echidna 0.9.0

//! Shared Taylor coefficient propagation functions.
//!
//! Convention: `c[k] = f^(k)(t₀) / k!` (scaled/normalized Taylor coefficients).
//! All functions operate on slices `&[F]` (inputs) and `&mut [F]` (outputs),
//! where `F: num_traits::Float`. The degree (number of coefficients) is
//! determined by the slice lengths.
//!
//! Used by both `Taylor<F, K>` (stack arrays) and `TaylorDyn<F>` (arena slices).

use num_traits::Float;

// ══════════════════════════════════════════════
//  Arithmetic
// ══════════════════════════════════════════════

/// `c = a + b`
#[inline]
pub fn taylor_add<F: Float>(a: &[F], b: &[F], c: &mut [F]) {
    for k in 0..c.len() {
        c[k] = a[k] + b[k];
    }
}

/// `c = a - b`
#[inline]
pub fn taylor_sub<F: Float>(a: &[F], b: &[F], c: &mut [F]) {
    for k in 0..c.len() {
        c[k] = a[k] - b[k];
    }
}

/// `c = -a`
#[inline]
pub fn taylor_neg<F: Float>(a: &[F], c: &mut [F]) {
    for k in 0..c.len() {
        c[k] = -a[k];
    }
}

/// `c = s * a` where `s` is a scalar.
#[inline]
pub fn taylor_scale<F: Float>(a: &[F], s: F, c: &mut [F]) {
    for k in 0..c.len() {
        c[k] = s * a[k];
    }
}

/// `c = a * b` — Cauchy product.
///
/// `c[k] = Σ_{j=0}^{k} a[j] * b[k-j]`
#[inline]
pub fn taylor_mul<F: Float>(a: &[F], b: &[F], c: &mut [F]) {
    let n = c.len();
    for k in 0..n {
        let mut sum = F::zero();
        for j in 0..=k {
            sum = sum + a[j] * b[k - j];
        }
        c[k] = sum;
    }
}

/// `c = a / b` — recursive Taylor division.
///
/// `c[k] = (a[k] - Σ_{j=1}^{k} b[j] * c[k-j]) / b[0]`
#[inline]
pub fn taylor_div<F: Float>(a: &[F], b: &[F], c: &mut [F]) {
    let n = c.len();
    let inv_b0 = F::one() / b[0];
    for k in 0..n {
        let mut sum = a[k];
        for j in 1..=k {
            sum = sum - b[j] * c[k - j];
        }
        c[k] = sum * inv_b0;
    }
}

/// `c = 1/a` — reciprocal via recursive division.
///
/// Special case of div with numerator = [1, 0, ..., 0].
#[inline]
pub fn taylor_recip<F: Float>(a: &[F], c: &mut [F]) {
    let n = c.len();
    let inv_a0 = F::one() / a[0];
    c[0] = inv_a0;
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + a[j] * c[k - j];
        }
        c[k] = -sum * inv_a0;
    }
}

// ══════════════════════════════════════════════
//  Transcendentals (Griewank Ch 13 logarithmic derivative technique)
// ══════════════════════════════════════════════

/// `c = exp(a)`
///
/// `c[0] = exp(a[0])`
/// `c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * c[k-j]`
#[inline]
pub fn taylor_exp<F: Float>(a: &[F], c: &mut [F]) {
    let n = c.len();
    c[0] = a[0].exp();
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * c[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

/// `c = ln(a)`
///
/// `c[0] = ln(a[0])`
/// `c[k] = (a[k] - (1/k) * Σ_{j=1}^{k-1} j * c[j] * a[k-j]) / a[0]`
#[inline]
pub fn taylor_ln<F: Float>(a: &[F], c: &mut [F]) {
    let n = c.len();
    let inv_a0 = F::one() / a[0];
    c[0] = a[0].ln();
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..k {
            sum = sum + F::from(j).unwrap() * c[j] * a[k - j];
        }
        c[k] = (a[k] - sum / F::from(k).unwrap()) * inv_a0;
    }
}

