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//! Explicitly bounded comparison of floating point numbers. //! //! Comparing floating point values for equality is *really hard*. To get it //! right requires careful thought and iteration based on the needs of each //! specific algorithm's inputs and error margins. This API provides a toolbox //! of components to make your options clear and your choices explicit to //! future maintainers. //! //! # Table of Contents //! //! - [Background](#background) //! - [Making comparisons](#making-comparisons) //! - [Absolute epsilon comparison](#absolute-epsilon-comparison) //! - [Relative epsilon comparison](#relative-epsilon-comparison) //! - [Units in the Last Place (ULPs) comparison](#units-in-the-last-place-ulps-comparison) //! - [Comparing composite types](#comparing-composite-types) //! - [Error messages](#error-messages) //! - [Comparing custom types](#comparing-custom-types) //! //! # Background //! //! Given how widely algorithmic requirements can vary, `float_eq` explores the //! idea that there are no generally sensible default margins for comparisons. //! This is in contrast to the approach taken by many other crates, which often //! provide default epsilon values in checks or implicitly favour particular //! algorithms. The author's hope is that by exposing the inherent complexity in //! a uniform way, programmers will find it easier to develop an intuition for how //! to write effective comparisons. The trade-off is that each individual //! comparison requires more iteration time and thought. //! //! And yes, this is yet another crate built on the principles described in *that* //! Random ASCII [floating point comparison] article, which is highly recommended //! background reading 🙂. //! //! # Making comparisons //! //! The [`float_eq!`] and [`float_ne!`] macros compare two floating point //! expressions for equality based on the result of one or more different kinds //! of check. Each check is invoked by name and an upper boundary, so for example //! `rel <= 0.1`, should be read as *"a [relative epsilon comparison](#relative-epsilon-comparison) //! with a maximum difference of less than or equal to `0.1`"*. If multiple checks //! are provided then they are executed in order from left to right, shortcutting //! to return early if one passes. The corresponding [`assert_float_eq!`] and //! [`assert_float_ne!`] use the same interface: //! //! ```rust //! use float_eq::{assert_float_eq, assert_float_ne, float_eq, float_ne}; //! //! assert!(float_eq!(1000.0f32, 1000.0002, ulps <= 4)); //! //! const ROUNDING_ERROR: f32 = 0.000_345_266_98; // f32::EPSILON.sqrt() //! assert!(float_ne!(4.0f32, 4.1, rel <= ROUNDING_ERROR)); //! //! const RECIP_REL_EPSILON: f32 = 0.000_366_210_94; // 1.5 * 2f32.powi(-12) //! assert_float_eq!(0.1f32.recip(), 10.0, rel <= RECIP_REL_EPSILON); //! //! assert_float_ne!(0.0f32, 0.000_1, abs <= 0.000_05, ulps <= 4); //! ``` //! //! The ideal choice of comparison will vary on a case by case basis, and depends //! on the input range and error margins of the expressions to be compared. For //! example, a test of the result of [finite difference approximation of //! derivatives] might use a relative epsilon check with a `max_diff` of the `sqrt` //! of machine epsilon, whereas a test of the SSE [`_mm_rcp_ps` operation] could //! instead opt for a maximum relative error of `1.5 * 2^(-12)` based on the //! available documentation. Algorithm stability can play a big part in the size //! of these margins, and it can be worth seeing if code might be rearranged to //! reduce loss of precision if you find yourself using large bounds. //! //! Relative comparisons (`ulps` and `rel`) are usually a good choice for comparing //! [normal floats] (e.g. when [`f32::is_normal`] is true). However, they become //! far too strict for comparisons of very small numbers with zero, where the //! relative differences are very large but the absolute