Adding upstream version 115.7.0esr.upstream/115.7.0esrupstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

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+# float-cmp
+
+[![Build Status](https://travis-ci.org/mikedilger/float-cmp.svg?branch=master)](https://travis-ci.org/mikedilger/float-cmp)
+[![MIT licensed](https://img.shields.io/badge/license-MIT-blue.svg)](./LICENSE)
+
+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":
+
+```rust
+  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:
+```rust
+  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
+
+You can use the `ApproxEq` trait directly like so:
+
+```rust
+    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:
+
+```rust
+    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:
+
+```rust
+    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.
+
+```rust
+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/](https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/)