/// `c = sqrt(a)`
///
/// `c[0] = sqrt(a[0])`
/// `c[k] = (a[k] - Σ_{j=1}^{k-1} c[j] * c[k-j]) / (2 * c[0])`
///
/// When `a[0] == 0`, returns `c[0] = 0` and `c[k] = Inf` for `k >= 1` (the
/// derivative is singular at a branch point). Use the `Laurent` type for
/// functions with branch points at the expansion point.
#[inline]
pub fn taylor_sqrt<F: Float>(a: &[F], c: &mut [F]) {
    let n = c.len();
    if a[0] == F::zero() {
        // sqrt(0) = 0, but sqrt'(0) = 1/(2*sqrt(0)) = Inf (vertical tangent).
        c[0] = F::zero();
        for ci in c.iter_mut().skip(1) {
            *ci = F::infinity();
        }
        return;
    }
    if a[0] < F::zero() {
        // sqrt is undefined on the negative reals. Relying on `.sqrt()`'s
        // silent NaN propagates the NaN into c[0] but leaves higher-order
        // coefficients computed from division by `2 * NaN` in a state that
        // looks like a normal recurrence. Make the degeneracy explicit so
        // downstream callers don't accidentally consume a mix of finite
        // and non-finite coefficients.
        for ci in c.iter_mut() {
            *ci = F::nan();
        }
        return;
    }
    c[0] = a[0].sqrt();
    let two_c0 = F::from(2.0).unwrap() * c[0];
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..k {
            sum = sum + c[j] * c[k - j];
        }
        c[k] = (a[k] - sum) / two_c0;
    }
}

/// `(s, co) = sin_cos(a)` — coupled recurrence.
///
/// `s[0] = sin(a[0])`, `co[0] = cos(a[0])`
/// `s[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * co[k-j]`
/// `co[k] = -(1/k) * Σ_{j=1}^{k} j * a[j] * s[k-j]`
#[inline]
pub fn taylor_sin_cos<F: Float>(a: &[F], s: &mut [F], co: &mut [F]) {
    let n = s.len();
    let (s0, c0) = a[0].sin_cos();
    s[0] = s0;
    co[0] = c0;
    for k in 1..n {
        let inv_k = F::one() / F::from(k).unwrap();
        let mut sum_s = F::zero();
        let mut sum_c = F::zero();
        for j in 1..=k {
            let jf = F::from(j).unwrap();
            sum_s = sum_s + jf * a[j] * co[k - j];
            sum_c = sum_c + jf * a[j] * s[k - j];
        }
        s[k] = sum_s * inv_k;
        co[k] = -sum_c * inv_k;
    }
}

/// `(sh, ch) = sinh_cosh(a)` — coupled recurrence (positive signs).
///
/// `sh[0] = sinh(a[0])`, `ch[0] = cosh(a[0])`
/// `sh[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * ch[k-j]`
/// `ch[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * sh[k-j]`
#[inline]
pub fn taylor_sinh_cosh<F: Float>(a: &[F], sh: &mut [F], ch: &mut [F]) {
    let n = sh.len();
    sh[0] = a[0].sinh();
    ch[0] = a[0].cosh();
    for k in 1..n {
        let inv_k = F::one() / F::from(k).unwrap();
        let mut sum_sh = F::zero();
        let mut sum_ch = F::zero();
        for j in 1..=k {
            let jf = F::from(j).unwrap();
            sum_sh = sum_sh + jf * a[j] * ch[k - j];
            sum_ch = sum_ch + jf * a[j] * sh[k - j];
        }
        sh[k] = sum_sh * inv_k;
        ch[k] = sum_ch * inv_k;
    }
}