difference is tiny. This //! is where you might choose to use an absolute epsilon (`abs`) comparison instead. //! There are also potential performance implications based on the target hardware. //! //! Be prepared to research, test, benchmark and iterate on your comparisons. The //! [floating point comparison] article which informed this crate's implementation //! is a good place to start. //! //! ## Absolute epsilon comparison //! //! A check to see how far apart two expressions are by comparing the absolute //! difference between them to an absolute, unscaled epsilon. Equivalent to, using //! `f32` as an example: //! //! ```rust //! fn float_eq_abs(a: f32, b: f32, max_diff: f32) -> bool { //! // the PartialEq check covers equality of infinities //! a == b || (a - b).abs() <= max_diff //! } //! # float_eq::assert_float_eq!(4f32, 4.000_002_5, abs <= 0.000_002_5); //! # assert!(float_eq_abs(4f32, 4.000_002_5, 0.000_002_5)); //! ``` //! //! Absolute epsilon tests *do not* work well for general floating point comparison, //! because they do not take into account that floating point values' precision //! changes with their magnitude. Thus `max_diff` must be very specific and //! dependent on the exact values being compared: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! let a = 1.0f32; //! let b = 1.000_000_1f32; // the next representable value above 1.0 //! assert_float_eq!(a, b, abs <= 0.000_000_2); // equal //! assert_float_ne!(a * 4.0, b * 4.0, abs <= 0.000_000_2); // not equal //! assert_float_eq!(a * 4.0, b * 4.0, abs <= 0.000_000_5); // equal //! ``` //! //! Whereas a relative epsilon comparison could cope with this since it scales by //! the size of the largest input parameter: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! # let a: f32 = 1.0; //! # let b: f32 = 1.000_000_1; //! assert_float_eq!(a, b, rel <= 0.000_000_2); //! assert_float_eq!(a * 4.0, b * 4.0, rel <= 0.000_000_2); //! ``` //! //! However, absolute epsilon comparison is often the best choice when comparing //! values directly against zero, especially when those values have undergone //! [catastrophic cancellation], like the subtractions below. In this case, the //! relative comparison methods break down due to the relative ratio between values //! being so high compared to their absolute difference: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! assert_float_eq!(1.0f32 - 1.000_000_1, 0.0, abs <= 0.000_000_2); // equal //! assert_float_ne!(1.0f32 - 1.000_000_1, 0.0, rel <= 0.000_000_2); // not equal //! assert_float_ne!(1.0f32 - 1.000_000_1, 0.0, ulps <= 1); // not equal //! ``` //! //! Absolute epsilon comparisons: //! - Are useful for checking if a float is equal to zero, especially if it has //! undergone an operation that suffers from [catastrophic cancellation] or is //! a [subnormal value]. //! - Are almost certainly not what you want to use when testing [normal floats] //! for equality. `rel` and `ulps` checks can be easier to parameterise and //! reason about. //! - Can be useful for testing equality of infinities. //! //! ## Relative epsilon comparison //! //! A check to see how far apart two expressions are by comparing the absolute //! difference between them to an epsilon that is scaled to the precision of the //! larger input. Equivalent to, using `f32` as an example: //! //! ```rust //! fn float_eq_rel(a: f32, b: f32, max_diff: f32) -> bool { //! // the PartialEq check covers equality of infinities //! a == b || { //! let largest = a.abs().max(b.abs()); //! (a - b).abs() <= (largest * max_diff) //! } //! } //! # float_eq::assert_float_eq!(4.0f32, 4.000_002_5, rel <= 0.000_000_6); //! # assert!(float_eq_rel(4.0f32, 4.000_002_5, 0.000_000_6)); //! ``` //! //! This makes it suitable for general comparison of values where the ratio between //! those values is relatively stable (e.g. [normal floats], excluding //! infinity): //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! let a: f32 = 1.0; //! let b: f32 = 1.000_000_1; // the next representable value above 1.0 //! assert_float_eq!