From 71db9b1d526488ffb8736252e8b80ad50cf65f1f Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Wed, 10 Apr 2024 23:28:28 +0200
Subject: [PATCH 1/6] fix link

---
 src/gleam_community/maths/metrics.gleam | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/gleam_community/maths/metrics.gleam b/src/gleam_community/maths/metrics.gleam
index d5ddc69..237b900 100644
--- a/src/gleam_community/maths/metrics.gleam
+++ b/src/gleam_community/maths/metrics.gleam
@@ -38,6 +38,7 @@
 ////   * [`sorensen_dice_coefficient`](#sorensen_dice_coefficient)
 ////   * [`tversky_index`](#tversky_index)
 ////   * [`overlap_coefficient`](#overlap_coefficient)
+////   * [`levenshtein_distance`](#levenshtein_distance)
 //// * **Basic statistical measures**
 ////   * [`mean`](#mean)
 ////   * [`median`](#median)
@@ -1014,8 +1015,8 @@ pub fn cosine_similarity(
 /// - deletions
 /// - substitutions
 /// 
-/// Note: The implementation is primarily based on the elixir implementation 
-/// [https://hex.pm/packages/levenshtein](levenshtein).
+/// Note: The implementation is primarily based on the Elixir implementation 
+/// [levenshtein](https://hex.pm/packages/levenshtein).
 /// 
 /// <details>
 ///     <summary>Example:</summary>

From d9c642062a1c3cc1007f41c47b268f902b0de8a2 Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Wed, 10 Apr 2024 23:32:42 +0200
Subject: [PATCH 2/6] Run gleam update

---
 manifest.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/manifest.toml b/manifest.toml
index 6c5a157..4a7c920 100644
--- a/manifest.toml
+++ b/manifest.toml
@@ -3,7 +3,7 @@
 
 packages = [
   { name = "gleam_stdlib", version = "0.36.0", build_tools = ["gleam"], requirements = [], otp_app = "gleam_stdlib", source = "hex", outer_checksum = "C0D14D807FEC6F8A08A7C9EF8DFDE6AE5C10E40E21325B2B29365965D82EB3D4" },
-  { name = "gleeunit", version = "1.0.2", build_tools = ["gleam"], requirements = ["gleam_stdlib"], otp_app = "gleeunit", source = "hex", outer_checksum = "D364C87AFEB26BDB4FB8A5ABDE67D635DC9FA52D6AB68416044C35B096C6882D" },
+  { name = "gleeunit", version = "1.1.2", build_tools = ["gleam"], requirements = ["gleam_stdlib"], otp_app = "gleeunit", source = "hex", outer_checksum = "72CDC3D3F719478F26C4E2C5FED3E657AC81EC14A47D2D2DEBB8693CA3220C3B" },
 ]
 
 [requirements]

From b7f7b29e48274b98136519e4b076812eb06acda8 Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Sat, 13 Apr 2024 23:38:39 +0200
Subject: [PATCH 3/6] Add canberra and bary-curtis distance

---
 src/gleam_community/maths/arithmetics.gleam   |  77 +++--
 src/gleam_community/maths/metrics.gleam       | 309 +++++++++++++-----
 .../maths/arithmetics_test.gleam              |  19 +-
 test/gleam_community/maths/metrics_test.gleam | 165 ++++++++--
 4 files changed, 432 insertions(+), 138 deletions(-)

diff --git a/src/gleam_community/maths/arithmetics.gleam b/src/gleam_community/maths/arithmetics.gleam
index dbd3ffe..b37a067 100644
--- a/src/gleam_community/maths/arithmetics.gleam
+++ b/src/gleam_community/maths/arithmetics.gleam
@@ -44,6 +44,9 @@
 
 import gleam/int
 import gleam/list
+import gleam/option
+import gleam/pair
+import gleam/result
 import gleam_community/maths/conversion
 import gleam_community/maths/elementary
 import gleam_community/maths/piecewise
@@ -289,29 +292,32 @@ pub fn proper_divisors(n: Int) -> List(Int) {
 ///     </a>
 /// </div>
 ///
-/// Calculate the sum of the elements in a list:
+/// Calculate the (weighted) sum of the elements in a list:
 ///
 /// \\[
-/// \sum_{i=1}^n x_i
+/// \sum_{i=1}^n w_i x_i
 /// \\]
 ///
-/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is the value in the input list indexed by $$i$$.
+/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
+/// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
+/// a corresponding weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/arithmetics
 ///
 ///     pub fn example () {
 ///       // An empty list returns an error
 ///       []
-///       |> arithmetics.float_sum()
+///       |> arithmetics.float_sum(option.None)
 ///       |> should.equal(0.0)
 ///
 ///       // Valid input returns a result
 ///       [1.0, 2.0, 3.0]
-///       |> arithmetics.float_sum()
+///       |> arithmetics.float_sum(option.None)
 ///       |> should.equal(6.0)
 ///     }
 /// </details>
@@ -322,12 +328,18 @@ pub fn proper_divisors(n: Int) -> List(Int) {
 ///     </a>
 /// </div>
 ///
-pub fn float_sum(arr: List(Float)) -> Float {
-  case arr {
-    [] -> 0.0
-    _ ->
+pub fn float_sum(arr: List(Float), weights: option.Option(List(Float))) -> Float {
+  case arr, weights {
+    [], _ -> 0.0
+    _, option.None ->
       arr
       |> list.fold(0.0, fn(acc: Float, a: Float) -> Float { a +. acc })
+    _, option.Some(warr) -> {
+      list.zip(arr, warr)
+      |> list.fold(0.0, fn(acc: Float, a: #(Float, Float)) -> Float {
+        pair.first(a) *. pair.second(a) +. acc
+      })
+    }
   }
 }
 
@@ -385,29 +397,32 @@ pub fn int_sum(arr: List(Int)) -> Int {
 ///     </a>
 /// </div>
 ///
-/// Calculate the product of the elements in a list:
+/// Calculate the (weighted) product of the elements in a list:
 ///
 /// \\[
-/// \prod_{i=1}^n x_i
+/// \prod_{i=1}^n x_i^{w_i}
 /// \\]
 ///
-/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is the value in the input list indexed by $$i$$.
-///
+/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
+/// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
+/// a corresponding weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// 
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/arithmetics
 ///
 ///     pub fn example () {
 ///       // An empty list returns 0.0
 ///       []
-///       |> arithmetics.float_product()
+///       |> arithmetics.float_product(option.None)
 ///       |> should.equal(0.0)
 ///
 ///       // Valid input returns a result
 ///       [1.0, 2.0, 3.0]
-///       |> arithmetics.float_product()
+///       |> arithmetics.float_product(option.None)
 ///       |> should.equal(6.0)
 ///     }
 /// </details>
@@ -418,12 +433,36 @@ pub fn int_sum(arr: List(Int)) -> Int {
 ///     </a>
 /// </div>
 ///
-pub fn float_product(arr: List(Float)) -> Float {
-  case arr {
-    [] -> 1.0
-    _ ->
+pub fn float_product(
+  arr: List(Float),
+  weights: option.Option(List(Float)),
+) -> Result(Float, String) {
+  case arr, weights {
+    [], _ ->
+      1.0
+      |> Ok
+    _, option.None ->
       arr
       |> list.fold(1.0, fn(acc: Float, a: Float) -> Float { a *. acc })
+      |> Ok
+    _, option.Some(warr) -> {
+      let results =
+        list.zip(arr, warr)
+        |> list.map(fn(a: #(Float, Float)) -> Result(Float, String) {
+          pair.first(a)
+          |> elementary.power(pair.second(a))
+        })
+        |> result.all
+      case results {
+        Ok(prods) ->
+          prods
+          |> list.fold(1.0, fn(acc: Float, a: Float) -> Float { a *. acc })
+          |> Ok
+        Error(msg) ->
+          msg
+          |> Error
+      }
+    }
   }
 }
 
diff --git a/src/gleam_community/maths/metrics.gleam b/src/gleam_community/maths/metrics.gleam
index 237b900..84cb8b2 100644
--- a/src/gleam_community/maths/metrics.gleam
+++ b/src/gleam_community/maths/metrics.gleam
@@ -33,6 +33,8 @@
 ////   * [`chebyshev_distance`](#chebyshev_distance)
 ////   * [`minkowski_distance`](#minkowski_distance)
 ////   * [`cosine_similarity`](#cosine_similarity)
+////   * [`canberra_distance`](#canberra_distance)
+////   * [`braycurtis_distance`](#braycurtis_distance)
 //// * **Set & string similarity measures**
 ////   * [`jaccard_index`](#jaccard_index)
 ////   * [`sorensen_dice_coefficient`](#sorensen_dice_coefficient)
@@ -57,6 +59,48 @@ import gleam/set
 import gleam/float
 import gleam/int
 import gleam/string
+import gleam/option
+
+/// Utility function that checks all lists have the expected length
+/// Primarily used by all distance measures taking List(Float) as input
+fn check_lists(
+  xarr: List(Float),
+  yarr: List(Float),
+  weights: option.Option(List(Float)),
+) -> Result(Bool, String) {
+  case xarr, yarr {
+    [], _ ->
+      "Invalid input argument: The list xarr is empty."
+      |> Error
+    _, [] ->
+      "Invalid input argument: The list yarr is empty."
+      |> Error
+    _, _ -> {
+      let xlen: Int = list.length(xarr)
+      let ylen: Int = list.length(yarr)
+      case xlen == ylen, weights {
+        False, _ ->
+          "Invalid input argument: length(xarr) != length(yarr). Valid input is when length(xarr) == length(yarr)."
+          |> Error
+        True, option.None -> {
+          True
+          |> Ok
+        }
+        True, option.Some(warr) -> {
+          let wlen: Int = list.length(warr)
+          case xlen == wlen {
+            True ->
+              True
+              |> Ok
+            False ->
+              "Invalid input argument: length(weights) != length(xarr) and length(weights) != length(yarr). Valid input is when length(weights) == length(xarr) == length(yarr)."
+              |> Error
+          }
+        }
+      }
+    }
+  }
+}
 
 /// <div style="text-align: right;">
 ///     <a href="https://github.com/gleam-community/maths/issues">
@@ -169,8 +213,9 @@ pub fn norm(arr: List(Float), p: Float) -> Float {
 pub fn manhattan_distance(
   xarr: List(Float),
   yarr: List(Float),
+  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  minkowski_distance(xarr, yarr, 1.0)
+  minkowski_distance(xarr, yarr, 1.0, weights)
 }
 
 /// <div style="text-align: right;">
@@ -231,34 +276,24 @@ pub fn minkowski_distance(
   xarr: List(Float),
   yarr: List(Float),
   p: Float,
+  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case xarr, yarr {
-    [], _ ->
-      "Invalid input argument: The list xarr is empty."
+  case check_lists(xarr, yarr, weights) {
+    Error(msg) ->
+      msg
       |> Error
-    _, [] ->
-      "Invalid input argument: The list yarr is empty."
-      |> Error
-    _, _ -> {
-      let xlen: Int = list.length(xarr)
-      let ylen: Int = list.length(yarr)
-      case xlen == ylen {
-        False ->
-          "Invalid input argument: length(xarr) != length(yarr). Valid input is when length(xarr) == length(yarr)."
-          |> Error
+    Ok(_) -> {
+      case p <. 1.0 {
         True ->
-          case p <. 1.0 {
-            True ->
-              "Invalid input argument: p < 1. Valid input is p >= 1."
-              |> Error
-            False ->
-              list.zip(xarr, yarr)
-              |> list.map(fn(tuple: #(Float, Float)) -> Float {
-                pair.first(tuple) -. pair.second(tuple)
-              })
-              |> norm(p)
-              |> Ok
-          }
+          "Invalid input argument: p < 1. Valid input is p >= 1."
+          |> Error
+        False ->
+          list.zip(xarr, yarr)
+          |> list.map(fn(tuple: #(Float, Float)) -> Float {
+            pair.first(tuple) -. pair.second(tuple)
+          })
+          |> norm(p)
+          |> Ok
       }
     }
   }
@@ -314,8 +349,9 @@ pub fn minkowski_distance(
 pub fn euclidean_distance(
   xarr: List(Float),
   yarr: List(Float),
+  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  minkowski_distance(xarr, yarr, 2.0)
+  minkowski_distance(xarr, yarr, 2.0, weights)
 }
 