/// `c = atan(a)` — via `c' = a' / (1 + a²)`, then integrate.
///
/// Uses `scratch` for the `1 + a²` denominator.
#[inline]
pub fn taylor_atan<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let n = c.len();
    // scratch1 = a²
    taylor_mul(a, a, scratch1);
    // scratch2 = 1 + a²
    scratch2[..n].copy_from_slice(&scratch1[..n]);
    scratch2[0] = F::one() + scratch1[0];
    // c[0] = atan(a[0])
    c[0] = a[0].atan();
    // c' = a' / (1 + a²), so c[k] via the division-integration recurrence:
    // Let d = 1/(1+a²). Then c'[k-1] = Σ d[j] * a'[k-j-1]... but simpler approach:
    // Since (1+a²) * c' = a', we have:
    // k * (1+a²)[0] * c[k] + Σ_{j=1}^{k-1} (1+a²)[j] * (k-j) * c[k-j]...
    // Actually the integration approach: if g = 1/(1+a²), then c' = a' * g
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * g[k-j]
    // First compute g = 1/(1+a²):
    // Reuse scratch1 for g = recip(1+a²)
    taylor_recip(scratch2, scratch1);
    // Now c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch1[k-j]
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch1[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

/// `c = asin(a)` — via `c' = a' / sqrt(1 - a²)`, then integrate.
///
/// Uses `scratch1` and `scratch2` as work space.
#[inline]
pub fn taylor_asin<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let n = c.len();
    c[0] = a[0].asin();
    // scratch1 = a²
    taylor_mul(a, a, scratch1);
    // scratch2 = 1 - a²  (use (1-a₀)(1+a₀) to avoid cancellation near |a₀|→1)
    scratch2[0] = (F::one() - a[0]) * (F::one() + a[0]);
    for k in 1..n {
        scratch2[k] = -scratch1[k];
    }
    // scratch1 = sqrt(1 - a²)
    taylor_sqrt(scratch2, scratch1);
    // scratch2 = 1/sqrt(1 - a²)
    taylor_recip(scratch1, scratch2);
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch2[k-j]
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch2[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

/// `c = acos(a) = π/2 - asin(a)`
#[inline]
pub fn taylor_acos<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    taylor_asin(a, c, scratch1, scratch2);
    c[0] = a[0].acos();
    for ck in c[1..].iter_mut() {
        *ck = -*ck;
    }
}

/// `c = tan(a)` — via `c' = a' * (1 + tan²(a))` = `a' * (1 + c²)`.
///
/// Uses `scratch` for `c²`.
#[inline]
pub fn taylor_tan<F: Float>(a: &[F], c: &mut [F], scratch: &mut [F]) {
    let n = c.len();
    c[0] = a[0].tan();
    // Recurrence: c' = a' * (1 + c²)
    // c[k] = a[k] + (1/k) * Σ_{j=1}^{k} j * a[j] * scratch[k-j]
    // where scratch holds running c² (partial)
    // Actually let's do it step by step: after computing c[0..k], update scratch = c[0..k]²
    // But this is circular. Better approach:
    // Let s = 1 + c². Then c' = a' * s.
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * s[k-j]
    // s[0] = 1 + c[0]², and for k>=1: s[k] = Σ_{j=0}^{k} c[j]*c[k-j]
    // But s[k] depends on c[k], which depends on s...
    // Expand: s[k] = 2*c[0]*c[k] + Σ_{j=1}^{k-1} c[j]*c[k-j]
    // And c[k] = (1/k) * [k*a[k]*s[0] + Σ_{j=1}^{k-1} j*a[j]*s[k-j]]
    // So substitute and solve for c[k]:
    // c[k] = a[k]*s[0] + (1/k) * Σ_{j=1}^{k-1} j*a[j]*s[k-j]  ... (*)
    // Then s[k] = 2*c[0]*c[k] + Σ_{j=1}^{k-1} c[j]*c[k-j]
    // Wait, (*) only uses s[0..k-1] which are known. So this works!

