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use super::{Expr, E, constant, sin, cos, cosh, sinh, tanh, exp, ln, sqrt, abs, pow};
/// Types that can be used as differentiation variables.
///
/// Implemented for `&str`, `String`, and [`E`] (when it wraps a `Sym` node),
/// so you can call `expr.diff("x")` or `expr.diff(&my_symbol)`.
pub trait DiffVar {
/// Return the variable name as a string slice.
fn var_name(&self) -> &str;
}
impl DiffVar for &str {
fn var_name(&self) -> &str { self }
}
impl DiffVar for &&str {
fn var_name(&self) -> &str { self }
}
impl DiffVar for str {
fn var_name(&self) -> &str { self }
}
impl DiffVar for String {
fn var_name(&self) -> &str { self.as_str() }
}
impl DiffVar for &E {
fn var_name(&self) -> &str {
(*self).var_name()
}
}
impl DiffVar for E {
fn var_name(&self) -> &str {
match self.as_ref() {
Expr::Sym(name) => name.as_str(),
_ => panic!("diff: expected a symbol, got `{self}`"),
}
}
}
impl Expr {
/// Symbolically differentiate this expression with respect to a variable.
///
/// Applies the chain rule, product rule, and quotient rule automatically.
/// The result is simplified.
pub fn diff(&self, var: impl DiffVar) -> E {
let var = var.var_name();
let zero = || constant(0.0);
let one = || constant(1.0);
let two = || constant(2.0);
match self {
Expr::Sym(name) => {
if name == var { one() } else { zero() }
}
Expr::Const(_) => zero(),
Expr::Neg(a) => {
-a.diff(var)
}
Expr::Add(a, b) => {
a.diff(var) + b.diff(var)
}
Expr::Sub(a, b) => {
a.diff(var) - b.diff(var)
}
Expr::Mul(a, b) => {
// product rule: a'*b + a*b'
let da = a.diff(var);
let db = b.diff(var);
da * b.clone() + a.clone() * db
}
Expr::Div(a, b) => {
// quotient rule: (a'*b - a*b') / b^2
let da = a.diff(var);
let db = b.diff(var);
(da * b.clone() - a.clone() * db) / pow(b.clone(), two())
}
Expr::Pow(a, b) => {
let da = a.diff(var);
let db = b.diff(var);
if matches!(b.as_ref(), Expr::Const(_)) {
// Power rule: n * a^(n-1) * a'
b.clone() * pow(a.clone(), b.clone() - constant(1.0)) * da
} else if matches!(a.as_ref(), Expr::Const(_)) {
// Constant base: a^b * ln(a) * b'
pow(a.clone(), b.clone()) * ln(a.clone()) * db
} else {
// General: a^b * (b' * ln(a) + b * a' / a)
let base = pow(a.clone(), b.clone());
base * (db * ln(a.clone()) + b.clone() * da / a.clone())
}
}
Expr::Sin(a) => {
cos(a.clone()) * a.diff(var)
}
Expr::Cos(a) => {
-(sin(a.clone()) * a.diff(var))
}
Expr::Tan(a) => {
// a' / cos(a)^2
a.diff(var) / pow(cos(a.clone()), two())
}
Expr::Asin(a) => {
// a' / sqrt(1 - a^2)
a.diff(var) / sqrt(one() - pow(a.clone(), two()))
}
Expr::Acos(a) => {
// -a' / sqrt(1 - a^2)
-(a.diff(var) / sqrt(one() - pow(a.clone(), two())))
}
Expr::Atan(a) => {
// a' / (1 + a^2)
a.diff(var) / (one() + pow(a.clone(), two()))
}
Expr::Atan2(y, x) => {
// (x*dy - y*dx) / (x^2 + y^2)
let dy = y.diff(var);
let dx = x.diff(var);
(x.clone() * dy - y.clone() * dx) / (pow(x.clone(), two()) + pow(y.clone(), two()))
}
Expr::Sinh(a) => {
cosh(a.clone()) * a.diff(var)
}
Expr::Cosh(a) => {
sinh(a.clone()) * a.diff(var)
}
Expr::Tanh(a) => {
// a' * (1 - tanh(a)^2)
a.diff(var) * (one() - pow(tanh(a.clone()), two()))
}
Expr::Exp(a) => {
exp(a.clone()) * a.diff(var)
}
Expr::Ln(a) => {
// a' / a
a.diff(var) / a.clone()
}
Expr::Log2(a) => {
// a' / (a * ln(2))
a.diff(var) / (a.clone() * ln(constant(2.0)))
}
Expr::Log10(a) => {
// a' / (a * ln(10))
a.diff(var) / (a.clone() * ln(constant(10.0)))
}
Expr::Sqrt(a) => {
// a' / (2 * sqrt(a))
a.diff(var) / (two() * sqrt(a.clone()))
}
Expr::Abs(a) => {
// a * a' / |a| (i.e., sign(a) * a')
a.clone() * a.diff(var) / abs(a.clone())
}
}
}
}