 /// <div style="text-align: right;">
@@ -364,32 +400,21 @@ pub fn euclidean_distance(
 pub fn chebyshev_distance(
   xarr: List(Float),
   yarr: List(Float),
+  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case xarr, yarr {
-    [], _ ->
-      "Invalid input argument: The list xarr is empty."
+  case check_lists(xarr, yarr, weights) {
+    Error(msg) ->
+      msg
       |> Error
-    _, [] ->
-      "Invalid input argument: The list yarr is empty."
-      |> Error
-    _, _ -> {
-      let xlen: Int = list.length(xarr)
-      let ylen: Int = list.length(yarr)
-      case xlen == ylen {
-        False ->
-          "Invalid input argument: length(xarr) != length(yarr). Valid input is when length(xarr) == length(yarr)."
-          |> Error
-        True -> {
-          let differences =
-            list.zip(xarr, yarr)
-            |> list.map(fn(tuple: #(Float, Float)) -> Float {
-              { pair.first(tuple) -. pair.second(tuple) }
-              |> piecewise.float_absolute_value()
-            })
-          differences
-          |> piecewise.list_maximum(float.compare)
-        }
-      }
+    Ok(_) -> {
+      let differences =
+        list.zip(xarr, yarr)
+        |> list.map(fn(tuple: #(Float, Float)) -> Float {
+          { pair.first(tuple) -. pair.second(tuple) }
+          |> piecewise.float_absolute_value()
+        })
+      differences
+      |> piecewise.list_maximum(float.compare)
     }
   }
 }
@@ -441,7 +466,7 @@ pub fn mean(arr: List(Float)) -> Result(Float, String) {
       |> Error
     _ ->
       arr
-      |> arithmetics.float_sum()
+      |> arithmetics.float_sum(option.None)
       |> fn(a: Float) -> Float {
         a /. conversion.int_to_float(list.length(arr))
       }
@@ -579,7 +604,7 @@ pub fn variance(arr: List(Float), ddof: Int) -> Result(Float, String) {
             let assert Ok(result) = elementary.power(a -. mean, 2.0)
             result
           })
-          |> arithmetics.float_sum()
+          |> arithmetics.float_sum(option.None)
           |> fn(a: Float) -> Float {
             a
             /. {
@@ -969,34 +994,23 @@ pub fn overlap_coefficient(xset: set.Set(a), yset: set.Set(a)) -> Float {
 pub fn cosine_similarity(
   xarr: List(Float),
   yarr: List(Float),
+  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case xarr, yarr {
-    [], _ ->
-      "Invalid input argument: The list xarr is empty."
-      |> Error
-    _, [] ->
-      "Invalid input argument: The list yarr is empty."
+  case check_lists(xarr, yarr, weights) {
+    Error(msg) ->
+      msg
       |> Error
-    _, _ -> {
-      let xlen: Int = list.length(xarr)
-      let ylen: Int = list.length(yarr)
-      case xlen == ylen {
-        False ->
-          "Invalid input argument: length(xarr) != length(yarr). Valid input is when length(xarr) == length(yarr)."
-          |> Error
-        True -> {
-          list.fold(
-            list.zip(xarr, yarr),
-            0.0,
-            fn(acc: Float, a: #(Float, Float)) -> Float {
-              let result: Float = pair.first(a) *. pair.second(a)
-              result +. acc
-            },
-          )
-          /. { norm(xarr, 2.0) *. norm(yarr, 2.0) }
-          |> Ok
-        }
-      }
+    Ok(_) -> {
+      list.fold(
+        list.zip(xarr, yarr),
+        0.0,
+        fn(acc: Float, a: #(Float, Float)) -> Float {
+          let result: Float = pair.first(a) *. pair.second(a)
+          result +. acc
+        },
+      )
+      /. { norm(xarr, 2.0) *. norm(yarr, 2.0) }
+      |> Ok
     }
   }
 }
@@ -1119,3 +1133,140 @@ fn distance_list_helper(
     }
   }
 }
+
+/// <div style="text-align: right;">
+///     <a href="https://github.com/gleam-community/maths/issues">
+///         <small>Spot a typo? Open an issue!</small>
+///     </a>
+/// </div>
+/// 
+/// <details>
+///     <summary>Example:</summary>
+///
+///     import gleeunit/should
+///     import gleam_community/maths/metrics
+///
+///     pub fn example () {
+///     }
+/// </details>
+///
+/// <div style="text-align: right;">
+///     <a href="#">
+///         <small>Back to top ↑</small>
+///     </a>
+/// </div>
+///
+///
+pub fn canberra_distance(
+  xarr: List(Float),
+  yarr: List(Float),
+  weights: option.Option(List(Float)),
+) -> Result(Float, String) {
+  case check_lists(xarr, yarr, weights) {
+    Error(msg) ->
+      msg
+      |> Error
+    Ok(_) -> {
+      let arr: List(Float) =
+        list.zip(xarr, yarr)
+        |> list.map(canberra_distance_helper)
+      case weights {
+        option.None -> {
+          arr
+          |> arithmetics.float_sum(option.None)
+          |> Ok
+        }
+        _ -> {
+          arr
+          |> arithmetics.float_sum(weights)
+          |> Ok
+        }
+      }
+    }
+  }
+}
+
+fn canberra_distance_helper(tuple: #(Float, Float)) -> Float {
+  let numerator: Float =
+    piecewise.float_absolute_value({ pair.first(tuple) -. pair.second(tuple) })
+  let denominator: Float = {
+    piecewise.float_absolute_value(pair.first(tuple))
+    +. piecewise.float_absolute_value(pair.second(tuple))
+  }
+  numerator /. denominator
+}
+
+/// <div style="text-align: right;">
+///     <a href="https://github.com/gleam-community/maths/issues">
+///         <small>Spot a typo? Open an issue!</small>
+///     </a>
+/// </div>
+/// 
+/// <details>
+///     <summary>Example:</summary>
+///
+///     import gleeunit/should
+///     import gleam_community/maths/metrics
+///
+///     pub fn example () {
+///     }
+/// </details>
+///
+/// <div style="text-align: right;">
+///     <a href="#">
+///         <small>Back to top ↑</small>
+///     </a>
+/// </div>
+///
+///
+pub fn braycurtis_distance(
+  xarr: List(Float),
+  yarr: List(Float),
+  weights: option.Option(List(Float)),
+) -> Result(Float, String) {
+  case check_lists(xarr, yarr, weights) {
+    Error(msg) ->
+      msg
+      |> Error
+    Ok(_) -> {
+      let zipped_arr: List(#(Float, Float)) = list.zip(xarr, yarr)
+      let numerator_elements: List(Float) =
+        zipped_arr
+        |> list.map(fn(tuple: #(Float, Float)) -> Float {
+          piecewise.float_absolute_value({
+            pair.first(tuple) -. pair.second(tuple)
+          })
+        })
+      let denominator_elements: List(Float) =
+        zipped_arr
+        |> list.map(fn(tuple: #(Float, Float)) -> Float {
+          piecewise.float_absolute_value({
+            pair.first(tuple) +. pair.second(tuple)
+          })
+        })
+
+      case weights {
+        option.None -> {
+          let numerator =
+            numerator_elements
+            |> arithmetics.float_sum(option.None)
+          let denominator =
+            denominator_elements
+            |> arithmetics.float_sum(option.None)
+          { numerator /. denominator }
+          |> Ok
+        }
+        _ -> {
+          let numerator =
+            numerator_elements
+            |> arithmetics.float_sum(weights)
+          let denominator =
+            denominator_elements
+            |> arithmetics.float_sum(weights)
+          { numerator /. denominator }
+          |> Ok
+        }
+      }
+    }
+  }
+}
diff --git a/test/gleam_community/maths/arithmetics_test.gleam b/test/gleam_community/maths/arithmetics_test.gleam
index aa694b7..7ec5ad5 100644
--- a/test/gleam_community/maths/arithmetics_test.gleam
+++ b/test/gleam_community/maths/arithmetics_test.gleam
@@ -1,5 +1,6 @@
 import gleam_community/maths/arithmetics
 import gleeunit/should
+import gleam/option
 
 pub fn int_gcd_test() {
   arithmetics.gcd(1, 1)
@@ -100,16 +101,16 @@ pub fn int_divisors_test() {
 pub fn float_list_sum_test() {
   // An empty list returns 0
   []
-  |> arithmetics.float_sum()
+  |> arithmetics.float_sum(option.None)
   |> should.equal(0.0)
 
   // Valid input returns a result
   [1.0, 2.0, 3.0]
-  |> arithmetics.float_sum()
+  |> arithmetics.float_sum(option.None)
   |> should.equal(6.0)
 
   [-2.0, 4.0, 6.0]
-  |> arithmetics.float_sum()
+  |> arithmetics.float_sum(option.None)
   |> should.equal(8.0)
 }
 
@@ -132,17 +133,17 @@ pub fn int_list_sum_test() {
 pub fn float_list_product_test() {
   // An empty list returns 0
   []
-  |> arithmetics.float_product()
-  |> should.equal(1.0)
+  |> arithmetics.float_product(option.None)
+  |> should.equal(Ok(1.0))
 
   // Valid input returns a result
   [1.0, 2.0, 3.0]
-  |> arithmetics.float_product()
-  |> should.equal(6.0)
+  |> arithmetics.float_product(option.None)
+  |> should.equal(Ok(6.0))
 
   [-2.0, 4.0, 6.0]
-  |> arithmetics.float_product()
-  |> should.equal(-48.0)
+  |> arithmetics.float_product(option.None)
+  |> should.equal(Ok(-48.0))
 }
 
 pub fn int_list_product_test() {
diff --git a/test/gleam_community/maths/metrics_test.gleam b/test/gleam_community/maths/metrics_test.gleam
index e5888a0..3ccf758 100644
--- a/test/gleam_community/maths/metrics_test.gleam
+++ b/test/gleam_community/maths/metrics_test.gleam
@@ -3,6 +3,7 @@ import gleam_community/maths/metrics
 import gleam_community/maths/predicates
 import gleeunit/should
 import gleam/set
+import gleam/option
 
 pub fn float_list_norm_test() {
   let assert Ok(tol) = elementary.power(-10.0, -6.0)
@@ -54,15 +55,16 @@ pub fn float_list_manhattan_test() {
   let assert Ok(tol) = elementary.power(-10.0, -6.0)
 
   // Empty lists returns an error
-  metrics.manhattan_distance([], [])
+  metrics.manhattan_distance([], [], option.None)
   |> should.be_error()
 
   // Differing lengths returns error
-  metrics.manhattan_distance([], [1.0])
+  metrics.manhattan_distance([], [1.0], option.None)
   |> should.be_error()
 
   // manhattan distance (p = 1)
-  let assert Ok(result) = metrics.manhattan_distance([0.0, 0.0], [1.0, 2.0])
+  let assert Ok(result) =
+    metrics.manhattan_distance([0.0, 0.0], [1.0, 2.0], option.None)
   result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
@@ -72,53 +74,53 @@ pub fn float_list_minkowski_test() {
   let assert Ok(tol) = elementary.power(-10.0, -6.0)
 
   // Empty lists returns an error
-  metrics.minkowski_distance([], [], 1.0)
+  metrics.minkowski_distance([], [], 1.0, option.None)
   |> should.be_error()
 
   // Differing lengths returns error
-  metrics.minkowski_distance([], [1.0], 1.0)
+  metrics.minkowski_distance([], [1.0], 1.0, option.None)
   |> should.be_error()
 
   // Test order < 1
-  metrics.minkowski_distance([0.0, 0.0], [0.0, 0.0], -1.0)
+  metrics.minkowski_distance([0.0, 0.0], [0.0, 0.0], -1.0, option.None)
   |> should.be_error()
 
   // Check that the function agrees, at some arbitrary input
   // points, with known function values
   let assert Ok(result) =
-    metrics.minkowski_distance([1.0, 1.0], [1.0, 1.0], 1.0)
+    metrics.minkowski_distance([1.0, 1.0], [1.0, 1.0], 1.0, option.None)
   result
   |> predicates.is_close(0.0, 0.0, tol)
   |> should.be_true()
 
   let assert Ok(result) =
-    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 10.0)
+    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 10.0, option.None)
   result
   |> predicates.is_close(1.0717734625362931, 0.0, tol)
   |> should.be_true()
 
   let assert Ok(result) =
-    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 100.0)
+    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 100.0, option.None)
   result
   |> predicates.is_close(1.0069555500567189, 0.0, tol)
   |> should.be_true()
 
   let assert Ok(result) =
-    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 10.0)
+    metrics.minkowski_distance([0.0, 0.0], [1.0, 1.0], 10.0, option.None)
   result
   |> predicates.is_close(1.0717734625362931, 0.0, tol)
   |> should.be_true()
 
