1// Copyright 2014-2020 Optimal Computing (NZ) Ltd.
2// Licensed under the MIT license. See LICENSE for details.
3
4//! # float-cmp
5//!
6//! float-cmp defines and implements traits for approximate comparison of floating point types
7//! which have fallen away from exact equality due to the limited precision available within
8//! floating point representations. Implementations of these traits are provided for `f32`
9//! and `f64` types.
10//!
11//! When I was a kid in the '80s, the programming rule was "Never compare floating point
12//! numbers". If you can follow that rule and still get the outcome you desire, then more
13//! power to you. However, if you really do need to compare them, this crate provides a
14//! reasonable way to do so.
15//!
16//! Another crate `efloat` offers another solution by providing a floating point type that
17//! tracks its error bounds as operations are performed on it, and thus can implement the
18//! `ApproxEq` trait in this crate more accurately, without specifying a `Margin`.
19//!
20//! The recommended go-to solution (although it may not be appropriate in all cases) is the
21//! `approx_eq()` function in the `ApproxEq` trait (or better yet, the macros). For `f32`
22//! and `f64`, the `F32Margin` and `F64Margin` types are provided for specifying margins as
23//! both an epsilon value and an ULPs value, and defaults are provided via `Default`
24//! (although there is no perfect default value that is always appropriate, so beware).
25//!
26//! Several other traits are provided including `Ulps`, `ApproxEqUlps`, `ApproxOrdUlps`, and
27//! `ApproxEqRatio`.
28//!
29//! ## The problem
30//!
31//! Floating point operations must round answers to the nearest representable number. Multiple
32//! operations may result in an answer different from what you expect. In the following example,
33//! the assert will fail, even though the printed output says "0.45 == 0.45":
34//!
35//! ```should_panic
36//! # extern crate float_cmp;
37//! # use float_cmp::ApproxEq;
38//! # fn main() {
39//! let a: f32 = 0.15 + 0.15 + 0.15;
40//! let b: f32 = 0.1 + 0.1 + 0.25;
41//! println!("{} == {}", a, b);
42//! assert!(a==b) // Fails, because they are not exactly equal
43//! # }
44//! ```
45//!
46//! This fails because the correct answer to most operations isn't exactly representable, and so
47//! your computer's processor chooses to represent the answer with the closest value it has
48//! available. This introduces error, and this error can accumulate as multiple operations are
49//! performed.
50//!
51//! ## The solution
52//!
53//! With `ApproxEq`, we can get the answer we intend:
54//!
55//! ```
56//! # #[macro_use]
57//! # extern crate float_cmp;
58//! # use float_cmp::{ApproxEq, F32Margin};
59//! # fn main() {
60//! let a: f32 = 0.15 + 0.15 + 0.15;
61//! let b: f32 = 0.1 + 0.1 + 0.25;
62//! println!("{} == {}", a, b);
63//! // They are equal, within 2 ulps
64//! assert!( approx_eq!(f32, a, b, ulps = 2) );
65//! # }
66//! ```
67//!
68//! ## Some explanation
69//!
70//! We use the term ULP (units of least precision, or units in the last place) to mean the
71//! difference between two adjacent floating point representations (adjacent meaning that there is
72//! no floating point number between them). This term is borrowed from prior work (personally I
73//! would have chosen "quanta"). The size of an ULP (measured as a float) varies
74//! depending on the exponents of the floating point numbers in question. That is a good thing,
75//! because as numbers fall away from equality due to the imprecise nature of their representation,
76//! they fall away in ULPs terms, not in absolute terms. Pure epsilon-based comparisons are
77//! absolute and thus don't map well to the nature of the additive error issue. They work fine
78//! for many ranges of numbers, but not for others (consider comparing -0.0000000028
79//! to +0.00000097).
80//!
81//! ## Using this crate
82//!
83//! By default this crate enables the `ratio` module providing the `ApproxEqRatio` trait. This
84//! feature pulls in `num-traits`. If you disable this feature, you'll need to either enable
85//! `num-traits` directly or else enable the `std` feature; otherwise it won't compile. This crate
86//! is `#![no_std]` unless you enable the `std` feature.
87//!
88//! You can use the `ApproxEq` trait directly like so:
89//!
