add Float and ExtFloat methods for calculus

- add `derivative`, `integrate`, `partial_derivative_x`, `partial_derivative_y`.

DOCS/CHANGELOG.md

@@ -48,7 +48,9 @@ The format is based on [Keep a Changelog], and this project adheres to
   - `Array` when storing `Option<T>`.
   - `BareBox`.
   - `ExtAny` method `type_id`.
-  - `Float` and `ExtFloat` method `eval_poly` to evaluate polynomials.
+  - `Float` and `ExtFloat`:
+    - `eval_poly` to evaluate polynomials.
+    - for calculus: `derivative`, `integrate`, `partial_derivative_[x|y]`.
   - `Float` and `ExtFloatConst` consts: `LOW_MARGIN`, `MEDIUM_MARGIN`, `HIGH_MARGIN`.
   - `Graph` methods: `edge_exists_unchecked`, `edge_remove`.
   - `NonValue*`: `is_max`, `is_min`, `[checked|strict|saturating|wrapping]_[add|sub]`.

src/num/float/ext_float.rs

@@ -438,6 +438,37 @@ pub trait ExtFloat: ExtFloatConst + Sized {
     /// [Horner's method]: https://en.wikipedia.org/wiki/Horner%27s_method#Polynomial_evaluation_and_long_division
     #[must_use]
     fn eval_poly(self, coefficients: &[Self]) -> Self;
+
+    /// Approximates the derivative of the 1D function `f` at `x` point using step size `h`.
+    ///
+    /// Uses the [finite difference method].
+    ///
+    /// See also the [`autodiff`] attr macro, enabled with the `nightly_autodiff` feature.
+    ///
+    /// [finite difference method]: https://en.wikipedia.org/wiki/Finite_difference_method
+    fn derivative<F>(f: F, x: Self, h: Self) -> Self
+    where
+        F: Fn(Self) -> Self;
+
+    /// Approximates the integral of the 1D function `f` over the range `[x, y]`
+    /// using `n` subdivisions.
+    ///
+    /// Uses the [Riemann Sum](https://en.wikipedia.org/wiki/Riemann_sum).
+    fn integrate<F>(f: F, x: Self, y: Self, n: usize) -> Self
+    where
+        F: Fn(Self) -> Self;
+
+    /// Approximates the partial derivative of the 2D function `f` at point (`x`, `y`)
+    /// with step size `h`, differentiating over `x`.
+    fn partial_derivative_x<F>(f: F, x: Self, y: Self, h: Self) -> Self
+    where
+        F: Fn(Self, Self) -> Self;
+
+    /// Approximates the partial derivative of the 2D function `f` at point (`x`, `y`)
+    /// with step size `h`, differentiating over `x`.
+    fn partial_derivative_y<F>(f: F, x: Self, y: Self, h: Self) -> Self
+    where
+        F: Fn(Self, Self) -> Self;
 }
 
 #[doc = crate::doc_private!()]
@@ -711,6 +742,29 @@ macro_rules! impl_float_ext {
             fn eval_poly(self, coefficients: &[Self]) -> Self {
                 Float(self).eval_poly(coefficients).0
             }
+
+            fn derivative<F>(f: F, x: Self, h: Self) -> Self
+            where F: Fn(Self) -> Self {
+                Float::<Self>::derivative(f, x, h).0
+            }
+            fn integrate<F>(f: F, x: Self, y: Self, n: usize) -> Self
+            where F: Fn(Self) -> Self {
+                Float::<Self>::integrate(f, x, y, n).0
+            }
+
+            fn partial_derivative_x<F>(f: F, x: Self, y: Self, h: Self) -> Self
+            where
+                F: Fn(Self, Self) -> Self,
+            {
+                Float::<Self>::partial_derivative_x(f, x, y, h).0
+            }
+
+            fn partial_derivative_y<F>(f: F, x: Self, y: Self, h: Self) -> Self
+            where
+                F: Fn(Self, Self) -> Self,
+            {
+                Float::<Self>::partial_derivative_y(f, x, y, h).0
+            }
         }
     }
 }

src/num/float/wrapper/shared.rs

@@ -986,6 +986,76 @@ macro_rules! custom_impls {
                     }
                 }
             }
+
+            /// Approximates the derivative of the 1D function `f` at `x` point using step size `h`.
+            ///
+            /// Uses the [finite difference method].
+            ///
+            /// # Formula
+            /// $$ f'(x) \approx \frac{f(x + h) - f(x)}{h} $$
+            ///
+            /// See also the [`autodiff`] attr macro, enabled with the `nightly_autodiff` feature.
+            ///
+            /// [finite difference method]: https://en.wikipedia.org/wiki/Finite_difference_method
+            pub fn derivative<F>(f: F, x: $f, h: $f) -> Float<$f>
+            where
+                F: Fn($f) -> $f,
+            {
+                Float((f(x + h) - f(x)) / h)
+            }
+
+            /// Approximates the integral of the 1D function `f` over the range `[x, y]`
+            /// using `n` subdivisions.
+            ///
+            /// Uses the [Riemann Sum](https://en.wikipedia.org/wiki/Riemann_sum).
+            ///
+            /// # Formula
+            /// $$
+            /// \int_a^b f(x) \, dx \approx \sum_{i=0}^{n-1} f(x_i) \cdot \Delta x
+            /// $$
+            /// where
+            /// $$
+            /// \Delta x = \frac{b-a}{n}
+            /// $$
+            pub fn integrate<F>(f: F, x: $f, y: $f, n: usize) -> Float<$f>
+            where
+                F: Fn($f) -> $f,
+            {
+                let dx = (y - x) / n as $f;
+                Float(
+                    (0..n)
+                    .map(|i| { let x = x + i as $f * dx; f(x) * dx })
+                    .sum()
+                )
+            }
+
+            /// Approximates the partial derivative of the 2D function `f` at point (`x`, `y`)
+            /// with step size `h`, differentiating over `x`.
+            ///
+            /// # Formula
+            /// $$
+            /// \frac{\partial f}{\partial x} \approx \frac{f(x + h, y) - f(x, y)}{h}
+            /// $$
+            pub fn partial_derivative_x<F>(f: F, x: $f, y: $f, h: $f) -> Float<$f>
+            where
+                F: Fn($f, $f) -> $f,
+            {
+                Float((f(x + h, y) - f(x, y)) / h)
+            }
+
+            /// Approximates the partial derivative of the 2D function `f` at point (`x`, `y`)
+            /// with step size `h`, differentiating over `x`.
+            ///
+            /// # Formula
+            /// $$
+            /// \frac{\partial f}{\partial x} \approx \frac{f(x + h, y) - f(x, y)}{h}
+            /// $$
+            pub fn partial_derivative_y<F>(f: F, x: $f, y: $f, h: $f) -> Float<$f>
+            where
+                F: Fn($f, $f) -> $f,
+            {
+                Float((f(x, y + h) - f(x, y)) / h)
+            }
         }
     };
 }