(a, b, rel <= 0.000_000_2); //! assert_float_eq!(a * 4.0, b * 4.0, rel <= 0.000_000_2); //! ``` //! //! However, relative epsilon comparison becomes far too strict when the numbers //! being checked are too close to zero, since the relative ratio between the values //! can be huge whilst the absolute difference remains tiny. In these circumstances, //! it is usually better to make an absolute epsilon check instead, especially if //! your algorithm contains some form of [catastrophic cancellation], like these //! subtractions: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! assert_float_ne!(1.0f32 - 1.000_000_1, 0.0, rel <= 0.000_000_2); // not equal //! assert_float_eq!(1.0f32 - 1.000_000_1, 0.0, abs <= 0.000_000_2); // equal //! ``` //! //! Relative epsilon comparisons: //! - Are useful for checking if two [normal floats] are equal. //! - Aren't a good choice when checking values against zero, where `abs` is often //! far better. //! - Have slightly counterintuitive results around powers of two values, where //! the relative precision ratio changes due to way the floating point exponent //! works. //! - Are not useful at infinity, where any comparison using a non-zero margin //! will compare true. //! //! ## Units in the Last Place (ULPs) comparison //! //! A check to see how far apart two expressions are by comparing the number of //! discrete values that can be expressed between them. This works by interpreting //! the bitwise representation of the input values as integers and comparing the //! absolute difference between those. Equivalent to, using `f32` as an example: //! //! ```rust //! fn float_eq_ulps(a: f32, b: f32, max_diff: u32) -> bool { //! if a.is_nan() || b.is_nan() { //! false // NaNs are never equal //! } else if a.is_sign_positive() != b.is_sign_positive() { //! a == b // values of different signs are only equal if both are zero. //! } else { //! let a_bits = a.to_bits(); //! let b_bits = b.to_bits(); //! let max = a_bits.max(b_bits); //! let min = a_bits.min(b_bits); //! (max - min) <= max_diff //! } //! } //! # float_eq::assert_float_eq!(4f32, 4.000_002_5, ulps <= 5); //! # assert!(float_eq_ulps(4f32, 4.000_002_5, 5)); //! ``` //! //! Thanks to a deliberate quirk in the way the [underlying format] of IEEE floats //! was designed, this is a good measure of how near two values are that scales with //! their relative precision: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! assert_float_eq!(1.0f32, 1.000_000_1, ulps <= 1); //! assert_float_eq!(4.0f32, 4.000_000_5, ulps <= 1); //! assert_float_eq!(-1_000_000.0f32, -1_000_000.06, ulps <= 1); //! ``` //! //! However, it becames far too strict when both expressions are close to zero, //! since the relative difference between them can be very large, whilst the //! absolute difference remains small. In these circumstances, it is usually better //! to make an absolute epsilon check instead, especially if your algorithm contains //! some form of [catastrophic cancellation], like these subtractions: //! //! ```rust //! # use float_eq::{assert_float_eq, assert_float_ne}; //! assert_float_ne!(1.0f32 - 1.000_000_1, 0.0, ulps <= 1); // not equal //! assert_float_eq!(1.0f32 - 1.000_000_1, 0.0, abs <= 0.000_000_2); // equal //! ``` //! //! ULPs based comparisons: //! - Are useful for checking if two [normal floats] are equal. //! - Aren't a good choice when checking values against zero, where `abs` is often //! a better choice. //! - Provide a way to precisely tweak `max_diff` margins, since they have a 1-to-1 //! correlation with the underlying representation. //! - Have slightly counterintuitive results around powers of two values, where //! the relative precision ratio changes due to way the floating point exponent //! works. //! - Do not work at all if the two values being checked have different signs. //! - Whilst slightly