90//! ```
91//! # extern crate float_cmp;
92//! # use float_cmp::{ApproxEq, F32Margin};
93//! # fn main() {
94//! # let a: f32 = 0.15 + 0.15 + 0.15;
95//! # let b: f32 = 0.1 + 0.1 + 0.25;
96//! assert!( a.approx_eq(b, F32Margin { ulps: 2, epsilon: 0.0 }) );
97//! # }
98//! ```
99//!
100//! We have implemented `From<(f32,i32)>` for `F32Margin` (and similarly for `F64Margin`)
101//! so you can use this shorthand:
102//!
103//! ```
104//! # extern crate float_cmp;
105//! # use float_cmp::{ApproxEq, F32Margin};
106//! # fn main() {
107//! # let a: f32 = 0.15 + 0.15 + 0.15;
108//! # let b: f32 = 0.1 + 0.1 + 0.25;
109//! assert!( a.approx_eq(b, (0.0, 2)) );
110//! # }
111//! ```
112//!
113//! With macros, it is easier to be explicit about which type of margin you wish to set,
114//! without mentioning the other one (the other one will be zero). But the downside is
115//! that you have to specify the type you are dealing with:
116//!
117//! ```
118//! # #[macro_use]
119//! # extern crate float_cmp;
120//! # use float_cmp::{ApproxEq, F32Margin};
121//! # fn main() {
122//! # let a: f32 = 0.15 + 0.15 + 0.15;
123//! # let b: f32 = 0.1 + 0.1 + 0.25;
124//! assert!( approx_eq!(f32, a, b, ulps = 2) );
125//! assert!( approx_eq!(f32, a, b, epsilon = 0.00000003) );
126//! assert!( approx_eq!(f32, a, b, epsilon = 0.00000003, ulps = 2) );
127//! assert!( approx_eq!(f32, a, b, (0.0, 2)) );
128//! assert!( approx_eq!(f32, a, b, F32Margin { epsilon: 0.0, ulps: 2 }) );
129//! assert!( approx_eq!(f32, a, b, F32Margin::default()) );
130//! assert!( approx_eq!(f32, a, b) ); // uses the default
131//! # }
132//! ```
133//!
134//! For most cases, I recommend you use a smallish integer for the `ulps` parameter (1 to 5
135//! or so), and a similar small multiple of the floating point's EPSILON constant (1.0 to 5.0
136//! or so), but there are *plenty* of cases where this is insufficient.
137//!
138//! ## Implementing these traits
139//!
140//! You can implement `ApproxEq` for your own complex types like shown below.
141//! The floating point type `F` must be `Copy`, but for large types you can implement
142//! it for references to your type as shown.
143//!
144//! ```
145//! use float_cmp::ApproxEq;
146//!
147//! pub struct Vec2<F> {
148//! pub x: F,
149//! pub y: F,
150//! }
151//!
152//! impl<'a, M: Copy + Default, F: Copy + ApproxEq<Margin=M>> ApproxEq for &'a Vec2<F> {
153//! type Margin = M;
154//!
155//! fn approx_eq<T: Into<Self::Margin>>(self, other: Self, margin: T) -> bool {
156//! let margin = margin.into();
157//! self.x.approx_eq(other.x, margin)
158//! && self.y.approx_eq(other.y, margin)
159//! }
160//! }
161//! ```
162//!
163//! ## Non floating-point types
164//!
165//! `ApproxEq` can be implemented for non floating-point types as well, since `Margin` is
166//! an associated type.
167//!
168//! The `efloat` crate implements (or soon will implement) `ApproxEq` for a compound type
169//! that tracks floating point error bounds by checking if the error bounds overlap.
170//! In that case `type Margin = ()`.
171//!
172//! ## Inspiration
173//!
174//! This crate was inspired by this Random ASCII blog post:
175//!
176//! [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/)
177
178#![cfg_attr(not(feature = "std"), no_std)]
179
180#[macro_use]
181mod macros;
182
183mod ulps;
184pub use self::ulps::Ulps;
185
186mod ulps_eq;
187pub use self::ulps_eq::ApproxEqUlps;
188
189mod eq;
190pub use self::eq::{ApproxEq, F32Margin, F64Margin};
191
192#[cfg(feature="ratio")]
193mod ratio;
194#[cfg(feature="ratio")]
195pub use self::ratio::ApproxEqRatio;
196