   // Euclidean distance (p = 2)
   let assert Ok(result) =
-    metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 2.0)
+    metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 2.0, option.None)
   result
   |> predicates.is_close(2.23606797749979, 0.0, tol)
   |> should.be_true()
 
   // Manhattan distance (p = 1)
   let assert Ok(result) =
-    metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 1.0)
+    metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 1.0, option.None)
   result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
@@ -128,15 +130,16 @@ pub fn float_list_euclidean_test() {
   let assert Ok(tol) = elementary.power(-10.0, -6.0)
 
   // Empty lists returns an error
-  metrics.euclidean_distance([], [])
+  metrics.euclidean_distance([], [], option.None)
   |> should.be_error()
 
   // Differing lengths returns error
-  metrics.euclidean_distance([], [1.0])
+  metrics.euclidean_distance([], [1.0], option.None)
   |> should.be_error()
 
   // Euclidean distance (p = 2)
-  let assert Ok(result) = metrics.euclidean_distance([0.0, 0.0], [1.0, 2.0])
+  let assert Ok(result) =
+    metrics.euclidean_distance([0.0, 0.0], [1.0, 2.0], option.None)
   result
   |> predicates.is_close(2.23606797749979, 0.0, tol)
   |> should.be_true()
@@ -266,65 +269,69 @@ pub fn overlap_coefficient_test() {
 
 pub fn cosine_similarity_test() {
   // Empty lists returns an error
-  metrics.cosine_similarity([], [])
+  metrics.cosine_similarity([], [], option.None)
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.cosine_similarity([1.0, 2.0, 3.0], [])
+  metrics.cosine_similarity([1.0, 2.0, 3.0], [], option.None)
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.cosine_similarity([], [1.0, 2.0, 3.0])
+  metrics.cosine_similarity([], [1.0, 2.0, 3.0], option.None)
   |> should.be_error()
 
   // Different sized lists returns an error
-  metrics.cosine_similarity([1.0, 2.0], [1.0, 2.0, 3.0, 4.0])
+  metrics.cosine_similarity([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
   |> should.be_error()
 
   // Two orthogonal vectors (represented by lists)
-  metrics.cosine_similarity([-1.0, 1.0, 0.0], [1.0, 1.0, -1.0])
+  metrics.cosine_similarity([-1.0, 1.0, 0.0], [1.0, 1.0, -1.0], option.None)
   |> should.equal(Ok(0.0))
 
   // Two identical (parallel) vectors (represented by lists)
-  metrics.cosine_similarity([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])
+  metrics.cosine_similarity([1.0, 2.0, 3.0], [1.0, 2.0, 3.0], option.None)
   |> should.equal(Ok(1.0))
 
   // Two parallel, but oppositely oriented vectors (represented by lists)
-  metrics.cosine_similarity([-1.0, -2.0, -3.0], [1.0, 2.0, 3.0])
+  metrics.cosine_similarity([-1.0, -2.0, -3.0], [1.0, 2.0, 3.0], option.None)
   |> should.equal(Ok(-1.0))
 }
 
 pub fn chebyshev_distance_test() {
   // Empty lists returns an error
-  metrics.chebyshev_distance([], [])
+  metrics.chebyshev_distance([], [], option.None)
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.chebyshev_distance([1.0, 2.0, 3.0], [])
+  metrics.chebyshev_distance([1.0, 2.0, 3.0], [], option.None)
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.chebyshev_distance([], [1.0, 2.0, 3.0])
+  metrics.chebyshev_distance([], [1.0, 2.0, 3.0], option.None)
   |> should.be_error()
 
   // Different sized lists returns an error
-  metrics.chebyshev_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0])
+  metrics.chebyshev_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
   |> should.be_error()
 
   // Try different types of valid input
-  metrics.chebyshev_distance([1.0, 0.0], [0.0, 2.0])
+  metrics.chebyshev_distance([1.0, 0.0], [0.0, 2.0], option.None)
   |> should.equal(Ok(2.0))
 
-  metrics.chebyshev_distance([1.0, 0.0], [2.0, 0.0])
+  metrics.chebyshev_distance([1.0, 0.0], [2.0, 0.0], option.None)
   |> should.equal(Ok(1.0))
 
-  metrics.chebyshev_distance([1.0, 0.0], [-2.0, 0.0])
+  metrics.chebyshev_distance([1.0, 0.0], [-2.0, 0.0], option.None)
   |> should.equal(Ok(3.0))
 
-  metrics.chebyshev_distance([-5.0, -10.0, -3.0], [-1.0, -12.0, -3.0])
+  metrics.chebyshev_distance(
+    [-5.0, -10.0, -3.0],
+    [-1.0, -12.0, -3.0],
+    option.None,
+  )
   |> should.equal(Ok(4.0))
 
-  metrics.chebyshev_distance([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])
+  metrics.chebyshev_distance([1.0, 2.0, 3.0], [1.0, 2.0, 3.0], option.None)
   |> should.equal(Ok(0.0))
 }
 
@@ -367,3 +374,99 @@ pub fn levenshtein_distance_test() {
   )
   |> should.equal(10)
 }
+
+pub fn canberra_distance_test() {
+  // Empty lists returns an error
+  metrics.canberra_distance([], [], option.None)
+  |> should.be_error()
+
+  // One empty list returns an error
+  metrics.canberra_distance([1.0, 2.0, 3.0], [], option.None)
+  |> should.be_error()
+
+  // One empty list returns an error
+  metrics.canberra_distance([], [1.0, 2.0, 3.0], option.None)
+  |> should.be_error()
+
+  // Different sized lists returns an error
+  metrics.canberra_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
+  |> should.be_error()
+
+  // Try different types of valid input
+  metrics.canberra_distance([0.0, 0.0], [0.0, 0.0], option.None)
+  |> should.equal(Ok(0.0))
+
+  metrics.canberra_distance([1.0, 2.0], [-2.0, -1.0], option.None)
+  |> should.equal(Ok(2.0))
+
+  metrics.canberra_distance([1.0, 0.0], [0.0, 2.0], option.None)
+  |> should.equal(Ok(2.0))
+
+  metrics.canberra_distance([1.0, 0.0], [2.0, 0.0], option.None)
+  |> should.equal(Ok(1.0 /. 3.0))
+
+  metrics.canberra_distance([1.0, 0.0], [0.0, 2.0], option.Some([1.0, 1.0]))
+  |> should.equal(Ok(2.0))
+
+  metrics.canberra_distance([1.0, 0.0], [0.0, 2.0], option.Some([1.0, 0.5]))
+  |> should.equal(Ok(1.5))
+
+  metrics.canberra_distance([1.0, 0.0], [0.0, 2.0], option.Some([0.5, 0.5]))
+  |> should.equal(Ok(1.0))
+
+  // Different sized lists (weights) returns an error
+  metrics.canberra_distance(
+    [1.0, 2.0, 3.0],
+    [1.0, 2.0, 3.0],
+    option.Some([1.0]),
+  )
+  |> should.be_error()
+}
+
+pub fn braycurtis_distance_test() {
+  // Empty lists returns an error
+  metrics.braycurtis_distance([], [], option.None)
+  |> should.be_error()
+
+  // One empty list returns an error
+  metrics.braycurtis_distance([1.0, 2.0, 3.0], [], option.None)
+  |> should.be_error()
+
+  // One empty list returns an error
+  metrics.braycurtis_distance([], [1.0, 2.0, 3.0], option.None)
+  |> should.be_error()
+
+  // Different sized lists returns an error
+  metrics.braycurtis_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
+  |> should.be_error()
+
+  // Try different types of valid input
+  metrics.braycurtis_distance([0.0, 0.0], [0.0, 0.0], option.None)
+  |> should.equal(Ok(0.0))
+
+  metrics.braycurtis_distance([1.0, 2.0], [-2.0, -1.0], option.None)
+  |> should.equal(Ok(3.0))
+
+  metrics.braycurtis_distance([1.0, 0.0], [0.0, 2.0], option.None)
+  |> should.equal(Ok(1.0))
+
+  metrics.braycurtis_distance([1.0, 2.0], [3.0, 4.0], option.None)
+  |> should.equal(Ok(0.4))
+
+  metrics.braycurtis_distance([1.0, 2.0], [3.0, 4.0], option.Some([1.0, 1.0]))
+  |> should.equal(Ok(0.4))
+
+  metrics.braycurtis_distance([1.0, 2.0], [3.0, 4.0], option.Some([0.5, 1.0]))
+  |> should.equal(Ok(0.375))
+
+  metrics.braycurtis_distance([1.0, 2.0], [3.0, 4.0], option.Some([0.25, 0.25]))
+  |> should.equal(Ok(0.4))
+
+  // Different sized lists (weights) returns an error
+  metrics.braycurtis_distance(
+    [1.0, 2.0, 3.0],
+    [1.0, 2.0, 3.0],
+    option.Some([1.0]),
+  )
+  |> should.be_error()
+}

From 2313363a9e921f11df2675f64226d566b81d1af2 Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Sun, 14 Apr 2024 16:54:10 +0200
Subject: [PATCH 4/6] Allow passing weights in distance calculations

---
 src/gleam_community/maths/arithmetics.gleam   |  91 +++--
 src/gleam_community/maths/combinatorics.gleam |   6 +-
 src/gleam_community/maths/conversion.gleam    |   6 +-
 src/gleam_community/maths/elementary.gleam    |   6 +-
 src/gleam_community/maths/metrics.gleam       | 343 +++++++++++++-----
 src/gleam_community/maths/piecewise.gleam     |   6 +-
 src/gleam_community/maths/predicates.gleam    |  36 +-
 src/gleam_community/maths/sequences.gleam     |  24 +-
 src/gleam_community/maths/special.gleam       |   4 +-
 .../maths/arithmetics_test.gleam              |   6 +-
 test/gleam_community/maths/metrics_test.gleam | 263 ++++++++++++--
 11 files changed, 584 insertions(+), 207 deletions(-)