    // scratch = s (1 + c²)
    scratch[0] = F::one() + c[0] * c[0];
    for k in 1..n {
        // First compute c[k] using s[0..k-1]
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
        // Now update scratch[k] = s[k] = Σ_{j=0}^{k} c[j]*c[k-j]
        let mut s_k = F::zero();
        for j in 0..=k {
            s_k = s_k + c[j] * c[k - j];
        }
        scratch[k] = s_k;
    }
}

/// `c = tanh(a)` — via `c' = a' * (1 - tanh²(a))` = `a' * (1 - c²)`.
///
/// Uses `scratch` for `1 - c²`.
#[inline]
pub fn taylor_tanh<F: Float>(a: &[F], c: &mut [F], scratch: &mut [F]) {
    let n = c.len();
    c[0] = a[0].tanh();
    // scratch = s = 1 - c²
    scratch[0] = F::one() - c[0] * c[0];
    for k in 1..n {
        // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch[k-j]
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
        // scratch[k] = -Σ_{j=0}^{k} c[j]*c[k-j]
        let mut s_k = F::zero();
        for j in 0..=k {
            s_k = s_k + c[j] * c[k - j];
        }
        scratch[k] = -s_k;
    }
}

/// `c = asinh(a)` — via `c' = a' / sqrt(1 + a²)`.
#[inline]
pub fn taylor_asinh<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let n = c.len();
    c[0] = a[0].asinh();
    // scratch1 = a²
    taylor_mul(a, a, scratch1);
    // scratch2 = 1 + a²
    scratch2[..n].copy_from_slice(&scratch1[..n]);
    scratch2[0] = F::one() + scratch1[0];
    // scratch1 = sqrt(1 + a²)
    taylor_sqrt(scratch2, scratch1);
    // scratch2 = 1/sqrt(1 + a²)
    taylor_recip(scratch1, scratch2);
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch2[k-j]
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch2[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

/// `c = acosh(a)` — via `c' = a' / sqrt(a² - 1)`.
#[inline]
pub fn taylor_acosh<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let n = c.len();
    c[0] = a[0].acosh();
    // scratch1 = a²
    taylor_mul(a, a, scratch1);
    // scratch2 = a² - 1  (factored form avoids cancellation near a[0]=1)
    scratch2[..n].copy_from_slice(&scratch1[..n]);
    scratch2[0] = (a[0] - F::one()) * (a[0] + F::one());
    // scratch1 = sqrt(a² - 1)
    taylor_sqrt(scratch2, scratch1);
    // scratch2 = 1/sqrt(a² - 1)
    taylor_recip(scratch1, scratch2);
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch2[k-j]
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch2[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

/// `c = atanh(a)` — via `c' = a' / (1 - a²)`.
#[inline]
pub fn taylor_atanh<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let n = c.len();
    c[0] = a[0].atanh();
    // scratch1 = a²
    taylor_mul(a, a, scratch1);
    // scratch2 = 1 - a²  (use (1-a₀)(1+a₀) to avoid cancellation near |a₀|→1)
    scratch2[0] = (F::one() - a[0]) * (F::one() + a[0]);
    for k in 1..n {
        scratch2[k] = -scratch1[k];
    }
    // scratch1 = 1/(1 - a²)
    taylor_recip(scratch2, scratch1);
    // c[k] = (1/k) * Σ_{j=1}^{k} j * a[j] * scratch1[k-j]
    for k in 1..n {
        let mut sum = F::zero();
        for j in 1..=k {
            sum = sum + F::from(j).unwrap() * a[j] * scratch1[k - j];
        }
        c[k] = sum / F::from(k).unwrap();
    }
}

// ══════════════════════════════════════════════
//  Derived functions
// ══════════════════════════════════════════════