counterintuitive at infinity (`MAX` is one ULP away from //! `INFINITY`), are more useful than `rel` checks for this. //! //! # Comparing composite types //! //! When comparing composite values, it can be helpful to specify thresholds //! separately for each individual field. The `abs`, `rel` and `ulps` checks //! expect this behaviour. Conversely, the `abs_all`, `rel_all` and `ulps_all` //! checks accept a single epsilon that is then used to compare across all fields. //! For example, arrays may be compared using an epsilon that covers each index //! separately: //! //! ``` //! # use float_eq::assert_float_eq; //! let a = [1.0, -2.0, 3.0]; //! let b = [-1.0, 2.0, 3.5]; //! assert_float_eq!(a, b, abs <= [2.0, 4.0, 0.5]); //! ``` //! //! Or with the same threshold across all values: //! //! ``` //! # use float_eq::assert_float_eq; //! # let a = [1.0, -2.0, 3.0]; //! # let b = [-1.0, 2.0, 3.5]; //! assert_float_eq!(a, b, abs_all <= 4.0); //! ``` //! //! Similarly, if [`FloatEq`] and [`FloatEqAll`] have been implemented for a //! struct type: //! //! ``` //! # use float_eq::{ //! # assert_float_eq, FloatDiff, FloatEq, FloatEqAll, FloatEqDebug, FloatEqAllDebug //! # }; //! # //! # #[derive(Debug, Clone, Copy, PartialEq)] //! # struct Complex32 { re: f32, im: f32 } //! # //! # #[derive(Debug, Clone, Copy, PartialEq)] //! # struct Complex32Ulps { re: u32, im: u32 } //! # //! # impl FloatDiff for Complex32 { //! # type AbsDiff = Complex32; //! # type UlpsDiff = Complex32Ulps; //! # fn abs_diff(&self, other: &Self) -> Complex32 { //! # Complex32 { //! # re: self.re.abs_diff(&other.re), //! # im: self.im.abs_diff(&other.im), //! # } //! # } //! # fn ulps_diff(&self, other: &Self) -> Option<Complex32Ulps> { //! # Some(Complex32Ulps { //! # re: self.re.ulps_diff(&other.re)?, //! # im: self.im.ulps_diff(&other.im)?, //! # }) //! # } //! # } //! # //! # impl FloatEq for Complex32 { //! # type Epsilon = Complex32; //! # type UlpsEpsilon = Complex32Ulps; //! # fn eq_abs(&self, other: &Self, max_diff: &Complex32) -> bool { //! # self.re.eq_abs(&other.re, &max_diff.re) && self.im.eq_abs(&other.im, &max_diff.im) //! # } //! # fn eq_rel(&self, other: &Self, max_diff: &Complex32) -> bool { //! # self.re.eq_rel(&other.re, &max_diff.re) && self.im.eq_rel(&other.im, &max_diff.im) //! # } //! # fn eq_ulps(&self, other: &Self, max_diff: &Complex32Ulps) -> bool { //! # self.re.eq_ulps(&other.re, &max_diff.re) && self.im.eq_ulps(&other.im, &max_diff.im) //! # } //! # } //! # //! # impl FloatEqAll for Complex32 { //! # type Epsilon = f32; //! # type UlpsEpsilon = u32; //! # fn eq_abs_all(&self, other: &Self, max_diff: &f32) -> bool { //! # self.re.eq_abs_all(&other.re, &max_diff) && self.im.eq_abs_all(&other.im, &max_diff) //! # } //! # fn eq_rel_all(&self, other: &Self, max_diff: &f32) -> bool { //! # self.re.eq_rel_all(&other.re, &max_diff) && self.im.eq_rel_all(&other.im, &max_diff) //! # } //! # fn eq_ulps_all(&self, other: &Self, max_diff: &u32) -> bool { //! # self.re.eq_ulps_all(&other.re, &max_diff) && self.im.eq_ulps_all(&other.im, &max_diff) //! # } //! # } //! # //! # impl FloatEqDebug for Complex32 { //! # type DebugEpsilon = Complex32; //! # type DebugUlpsEpsilon = Complex32Ulps; //! # fn debug_abs_epsilon(&self, other: &Self, max_diff: &Complex32) -> Complex32 { //! # Complex32 { //! # re: self.re.debug_abs_epsilon(&other.re, &max_diff.re), //! # im: self.im.debug_abs_epsilon(&other.re, &max_diff.im), //! # } //! # } //! # fn debug_rel_epsilon(&self, other: &Self, max_diff: &Complex32) -> Complex32 { //! # Complex32 { //! # re: self.re.debug_rel_epsilon(&other.re, &max_diff.re), //! # im: self.im.debug_rel_epsilon(&other.re, &max_diff.im), //! # } //! # } //! # fn debug_ulps_epsilon(&self, other: &Self, max_diff: &Complex32Ulps) -> Complex32Ulps { //! # Complex32Ulps { //! # re: self.re.debug_ulps_epsilon(&other.re, &max_diff.re), //! # im: self.im.debug_ulps_epsilon(&other.re, &max_diff.im), //! # } //! # } //! # } //! # //! # impl