diff --git a/src/gleam_community/maths/arithmetics.gleam b/src/gleam_community/maths/arithmetics.gleam
index b37a067..ebffe30 100644
--- a/src/gleam_community/maths/arithmetics.gleam
+++ b/src/gleam_community/maths/arithmetics.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
@@ -57,8 +57,9 @@ import gleam_community/maths/piecewise
 ///     </a>
 /// </div>
 ///
-/// The function calculates the greatest common multiple of two integers $$x, y \in \mathbb{Z}$$.
-/// The greatest common multiple is the largest positive integer that is divisible by both $$x$$ and $$y$$.
+/// The function calculates the greatest common divisor of two integers 
+/// $$x, y \in \mathbb{Z}$$. The greatest common divisor is the largest positive
+/// integer that is divisible by both $$x$$ and $$y$$.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -67,10 +68,10 @@ import gleam_community/maths/piecewise
 ///     import gleam_community/maths/arithmetics
 ///
 ///     pub fn example() {
-///       arithmetics.lcm(1, 1)
+///       arithmetics.gcd(1, 1)
 ///       |> should.equal(1)
 ///   
-///       arithmetics.lcm(100, 10)
+///       arithmetics.gcd(100, 10)
 ///       |> should.equal(10)
 ///
 ///       arithmetics.gcd(-36, -17)
@@ -107,8 +108,9 @@ fn do_gcd(x: Int, y: Int) -> Int {
 /// </div>
 ///
 /// 
-/// Given two integers, $$x$$ (dividend) and $$y$$ (divisor), the Euclidean modulo of $$x$$ by $$y$$,
-/// denoted as $$x \mod y$$, is the remainder $$r$$ of the division of $$x$$ by $$y$$, such that:
+/// Given two integers, $$x$$ (dividend) and $$y$$ (divisor), the Euclidean modulo
+/// of $$x$$ by $$y$$, denoted as $$x \mod y$$, is the remainder $$r$$ of the 
+/// division of $$x$$ by $$y$$, such that:
 /// 
 /// \\[
 /// x = q \cdot y + r \quad \text{and} \quad 0 \leq r < |y|,
@@ -116,13 +118,15 @@ fn do_gcd(x: Int, y: Int) -> Int {
 /// 
 /// where $$q$$ is an integer that represents the quotient of the division.
 ///
-/// The Euclidean modulo function of two numbers, is the remainder operation most commonly utilized in 
-/// mathematics. This differs from the standard truncating modulo operation frequently employed in 
-/// programming via the `%` operator. Unlike the `%` operator, which may return negative results 
-/// depending on the divisor's sign, the Euclidean modulo function is designed to
-/// always yield a positive outcome, ensuring consistency with mathematical conventions.
+/// The Euclidean modulo function of two numbers, is the remainder operation most 
+/// commonly utilized in mathematics. This differs from the standard truncating 
+/// modulo operation frequently employed in programming via the `%` operator. 
+/// Unlike the `%` operator, which may return negative results depending on the 
+/// divisor's sign, the Euclidean modulo function is designed to always yield a 
+/// positive outcome, ensuring consistency with mathematical conventions.
 /// 
-/// Note that like the Gleam division operator `/` this will return `0` if one of the arguments is `0`.
+/// Note that like the Gleam division operator `/` this will return `0` if one of
+/// the arguments is `0`.
 ///
 ///
 /// <details>
@@ -164,8 +168,9 @@ pub fn int_euclidean_modulo(x: Int, y: Int) -> Int {
 ///     </a>
 /// </div>
 ///
-/// The function calculates the least common multiple of two integers $$x, y \in \mathbb{Z}$$.
-/// The least common multiple is the smallest positive integer that has both $$x$$ and $$y$$ as factors.
+/// The function calculates the least common multiple of two integers 
+/// $$x, y \in \mathbb{Z}$$. The least common multiple is the smallest positive
+/// integer that has both $$x$$ and $$y$$ as factors.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -203,7 +208,8 @@ pub fn lcm(x: Int, y: Int) -> Int {
 ///     </a>
 /// </div>
 ///
-/// The function returns all the positive divisors of an integer, including the number iteself.
+/// The function returns all the positive divisors of an integer, including the 
+/// number itself.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -254,7 +260,8 @@ fn find_divisors(n: Int) -> List(Int) {
 ///     </a>
 /// </div>
 ///
-/// The function returns all the positive divisors of an integer, excluding the number iteself.
+/// The function returns all the positive divisors of an integer, excluding the 
+/// number iteself.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -355,7 +362,8 @@ pub fn float_sum(arr: List(Float), weights: option.Option(List(Float))) -> Float
 /// \sum_{i=1}^n x_i
 /// \\]
 ///
-/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is the value in the input list indexed by $$i$$.
+/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is 
+/// the value in the input list indexed by $$i$$.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -415,10 +423,10 @@ pub fn int_sum(arr: List(Int)) -> Int {
 ///     import gleam_community/maths/arithmetics
 ///
 ///     pub fn example () {
-///       // An empty list returns 0.0
+///       // An empty list returns 1.0
 ///       []
 ///       |> arithmetics.float_product(option.None)
-///       |> should.equal(0.0)
+///       |> should.equal(1.0)
 ///
 ///       // Valid input returns a result
 ///       [1.0, 2.0, 3.0]
@@ -478,7 +486,8 @@ pub fn float_product(
 /// \prod_{i=1}^n x_i
 /// \\]
 ///
-/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is the value in the input list indexed by $$i$$.
+/// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is 
+/// the value in the input list indexed by $$i$$.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -487,10 +496,10 @@ pub fn float_product(
 ///     import gleam_community/maths/arithmetics
 ///
 ///     pub fn example () {
-///       // An empty list returns 0
+///       // An empty list returns 1
 ///       []
 ///       |> arithmetics.int_product()
-///       |> should.equal(0)
+///       |> should.equal(1)
 ///
 ///       // Valid input returns a result
 ///       [1, 2, 3]
@@ -526,9 +535,10 @@ pub fn int_product(arr: List(Int)) -> Int {
 /// v_j = \sum_{i=1}^j x_i \\;\\; \forall j = 1,\dots, n
 /// \\]
 ///
-/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative sum of $$n$$ elements.
-/// That is, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is the value in the input list indexed by $$i$$. 
-/// The value $$v_j$$ is thus the sum of the $$1$$ to $$j$$ first elements in the given list.
+/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative sum of $$n$$
+/// elements. That is, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ 
+/// is the value in the input list indexed by $$i$$. The value $$v_j$$ is thus the
+/// sum of the $$1$$ to $$j$$ first elements in the given list.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -575,9 +585,10 @@ pub fn float_cumulative_sum(arr: List(Float)) -> List(Float) {
 /// v_j = \sum_{i=1}^j x_i \\;\\; \forall j = 1,\dots, n
 /// \\]
 ///
-/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative sum of $$n$$ elements.
-/// That is, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is the value in the input list indexed by $$i$$. 
-/// The value $$v_j$$ is thus the sum of the $$1$$ to $$j$$ first elements in the given list.
+/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative sum of $$n$$
+/// elements. That is, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ 
+/// is the value in the input list indexed by $$i$$. The value $$v_j$$ is thus the
+/// sum of the $$1$$ to $$j$$ first elements in the given list.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -624,9 +635,11 @@ pub fn int_cumulative_sum(arr: List(Int)) -> List(Int) {
 /// v_j = \prod_{i=1}^j x_i \\;\\; \forall j = 1,\dots, n
 /// \\]
 ///
-/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative product of $$n$$ elements.
-/// That is, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is the value in the input list indexed by $$i$$. 
-/// The value $$v_j$$ is thus the sum of the $$1$$ to $$j$$ first elements in the given list.
+/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative product of 
+/// $$n$$ elements. That is, $$n$$ is the length of the list and 
+/// $$x_i \in \mathbb{R}$$ is the value in the input list indexed by $$i$$. The 
+/// value $$v_j$$ is thus the sum of the $$1$$ to $$j$$ first elements in the 
+/// given list.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -653,7 +666,7 @@ pub fn int_cumulative_sum(arr: List(Int)) -> List(Int) {
 ///     </a>
 /// </div>
 ///
-pub fn float_cumumlative_product(arr: List(Float)) -> List(Float) {
+pub fn float_cumulative_product(arr: List(Float)) -> List(Float) {
   case arr {
     [] -> []
     _ ->
@@ -674,9 +687,11 @@ pub fn float_cumumlative_product(arr: List(Float)) -> List(Float) {
 /// v_j = \prod_{i=1}^j x_i \\;\\; \forall j = 1,\dots, n
 /// \\]
 ///
-/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative product of $$n$$ elements.
-/// That is, $$n$$ is the length of the list and $$x_i \in \mathbb{Z}$$ is the value in the input list indexed by $$i$$. 
-/// The value $$v_j$$ is thus the product of the $$1$$ to $$j$$ first elements in the given list.
+/// In the formula, $$v_j$$ is the $$j$$'th element in the cumulative product of 
+/// $$n$$ elements. That is, $$n$$ is the length of the list and 
+/// $$x_i \in \mathbb{Z}$$ is the value in the input list indexed by $$i$$. The 
+/// value $$v_j$$ is thus the product of the $$1$$ to $$j$$ first elements in the
+/// given list.
 ///
 /// <details>
 ///     <summary>Example:</summary>
diff --git a/src/gleam_community/maths/combinatorics.gleam b/src/gleam_community/maths/combinatorics.gleam
index b16d0ad..f459ef6 100644
--- a/src/gleam_community/maths/combinatorics.gleam
+++ b/src/gleam_community/maths/combinatorics.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
diff --git a/src/gleam_community/maths/conversion.gleam b/src/gleam_community/maths/conversion.gleam
index 017aabd..d0eb040 100644
--- a/src/gleam_community/maths/conversion.gleam
+++ b/src/gleam_community/maths/conversion.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
diff --git a/src/gleam_community/maths/elementary.gleam b/src/gleam_community/maths/elementary.gleam
index 1b518a4..c71ebf6 100644
--- a/src/gleam_community/maths/elementary.gleam
+++ b/src/gleam_community/maths/elementary.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
diff --git a/src/gleam_community/maths/metrics.gleam b/src/gleam_community/maths/metrics.gleam
index 84cb8b2..2cacfb5 100644
--- a/src/gleam_community/maths/metrics.gleam
+++ b/src/gleam_community/maths/metrics.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
@@ -61,9 +61,10 @@ import gleam/int
 import gleam/string
 import gleam/option
 
-/// Utility function that checks all lists have the expected length
-/// Primarily used by all distance measures taking List(Float) as input
-fn check_lists(
+/// Utility function that checks all lists have the expected length and contents
+/// The function is primarily used by all distance measures taking 'List(Float)'
+/// as input
+fn validate_lists(
   xarr: List(Float),
   yarr: List(Float),
   weights: option.Option(List(Float)),
@@ -76,9 +77,9 @@ fn check_lists(
       "Invalid input argument: The list yarr is empty."
       |> Error
     _, _ -> {
-      let xlen: Int = list.length(xarr)
-      let ylen: Int = list.length(yarr)
-      case xlen == ylen, weights {
+      let xarr_length: Int = list.length(xarr)
+      let yarr_length: Int = list.length(yarr)
+      case xarr_length == yarr_length, weights {
         False, _ ->
           "Invalid input argument: length(xarr) != length(yarr). Valid input is when length(xarr) == length(yarr)."
           |> Error
@@ -87,11 +88,11 @@ fn check_lists(
           |> Ok
         }
         True, option.Some(warr) -> {
-          let wlen: Int = list.length(warr)
-          case xlen == wlen {
-            True ->
-              True
-              |> Ok
+          let warr_length: Int = list.length(warr)
+          case xarr_length == warr_length {
+            True -> {
+              validate_weights(warr)
+            }
             False ->
               "Invalid input argument: length(weights) != length(xarr) and length(weights) != length(yarr). Valid input is when length(weights) == length(xarr) == length(yarr)."
               |> Error
@@ -102,39 +103,58 @@ fn check_lists(
   }
 }
 