/// `c = a^b` (powf) = `exp(b * ln(a))`.
///
/// Uses `scratch1` for `ln(a)` and `scratch2` for `b * ln(a)`.
#[inline]
pub fn taylor_powf<F: Float>(
    a: &[F],
    b: &[F],
    c: &mut [F],
    scratch1: &mut [F],
    scratch2: &mut [F],
) {
    // Constant integer exponent fast path: if `b` is a plain scalar (higher
    // coefficients are zero) and that scalar is an integer, route to
    // `taylor_powi`. Otherwise `taylor_ln(a)` returns NaN for `a[0] <= 0`,
    // poisoning the entire result — even for negative-base integer powers
    // that have well-defined Taylor coefficients.
    if b[1..].iter().all(|&bk| bk == F::zero()) {
        let b0 = b[0];
        if let Some(ni) = b0.to_i32() {
            if F::from(ni).unwrap() == b0 {
                taylor_powi(a, ni, c, scratch1, scratch2);
                return;
            }
        }
    }
    // scratch1 = ln(a)
    taylor_ln(a, scratch1);
    // scratch2 = b * ln(a)
    taylor_mul(b, scratch1, scratch2);
    // c = exp(b * ln(a))
    // But we can't use scratch1 anymore since taylor_exp needs its own output...
    // Actually taylor_exp writes to c, so we just need scratch2 as input.
    taylor_exp(scratch2, c);
    // Fix c[0] for better primal accuracy (use direct powf instead of exp(b*ln(a))).
    // Higher coefficients c[1..] were computed using the exp-ln path's c[0], which
    // may differ from the patched value by sub-ULP rounding. This is a deliberate
    // Intentional precision tradeoff: patching c[0] with direct powf is more accurate
    // than the exp-ln roundtrip, but c[1..K] were computed using the exp-ln c[0] value.
    // The inconsistency is O(ULP) for well-conditioned inputs and does not affect
    // derivative correctness beyond rounding.
    c[0] = a[0].powf(b[0]);
}

/// `c = a^n` (powi) — integer power.
///
/// Dispatches between two strategies:
/// - **Repeated squaring** (binary exponentiation via `taylor_mul`): used when
///   `a[0] < 0` (where `ln` would produce NaN) or `|n| <= 8` (at most 3
///   multiplications, competitive with exp-ln).
/// - **exp(n * ln(a))**: used for positive base with large exponents.
#[inline]
pub fn taylor_powi<F: Float>(a: &[F], n: i32, c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let deg = c.len();
    if n == 0 {
        c[0] = F::one();
        for ck in c[1..deg].iter_mut() {
            *ck = F::zero();
        }
        return;
    }
    if n == 1 {
        c.copy_from_slice(a);
        return;
    }
    if n == -1 {
        taylor_recip(a, c);
        return;
    }
    if a[0] <= F::zero() || n.unsigned_abs() <= 8 {
        taylor_powi_squaring(a, n, c, scratch1, scratch2);
    } else {
        // scratch1 = ln(a)
        taylor_ln(a, scratch1);
        // scratch2 = n * ln(a)
        let nf = F::from(n).unwrap();
        taylor_scale(scratch1, nf, scratch2);
        // c = exp(n * ln(a))
        taylor_exp(scratch2, c);
        c[0] = a[0].powi(n);
    }
}

/// Integer power via binary exponentiation on Taylor coefficient arrays.
///
/// Computes `a^n` using repeated squaring with `taylor_mul`. Works correctly
/// for negative base values (unlike the exp-ln path). For negative `n`,
/// computes `a^|n|` then takes the reciprocal.
fn taylor_powi_squaring<F: Float>(
    a: &[F],
    n: i32,
    c: &mut [F],
    scratch1: &mut [F],
    scratch2: &mut [F],
) {
    let deg = c.len();
    let abs_n = n.unsigned_abs();

    // result (c) = 1
    c[0] = F::one();
    for ck in c[1..deg].iter_mut() {
        *ck = F::zero();
    }

