copp 0.2.2

Convex-objective path parameterization for robotic trajectory planning.
Documentation 
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//! Third-order forward-mode automatic differentiation primitives.
//!
//! [`Jet3`] carries value and derivatives up to the 3rd order with respect
//! to a single scalar seed variable, and supports common arithmetic plus
//! elementary functions ([`sin`], [`cos`], [`exp`], [`ln`], [`sqrt`], [`powi`]).

use std::ops::{Add, Div, Mul, Neg, Sub};

/// Third-order forward-mode automatic differentiation scalar.
#[derive(Clone, Copy, Debug, Default)]
pub struct Jet3 {
    /// Function value.
    pub v: f64,
    /// First derivative with respect to the seed variable.
    pub d1: f64,
    /// Second derivative with respect to the seed variable.
    pub d2: f64,
    /// Third derivative with respect to the seed variable.
    pub d3: f64,
}

impl Jet3 {
    /// Construct a constant scalar with zero derivatives.
    #[inline(always)]
    pub fn constant(v: f64) -> Self {
        Self {
            v,
            d1: 0.0,
            d2: 0.0,
            d3: 0.0,
        }
    }

    /// Construct an independent variable seed (`d/ds = 1`).
    #[inline(always)]
    pub fn seed(v: f64) -> Self {
        Self {
            v,
            d1: 1.0,
            d2: 0.0,
            d3: 0.0,
        }
    }

    /// Apply sine and propagate derivatives up to third order.
    #[inline(always)]
    pub fn sin(self) -> Self {
        let sv = self.v.sin();
        let cv = self.v.cos();
        let d1sq = self.d1 * self.d1;
        let d1 = cv * self.d1;
        let d2 = -sv * d1sq + cv * self.d2;
        let d3 = -cv * d1sq * self.d1 - 3.0 * sv * self.d1 * self.d2 + cv * self.d3;
        Self { v: sv, d1, d2, d3 }
    }

    /// Apply cosine and propagate derivatives up to third order.
    #[inline(always)]
    pub fn cos(self) -> Self {
        let sv = self.v.sin();
        let cv = self.v.cos();
        let d1sq = self.d1 * self.d1;
        let d1 = -sv * self.d1;
        let d2 = -cv * d1sq - sv * self.d2;
        let d3 = sv * d1sq * self.d1 - 3.0 * cv * self.d1 * self.d2 - sv * self.d3;
        Self { v: cv, d1, d2, d3 }
    }

    /// Apply exponential and propagate derivatives up to third order.
    #[inline(always)]
    pub fn exp(self) -> Self {
        let ev = self.v.exp();
        let d1sq = self.d1 * self.d1;
        let d1 = ev * self.d1;
        let d2 = ev * (d1sq + self.d2);
        let d3 = ev * (d1sq * self.d1 + 3.0 * self.d1 * self.d2 + self.d3);
        Self { v: ev, d1, d2, d3 }
    }

    /// Apply natural logarithm and propagate derivatives up to third order.
    #[inline(always)]
    pub fn ln(self) -> Self {
        let v = self.v.ln();
        let inv_x = 1.0 / self.v;
        let inv_x2 = inv_x * inv_x;
        let inv_x3 = inv_x2 * inv_x;
        let d1sq = self.d1 * self.d1;
        let d1 = inv_x * self.d1;
        let d2 = -inv_x2 * d1sq + inv_x * self.d2;
        let d3 = 2.0 * inv_x3 * d1sq * self.d1 - 3.0 * inv_x2 * self.d1 * self.d2 + inv_x * self.d3;
        Self { v, d1, d2, d3 }
    }

    /// Apply square root and propagate derivatives up to third order.
    ///
    /// The value component should be positive for finite real derivatives.
    #[inline(always)]
    pub fn sqrt(self) -> Self {
        let sqrtv = self.v.sqrt();
        let inv_sqrt = 1.0 / sqrtv;
        let inv_v_sqrt = inv_sqrt / self.v;
        let inv_v2_sqrt = inv_v_sqrt / self.v;
        let d1sq = self.d1 * self.d1;
        let d1 = 0.5 * inv_sqrt * self.d1;
        let d2 = -0.25 * inv_v_sqrt * d1sq + 0.5 * inv_sqrt * self.d2;
        let d3 = 0.375 * inv_v2_sqrt * d1sq * self.d1 - 0.75 * inv_v_sqrt * self.d1 * self.d2
            + 0.5 * inv_sqrt * self.d3;
        Self {
            v: sqrtv,
            d1,
            d2,
            d3,
        }
    }

    /// Raise the value to an integer power and propagate derivatives.
    #[inline(always)]
    pub fn powi(self, n: i32) -> Self {
        if n == 0 {
            return Self::constant(1.0);
        }
        let nf = n as f64;
        // Compute v^(n-3) once; derive v^(n-2), v^(n-1), v^n by repeated multiplication.
        // This replaces 4 independent `f64::powi` calls with 1 call + 3 multiplications,
        // avoiding redundant repeated-squaring work for the same base.
        let vn3 = self.v.powi(n - 3);
        let vn2 = vn3 * self.v;
        let vn1 = vn2 * self.v;
        let vn = vn1 * self.v;
        let dv = nf * vn1;
        let ddv = nf * (nf - 1.0) * vn2;
        let dddv = nf * (nf - 1.0) * (nf - 2.0) * vn3;
        let d1sq = self.d1 * self.d1;
        let d1 = dv * self.d1;
        let d2 = ddv * d1sq + dv * self.d2;
        let d3 = dddv * d1sq * self.d1 + 3.0 * ddv * self.d1 * self.d2 + dv * self.d3;
        Self { v: vn, d1, d2, d3 }
    }