+fn validate_weights(warr: List(Float)) -> Result(Bool, String) {
+  // Check that all the given weights are positive
+  let assert Ok(minimum) = piecewise.list_minimum(warr, float.compare)
+  case minimum >=. 0.0 {
+    False ->
+      "Invalid input argument: One or more weights are negative. Valid input is when all weights are >= 0."
+      |> Error
+    True ->
+      True
+      |> Ok
+  }
+}
+
 /// <div style="text-align: right;">
 ///     <a href="https://github.com/gleam-community/maths/issues">
 ///         <small>Spot a typo? Open an issue!</small>
 ///     </a>
 /// </div>
 ///
-/// Calculate the $$p$$-norm of a list (representing a vector):
+/// Calculate the (weighted) $$p$$-norm of a list (representing a vector):
 ///
 /// \\[
-/// \left( \sum_{i=1}^n \left|x_i\right|^{p} \right)^{\frac{1}{p}}
+/// \left( \sum_{i=1}^n w_{i} \left|x_{i}\right|^{p} \right)^{\frac{1}{p}}
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i$$ is the value in 
-/// the input list indexed by $$i$$.
+/// the input list indexed by $$i$$, while $$w_i \in \mathbb{R}_{+}$$ is
+/// a corresponding positive weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/elementary
 ///     import gleam_community/maths/metrics
 ///     import gleam_community/maths/predicates
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       let assert Ok(tol) = elementary.power(-10.0, -6.0)
-///
-///       [1.0, 1.0, 1.0]
-///       |> metrics.norm(1.0)
+///     
+///       let assert Ok(result) =
+///         [1.0, 1.0, 1.0]
+///         |> metrics.norm(1.0, option.None)
+///       result
 ///       |> predicates.is_close(3.0, 0.0, tol)
 ///       |> should.be_true()
-///
-///       [1.0, 1.0, 1.0]
-///       |> metrics.norm(-1.0)
+///     
+///       let assert Ok(result) =
+///         [1.0, 1.0, 1.0]
+///         |> metrics.norm(-1.0, option.None)
+///       result
 ///       |> predicates.is_close(0.3333333333333333, 0.0, tol)
 ///       |> should.be_true()
 ///     }
@@ -146,19 +166,65 @@ fn check_lists(
 ///     </a>
 /// </div>
 ///
-pub fn norm(arr: List(Float), p: Float) -> Float {
-  case arr {
-    [] -> 0.0
-    _ -> {
-      let agg: Float =
+pub fn norm(
+  arr: List(Float),
+  p: Float,
+  weights: option.Option(List(Float)),
+) -> Result(Float, String) {
+  case arr, weights {
+    [], _ ->
+      0.0
+      |> Ok
+    _, option.None -> {
+      let aggregate: Float =
         arr
-        |> list.fold(0.0, fn(acc: Float, a: Float) -> Float {
+        |> list.fold(0.0, fn(accumulator: Float, element: Float) -> Float {
           let assert Ok(result) =
-            elementary.power(piecewise.float_absolute_value(a), p)
-          result +. acc
+            piecewise.float_absolute_value(element)
+            |> elementary.power(p)
+          result +. accumulator
         })
-      let assert Ok(result) = elementary.power(agg, 1.0 /. p)
+      let assert Ok(result) = elementary.power(aggregate, 1.0 /. p)
       result
+      |> Ok
+    }
+    _, option.Some(warr) -> {
+      let arr_length: Int = list.length(arr)
+      let warr_length: Int = list.length(warr)
+      case arr_length == warr_length {
+        True -> {
+          case validate_weights(warr) {
+            Ok(_) -> {
+              let tuples: List(#(Float, Float)) = list.zip(arr, warr)
+              let aggregate: Float =
+                tuples
+                |> list.fold(
+                  0.0,
+                  fn(accumulator: Float, tuple: #(Float, Float)) -> Float {
+                    let first_element: Float = pair.first(tuple)
+                    let second_element: Float = pair.second(tuple)
+                    let assert Ok(result) =
+                      elementary.power(
+                        piecewise.float_absolute_value(first_element),
+                        p,
+                      )
+                    second_element *. result +. accumulator
+                  },
+                )
+              let assert Ok(result) = elementary.power(aggregate, 1.0 /. p)
+              result
+              |> Ok
+            }
+            Error(msg) ->
+              msg
+              |> Error
+          }
+        }
+        False -> {
+          "Invalid input argument: length(weights) != length(arr). Valid input is when length(weights) == length(arr)."
+          |> Error
+        }
+      }
     }
   }
 }
@@ -169,35 +235,40 @@ pub fn norm(arr: List(Float), p: Float) -> Float {
 ///     </a>
 /// </div>
 ///
-/// Calculate the Manhattan distance between two lists (representing vectors):
+/// Calculate the (weighted) Manhattan distance between two lists (representing 
+/// vectors):
 ///
 /// \\[
-/// \sum_{i=1}^n \left|x_i - y_i \right|
+/// \sum_{i=1}^n w_{i} \left|x_i - y_i \right|
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the 
-/// values in the respective input lists indexed by $$i$$.
+/// values in the respective input lists indexed by $$i$$, while 
+/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/elementary
 ///     import gleam_community/maths/metrics
 ///     import gleam_community/maths/predicates
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       let assert Ok(tol) = elementary.power(-10.0, -6.0)
 ///     
 ///       // Empty lists returns an error
-///       metrics.manhattan_distance([], [])
+///       metrics.manhattan_distance([], [], option.None)
 ///       |> should.be_error()
 ///     
 ///       // Differing lengths returns error
-///       metrics.manhattan_distance([], [1.0])
+///       metrics.manhattan_distance([], [1.0], option.None)
 ///       |> should.be_error()
 ///     
-///       let assert Ok(result) = metrics.manhattan_distance([0.0, 0.0], [1.0, 2.0])
+///       let assert Ok(result) =
+///         metrics.manhattan_distance([0.0, 0.0], [1.0, 2.0], option.None)
 ///       result
 ///       |> predicates.is_close(3.0, 0.0, tol)
 ///       |> should.be_true()
@@ -224,14 +295,17 @@ pub fn manhattan_distance(
 ///     </a>
 /// </div>
 ///
-/// Calculate the Minkowski distance between two lists (representing vectors):
+/// Calculate the (weighted) Minkowski distance between two lists (representing
+/// vectors):
 ///
 /// \\[
-/// \left( \sum_{i=1}^n \left|x_i - y_i \right|^{p} \right)^{\frac{1}{p}}
+/// \left( \sum_{i=1}^n w_{i} \left|x_i - y_i \right|^{p} \right)^{\frac{1}{p}}
 /// \\]
 ///
 /// In the formula, $$p >= 1$$ is the order, $$n$$ is the length of the two lists 
 /// and $$x_i, y_i$$ are the values in the respective input lists indexed by $$i$$.
+/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
 ///
 /// The Minkowski distance is a generalization of both the Euclidean distance 
 /// ($$p=2$$) and the Manhattan distance ($$p = 1$$).
@@ -240,29 +314,31 @@ pub fn manhattan_distance(
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/elementary
 ///     import gleam_community/maths/metrics
 ///     import gleam_community/maths/predicates
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       let assert Ok(tol) = elementary.power(-10.0, -6.0)
 ///     
 ///       // Empty lists returns an error
-///       metrics.minkowski_distance([], [], 1.0)
+///       metrics.minkowski_distance([], [], 1.0, option.None)
 ///       |> should.be_error()
 ///     
 ///       // Differing lengths returns error
-///       metrics.minkowski_distance([], [1.0], 1.0)
+///       metrics.minkowski_distance([], [1.0], 1.0, option.None)
 ///       |> should.be_error()
 ///     
 ///       // Test order < 1
-///       metrics.minkowski_distance([0.0, 0.0], [0.0, 0.0], -1.0)
+///       metrics.minkowski_distance([0.0, 0.0], [0.0, 0.0], -1.0, option.None)
 ///       |> should.be_error()
 ///     
-///       let assert Ok(result) = metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 1.0)
+///       let assert Ok(result) =
+///         metrics.minkowski_distance([0.0, 0.0], [1.0, 2.0], 1.0, option.None)
 ///       result
 ///       |> predicates.is_close(3.0, 0.0, tol)
-///       |> should.be_true()  
+///       |> should.be_true()
 ///     }
 /// </details>
 ///
@@ -278,7 +354,7 @@ pub fn minkowski_distance(
   p: Float,
   weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case check_lists(xarr, yarr, weights) {
+  case validate_lists(xarr, yarr, weights) {
     Error(msg) ->
       msg
       |> Error
@@ -287,13 +363,17 @@ pub fn minkowski_distance(
         True ->
           "Invalid input argument: p < 1. Valid input is p >= 1."
           |> Error
-        False ->
-          list.zip(xarr, yarr)
-          |> list.map(fn(tuple: #(Float, Float)) -> Float {
-            pair.first(tuple) -. pair.second(tuple)
-          })
-          |> norm(p)
+        False -> {
+          let differences: List(Float) =
+            list.zip(xarr, yarr)
+            |> list.map(fn(tuple: #(Float, Float)) -> Float {
+              pair.first(tuple) -. pair.second(tuple)
+            })
+
+          let assert Ok(result) = norm(differences, p, weights)
+          result
           |> Ok
+        }
       }
     }
   }
@@ -305,35 +385,40 @@ pub fn minkowski_distance(
 ///     </a>
 /// </div>
 ///
-/// Calculate the Euclidean distance between two lists (representing vectors):
+/// Calculate the (weighted) Euclidean distance between two lists (representing 
+/// vectors):
 ///
 /// \\[
-/// \left( \sum_{i=1}^n \left|x_i - y_i \right|^{2} \right)^{\frac{1}{2}}
+/// \left( \sum_{i=1}^n w_{i} \left|x_i - y_i \right|^{2} \right)^{\frac{1}{2}}
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the
-/// values in the respective input lists indexed by $$i$$.
+/// values in the respective input lists indexed by $$i$$, while
+/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/elementary
 ///     import gleam_community/maths/metrics
 ///     import gleam_community/maths/predicates
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       let assert Ok(tol) = elementary.power(-10.0, -6.0)
 ///     
 ///       // Empty lists returns an error
-///       metrics.euclidean_distance([], [])
+///       metrics.euclidean_distance([], [], option.None)
 ///       |> should.be_error()
 ///     
 ///       // Differing lengths returns an error
-///       metrics.euclidean_distance([], [1.0])
+///       metrics.euclidean_distance([], [1.0], option.None)
 ///       |> should.be_error()
 ///     
-///       let assert Ok(result) = metrics.euclidean_distance([0.0, 0.0], [1.0, 2.0])
+///       let assert Ok(result) =
+///         metrics.euclidean_distance([0.0, 0.0], [1.0, 2.0], option.None)
 ///       result
 ///       |> predicates.is_close(2.23606797749979, 0.0, tol)
 ///       |> should.be_true()
@@ -377,7 +462,7 @@ pub fn euclidean_distance(
 ///     import gleam_community/maths/metrics
 ///     import gleam_community/maths/predicates
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       // Empty lists returns an error
 ///       metrics.chebyshev_distance([], [])
 ///       |> should.be_error()
@@ -400,20 +485,17 @@ pub fn euclidean_distance(
 pub fn chebyshev_distance(
   xarr: List(Float),
   yarr: List(Float),
-  weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case check_lists(xarr, yarr, weights) {
+  case validate_lists(xarr, yarr, option.None) {
     Error(msg) ->
       msg
       |> Error
     Ok(_) -> {
-      let differences =
-        list.zip(xarr, yarr)
-        |> list.map(fn(tuple: #(Float, Float)) -> Float {
-          { pair.first(tuple) -. pair.second(tuple) }
-          |> piecewise.float_absolute_value()
-        })
-      differences
+      list.zip(xarr, yarr)
+      |> list.map(fn(tuple: #(Float, Float)) -> Float {
+        { pair.first(tuple) -. pair.second(tuple) }
+        |> piecewise.float_absolute_value()
+      })
       |> piecewise.list_maximum(float.compare)
     }
   }
@@ -948,39 +1030,44 @@ pub fn overlap_coefficient(xset: set.Set(a), yset: set.Set(a)) -> Float {
 ///     </a>
 /// </div>
 /// 
-/// Calculate the cosine similarity between two lists (representing vectors):
+/// Calculate the (weighted) cosine similarity between two lists (representing
+/// vectors):
 ///
 /// \\[
-/// \frac{\sum_{i=1}^n x_i \cdot y_i}{\left(\sum_{i=1}^n x_i^2\right)^{\frac{1}{2}}
-/// \cdot \left(\sum_{i=1}^n y_i^2\right)^{\frac{1}{2}}} 
+/// \frac{\sum_{i=1}^n w_i \cdot x_i \cdot y_i}
+/// {\left(\sum_{i=1}^n x_i^2\right)^{\frac{1}{2}}
+/// \cdot 
+/// \left(\sum_{i=1}^n w_i y_i^2\right)^{\frac{1}{2}}} 
 /// \\; \in \\; \left[-1, 1\right]
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i$$, $$y_i$$ are
-/// the values in the respective input lists indexed by $$i$$. The numerator 
-/// represents the dot product of the two vectors, while the denominator is the
-/// product of the magnitudes (Euclidean norms) of the two vectors. The cosine
-/// similarity provides a value between -1 and 1, where 1 means the vectors are
-/// in the same direction, -1 means they are in exactly opposite directions, 
-/// and 0 indicates orthogonality. 
+/// the values in the respective input lists indexed by $$i$$, while 
+/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// ($$w_i = 1.0\;\forall i=1...n$$ by default). 
+/// 
+/// The cosine similarity provides a value between -1 and 1, where 1 means the 
+/// vectors are in the same direction, -1 means they are in exactly opposite 
+/// directions, and 0 indicates orthogonality. 
 /// 
 /// <details>
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/metrics
 ///
-///     pub fn example () {
+///     pub fn example() {
 ///       // Two orthogonal vectors
-///       metrics.cosine_similarity([-1.0, 1.0, 0.0], [1.0, 1.0, -1.0])
+///       metrics.cosine_similarity([-1.0, 1.0, 0.0], [1.0, 1.0, -1.0], option.None)
 ///       |> should.equal(Ok(0.0))
 ///     
 ///       // Two identical (parallel) vectors
-///       metrics.cosine_similarity([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])
+///       metrics.cosine_similarity([1.0, 2.0, 3.0], [1.0, 2.0, 3.0], option.None)
 ///       |> should.equal(Ok(1.0))
 ///     