    // base (scratch1) = a
    scratch1[..deg].copy_from_slice(&a[..deg]);

    let mut power = abs_n;
    while power > 0 {
        if power & 1 == 1 {
            // result = result * base
            taylor_mul(c, &*scratch1, scratch2);
            c[..deg].copy_from_slice(&scratch2[..deg]);
        }
        power >>= 1;
        if power > 0 {
            // base = base * base
            let base_ref: &[F] = &*scratch1;
            // Inline squaring to avoid borrow conflict (scratch1 is both source and dest)
            for k in 0..deg {
                let mut sum = F::zero();
                for j in 0..=k {
                    sum = sum + base_ref[j] * base_ref[k - j];
                }
                scratch2[k] = sum;
            }
            scratch1[..deg].copy_from_slice(&scratch2[..deg]);
        }
    }

    if n < 0 {
        // c = 1/c: copy c into scratch1, then compute recip into c
        scratch1[..deg].copy_from_slice(&c[..deg]);
        taylor_recip(scratch1, c);
    }
}

/// `c = cbrt(a) = a^(1/3)`.
///
/// Uses `scratch1` and `scratch2`.
#[inline]
pub fn taylor_cbrt<F: Float>(a: &[F], c: &mut [F], scratch1: &mut [F], scratch2: &mut [F]) {
    let deg = c.len();
    debug_assert_eq!(a.len(), c.len());
    if a[0] == F::zero() {
        // cbrt(0) = 0, but cbrt'(0) = 1/(3*cbrt(0)^2) = Inf (vertical tangent).
        c[0] = F::zero();
        for ci in c.iter_mut().skip(1) {
            *ci = F::infinity();
        }
        return;
    }
    if a[0] < F::zero() {
        // cbrt(-x) = -cbrt(x): negate input, compute cbrt on positive, negate output.
        // Use c as temporary for negated input (safe: taylor_ln reads before writing).
        for i in 0..deg {
            c[i] = -a[i];
        }
        let three = F::from(3.0).unwrap();
        let third = F::one() / three;
        taylor_ln(c, scratch1);
        taylor_scale(scratch1, third, scratch2);
        taylor_exp(scratch2, c);
        // Same O(ULP) primal-patch tradeoff as taylor_powf (see comment there).
        // The negate-all-coefficients approach uses cbrt(a) = -cbrt(-a), which is exact.
        c[0] = a[0].cbrt();
        for ci in c.iter_mut().skip(1) {
            *ci = -*ci;
        }
    } else {
        let three = F::from(3.0).unwrap();
        let third = F::one() / three;
        taylor_ln(a, scratch1);
        taylor_scale(scratch1, third, scratch2);
        taylor_exp(scratch2, c);
        // Primal patch: same O(ULP) tradeoff as taylor_powf (see comment there).
        c[0] = a[0].cbrt();
    }
}

/// `c = exp2(a) = 2^a = exp(a * ln(2))`.
#[inline]
pub fn taylor_exp2<F: Float>(a: &[F], c: &mut [F], scratch: &mut [F]) {
    let ln2 = F::from(2.0).unwrap().ln();
    taylor_scale(a, ln2, scratch);
    taylor_exp(scratch, c);
    // Primal patch: same O(ULP) tradeoff as taylor_powf (see comment there).
    c[0] = a[0].exp2();
}

/// `c = exp(a) - 1` (exp_m1).
///
/// NOTE (verified correct): The recurrence uses `exp(a\[0\])` for `c\[1..\]` (via `taylor_exp`),
/// then patches `c\[0\]` to `exp_m1(a\[0\])`. This is correct because derivatives of `exp(x)-1`
/// and `exp(x)` are identical for k>=1 (the -1 is a constant offset).
///
/// The recurrence correctly uses `c[0] = exp(a[0])` (not `exp_m1(a[0])`) to compute
/// `c[1..K]`, because `d/dx[exp(x)-1] = exp(x)` — the higher-order Taylor coefficients
/// of `exp(x)-1` are identical to those of `exp(x)`. Only `c[0]` is patched afterward.
#[inline]
pub fn taylor_exp_m1<F: Float>(a: &[F], c: &mut [F]) {
    taylor_exp(a, c);
    c[0] = a[0].exp_m1();
}

