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```
```// Copyright 2014-2020 Optimal Computing (NZ) Ltd.
// Licensed under the MIT license.  See LICENSE for details.

#[cfg(feature = "num-traits")]
#[allow(unused_imports)]
use num_traits::float::FloatCore;
use super::Ulps;

/// ApproxEqUlps is a trait for approximate equality comparisons.
/// The associated type Flt is a floating point type which implements Ulps, and is
/// required so that this trait can be implemented for compound types (e.g. vectors),
/// not just for the floats themselves.
pub trait ApproxEqUlps {
type Flt: Ulps;

/// This method tests for `self` and `other` values to be approximately equal
/// within ULPs (Units of Least Precision) floating point representations.
/// Differing signs are always unequal with this method, and zeroes are only
/// equal to zeroes. Use approx_eq() from the ApproxEq trait if that is more
/// appropriate.
fn approx_eq_ulps(&self, other: &Self, ulps: <Self::Flt as Ulps>::U) -> bool;

/// This method tests for `self` and `other` values to be not approximately
/// equal within ULPs (Units of Least Precision) floating point representations.
/// Differing signs are always unequal with this method, and zeroes are only
/// equal to zeroes. Use approx_eq() from the ApproxEq trait if that is more
/// appropriate.
#[inline]
fn approx_ne_ulps(&self, other: &Self, ulps: <Self::Flt as Ulps>::U) -> bool {
!self.approx_eq_ulps(other, ulps)
}
}

impl ApproxEqUlps for f32 {
type Flt = f32;

fn approx_eq_ulps(&self, other: &f32, ulps: i32) -> bool {
// -0 and +0 are drastically far in ulps terms, so
// we need a special case for that.
if *self==*other { return true; }

// Handle differing signs as a special case, even if
// they are very close, most people consider them
// unequal.
if self.is_sign_positive() != other.is_sign_positive() { return false; }

let diff: i32 = self.ulps(other);
diff >= -ulps && diff <= ulps
}
}

#[test]
fn f32_approx_eq_ulps_test1() {
let f: f32 = 0.1_f32;
let mut sum: f32 = 0.0_f32;
for _ in 0_isize..10_isize { sum += f; }
let product: f32 = f * 10.0_f32;
assert!(sum != product); // Should not be directly equal:
assert!(sum.approx_eq_ulps(&product,1) == true); // But should be close
assert!(sum.approx_eq_ulps(&product,0) == false);
}
#[test]
fn f32_approx_eq_ulps_test2() {
let x: f32 = 1000000_f32;
let y: f32 = 1000000.1_f32;
assert!(x != y); // Should not be directly equal
assert!(x.approx_eq_ulps(&y,2) == true);
assert!(x.approx_eq_ulps(&y,1) == false);
}
#[test]
fn f32_approx_eq_ulps_test_zeroes() {
let x: f32 = 0.0_f32;
let y: f32 = -0.0_f32;
assert!(x.approx_eq_ulps(&y,0) == true);
}

impl ApproxEqUlps for f64 {
type Flt = f64;

fn approx_eq_ulps(&self, other: &f64, ulps: i64) -> bool {
// -0 and +0 are drastically far in ulps terms, so
// we need a special case for that.
if *self==*other { return true; }

// Handle differing signs as a special case, even if
// they are very close, most people consider them
// unequal.
if self.is_sign_positive() != other.is_sign_positive() { return false; }

let diff: i64 = self.ulps(other);
diff >= -ulps && diff <= ulps
}
}

#[test]
fn f64_approx_eq_ulps_test1() {
let f: f64 = 0.1_f64;
let mut sum: f64 = 0.0_f64;
for _ in 0_isize..10_isize { sum += f; }
let product: f64 = f * 10.0_f64;
assert!(sum != product); // Should not be directly equal:
assert!(sum.approx_eq_ulps(&product,1) == true); // But should be close
assert!(sum.approx_eq_ulps(&product,0) == false);
}
#[test]
fn f64_approx_eq_ulps_test2() {
let x: f64 = 1000000_f64;
let y: f64 = 1000000.0000000003_f64;
assert!(x != y); // Should not be directly equal
assert!(x.approx_eq_ulps(&y,3) == true);
assert!(x.approx_eq_ulps(&y,2) == false);
}
#[test]
fn f64_approx_eq_ulps_test_zeroes() {
let x: f64 = 0.0_f64;
let y: f64 = -0.0_f64;
assert!(x.approx_eq_ulps(&y,0) == true);
}
```