//! The Ziv retry loop — guaranteed-correct rounding for transcendentals.
//!
//! Transcendentals (`exp`, `ln`, …) cannot compute an exact result the way arithmetic can,
//! so they approximate at a working precision `p + guard` and round down. A single guard-digit
//! heuristic is only *near*-correct: for an input whose true value sits on a rounding boundary,
//! the rounded result can be off by one ULP. The Ziv loop closes that gap.
//!
//! Each transcendental reports its approximation together with a provable absolute error radius
//! `E` (a true upper bound on `|approx − true|`, built from the fact that every `+`/`-`/`*`/`/`
//! in the algorithm is itself correctly rounded). The driver rounds the approximation to the
//! target precision and checks whether the entire error interval `[ã − E, ã + E]` lies inside
//! the candidate's [`ErrorBounds`] preimage — the set of reals that round to it. If it does,
//! the rounded value is *guaranteed* correct; otherwise the driver retries with more guard
//! digits. The loop provably terminates (a true tie is resolved deterministically by the mode),
//! with a large sanity cap as an unreachable backstop.

use core::cmp::Ordering;

use dashu_base::Approximation::*;

use crate::{
    fbig::FBig,
    repr::{Context, Repr},
    round::ErrorBounds,
    round::Rounded,
};
use dashu_int::Word;

/// Maximum number of Ziv retries before falling back to the best-effort rounded value.
///
/// This is a sanity backstop only — the loop converges as soon as the working precision is
/// large enough that the error interval no longer straddles a rounding boundary, which happens
/// in one attempt for essentially all inputs (the guard-digit heuristic is sized for that) and
/// in a handful of attempts only for inputs pathological close to a tie. The cap exists so a
/// bug in an error-radius bound can never produce an infinite loop.
const MAX_ZIV_RETRIES: usize = 32;

// A test-only retry counter, so tests can assert the loop converges on the first attempt for
// typical inputs (validating that the guard-digit heuristic wasn't over-tightened). Reads as
// the number of *extra* attempts beyond the first, i.e. `0` means first-attempt success.
#[cfg(all(test, feature = "std"))]
thread_local! {
    pub(crate) static LAST_ZIV_RETRIES: core::cell::Cell<usize> = const { core::cell::Cell::new(0) };
}

impl<R: ErrorBounds> Context<R> {
    /// Correctly round a transcendental approximation to this context's precision using a Ziv
    /// retry loop.
    ///
    /// `approx(guard)` computes the function at working precision `self.precision + guard` and
    /// returns `(value, error_radius)` — the value and a provable upper bound on its absolute
    /// error, both as [`FBig`]s at the working context. The closure is expected to capture and
    /// reborrow any [`ConstCache`](crate::ConstCache) from the enclosing scope; the driver calls
    /// it once per attempt and grows `guard` when the result cannot be certified.
    ///
    /// The loop preserves the [`Exact`](Rounded)/[`Inexact`](Rounded) flag from rounding the
    /// approximation to the target precision.
    pub(crate) fn ziv<const B: Word>(
        &self,
        initial_guard: usize,
        mut approx: impl FnMut(usize) -> (FBig<R, B>, FBig<R, B>),
    ) -> Rounded<FBig<R, B>> {
        // Unlimited precision: the approximation is exact, so report it as-is.
        if !self.is_limited() {
            let (value, _err) = approx(0);
            return Exact(value);
        }

        let mut guard = initial_guard;
        let mut last = None;
        #[cfg(all(test, feature = "std"))]
        LAST_ZIV_RETRIES.with(|c| c.set(0));
        for _ in 0..MAX_ZIV_RETRIES {
            let (a, e) = approx(guard);
            // `with_precision` consumes `a`, but the containment test still needs it, so round a
            // clone and keep the original for the interval check.
            let candidate = a.clone().with_precision(self.precision);
            if Self::contained::<B>(&a.repr, &e.repr, candidate.value_ref()) {
                return candidate;
            }
            last = Some(candidate);

            // Grow the guard aggressively so a near-tie resolves in a couple of retries, while
            // the first attempt (with the heuristic guard) handles the common case.
            let step = core::cmp::max(guard, self.precision / 2).max(1);
            guard += step;
            #[cfg(all(test, feature = "std"))]
            LAST_ZIV_RETRIES.with(|c| c.set(c.get() + 1));
        }

        // Unreachable in practice: return the best-effort candidate from the last attempt,
        // matching the pre-Ziv near-correct behavior rather than looping forever.
        last.expect("MAX_ZIV_RETRIES is non-zero")
    }

