decimal-scaled 0.4.1

//! D38 trigonometric kernels — sin / cos / tan / atan / asin / acos /
//! atan2 plus the hyperbolic family on the `Fixed` 256-bit
//! intermediate via `sin_fixed` / `atan_fixed`.
//!
//! Width-level specialisation for D38. Hosts the shared `Fixed`
//! primitives (`sin_fixed`, `atan_fixed`, `atan2_kernel`, `to_fixed`,
//! `wide_pi`, `wide_half_pi`, `small_x_linear_threshold`) plus every
//! D38 strict-trig + hyperbolic kernel. The typed-shell file in
//! `src/types/trig.rs` has no `crate::algos::*` or
//! `crate::algos::fixed_d38::*` references left — each public method
//! delegates one line through `policy::trig::TrigPolicy::*_impl`,
//! whose default body lives here.
//!
//! Fast paths preserved:
//! - `raw == 0` returns the appropriate identity (sin: 0, cos: 1, tan: 0).
//! - `|raw| <= small_x_linear_threshold` returns `raw` itself
//!   (linear band where the result is exact at storage precision).

use crate::algos::exp::fixed_d38::exp_fixed;
use crate::algos::ln::fixed_d38::{STRICT_GUARD, ln_fixed};
use crate::types::consts::DecimalConstants;
use crate::types::widths::D38;
use crate::algos::fixed_d38::Fixed;
use crate::support::rounding::RoundingMode;

// ── Shared Fixed primitives ────────────────────────────────────────

/// Threshold below which the linear small-x fast paths fire for the
/// odd trig functions (`atan`, `sin`, `tan`, `sinh`, `tanh`, `asin`,
/// `asinh`, `atanh`).
///
/// All these functions have a Taylor series `f(x) = x + c·x³ + …`
/// where `|c| ≤ 1/3`. For `|x| < (1.5·10⁻ˢᶜᴬᴸᴱ)^(1/3) ≈ 10^(−⌈SCALE/3⌉)`
/// the cubic correction is bounded by `0.5·ULP` and `f(x) == x`
/// exactly at the storage scale. The threshold returned here is the
/// conservative integer `10^(SCALE − ⌈(SCALE+2)/3⌉)` in storage
/// units (one decimal digit safety margin from the exact bound).
#[inline]
pub(crate) const fn small_x_linear_threshold<const SCALE: u32>() -> i128 {
    let thresh_exp = SCALE.saturating_sub((SCALE + 2) / 3);
    10_i128.pow(thresh_exp)
}

/// π at working scale `w`, sourced from the crate-wide 75-digit
/// `consts::PI_RAW` (Int256 holding `π × 10^75`).
pub(crate) fn wide_pi(w: u32) -> Fixed {
    debug_assert!(w <= 75, "wide_pi: working scale {w} exceeds embedded 75-digit π");
    let words = crate::types::consts::PI_RAW.0;
    let pi_at_75 = Fixed {
        negative: false,
        mag: [
            (words[0] as u128) | ((words[1] as u128) << 64),
            (words[2] as u128) | ((words[3] as u128) << 64),
        ],
    };
    if w == 75 {
        pi_at_75
    } else {
        pi_at_75.rescale_down(75, w)
    }
}

/// τ = 2π at working scale `w`.
fn wide_tau(w: u32) -> Fixed {
    wide_pi(w).double()
}

/// π/2 at working scale `w`.
pub(crate) fn wide_half_pi(w: u32) -> Fixed {
    wide_pi(w).halve()
}

/// Builds a working-scale `Fixed` from a signed `D38` raw value `r`:
/// `r · 10^STRICT_GUARD`, carrying the sign.
pub(crate) fn to_fixed(raw: i128) -> Fixed {
    to_fixed_w(raw, STRICT_GUARD)
}

/// Builds a working-scale `Fixed` from a signed `D38` raw value `r`:
/// `r · 10^working_digits`, carrying the sign. Used by the `_approx`
/// variants where the guard width is chosen at runtime.
pub(crate) fn to_fixed_w(raw: i128, working_digits: u32) -> Fixed {
    let m = Fixed::from_u128_mag(raw.unsigned_abs(), false)
        .mul_u128(10u128.pow(working_digits));
    if raw < 0 {
        m.neg()
    } else {
        m
    }
}