 ///       // Two parallel, but oppositely oriented vectors
-///       metrics.cosine_similarity([-1.0, -2.0, -3.0], [1.0, 2.0, 3.0])
+///       metrics.cosine_similarity([-1.0, -2.0, -3.0], [1.0, 2.0, 3.0], option.None)
 ///       |> should.equal(Ok(-1.0))
 ///     }
 /// </details>
@@ -996,21 +1083,47 @@ pub fn cosine_similarity(
   yarr: List(Float),
   weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case check_lists(xarr, yarr, weights) {
+  case validate_lists(xarr, yarr, weights) {
     Error(msg) ->
       msg
       |> Error
     Ok(_) -> {
-      list.fold(
-        list.zip(xarr, yarr),
-        0.0,
-        fn(acc: Float, a: #(Float, Float)) -> Float {
-          let result: Float = pair.first(a) *. pair.second(a)
-          result +. acc
-        },
-      )
-      /. { norm(xarr, 2.0) *. norm(yarr, 2.0) }
-      |> Ok
+      let zipped_arr: List(#(Float, Float)) = list.zip(xarr, yarr)
+
+      let numerator_elements: List(Float) =
+        zipped_arr
+        |> list.map(fn(tuple: #(Float, Float)) -> Float {
+          pair.first(tuple) *. pair.second(tuple)
+        })
+
+      case weights {
+        option.None -> {
+          let numerator: Float =
+            numerator_elements
+            |> arithmetics.float_sum(option.None)
+
+          let assert Ok(xarr_norm) = norm(xarr, 2.0, option.None)
+          let assert Ok(yarr_norm) = norm(yarr, 2.0, option.None)
+          let denominator: Float = {
+            xarr_norm *. yarr_norm
+          }
+          numerator /. denominator
+          |> Ok
+        }
+        _ -> {
+          let numerator: Float =
+            numerator_elements
+            |> arithmetics.float_sum(weights)
+
+          let assert Ok(xarr_norm) = norm(xarr, 2.0, weights)
+          let assert Ok(yarr_norm) = norm(yarr, 2.0, weights)
+          let denominator: Float = {
+            xarr_norm *. yarr_norm
+          }
+          numerator /. denominator
+          |> Ok
+        }
+      }
     }
   }
 }
@@ -1144,9 +1257,23 @@ fn distance_list_helper(
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/metrics
 ///
-///     pub fn example () {
+///     pub fn example() {
+///       // Empty lists returns an error
+///       metrics.canberra_distance([], [], option.None)
+///       |> should.be_error()
+///     
+///       // Different sized lists returns an error
+///       metrics.canberra_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
+///       |> should.be_error()
+///     
+///       // Valid inputs
+///       metrics.canberra_distance([1.0, 2.0], [-2.0, -1.0], option.None)
+///       |> should.equal(Ok(2.0))
+///     
+///       metrics.canberra_distance([1.0, 0.0], [0.0, 2.0], option.Some([1.0, 0.5]))
 ///     }
 /// </details>
 ///
@@ -1162,7 +1289,7 @@ pub fn canberra_distance(
   yarr: List(Float),
   weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case check_lists(xarr, yarr, weights) {
+  case validate_lists(xarr, yarr, weights) {
     Error(msg) ->
       msg
       |> Error
@@ -1170,6 +1297,7 @@ pub fn canberra_distance(
       let arr: List(Float) =
         list.zip(xarr, yarr)
         |> list.map(canberra_distance_helper)
+
       case weights {
         option.None -> {
           arr
@@ -1206,9 +1334,24 @@ fn canberra_distance_helper(tuple: #(Float, Float)) -> Float {
 ///     <summary>Example:</summary>
 ///
 ///     import gleeunit/should
+///     import gleam/option
 ///     import gleam_community/maths/metrics
 ///
-///     pub fn example () {
+///     pub fn example() {
+///       // Empty lists returns an error
+///       metrics.braycurtis_distance([], [], option.None)
+///       |> should.be_error()
+///     
+///       // Different sized lists returns an error
+///       metrics.braycurtis_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
+///       |> should.be_error()
+///     
+///       // Valid inputs
+///       metrics.braycurtis_distance([1.0, 0.0], [0.0, 2.0], option.None)
+///       |> should.equal(Ok(1.0))
+///     
+///       metrics.braycurtis_distance([1.0, 2.0], [3.0, 4.0], option.Some([0.5, 1.0]))
+///       |> should.equal(Ok(0.375))
 ///     }
 /// </details>
 ///
@@ -1224,7 +1367,7 @@ pub fn braycurtis_distance(
   yarr: List(Float),
   weights: option.Option(List(Float)),
 ) -> Result(Float, String) {
-  case check_lists(xarr, yarr, weights) {
+  case validate_lists(xarr, yarr, weights) {
     Error(msg) ->
       msg
       |> Error
diff --git a/src/gleam_community/maths/piecewise.gleam b/src/gleam_community/maths/piecewise.gleam
index c8492da..c840f25 100644
--- a/src/gleam_community/maths/piecewise.gleam
+++ b/src/gleam_community/maths/piecewise.gleam
@@ -1,6 +1,6 @@
-////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css" integrity="sha384-GvrOXuhMATgEsSwCs4smul74iXGOixntILdUW9XmUC6+HX0sLNAK3q71HotJqlAn" crossorigin="anonymous">
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js" integrity="sha384-cpW21h6RZv/phavutF+AuVYrr+dA8xD9zs6FwLpaCct6O9ctzYFfFr4dgmgccOTx" crossorigin="anonymous"></script>
-////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
+////<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous">
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script>
+////<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script>
 ////<script>
 ////    document.addEventListener("DOMContentLoaded", function() {
 ////        renderMathInElement(document.body, {
diff --git a/src/gleam_community/maths/predicates.gleam b/src/gleam_community/maths/predicates.gleam
index c5d5f01..e13f41d 100644
--- a/src/gleam_community/maths/predicates.gleam
+++ b/src/gleam_community/maths/predicates.gleam
@@ -23,7 +23,8 @@
 //// 
 //// ---
 //// 
-//// Predicates: A module containing functions for testing various mathematical properties of numbers.
+//// Predicates: A module containing functions for testing various mathematical 
+//// properties of numbers.
 //// 
 //// * **Tests**
 ////   * [`is_close`](#is_close)
@@ -50,8 +51,9 @@ import gleam_community/maths/arithmetics
 /// </div>
 ///
 /// Determine if a given value $$a$$ is close to or equivalent to a reference value 
-/// $$b$$ based on supplied relative $$r_{tol}$$ and absolute $$a_{tol}$$ tolerance values.
-/// The equivalance of the two given values are then determined based on the equation:
+/// $$b$$ based on supplied relative $$r_{tol}$$ and absolute $$a_{tol}$$ tolerance
+/// values. The equivalance of the two given values are then determined based on 
+/// the equation:
 ///
 /// \\[
 ///     \|a - b\| \leq (a_{tol} + r_{tol} \cdot \|b\|)
@@ -109,7 +111,8 @@ fn float_absolute_difference(a: Float, b: Float) -> Float {
 ///     </a>
 /// </div>
 ///
-/// Determine if a list of values are close to or equivalent to a another list of reference values.
+/// Determine if a list of values are close to or equivalent to a another list of
+/// reference values.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -176,7 +179,8 @@ pub fn all_close(
 ///
 /// Determine if a given value is fractional.
 /// 
-/// `True` is returned if the given value is fractional, otherwise `False` is returned. 
+/// `True` is returned if the given value is fractional, otherwise `False` is 
+/// returned. 
 /// 
 /// <details>
 ///     <summary>Example</summary>
@@ -213,7 +217,8 @@ fn do_ceiling(a: Float) -> Float
 ///     </a>
 /// </div>
 ///
-/// A function that tests whether a given integer value $$x \in \mathbb{Z}$$ is a power of another integer value $$y \in \mathbb{Z}$$.  
+/// A function that tests whether a given integer value $$x \in \mathbb{Z}$$ is a
+/// power of another integer value $$y \in \mathbb{Z}$$.  
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -252,7 +257,9 @@ pub fn is_power(x: Int, y: Int) -> Bool {
 ///     </a>
 /// </div>
 ///
-/// A function that tests whether a given integer value $$n \in \mathbb{Z}$$ is a perfect number. A number is perfect if it is equal to the sum of its proper positive divisors.
+/// A function that tests whether a given integer value $$n \in \mathbb{Z}$$ is a
+/// perfect number. A number is perfect if it is equal to the sum of its proper 
+/// positive divisors.
 /// 
 /// <details>
 ///     <summary>Details</summary>
@@ -369,13 +376,16 @@ pub fn is_odd(x: Int) -> Bool {
 ///     </a>
 /// </div>
 ///
-/// A function that tests whether a given integer value $$x \in \mathbb{Z}$$ is a prime number. 
-/// A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
+/// A function that tests whether a given integer value $$x \in \mathbb{Z}$$ is a 
+/// prime number. A prime number is a natural number greater than 1 that has no 
+/// positive divisors other than 1 and itself.
 /// 
-/// The function uses the Miller-Rabin primality test to assess if $$x$$ is prime. It is a probabilistic
-/// test, so it can mistakenly identify a composite number as prime. However, the probability of such errors decreases
-/// with more testing iterations (the function uses 64 iterations internally, which is typically more than sufficient). 
-/// The Miller-Rabin test is particularly useful for large numbers.
+/// The function uses the Miller-Rabin primality test to assess if $$x$$ is prime. 
+/// It is a probabilistic test, so it can mistakenly identify a composite number 
+/// as prime. However, the probability of such errors decreases with more testing
+/// iterations (the function uses 64 iterations internally, which is typically 
+/// more than sufficient). The Miller-Rabin test is particularly useful for large
+/// numbers.
 /// 
 /// <details>
 ///     <summary>Details</summary>
diff --git a/src/gleam_community/maths/sequences.gleam b/src/gleam_community/maths/sequences.gleam
index 9a5cb56..0ec8118 100644
--- a/src/gleam_community/maths/sequences.gleam
+++ b/src/gleam_community/maths/sequences.gleam
@@ -23,7 +23,8 @@
 //// 
 //// ---
 //// 
-//// Sequences: A module containing functions for generating various types of sequences, ranges and intervals.
+//// Sequences: A module containing functions for generating various types of 
+//// sequences, ranges and intervals.
 //// 
 //// * **Ranges and intervals**
 ////   * [`arange`](#arange)
@@ -42,8 +43,10 @@ import gleam/list
 ///     </a>
 /// </div>
 ///
-/// The function returns a list with evenly spaced values within a given interval based on a start, stop value and a given increment (step-length) between consecutive values. 
-/// The list returned includes the given start value but excludes the stop value.
+/// The function returns a list with evenly spaced values within a given interval 
+/// based on a start, stop value and a given increment (step-length) between 
+/// consecutive values. The list returned includes the given start value but 
+/// excludes the stop value.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -98,7 +101,8 @@ pub fn arange(start: Float, stop: Float, step: Float) -> List(Float) {
 ///     </a>
 /// </div>
 ///
-/// Generate a linearly spaced list of points over a specified interval. The endpoint of the interval can optionally be included/excluded.
+/// Generate a linearly spaced list of points over a specified interval. The 
+/// endpoint of the interval can optionally be included/excluded.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -176,7 +180,8 @@ pub fn linear_space(
 ///     </a>
 /// </div>
 ///
-/// Generate a logarithmically spaced list of points over a specified interval. The endpoint of the interval can optionally be included/excluded.
+/// Generate a logarithmically spaced list of points over a specified interval. The 
+/// endpoint of the interval can optionally be included/excluded.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -236,8 +241,11 @@ pub fn logarithmic_space(
 ///     </a>
 /// </div>
 ///
-/// The function returns a list of numbers spaced evenly on a log scale (a geometric progression). Each point in the list is a constant multiple of the previous.
-/// The function is similar to the [`logarithmic_space`](#logarithmic_space) function, but with endpoints specified directly.
+/// The function returns a list of numbers spaced evenly on a log scale (a 
+/// geometric progression). Each point in the list is a constant multiple of the 
+/// previous. The function is similar to the 
+/// [`logarithmic_space`](#logarithmic_space) function, but with endpoints 
+/// specified directly.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -283,7 +291,7 @@ pub fn geometric_space(
 ) -> Result(List(Float), String) {
   case start == 0.0 || stop == 0.0 {
     True ->
-      ""
+      "Invalid input: Neither 'start' nor 'stop' can be zero, as they must be non-zero for logarithmic calculations."
       |> Error
     False ->
       case num > 0 {
diff --git a/src/gleam_community/maths/special.gleam b/src/gleam_community/maths/special.gleam
index 318511d..6b86e5a 100644
--- a/src/gleam_community/maths/special.gleam
+++ b/src/gleam_community/maths/special.gleam
@@ -100,8 +100,8 @@ pub fn erf(x: Float) -> Float {
 ///     </a>
 /// </div>
 ///
-/// The gamma function over the real numbers. The function is essentially equal to the
-/// factorial for any positive integer argument: $$\Gamma(n) = (n - 1)!$$
+/// The gamma function over the real numbers. The function is essentially equal to 
+/// the factorial for any positive integer argument: $$\Gamma(n) = (n - 1)!$$
 ///
 /// The implemented gamma function is approximated through Lanczos approximation
 /// using the same coefficients used by the GNU Scientific Library.
diff --git a/test/gleam_community/maths/arithmetics_test.gleam b/test/gleam_community/maths/arithmetics_test.gleam
index 7ec5ad5..990fce6 100644
--- a/test/gleam_community/maths/arithmetics_test.gleam
+++ b/test/gleam_community/maths/arithmetics_test.gleam
@@ -197,16 +197,16 @@ pub fn int_list_cumulative_sum_test() {
 pub fn float_list_cumulative_product_test() {
   // An empty lists returns an empty list
   []
-  |> arithmetics.float_cumumlative_product()
+  |> arithmetics.float_cumulative_product()
   |> should.equal([])
 
