From 4caae578d26c89fc8eb1880c750104746cf15b2b Mon Sep 17 00:00:00 2001
From: Killua Zoldyck <junyixu0@gmail.com>
Date: Mon, 25 Mar 2024 14:52:50 +0800
Subject: [PATCH 1/2] fix: Improve formula presentation

---
 docs/src/dev/how_it_works.md | 16 ++++++++++------
 1 file changed, 10 insertions(+), 6 deletions(-)

diff --git a/docs/src/dev/how_it_works.md b/docs/src/dev/how_it_works.md
index fc923d2d..753602e8 100644
--- a/docs/src/dev/how_it_works.md
+++ b/docs/src/dev/how_it_works.md
@@ -102,24 +102,28 @@ two calls to ``f`` to evaluate the gradient:
                x_3 \\
                x_4
            \end{bmatrix}
-
-\vec{x}_{\epsilon} = \begin{bmatrix}
+```
+1st call:
+```math
+\to \vec{x}_{\epsilon} = \begin{bmatrix}
                         x_1 + \epsilon_1 \\
                         x_2 + \epsilon_2 \\
                         x_3 \\
                         x_4
                      \end{bmatrix}
 \to
-f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_2} \epsilon_2
-
-\vec{x}_{\epsilon} = \begin{bmatrix}
+f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_2} \epsilon_2,
+```
+2nd call:
+```math
+\to \vec{x}_{\epsilon} = \begin{bmatrix}
                         x_1 \\
                         x_2 \\
                         x_3 + \epsilon_1 \\
                         x_4 + \epsilon_2
                      \end{bmatrix}
 \to
-f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2
+f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2.
 ```
 
 This seeding process is similar for Jacobians, so we won't rehash it here.

From 27b4d09a5101ebe6eb5e13c5f7366bdd8ea7419d Mon Sep 17 00:00:00 2001
From: Killua Zoldyck <junyixu0@gmail.com>
Date: Mon, 25 Mar 2024 20:43:53 +0800
Subject: [PATCH 2/2] fix: Replace `\to` with `\mapsto` due to the nonstandard
 meaning of `\to`

Co-authored-by: Michael Abbott <32575566+mcabbott@users.noreply.github.com>
---
 docs/src/dev/how_it_works.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/docs/src/dev/how_it_works.md b/docs/src/dev/how_it_works.md
index 753602e8..bcf1c6bd 100644
--- a/docs/src/dev/how_it_works.md
+++ b/docs/src/dev/how_it_works.md
@@ -86,7 +86,7 @@ the result. For example, to calculate the partial derivatives for the gradient o
                          \vdots \\
                          x_N + \epsilon_N
                      \end{bmatrix}
-\to
+\mapsto
 f(\vec{x}_{\epsilon}) = f(\vec{x}) + \sum_{i=1}^N \frac{\delta f(\vec{x})}{\delta x_i} \epsilon_i
 ```
 
@@ -111,7 +111,7 @@ two calls to ``f`` to evaluate the gradient:
                         x_3 \\
                         x_4
                      \end{bmatrix}
-\to
+\mapsto
 f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_2} \epsilon_2,
 ```
 2nd call:
@@ -122,7 +122,7 @@ f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsil
                         x_3 + \epsilon_1 \\
                         x_4 + \epsilon_2
                      \end{bmatrix}
-\to
+\mapsto
 f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2.
 ```