/// Shared `atan2` body factored out so the `_strict` and `_approx`
/// dispatchers can compose it at their chosen working scale `w`.
/// `y_raw` keeps the original sign of the y-argument for the x-zero
/// branch where the wide y value would have been signed-zero.
pub(crate) fn atan2_kernel(
    y: Fixed,
    x: Fixed,
    y_raw: i128,
    w: u32,
) -> Fixed {
    if x.is_zero() {
        return if y_raw > 0 {
            wide_half_pi(w)
        } else if y_raw < 0 {
            wide_half_pi(w).neg()
        } else {
            Fixed::ZERO
        };
    }
    // Max-branch: feed atan_fixed the |smaller|/|larger| ratio so the
    // argument-halving cascade doesn't blow up when |y| ≫ |x|.
    let abs_y_ge_abs_x = y.ge_mag(x);
    let base = if !abs_y_ge_abs_x {
        atan_fixed(y.div(x, w), w)
    } else {
        let inv = atan_fixed(x.div(y, w), w);
        let hp = wide_half_pi(w);
        let same_sign = y.negative == x.negative;
        if same_sign { hp.sub(inv) } else { hp.neg().sub(inv) }
    };
    if !x.negative {
        base
    } else if !y.negative {
        base.add(wide_pi(w))
    } else {
        base.sub(wide_pi(w))
    }
}

/// Taylor series for `sin` on a reduced non-negative argument
/// `r ∈ [0, π/2]`, evaluated at working scale `w`.
fn sin_taylor(r: Fixed, w: u32) -> Fixed {
    let r2 = r.mul(r, w);
    let mut sum = r;
    let mut term = r; // term = r^(2k-1)
    let mut k: u128 = 1;
    loop {
        // term_k = term_{k-1} · r² / ((2k)(2k+1)); sign alternates.
        term = term.mul(r2, w).div_small((2 * k) * (2 * k + 1));
        if term.is_zero() {
            break;
        }
        if k % 2 == 1 {
            sum = sum.sub(term);
        } else {
            sum = sum.add(term);
        }
        k += 1;
        if k > 200 {
            break;
        }
    }
    sum
}

/// Sine of a working-scale value `v_w`, at working scale `w`.
///
/// Reduces `v` modulo τ via `q = round(v/τ)`, folds the remainder into
/// `[0, π/2]` tracking sign and the `π − x` reflection, then evaluates
/// the Taylor series.
pub(crate) fn sin_fixed(v_w: Fixed, w: u32) -> Fixed {
    let tau = wide_tau(w);
    let pi = wide_pi(w);
    let half_pi = wide_half_pi(w);

    // r = v - round(v/τ)·τ ∈ [-π, π].
    let q = v_w.div(tau, w).round_to_nearest_int(w);
    let q_tau = if q >= 0 {
        tau.mul_u128(q as u128)
    } else {
        tau.mul_u128((-q) as u128).neg()
    };
    let r = v_w.sub(q_tau);

    // Fold |r| ∈ [0, π] into [0, π/2] via sin(π − x) = sin(x).
    let sign = r.negative;
    let abs_r = Fixed { negative: false, mag: r.mag };
    let reduced = if abs_r.ge_mag(half_pi) {
        pi.sub(abs_r)
    } else {
        abs_r
    };
    let s = sin_taylor(reduced, w);
    if sign {
        s.neg()
    } else {
        s
    }
}

/// Taylor series for `atan` on a reduced non-negative argument
/// `x ∈ [0, ~1/8]`, evaluated at working scale `w`.
fn atan_taylor(x: Fixed, w: u32) -> Fixed {
    let x2 = x.mul(x, w);
    let mut sum = x;
    let mut term = x; // term = x^(2k-1)
    let mut k: u128 = 1;
    loop {
        term = term.mul(x2, w);
        let contrib = term.div_small(2 * k + 1);
        if contrib.is_zero() {
            break;
        }
        if k % 2 == 1 {
            sum = sum.sub(contrib);
        } else {
            sum = sum.add(contrib);
        }
        k += 1;
        if k > 300 {
            break;
        }
    }
    sum
}