   // Valid input returns a result
   [1.0, 2.0, 3.0]
-  |> arithmetics.float_cumumlative_product()
+  |> arithmetics.float_cumulative_product()
   |> should.equal([1.0, 2.0, 6.0])
 
   [-2.0, 4.0, 6.0]
-  |> arithmetics.float_cumumlative_product()
+  |> arithmetics.float_cumulative_product()
   |> should.equal([-2.0, -8.0, -48.0])
 }
 
diff --git a/test/gleam_community/maths/metrics_test.gleam b/test/gleam_community/maths/metrics_test.gleam
index 3ccf758..43263cb 100644
--- a/test/gleam_community/maths/metrics_test.gleam
+++ b/test/gleam_community/maths/metrics_test.gleam
@@ -10,43 +10,57 @@ pub fn float_list_norm_test() {
 
   // An empty lists returns 0.0
   []
-  |> metrics.norm(1.0)
-  |> should.equal(0.0)
+  |> metrics.norm(1.0, option.None)
+  |> should.equal(Ok(0.0))
 
   // Check that the function agrees, at some arbitrary input
   // points, with known function values
-  [1.0, 1.0, 1.0]
-  |> metrics.norm(1.0)
+  let assert Ok(result) =
+    [1.0, 1.0, 1.0]
+    |> metrics.norm(1.0, option.None)
+  result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
 
-  [1.0, 1.0, 1.0]
-  |> metrics.norm(-1.0)
+  let assert Ok(result) =
+    [1.0, 1.0, 1.0]
+    |> metrics.norm(-1.0, option.None)
+  result
   |> predicates.is_close(0.3333333333333333, 0.0, tol)
   |> should.be_true()
 
-  [-1.0, -1.0, -1.0]
-  |> metrics.norm(-1.0)
+  let assert Ok(result) =
+    [-1.0, -1.0, -1.0]
+    |> metrics.norm(-1.0, option.None)
+  result
   |> predicates.is_close(0.3333333333333333, 0.0, tol)
   |> should.be_true()
 
-  [-1.0, -1.0, -1.0]
-  |> metrics.norm(1.0)
+  let assert Ok(result) =
+    [-1.0, -1.0, -1.0]
+    |> metrics.norm(1.0, option.None)
+  result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
 
-  [-1.0, -2.0, -3.0]
-  |> metrics.norm(-10.0)
+  let assert Ok(result) =
+    [-1.0, -2.0, -3.0]
+    |> metrics.norm(-10.0, option.None)
+  result
   |> predicates.is_close(0.9999007044905545, 0.0, tol)
   |> should.be_true()
 
-  [-1.0, -2.0, -3.0]
-  |> metrics.norm(-100.0)
+  let assert Ok(result) =
+    [-1.0, -2.0, -3.0]
+    |> metrics.norm(-100.0, option.None)
+  result
   |> predicates.is_close(1.0, 0.0, tol)
   |> should.be_true()
 
-  [-1.0, -2.0, -3.0]
-  |> metrics.norm(2.0)
+  let assert Ok(result) =
+    [-1.0, -2.0, -3.0]
+    |> metrics.norm(2.0, option.None)
+  result
   |> predicates.is_close(3.7416573867739413, 0.0, tol)
   |> should.be_true()
 }
@@ -62,12 +76,45 @@ pub fn float_list_manhattan_test() {
   metrics.manhattan_distance([], [1.0], option.None)
   |> should.be_error()
 
-  // manhattan distance (p = 1)
+  // Try with valid input (same as Minkowski distance with p = 1)
   let assert Ok(result) =
     metrics.manhattan_distance([0.0, 0.0], [1.0, 2.0], option.None)
   result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
+
+  metrics.manhattan_distance([1.0, 2.0, 3.0], [4.0, 5.0, 6.0], option.None)
+  |> should.equal(Ok(9.0))
+
+  metrics.manhattan_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([1.0, 1.0, 1.0]),
+  )
+  |> should.equal(Ok(9.0))
+
+  metrics.manhattan_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([1.0, 2.0, 3.0]),
+  )
+  |> should.equal(Ok(18.0))
+
+  // Try invalid input with weights (different sized lists returns an error)
+  metrics.manhattan_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([7.0, 8.0]),
+  )
+  |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.manhattan_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }
 
 pub fn float_list_minkowski_test() {
@@ -124,6 +171,58 @@ pub fn float_list_minkowski_test() {
   result
   |> predicates.is_close(3.0, 0.0, tol)
   |> should.be_true()
+
+  // Try different valid input
+  let assert Ok(result) =
+    metrics.minkowski_distance(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      4.0,
+      option.None,
+    )
+  result
+  |> predicates.is_close(3.9482220388574776, 0.0, tol)
+  |> should.be_true()
+
+  let assert Ok(result) =
+    metrics.minkowski_distance(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      4.0,
+      option.Some([1.0, 1.0, 1.0]),
+    )
+  result
+  |> predicates.is_close(3.9482220388574776, 0.0, tol)
+  |> should.be_true()
+
+  let assert Ok(result) =
+    metrics.minkowski_distance(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      4.0,
+      option.Some([1.0, 2.0, 3.0]),
+    )
+  result
+  |> predicates.is_close(4.6952537402198615, 0.0, tol)
+  |> should.be_true()
+
+  // Try invalid input with weights (different sized lists returns an error)
+  metrics.minkowski_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    2.0,
+    option.Some([7.0, 8.0]),
+  )
+  |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.minkowski_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    2.0,
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }
 
 pub fn float_list_euclidean_test() {
@@ -137,12 +236,55 @@ pub fn float_list_euclidean_test() {
   metrics.euclidean_distance([], [1.0], option.None)
   |> should.be_error()
 
-  // Euclidean distance (p = 2)
+  // Try with valid input (same as Minkowski distance with p = 2)
   let assert Ok(result) =
     metrics.euclidean_distance([0.0, 0.0], [1.0, 2.0], option.None)
   result
   |> predicates.is_close(2.23606797749979, 0.0, tol)
   |> should.be_true()
+
+  // Try different valid input
+  let assert Ok(result) =
+    metrics.euclidean_distance([1.0, 2.0, 3.0], [4.0, 5.0, 6.0], option.None)
+  result
+  |> predicates.is_close(5.196152422706632, 0.0, tol)
+  |> should.be_true()
+
+  let assert Ok(result) =
+    metrics.euclidean_distance(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      option.Some([1.0, 1.0, 1.0]),
+    )
+  result
+  |> predicates.is_close(5.196152422706632, 0.0, tol)
+  |> should.be_true()
+
+  let assert Ok(result) =
+    metrics.euclidean_distance(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      option.Some([1.0, 2.0, 3.0]),
+    )
+  result
+  |> predicates.is_close(7.3484692283495345, 0.0, tol)
+  |> should.be_true()
+
+  // Try invalid input with weights (different sized lists returns an error)
+  metrics.euclidean_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([7.0, 8.0]),
+  )
+  |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.euclidean_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }
 
 pub fn mean_test() {
@@ -268,6 +410,8 @@ pub fn overlap_coefficient_test() {
 }
 
 pub fn cosine_similarity_test() {
+  let assert Ok(tol) = elementary.power(-10.0, -6.0)
+
   // Empty lists returns an error
   metrics.cosine_similarity([], [], option.None)
   |> should.be_error()
@@ -295,43 +439,84 @@ pub fn cosine_similarity_test() {
   // Two parallel, but oppositely oriented vectors (represented by lists)
   metrics.cosine_similarity([-1.0, -2.0, -3.0], [1.0, 2.0, 3.0], option.None)
   |> should.equal(Ok(-1.0))
+
+  // Try with arbitrary valid input 
+  let assert Ok(result) =
+    metrics.cosine_similarity([1.0, 2.0, 3.0], [4.0, 5.0, 6.0], option.None)
+  result
+  |> predicates.is_close(0.9746318461970762, 0.0, tol)
+  |> should.be_true()
+
+  // Try valid input with weights
+  let assert Ok(result) =
+    metrics.cosine_similarity(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      option.Some([1.0, 1.0, 1.0]),
+    )
+  result
+  |> predicates.is_close(0.9746318461970762, 0.0, tol)
+  |> should.be_true()
+
+  // Try with different weights
+  let assert Ok(result) =
+    metrics.cosine_similarity(
+      [1.0, 2.0, 3.0],
+      [4.0, 5.0, 6.0],
+      option.Some([1.0, 2.0, 3.0]),
+    )
+  result
+  |> predicates.is_close(0.9855274566525745, 0.0, tol)
+  |> should.be_true()
+
+  // Try invalid input with weights (different sized lists returns an error)
+  metrics.cosine_similarity(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([7.0, 8.0]),
+  )
+  |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.cosine_similarity(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }
 
 pub fn chebyshev_distance_test() {
   // Empty lists returns an error
-  metrics.chebyshev_distance([], [], option.None)
+  metrics.chebyshev_distance([], [])
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.chebyshev_distance([1.0, 2.0, 3.0], [], option.None)
+  metrics.chebyshev_distance([1.0, 2.0, 3.0], [])
   |> should.be_error()
 
   // One empty list returns an error
-  metrics.chebyshev_distance([], [1.0, 2.0, 3.0], option.None)
+  metrics.chebyshev_distance([], [1.0, 2.0, 3.0])
   |> should.be_error()
 
   // Different sized lists returns an error
-  metrics.chebyshev_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0], option.None)
+  metrics.chebyshev_distance([1.0, 2.0], [1.0, 2.0, 3.0, 4.0])
   |> should.be_error()
 
   // Try different types of valid input
-  metrics.chebyshev_distance([1.0, 0.0], [0.0, 2.0], option.None)
+  metrics.chebyshev_distance([1.0, 0.0], [0.0, 2.0])
   |> should.equal(Ok(2.0))
 
-  metrics.chebyshev_distance([1.0, 0.0], [2.0, 0.0], option.None)
+  metrics.chebyshev_distance([1.0, 0.0], [2.0, 0.0])
   |> should.equal(Ok(1.0))
 
-  metrics.chebyshev_distance([1.0, 0.0], [-2.0, 0.0], option.None)
+  metrics.chebyshev_distance([1.0, 0.0], [-2.0, 0.0])
   |> should.equal(Ok(3.0))
 
-  metrics.chebyshev_distance(
-    [-5.0, -10.0, -3.0],
-    [-1.0, -12.0, -3.0],
-    option.None,
-  )
+  metrics.chebyshev_distance([-5.0, -10.0, -3.0], [-1.0, -12.0, -3.0])
   |> should.equal(Ok(4.0))
 
-  metrics.chebyshev_distance([1.0, 2.0, 3.0], [1.0, 2.0, 3.0], option.None)
+  metrics.chebyshev_distance([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])
   |> should.equal(Ok(0.0))
 }
 