    #[inline(always)]
    fn inv(self) -> Self {
        let v = 1.0 / self.v;
        let inv_x2 = v * v;
        let inv_x3 = inv_x2 * v;
        let inv_x4 = inv_x3 * v;
        let d1sq = self.d1 * self.d1;
        let d1 = -inv_x2 * self.d1;
        let d2 = 2.0 * inv_x3 * d1sq - inv_x2 * self.d2;
        let d3 =
            -6.0 * inv_x4 * d1sq * self.d1 + 6.0 * inv_x3 * self.d1 * self.d2 - inv_x2 * self.d3;
        Self { v, d1, d2, d3 }
    }
}

/// Function-style wrapper for [`Jet3::sin`].
#[inline(always)]
pub fn sin(x: Jet3) -> Jet3 {
    x.sin()
}

/// Function-style wrapper for [`Jet3::cos`].
#[inline(always)]
pub fn cos(x: Jet3) -> Jet3 {
    x.cos()
}

/// Function-style wrapper for [`Jet3::exp`].
#[inline(always)]
pub fn exp(x: Jet3) -> Jet3 {
    x.exp()
}

/// Function-style wrapper for [`Jet3::ln`].
#[inline(always)]
pub fn ln(x: Jet3) -> Jet3 {
    x.ln()
}

/// Function-style wrapper for [`Jet3::sqrt`].
#[inline(always)]
pub fn sqrt(x: Jet3) -> Jet3 {
    x.sqrt()
}

/// Function-style wrapper for [`Jet3::powi`].
#[inline(always)]
pub fn powi(x: Jet3, n: i32) -> Jet3 {
    x.powi(n)
}

impl From<f64> for Jet3 {
    #[inline(always)]
    fn from(value: f64) -> Self {
        Self::constant(value)
    }
}

impl Add for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn add(self, rhs: Self) -> Self::Output {
        Self {
            v: self.v + rhs.v,
            d1: self.d1 + rhs.d1,
            d2: self.d2 + rhs.d2,
            d3: self.d3 + rhs.d3,
        }
    }
}

impl Add<f64> for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn add(self, rhs: f64) -> Self::Output {
        Self {
            v: self.v + rhs,
            ..self
        }
    }
}

impl Add<Jet3> for f64 {
    type Output = Jet3;

    #[inline(always)]
    fn add(self, rhs: Jet3) -> Self::Output {
        rhs + self
    }
}

impl Sub for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn sub(self, rhs: Self) -> Self::Output {
        Self {
            v: self.v - rhs.v,
            d1: self.d1 - rhs.d1,
            d2: self.d2 - rhs.d2,
            d3: self.d3 - rhs.d3,
        }
    }
}

impl Sub<f64> for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn sub(self, rhs: f64) -> Self::Output {
        Self {
            v: self.v - rhs,
            ..self
        }
    }
}

impl Sub<Jet3> for f64 {
    type Output = Jet3;

    #[inline(always)]
    fn sub(self, rhs: Jet3) -> Self::Output {
        Jet3::constant(self) - rhs
    }
}

impl Mul for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn mul(self, rhs: Self) -> Self::Output {
        Self {
            v: self.v * rhs.v,
            d1: self.d1 * rhs.v + self.v * rhs.d1,
            d2: self.d2 * rhs.v + 2.0 * self.d1 * rhs.d1 + self.v * rhs.d2,
            d3: self.d3 * rhs.v + 3.0 * (self.d2 * rhs.d1 + self.d1 * rhs.d2) + self.v * rhs.d3,
        }
    }
}

impl Mul<f64> for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn mul(self, rhs: f64) -> Self::Output {
        Self {
            v: self.v * rhs,
            d1: self.d1 * rhs,
            d2: self.d2 * rhs,
            d3: self.d3 * rhs,
        }
    }
}

impl Mul<Jet3> for f64 {
    type Output = Jet3;

    #[inline(always)]
    fn mul(self, rhs: Jet3) -> Self::Output {
        rhs * self
    }
}

impl Div for Jet3 {
    type Output = Self;

    #[allow(clippy::suspicious_arithmetic_impl)]
    #[inline(always)]
    fn div(self, rhs: Self) -> Self::Output {
        self * rhs.inv()
    }
}

impl Div<f64> for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn div(self, rhs: f64) -> Self::Output {
        self * (1.0 / rhs)
    }
}

impl Div<Jet3> for f64 {
    type Output = Jet3;

    #[inline(always)]
    fn div(self, rhs: Jet3) -> Self::Output {
        Jet3::constant(self) / rhs
    }
}

impl Neg for Jet3 {
    type Output = Self;

    #[inline(always)]
    fn neg(self) -> Self::Output {
        Self {
            v: -self.v,
            d1: -self.d1,
            d2: -self.d2,
            d3: -self.d3,
        }
    }
}