/// Arctangent of a working-scale value `v_w`, at working scale `w`,
/// result in `(−π/2, π/2)`.
///
/// Odd-function fold to `x ≥ 0`; reciprocal reduction
/// `atan(x) = π/2 − atan(1/x)` for `x > 1`; up to 8 rounds of
/// argument halving `atan(x) = 2·atan(x / (1 + √(1+x²)))`; then the
/// series.
pub(crate) fn atan_fixed(v_w: Fixed, w: u32) -> Fixed {
    #[cfg(feature = "perf-trace")]
    let _atan_span = ::tracing::info_span!("atan_fixed").entered();

    #[cfg(feature = "perf-trace")]
    let _setup_span = ::tracing::info_span!("setup").entered();
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let sign = v_w.negative;
    let mut x = Fixed { negative: false, mag: v_w.mag };
    let mut add_half_pi = false;
    if x.ge_mag(one_w) && x != one_w {
        x = one_w.div(x, w); // atan(x) = π/2 − atan(1/x)
        add_half_pi = true;
    }
    #[cfg(feature = "perf-trace")]
    drop(_setup_span);

    // Adaptive argument halvings: atan(x) = 2·atan(x / (1 + √(1+x²))).
    // Halve only while |x| > ~0.2 (the Taylor convergence target);
    // matches g_math's adaptive-halvings approach. Skips halvings
    // entirely when the input is already small. Hard cap at 8 halvings
    // as a safety net against pathological edge cases.
    #[cfg(feature = "perf-trace")]
    let _halvings_span = ::tracing::info_span!("halvings").entered();
    let halving_threshold = one_w.div_small(5); // 0.2 at scale w
    let mut halvings: u32 = 0;
    while x.ge_mag(halving_threshold) && halvings < 8 {
        let x2 = x.mul(x, w);
        let denom = one_w.add(one_w.add(x2).sqrt(w));
        x = x.div(denom, w);
        halvings += 1;
    }
    #[cfg(feature = "perf-trace")]
    drop(_halvings_span);

    #[cfg(feature = "perf-trace")]
    let _taylor_span = ::tracing::info_span!("taylor").entered();
    let mut result = atan_taylor(x, w);
    #[cfg(feature = "perf-trace")]
    drop(_taylor_span);

    #[cfg(feature = "perf-trace")]
    let _reasm_span = ::tracing::info_span!("reassemble").entered();
    result = result.shl(halvings);
    if add_half_pi {
        result = wide_half_pi(w).sub(result);
    }
    if sign {
        result.neg()
    } else {
        result
    }
}

// ── sin ────────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn sin_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + STRICT_GUARD;
    sin_fixed(to_fixed(raw), w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("sin", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn sin_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + working_digits;
    sin_fixed(to_fixed_w(raw, working_digits), w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("sin", SCALE))
}

// ── cos ────────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn cos_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return 10_i128.pow(SCALE);
    }
    let w = SCALE + STRICT_GUARD;
    let arg = to_fixed(raw).add(wide_half_pi(w));
    sin_fixed(arg, w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("cos", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn cos_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 10_i128.pow(SCALE);
    }
    let w = SCALE + working_digits;
    let arg = to_fixed_w(raw, working_digits).add(wide_half_pi(w));
    sin_fixed(arg, w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("cos", SCALE))
}

// ── tan ────────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn tan_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + STRICT_GUARD;
    let v = to_fixed(raw);
    let sin_w = sin_fixed(v, w);
    let cos_w = sin_fixed(v.add(wide_half_pi(w)), w);
    assert!(
        !cos_w.is_zero(),
        "tan: cosine is zero (argument is an odd multiple of pi/2)"
    );
    sin_w
        .div(cos_w, w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("tan", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn tan_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + working_digits;
    let v = to_fixed_w(raw, working_digits);
    let sin_w = sin_fixed(v, w);
    let cos_w = sin_fixed(v.add(wide_half_pi(w)), w);
    assert!(
        !cos_w.is_zero(),
        "tan: cosine is zero (argument is an odd multiple of pi/2)"
    );
    sin_w
        .div(cos_w, w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("tan", SCALE))
}