@@ -421,6 +606,14 @@ pub fn canberra_distance_test() {
     option.Some([1.0]),
   )
   |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.canberra_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }
 
 pub fn braycurtis_distance_test() {
@@ -469,4 +662,12 @@ pub fn braycurtis_distance_test() {
     option.Some([1.0]),
   )
   |> should.be_error()
+
+  // Try invalid input with weights that are negative
+  metrics.braycurtis_distance(
+    [1.0, 2.0, 3.0],
+    [4.0, 5.0, 6.0],
+    option.Some([-7.0, -8.0, -9.0]),
+  )
+  |> should.be_error()
 }

From d2485d4e88a475d3031043c04330c74463e58fad Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Sun, 14 Apr 2024 17:04:44 +0200
Subject: [PATCH 5/6] Fix typos

---
 src/gleam_community/maths/arithmetics.gleam |  4 ++--
 src/gleam_community/maths/metrics.gleam     | 16 ++++++++--------
 src/gleam_community/maths/piecewise.gleam   | 13 ++++++++-----
 3 files changed, 18 insertions(+), 15 deletions(-)

diff --git a/src/gleam_community/maths/arithmetics.gleam b/src/gleam_community/maths/arithmetics.gleam
index ebffe30..6e46671 100644
--- a/src/gleam_community/maths/arithmetics.gleam
+++ b/src/gleam_community/maths/arithmetics.gleam
@@ -307,7 +307,7 @@ pub fn proper_divisors(n: Int) -> List(Int) {
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
 /// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
-/// a corresponding weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// a corresponding weight ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -413,7 +413,7 @@ pub fn int_sum(arr: List(Int)) -> Int {
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
 /// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
-/// a corresponding weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// a corresponding weight ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 /// 
 /// <details>
 ///     <summary>Example:</summary>
diff --git a/src/gleam_community/maths/metrics.gleam b/src/gleam_community/maths/metrics.gleam
index 2cacfb5..e722d03 100644
--- a/src/gleam_community/maths/metrics.gleam
+++ b/src/gleam_community/maths/metrics.gleam
@@ -130,7 +130,7 @@ fn validate_weights(warr: List(Float)) -> Result(Bool, String) {
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i$$ is the value in 
 /// the input list indexed by $$i$$, while $$w_i \in \mathbb{R}_{+}$$ is
-/// a corresponding positive weight ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// a corresponding positive weight ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -245,7 +245,7 @@ pub fn norm(
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the 
 /// values in the respective input lists indexed by $$i$$, while 
 /// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
-/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -305,7 +305,7 @@ pub fn manhattan_distance(
 /// In the formula, $$p >= 1$$ is the order, $$n$$ is the length of the two lists 
 /// and $$x_i, y_i$$ are the values in the respective input lists indexed by $$i$$.
 /// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
-/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// The Minkowski distance is a generalization of both the Euclidean distance 
 /// ($$p=2$$) and the Manhattan distance ($$p = 1$$).
@@ -395,7 +395,7 @@ pub fn minkowski_distance(
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the
 /// values in the respective input lists indexed by $$i$$, while
 /// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
-/// ($$w_i = 1.0\;\forall i=1...n$$ by default).
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -1034,17 +1034,17 @@ pub fn overlap_coefficient(xset: set.Set(a), yset: set.Set(a)) -> Float {
 /// vectors):
 ///
 /// \\[
-/// \frac{\sum_{i=1}^n w_i \cdot x_i \cdot y_i}
-/// {\left(\sum_{i=1}^n x_i^2\right)^{\frac{1}{2}}
+/// \frac{\sum_{i=1}^n w_{i} \cdot x_i \cdot y_i}
+/// {\left(\sum_{i=1}^n w_{i} \cdot x_i^2\right)^{\frac{1}{2}}
 /// \cdot 
-/// \left(\sum_{i=1}^n w_i y_i^2\right)^{\frac{1}{2}}} 
+/// \left(\sum_{i=1}^n w_{i} \cdot y_i^2\right)^{\frac{1}{2}}} 
 /// \\; \in \\; \left[-1, 1\right]
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i$$, $$y_i$$ are
 /// the values in the respective input lists indexed by $$i$$, while 
 /// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
-/// ($$w_i = 1.0\;\forall i=1...n$$ by default). 
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default). 
 /// 
 /// The cosine similarity provides a value between -1 and 1, where 1 means the 
 /// vectors are in the same direction, -1 means they are in exactly opposite 
diff --git a/src/gleam_community/maths/piecewise.gleam b/src/gleam_community/maths/piecewise.gleam
index c840f25..8939dc7 100644
--- a/src/gleam_community/maths/piecewise.gleam
+++ b/src/gleam_community/maths/piecewise.gleam
@@ -490,7 +490,7 @@ fn do_ceiling(a: Float) -> Float
 /// The absolute value:
 ///
 /// \\[
-///  \forall x, y \in \mathbb{R}, \\; |x|  \in \mathbb{R}_{+}. 
+///  \forall x \in \mathbb{R}, \\; |x|  \in \mathbb{R}_{+}. 
 /// \\]
 ///
 /// The function takes an input $$x$$ and returns a positive float value.
@@ -519,7 +519,7 @@ pub fn float_absolute_value(x: Float) -> Float {
 /// The absolute value:
 ///
 /// \\[
-///  \forall x, y \in \mathbb{Z}, \\; |x|  \in \mathbb{Z}_{+}. 
+///  \forall x \in \mathbb{Z}, \\; |x|  \in \mathbb{Z}_{+}. 
 /// \\]
 ///
 /// The function takes an input $$x$$ and returns a positive integer value.
@@ -592,7 +592,8 @@ pub fn float_absolute_difference(a: Float, b: Float) -> Float {
 ///  \forall x, y \in \mathbb{Z}, \\; |x - y|  \in \mathbb{Z}_{+}. 
 /// \\]
 ///
-/// The function takes two inputs $$x$$ and $$y$$ and returns a positive integer value which is the the absolute difference of the inputs.
+/// The function takes two inputs $$x$$ and $$y$$ and returns a positive integer
+/// value which is the the absolute difference of the inputs.
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -698,7 +699,8 @@ fn do_int_sign(a: Int) -> Int
 ///     </a>
 /// </div>
 ///
-/// The function takes two arguments $$x, y \in \mathbb{R}$$ and returns $$x$$ such that it has the same sign as $$y$$.
+/// The function takes two arguments $$x, y \in \mathbb{R}$$ and returns $$x$$ 
+/// such that it has the same sign as $$y$$.
 ///
 /// <div style="text-align: right;">
 ///     <a href="#">
@@ -723,7 +725,8 @@ pub fn float_copy_sign(x: Float, y: Float) -> Float {
 ///     </a>
 /// </div>
 ///
-/// The function takes two arguments $$x, y \in \mathbb{Z}$$ and returns $$x$$ such that it has the same sign as $$y$$.
+/// The function takes two arguments $$x, y \in \mathbb{Z}$$ and returns $$x$$ 
+/// such that it has the same sign as $$y$$.
 ///
 /// <div style="text-align: right;">
 ///     <a href="#">

From ef8844d9910e3379fc622466c4d22e7e5e6aa4b5 Mon Sep 17 00:00:00 2001
From: NicklasXYZ <18580183+NicklasXYZ@users.noreply.github.com>
Date: Sun, 14 Apr 2024 17:30:39 +0200
Subject: [PATCH 6/6] Add function descriptions and fix typos

---
 src/gleam_community/maths/arithmetics.gleam |  8 ++--
 src/gleam_community/maths/metrics.gleam     | 41 +++++++++++++++++----
 2 files changed, 38 insertions(+), 11 deletions(-)

diff --git a/src/gleam_community/maths/arithmetics.gleam b/src/gleam_community/maths/arithmetics.gleam
index 6e46671..a205116 100644
--- a/src/gleam_community/maths/arithmetics.gleam
+++ b/src/gleam_community/maths/arithmetics.gleam
@@ -306,8 +306,8 @@ pub fn proper_divisors(n: Int) -> List(Int) {
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
-/// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
-/// a corresponding weight ($$w_i = 1.0\\;\forall i=1...n$$ by default).
+/// the value in the input list indexed by $$i$$, while the $$w_i \in \mathbb{R}$$
+/// are corresponding weights ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
 ///     <summary>Example:</summary>
@@ -412,8 +412,8 @@ pub fn int_sum(arr: List(Int)) -> Int {
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the list and $$x_i \in \mathbb{R}$$ is
-/// the value in the input list indexed by $$i$$, while $$w_i \in \mathbb{R}$$ is
-/// a corresponding weight ($$w_i = 1.0\\;\forall i=1...n$$ by default).
+/// the value in the input list indexed by $$i$$, while the $$w_i \in \mathbb{R}$$
+/// are corresponding weights ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 /// 
 /// <details>
 ///     <summary>Example:</summary>
diff --git a/src/gleam_community/maths/metrics.gleam b/src/gleam_community/maths/metrics.gleam
index e722d03..7c691d3 100644
--- a/src/gleam_community/maths/metrics.gleam
+++ b/src/gleam_community/maths/metrics.gleam
@@ -243,8 +243,8 @@ pub fn norm(
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the 
-/// values in the respective input lists indexed by $$i$$, while 
-/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// values in the respective input lists indexed by $$i$$, while the
+/// $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
 /// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
@@ -304,7 +304,7 @@ pub fn manhattan_distance(
 ///
 /// In the formula, $$p >= 1$$ is the order, $$n$$ is the length of the two lists 
 /// and $$x_i, y_i$$ are the values in the respective input lists indexed by $$i$$.
-/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// The $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
 /// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// The Minkowski distance is a generalization of both the Euclidean distance 
@@ -393,8 +393,8 @@ pub fn minkowski_distance(
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i, y_i$$ are the
-/// values in the respective input lists indexed by $$i$$, while
-/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// values in the respective input lists indexed by $$i$$, while the
+/// $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
 /// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
 ///
 /// <details>
@@ -1042,8 +1042,8 @@ pub fn overlap_coefficient(xset: set.Set(a), yset: set.Set(a)) -> Float {
 /// \\]
 ///
 /// In the formula, $$n$$ is the length of the two lists and $$x_i$$, $$y_i$$ are
-/// the values in the respective input lists indexed by $$i$$, while 
-/// $$w_i \in \mathbb{R}_{+}$$ is a corresponding positive weight 
+/// the values in the respective input lists indexed by $$i$$, while the
+/// $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
 /// ($$w_i = 1.0\\;\forall i=1...n$$ by default). 
 /// 
 /// The cosine similarity provides a value between -1 and 1, where 1 means the 
@@ -1253,6 +1253,18 @@ fn distance_list_helper(
 ///     </a>
 /// </div>
 /// 
+/// Calculate the (weighted) Canberra distance between two lists:
+///
+/// \\[
+/// \sum_{i=1}^n w_{i}\frac{\left| x_i - y_i \right|}
+/// {\left| x_i \right| + \left| y_i \right|}
+/// \\]
+///
+/// In the formula, $$n$$ is the length of the two lists, and $$x_i, y_i$$ are the 
+/// values in the respective input lists indexed by $$i$$, while the 
+/// $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default). 
+///
 /// <details>
 ///     <summary>Example:</summary>
 ///
@@ -1330,6 +1342,21 @@ fn canberra_distance_helper(tuple: #(Float, Float)) -> Float {
 ///     </a>
 /// </div>
 /// 
+/// Calculate the (weighted) Bray-Curtis distance between two lists:
+///
+/// \\[
+/// \frac{\sum_{i=1}^n w_{i} \left| x_i - y_i \right|}
+/// {\sum_{i=1}^n w_{i}\left| x_i + y_i \right|}
+/// \\]
+///
+/// In the formula, $$n$$ is the length of the two lists, and $$x_i, y_i$$ are the values 
+/// in the respective input lists indexed by $$i$$, while the 
+/// $$w_i \in \mathbb{R}_{+}$$ are corresponding positive weights 
+/// ($$w_i = 1.0\\;\forall i=1...n$$ by default).
+/// 
+/// The Bray-Curtis distance is in the range $$[0, 1]$$ if all entries $$x_i, y_i$$ are
+/// positive.
+///
 /// <details>
 ///     <summary>Example:</summary>
 ///