/// `c = log2(a) = ln(a) / ln(2)`.
#[inline]
pub fn taylor_log2<F: Float>(a: &[F], c: &mut [F]) {
    taylor_ln(a, c);
    let inv_ln2 = F::one() / F::from(2.0).unwrap().ln();
    // Primal patch: same O(ULP) tradeoff as taylor_powf (see comment there).
    c[0] = a[0].log2();
    for ck in c[1..].iter_mut() {
        *ck = *ck * inv_ln2;
    }
}

/// `c = log10(a) = ln(a) / ln(10)`.
#[inline]
pub fn taylor_log10<F: Float>(a: &[F], c: &mut [F]) {
    taylor_ln(a, c);
    let inv_ln10 = F::one() / F::from(10.0).unwrap().ln();
    // Primal patch: same O(ULP) tradeoff as taylor_powf (see comment there).
    c[0] = a[0].log10();
    for ck in c[1..].iter_mut() {
        *ck = *ck * inv_ln10;
    }
}

/// `c = ln(1 + a)`.
///
/// Uses `scratch` for `1 + a`.
#[inline]
pub fn taylor_ln_1p<F: Float>(a: &[F], c: &mut [F], scratch: &mut [F]) {
    let n = c.len();
    // All call sites are already guarded by the compile-time `K >= 1` const
    // assert in bytecode_tape/taylor.rs, but `taylor_ln_1p` is also callable
    // directly. `c[0] = a[0].ln_1p()` below unconditionally indexes [0],
    // so n=0 would panic; surface it as a debug assert with an actionable
    // message rather than a raw slice panic.
    debug_assert!(n >= 1, "taylor_ln_1p requires c.len() >= 1");
    scratch[1..n].copy_from_slice(&a[1..n]);
    scratch[0] = F::one() + a[0];
    taylor_ln(scratch, c);
    // Primal patch: same O(ULP) tradeoff as taylor_powf (see comment there).
    c[0] = a[0].ln_1p();
}