// ── atan ───────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn atan_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return 0;
    }
    let one_bits: i128 = 10_i128.pow(SCALE);
    if raw == one_bits {
        return <D38<SCALE> as DecimalConstants>::quarter_pi().0;
    }
    if raw == -one_bits {
        return -<D38<SCALE> as DecimalConstants>::quarter_pi().0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + STRICT_GUARD;
    atan_fixed(to_fixed(raw), w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("atan", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn atan_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    let one_bits: i128 = 10_i128.pow(SCALE);
    if raw == one_bits {
        return <D38<SCALE> as DecimalConstants>::quarter_pi().0;
    }
    if raw == -one_bits {
        return -<D38<SCALE> as DecimalConstants>::quarter_pi().0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + working_digits;
    atan_fixed(to_fixed_w(raw, working_digits), w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("atan", SCALE))
}

// ── asin ───────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn asin_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + STRICT_GUARD;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed(raw);
    let abs_v = Fixed { negative: false, mag: v.mag };
    assert!(!(abs_v.ge_mag(one_w) && abs_v != one_w), "asin: argument out of domain [-1, 1]");
    if abs_v == one_w {
        let hp = wide_half_pi(w);
        let hp = if v.negative { hp.neg() } else { hp };
        return hp
            .round_to_i128_with(w, SCALE, mode)
            .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("asin", SCALE));
    }
    let half_w = one_w.halve();
    let asin_w = if !abs_v.ge_mag(half_w) {
        let denom = one_w.sub(v.mul(v, w)).sqrt(w);
        atan_fixed(v.div(denom, w), w)
    } else {
        let inner = one_w.sub(abs_v).halve();
        let inner_sqrt = inner.sqrt(w);
        let inner_denom = one_w.sub(inner_sqrt.mul(inner_sqrt, w)).sqrt(w);
        let inner_asin = atan_fixed(inner_sqrt.div(inner_denom, w), w);
        let result_abs = wide_half_pi(w).sub(inner_asin).sub(inner_asin);
        if v.negative { result_abs.neg() } else { result_abs }
    };
    asin_w
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("asin", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn asin_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold::<SCALE>() {
        return raw;
    }
    let w = SCALE + working_digits;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed_w(raw, working_digits);
    let abs_v = Fixed { negative: false, mag: v.mag };
    assert!(!(abs_v.ge_mag(one_w) && abs_v != one_w), "asin: argument out of domain [-1, 1]");
    if abs_v == one_w {
        let hp = wide_half_pi(w);
        let hp = if v.negative { hp.neg() } else { hp };
        return hp
            .round_to_i128_with(w, SCALE, mode)
            .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("asin", SCALE));
    }
    let half_w = one_w.halve();
    let asin_w = if !abs_v.ge_mag(half_w) {
        let denom = one_w.sub(v.mul(v, w)).sqrt(w);
        atan_fixed(v.div(denom, w), w)
    } else {
        let inner = one_w.sub(abs_v).halve();
        let inner_sqrt = inner.sqrt(w);
        let inner_denom = one_w.sub(inner_sqrt.mul(inner_sqrt, w)).sqrt(w);
        let inner_asin = atan_fixed(inner_sqrt.div(inner_denom, w), w);
        let result_abs = wide_half_pi(w).sub(inner_asin).sub(inner_asin);
        if v.negative { result_abs.neg() } else { result_abs }
    };
    asin_w
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("asin", SCALE))
}

