(f32, a, b) ); // uses the default
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//! # }
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//! ```
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//!
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//! For most cases, I recommend you use a smallish integer for the `ulps` parameter (1 to 5
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//! or so), and a similar small multiple of the floating point's EPSILON constant (1.0 to 5.0
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//! or so), but there are *plenty* of cases where this is insufficient.
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//!
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//! ## Implementing these traits
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//!
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//! You can implement `ApproxEq` for your own complex types like shown below.
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//! The floating point type `F` must be `Copy`, but for large types you can implement
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//! it for references to your type as shown.
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//!
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//! ```
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//! use float_cmp::ApproxEq;
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//!
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//! pub struct Vec2<F> {
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//! pub x: F,
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//! pub y: F,
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//! }
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//!
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//! impl<'a, M: Copy + Default, F: Copy + ApproxEq<Margin=M>> ApproxEq for &'a Vec2<F> {
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//! type Margin = M;
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//!
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//! fn approx_eq<T: Into<Self::Margin>>(self, other: Self, margin: T) -> bool {
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//! let margin = margin.into();
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//! self.x.approx_eq(other.x, margin)
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//! && self.y.approx_eq(other.y, margin)
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//! }
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//! }
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//! ```
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//!
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//! ## Non floating-point types
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//!
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//! `ApproxEq` can be implemented for non floating-point types as well, since `Margin` is
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//! an associated type.
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//!
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//! The `efloat` crate implements (or soon will implement) `ApproxEq` for a compound type
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//! that tracks floating point error bounds by checking if the error bounds overlap.
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//! In that case `type Margin = ()`.
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//!
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//! ## Inspiration
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//!
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//! This crate was inspired by this Random ASCII blog post:
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//!
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//! [https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/](https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/)
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#[cfg(feature="num-traits")]
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extern crate num_traits;
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#[macro_use]
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mod macros;
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pub fn trials() {
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println!("are they approximately equal?: {:?}",
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approx_eq!(f32, 1.0, 1.0000001));
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}
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mod ulps;
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pub use self::ulps::Ulps;
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mod ulps_eq;
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pub use self::ulps_eq::ApproxEqUlps;
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mod eq;
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pub use self::eq::{ApproxEq, F32Margin, F64Margin};
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#[cfg(feature="num-traits")]
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mod ratio;
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#[cfg(feature="num-traits")]
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pub use self::ratio::ApproxEqRatio;
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