/// `c = hypot(a, b) = sqrt(a² + b²)`.
///
/// Shared CPU HYPOT kernel for jet-coefficient arrays. Used by
/// [`crate::Taylor::hypot`], [`crate::TaylorDyn::hypot`], and
/// [`crate::Laurent::hypot`]. The Laurent caller first rebases
/// operands to a common pole order so the coefficient arrays
/// become directly comparable, then calls through here.
///
/// Uses `scratch1` for a², `scratch2` for b², and the result scratch
/// for a²+b². Rescales inputs by `max(|a[0]|, |b[0]|)` (leading
/// coefficients only) to avoid overflow/underflow in the intermediate
/// a²+b². At `scale == 0` with non-zero higher-order seeds, performs
/// a recursive shift-and-square: hypot(a, b) near t = 0 with both
/// leading-zero operands equals `|t|·hypot(a(t)/t, b(t)/t)`, so the
/// body unwinds to the correct Taylor expansion of that composed
/// function. Mirrors `Taylor::abs` at the function-domain boundary.
#[inline]
pub fn taylor_hypot<F: Float>(
    a: &[F],
    b: &[F],
    c: &mut [F],
    scratch1: &mut [F],
    scratch2: &mut [F],
) {
    let n = c.len();
    let scale = a[0].abs().max(b[0].abs());
    if scale == F::zero() {
        // Both leading primals are zero. If the *next* order has any signal,
        // the composite function t ↦ hypot(a(t), b(t)) is smoothly
        // `|t|·hypot(a(t)/t, b(t)/t)` near t=0. Recursively compute on the
        // shifted series, then shift the result back by one to represent the
        // `|t|·…` factor. This mirrors CPU `Taylor::abs` and gives the true
        // Taylor expansion rather than the `log(0)·exp` path's NaN/Inf.
        if n > 1 && (a[1] != F::zero() || b[1] != F::zero()) {
            let mut a_shifted = vec![F::zero(); n];
            let mut b_shifted = vec![F::zero(); n];
            a_shifted[..(n - 1)].copy_from_slice(&a[1..n]);
            b_shifted[..(n - 1)].copy_from_slice(&b[1..n]);
            let mut inner_c = vec![F::zero(); n];
            let mut inner_s1 = vec![F::zero(); n];
            let mut inner_s2 = vec![F::zero(); n];
            taylor_hypot(
                &a_shifted,
                &b_shifted,
                &mut inner_c,
                &mut inner_s1,
                &mut inner_s2,
            );
            c[0] = F::zero();
            c[1..n].copy_from_slice(&inner_c[..(n - 1)]);
            return;
        }
        // Deeper-order zero (a and b both vanish past the first-non-zero
        // we can detect in the shifted series) or identically-zero — fall
        // through to the direct square-then-sqrt path, which gives 0 for
        // the primal and Inf for higher derivatives (the latter matches
        // the singular-derivative convention at a true zero).
        taylor_mul(a, a, scratch1);
        taylor_mul(b, b, scratch2);
        for k in 0..n {
            scratch1[k] = scratch1[k] + scratch2[k];
        }
        taylor_sqrt(scratch1, c);
        c[0] = a[0].hypot(b[0]);
        return;
    }
    let inv_scale = F::one() / scale;
    // scratch1 = (a/scale)  -- temporarily store rescaled a
    for k in 0..n {
        scratch1[k] = a[k] * inv_scale;
    }
    // scratch2 = (b/scale)  -- temporarily store rescaled b
    for k in 0..n {
        scratch2[k] = b[k] * inv_scale;
    }
    // c = (a/scale)²  -- reuse c as temp
    taylor_mul(scratch1, scratch1, c);
    // scratch1 = (b/scale)²  -- reuse scratch1
    taylor_mul(scratch2, scratch2, scratch1);
    // c = (a/scale)² + (b/scale)²
    for k in 0..n {
        c[k] = c[k] + scratch1[k];
    }
    // scratch1 = sqrt((a/scale)² + (b/scale)²)
    taylor_sqrt(c, scratch1);
    // c = scale * sqrt(...)  — undo rescaling
    for k in 0..n {
        c[k] = scratch1[k] * scale;
    }
    c[0] = a[0].hypot(b[0]);
}

/// `c = atan2(a, b)` = atan(a/b) with quadrant handling.
///
/// Uses scratch arrays for intermediate computation.
#[inline]
pub fn taylor_atan2<F: Float>(
    a: &[F],
    b: &[F],
    c: &mut [F],
    scratch1: &mut [F],
    scratch2: &mut [F],
    scratch3: &mut [F],
) {
    if b[0] != F::zero() {
        // Standard path: atan(a/b) with quadrant correction on c[0].
        // Derivatives of atan2(a,b) and atan(a/b) are identical where both defined.
        taylor_div(a, b, scratch1);
        taylor_atan(scratch1, c, scratch2, scratch3);
        c[0] = a[0].atan2(b[0]);
    } else if a[0] != F::zero() {
        // b[0]==0, a[0]!=0: use atan2(a,b) = sign(a)*pi/2 - atan(b/a)
        taylor_div(b, a, scratch1);
        taylor_atan(scratch1, c, scratch2, scratch3);
        // c = -atan(b/a)
        for ck in c.iter_mut() {
            *ck = -*ck;
        }
        // Fix c[0] to the correct atan2 value
        c[0] = a[0].atan2(b[0]);
    } else {
        // Both zero: mathematically undefined; return discontinuous zero
        taylor_discontinuous(F::zero(), c);
    }
}

/// Discontinuous function: `c[0] = f(a[0])`, `c[k>=1] = 0`.
#[inline]
pub fn taylor_discontinuous<F: Float>(val: F, c: &mut [F]) {
    c[0] = val;
    for ck in c[1..].iter_mut() {
        *ck = F::zero();
    }
}