// ── acos ───────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn acos_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    if raw == 0 {
        return <D38<SCALE> as DecimalConstants>::half_pi().0;
    }
    let one_bits: i128 = 10_i128.pow(SCALE);
    if raw == one_bits {
        return 0;
    }
    if raw == -one_bits {
        return <D38<SCALE> as DecimalConstants>::pi().0;
    }
    let w = SCALE + STRICT_GUARD;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed(raw);
    let abs_v = Fixed { negative: false, mag: v.mag };
    assert!(!(abs_v.ge_mag(one_w) && abs_v != one_w), "acos: argument out of domain [-1, 1]");
    let half_w = one_w.halve();
    let asin_w = if abs_v == one_w {
        let hp = wide_half_pi(w);
        if v.negative { hp.neg() } else { hp }
    } else if !abs_v.ge_mag(half_w) {
        let denom = one_w.sub(v.mul(v, w)).sqrt(w);
        atan_fixed(v.div(denom, w), w)
    } else {
        let inner = one_w.sub(abs_v).halve();
        let inner_sqrt = inner.sqrt(w);
        let inner_denom = one_w.sub(inner_sqrt.mul(inner_sqrt, w)).sqrt(w);
        let inner_asin = atan_fixed(inner_sqrt.div(inner_denom, w), w);
        let result_abs = wide_half_pi(w).sub(inner_asin).sub(inner_asin);
        if v.negative { result_abs.neg() } else { result_abs }
    };
    wide_half_pi(w)
        .sub(asin_w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("acos", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn acos_with<const SCALE: u32>(
    raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return <D38<SCALE> as DecimalConstants>::half_pi().0;
    }
    let one_bits: i128 = 10_i128.pow(SCALE);
    if raw == one_bits {
        return 0;
    }
    if raw == -one_bits {
        return <D38<SCALE> as DecimalConstants>::pi().0;
    }
    let w = SCALE + working_digits;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed_w(raw, working_digits);
    let abs_v = Fixed { negative: false, mag: v.mag };
    assert!(!(abs_v.ge_mag(one_w) && abs_v != one_w), "acos: argument out of domain [-1, 1]");
    let half_w = one_w.halve();
    let asin_w = if abs_v == one_w {
        let hp = wide_half_pi(w);
        if v.negative { hp.neg() } else { hp }
    } else if !abs_v.ge_mag(half_w) {
        let denom = one_w.sub(v.mul(v, w)).sqrt(w);
        atan_fixed(v.div(denom, w), w)
    } else {
        let inner = one_w.sub(abs_v).halve();
        let inner_sqrt = inner.sqrt(w);
        let inner_denom = one_w.sub(inner_sqrt.mul(inner_sqrt, w)).sqrt(w);
        let inner_asin = atan_fixed(inner_sqrt.div(inner_denom, w), w);
        let result_abs = wide_half_pi(w).sub(inner_asin).sub(inner_asin);
        if v.negative { result_abs.neg() } else { result_abs }
    };
    wide_half_pi(w)
        .sub(asin_w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("acos", SCALE))
}

// ── atan2 ──────────────────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn atan2_strict<const SCALE: u32>(
    y_raw: i128,
    x_raw: i128,
    mode: RoundingMode,
) -> i128 {
    let w = SCALE + STRICT_GUARD;
    atan2_kernel(to_fixed(y_raw), to_fixed(x_raw), y_raw, w)
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("atan2", SCALE))
}

#[inline]
#[must_use]
pub(crate) fn atan2_with<const SCALE: u32>(
    y_raw: i128,
    x_raw: i128,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    let w = SCALE + working_digits;
    atan2_kernel(
        to_fixed_w(y_raw, working_digits),
        to_fixed_w(x_raw, working_digits),
        y_raw,
        w,
    )
        .round_to_i128_with(w, SCALE, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("atan2", SCALE))
}

// ── Hyperbolic family ─────────────────────────────────────────────
//
// sinh / cosh / tanh / asinh / acosh / atanh kernels on the `Fixed`
// 256-bit intermediate. The typed-shell file in `src/types/trig.rs`
// delegates one line to each `TrigPolicy::*_impl`, whose default body
// resolves to the matching kernel below.

#[inline]
#[must_use]
pub(crate) fn sinh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    sinh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn sinh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold_scale(scale) {
        return raw;
    }
    let w = scale + working_digits;
    let v = to_fixed_w(raw, working_digits);
    let ex = exp_fixed(v, w);
    let enx = exp_fixed(v.neg(), w);
    ex.sub(enx)
        .halve()
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::sinh", scale))
}

#[inline]
#[must_use]
pub(crate) fn cosh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    cosh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn cosh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 10_i128.pow(scale);
    }
    let w = scale + working_digits;
    let v = to_fixed_w(raw, working_digits);
    let ex = exp_fixed(v, w);
    let enx = exp_fixed(v.neg(), w);
    ex.add(enx)
        .halve()
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::cosh", scale))
}