FloatEqAllDebug for Complex32 { //! # type DebugEpsilon = Complex32; //! # type DebugUlpsEpsilon = Complex32Ulps; //! # fn debug_abs_all_epsilon(&self, other: &Self, max_diff: &f32) -> Complex32 { //! # Complex32 { //! # re: self.re.debug_abs_all_epsilon(&other.re, &max_diff), //! # im: self.im.debug_abs_all_epsilon(&other.re, &max_diff), //! # } //! # } //! # fn debug_rel_all_epsilon(&self, other: &Self, max_diff: &f32) -> Complex32 { //! # Complex32 { //! # re: self.re.debug_rel_all_epsilon(&other.re, &max_diff), //! # im: self.im.debug_rel_all_epsilon(&other.re, &max_diff), //! # } //! # } //! # fn debug_ulps_all_epsilon(&self, other: &Self, max_diff: &u32) -> Complex32Ulps { //! # Complex32Ulps { //! # re: self.re.debug_ulps_all_epsilon(&other.re, &max_diff), //! # im: self.im.debug_ulps_all_epsilon(&other.re, &max_diff), //! # } //! # } //! # } //! let a = Complex32 { re: 2.0, im: 4.000_002 }; //! let b = Complex32 { re: 2.000_000_5, im: 4.0 }; //! //! assert_float_eq!(a, b, rel <= Complex32 { re: 0.000_000_25, im: 0.000_000_5 }); //! assert_float_eq!(a, b, rel_all <= 0.000_000_5); //! //! assert_float_eq!(a, b, ulps <= Complex32Ulps { re: 2, im: 4 }); //! assert_float_eq!(a, b, ulps_all <= 4); //! ``` //! //! # Error messages //! //! Assertion failure messages provide context information that hopefully helps //! in determining how a check failed. The absolute difference (`abs_diff`) and //! ULPs difference (`ulps_diff`) between the values are always provided, and //! then the epsilon values used in the check are listed afterwards. For example, //! this line: //! //! ```should_panic //! # use float_eq::assert_float_eq; //! assert_float_eq!(4.0f32, 4.000_008, rel <= 0.000_001); //! ``` //! //! Panics with this error message, where the relative epsilon, `[rel] ε`, has //! been scaled based on the size of the inputs (ε is the greek letter epsilon): //! //! ```text //! thread 'test' panicked at 'assertion failed: `float_eq!(left, right, rel <= ε)` //! left: `4.0`, //! right: `4.000008`, //! abs_diff: `0.000008106232`, //! ulps_diff: `Some(17)`, //! [rel] ε: `0.000004000008`', assert_failure.rs:15:5 //! ``` //! //! # Comparing custom types //! //! Comparison of new types is supported by implementing [`FloatEq`] and [`FloatEqAll`]. //! If assert support is required, then [`FloatDiff`] and [`FloatEqDebug`]/[`FloatEqAllDebug`] //! should also be implemented, as they provide important context information on //! failure. //! //! [`assert_float_eq!`]: macro.assert_float_eq.html //! [`assert_float_ne!`]: macro.assert_float_ne.html //! [`float_eq!`]: macro.float_eq.html //! [`float_ne!`]: macro.float_ne.html //! [`FloatEq`]: trait.FloatEq.html //! [`FloatEqAll`]: trait.FloatEqAll.html //! [`FloatDiff`]: trait.FloatDiff.html //! [`FloatEqDebug`]: trait.FloatEqDebug.html //! [`FloatEqAllDebug`]: trait.FloatEqAllDebug.html //! //! [catastrophic cancellation]: https://en.wikipedia.org/wiki/Loss_of_significance //! [subnormal value]: https://en.wikipedia.org/wiki/Denormal_number //! [finite difference approximation of derivatives]: https://scicomp.stackexchange.com/questions/14355/choosing-epsilons //! [floating point comparison]: https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/ //! [normal floats]: https://en.wikipedia.org/wiki/Normal_number_(computing) //! [underlying format]: https://randomascii.wordpress.com/2012/01/23/stupid-float-tricks-2/ //! [`_mm_rcp_ps` operation]: https://software.intel.com/sites/landingpage/IntrinsicsGuide/#text=_mm_rcp_ps&expand=4482 //! [`f32::is_normal`]: https://doc.rust-lang.org/std/primitive.f32.html#method.is_normal #![warn(missing_docs)] #![cfg_attr(not(feature = "std"), no_std)] #[macro_use] mod macros; pub use crate::macros::*; mod traits; pub use crate::traits::*; // implementations of traits mod arrays; mod primitives; mod tuples; #[cfg(feature = "num")] mod num_complex; #[cfg(feature = "num")] pub use self::num_complex::*;