#[inline]
#[must_use]
pub(crate) fn tanh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    tanh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn tanh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold_scale(scale) {
        return raw;
    }
    let w = scale + working_digits;
    let v = to_fixed_w(raw, working_digits);
    let ex = exp_fixed(v, w);
    let enx = exp_fixed(v.neg(), w);
    ex.sub(enx)
        .div(ex.add(enx), w)
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::tanh", scale))
}

#[inline]
#[must_use]
pub(crate) fn asinh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    asinh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn asinh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold_scale(scale) {
        return raw;
    }
    let w = scale + working_digits;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed_w(raw, working_digits);
    let ax = Fixed { negative: false, mag: v.mag };
    let inner = if ax.ge_mag(one_w) {
        let inv = one_w.div(ax, w);
        let root = one_w.add(inv.mul(inv, w)).sqrt(w);
        ln_fixed(ax, w).add(ln_fixed(one_w.add(root), w))
    } else {
        let root = ax.mul(ax, w).add(one_w).sqrt(w);
        ln_fixed(ax.add(root), w)
    };
    let signed = if raw < 0 { inner.neg() } else { inner };
    signed
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::asinh", scale))
}

#[inline]
#[must_use]
pub(crate) fn acosh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    acosh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn acosh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    let one_bits: i128 = 10_i128.pow(scale);
    if raw == one_bits {
        return 0;
    }
    let w = scale + working_digits;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed_w(raw, working_digits);
    assert!(!v.negative && v.ge_mag(one_w), "D38::acosh: argument must be >= 1");
    let two_w = one_w.double();
    let inner = if v.ge_mag(two_w) {
        let inv = one_w.div(v, w);
        let root = one_w.sub(inv.mul(inv, w)).sqrt(w);
        ln_fixed(v, w).add(ln_fixed(one_w.add(root), w))
    } else {
        let root = v.mul(v, w).sub(one_w).sqrt(w);
        ln_fixed(v.add(root), w)
    };
    inner
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::acosh", scale))
}

#[inline]
#[must_use]
pub(crate) fn atanh_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    atanh_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn atanh_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    if raw.abs() <= small_x_linear_threshold_scale(scale) {
        return raw;
    }
    let w = scale + working_digits;
    let one_w = Fixed { negative: false, mag: Fixed::pow10(w) };
    let v = to_fixed_w(raw, working_digits);
    let ax = Fixed { negative: false, mag: v.mag };
    assert!(!ax.ge_mag(one_w), "D38::atanh: argument out of domain (-1, 1)");
    let ratio = one_w.add(v).div(one_w.sub(v), w);
    ln_fixed(ratio, w)
        .halve()
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::atanh", scale))
}

// ── Angle conversions ─────────────────────────────────────────────

#[inline]
#[must_use]
pub(crate) fn to_degrees_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    to_degrees_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn to_degrees_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    let w = scale + working_digits;
    to_fixed_w(raw, working_digits)
        .mul_u128(180)
        .div(wide_pi(w), w)
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::to_degrees", scale))
}

#[inline]
#[must_use]
pub(crate) fn to_radians_strict<const SCALE: u32>(raw: i128, mode: RoundingMode) -> i128 {
    to_radians_with(raw, SCALE, STRICT_GUARD, mode)
}

#[inline]
#[must_use]
pub(crate) fn to_radians_with(
    raw: i128,
    scale: u32,
    working_digits: u32,
    mode: RoundingMode,
) -> i128 {
    if raw == 0 {
        return 0;
    }
    let w = scale + working_digits;
    to_fixed_w(raw, working_digits)
        .mul(wide_pi(w), w)
        .div_small(180)
        .round_to_i128_with(w, scale, mode)
        .unwrap_or_else(|| crate::support::diagnostics::overflow_panic_with_scale("D38::to_radians", scale))
}

// ── Runtime-scale small-x threshold ───────────────────────────────

/// Runtime-scale companion to [`small_x_linear_threshold`]. Used by
/// the `_with` kernels where `scale` is a runtime value rather than
/// a const generic.
#[inline]
fn small_x_linear_threshold_scale(scale: u32) -> i128 {
    let thresh_exp = scale.saturating_sub((scale + 2) / 3);
    10_i128.pow(thresh_exp)
}