    /// Pair variant of [`ziv`](Self::ziv) for functions that return two values (e.g. `sin_cos`,
    /// `sinh_cosh`). `approx(guard)` returns `((v1, e1), (v2, e2))` — both values and their
    /// provable radii at the working context, sharing whatever computation is common. The driver
    /// certifies **both** values: it retries while *either* containment test fails, and returns
    /// both only when both fit their rounding preimages. Shares the guard-growth loop and retry
    /// counter with [`ziv`](Self::ziv).
    pub(crate) fn ziv_pair<const B: Word>(
        &self,
        initial_guard: usize,
        mut approx: impl FnMut(usize) -> ((FBig<R, B>, FBig<R, B>), (FBig<R, B>, FBig<R, B>)),
    ) -> (Rounded<FBig<R, B>>, Rounded<FBig<R, B>>) {
        // Unlimited precision: both approximations are exact, report them as-is.
        if !self.is_limited() {
            let ((v1, _), (v2, _)) = approx(0);
            return (Exact(v1), Exact(v2));
        }

        let mut guard = initial_guard;
        let mut last = None;
        #[cfg(all(test, feature = "std"))]
        LAST_ZIV_RETRIES.with(|c| c.set(0));
        for _ in 0..MAX_ZIV_RETRIES {
            let ((a1, e1), (a2, e2)) = approx(guard);
            let c1 = a1.clone().with_precision(self.precision);
            let c2 = a2.clone().with_precision(self.precision);
            if Self::contained::<B>(&a1.repr, &e1.repr, c1.value_ref())
                && Self::contained::<B>(&a2.repr, &e2.repr, c2.value_ref())
            {
                return (c1, c2);
            }
            last = Some((c1, c2));

            // Grow the guard aggressively so a near-tie resolves in a couple of retries.
            let step = core::cmp::max(guard, self.precision / 2).max(1);
            guard += step;
            #[cfg(all(test, feature = "std"))]
            LAST_ZIV_RETRIES.with(|c| c.set(c.get() + 1));
        }

        // Unreachable in practice: return the best-effort pair from the last attempt.
        last.expect("MAX_ZIV_RETRIES is non-zero")
    }

    /// Containment test: is the approximation's error interval `[a − e, a + e]` entirely inside
    /// the rounding preimage of `y` (every real in `[y − lb, y + rb]` rounds to `y` under `R`)?
    ///
    /// `a` and `e` are the working-precision approximation and its provable error radius; `y` is
    /// the candidate rounded to the target precision (kept as an [`FBig`] only because
    /// [`ErrorBounds::error_bounds`] is defined on [`FBig`]). The interval arithmetic runs on the
    /// raw [`Repr`]s, which carry no precision limit, so the additions are lossless — there is no
    /// rounding that could drop a guard digit and mis-decide (a wrong call here yields a wrong
    /// ULP). The old path promoted every value to unlimited precision via `with_precision(0)`;
    /// the [`Repr`]s are already exact, so that was a chain of no-op clones, now removed.
    ///
    /// The test compares sums rather than differences — algebraically identical for exact
    /// arithmetic, and it reads as a single shared inequality per endpoint:
    ///   `a − e ≥ y − lb  ⟺  a + lb ≥ y + e`
    ///   `a + e ≤ y + rb  ⟺  y + rb ≥ a + e`
    fn contained<const B: Word>(a: &Repr<B>, e: &Repr<B>, y: &FBig<R, B>) -> bool {
        let (lb, rb, incl_l, incl_r) = R::error_bounds::<B>(y);

        let y = &y.repr;
        let lb = lb.into_repr();
        let rb = rb.into_repr();

        let left = (a + &lb).cmp(&(y + e));
        let right = (y + &rb).cmp(&(a + e));
        let left_ok = if incl_l {
            left != Ordering::Less
        } else {
            left == Ordering::Greater
        };
        let right_ok = if incl_r {
            right != Ordering::Less
        } else {
            right == Ordering::Greater
        };
        left_ok && right_ok
    }
}

#[cfg(all(test, feature = "std"))]
mod tests {
    use super::*;
    use crate::round::mode;

    type F = crate::FBig<mode::HalfEven>;

    // An exact approximation (radius 0) is accepted on the first attempt as Exact.
    #[test]
    fn ziv_accepts_exact_first_attempt() {
        let ctx: Context<mode::HalfEven> = Context::new(10);
        LAST_ZIV_RETRIES.with(|c| c.set(usize::MAX));
        let r = ctx.ziv(4, |_| (F::ONE, F::ZERO));
        assert!(matches!(r, Exact(_)));
        assert_eq!(LAST_ZIV_RETRIES.with(|c| c.get()), 0);
    }

    // An approximation whose error interval straddles a rounding boundary must retry; once the
    // shrinking radius makes the interval unambiguous, ziv accepts.
    #[test]
    fn ziv_retries_until_contained() {
        let ctx: Context<mode::HalfEven> = Context::new(4);
        let r = ctx.ziv(2, |guard| {
            // value 1.0, radius 2^(-guard): large on the first attempt, tiny later.
            (F::ONE, F::ONE >> guard as isize)
        });
        let _ = r.value();
        assert!(LAST_ZIV_RETRIES.with(|c| c.get()) >= 1);
    }

    // Unlimited-precision context short-circuits to a single Exact call (counter untouched).
    #[test]
    fn ziv_unlimited_short_circuits() {
        let ctx: Context<mode::HalfEven> = Context::new(0);
        LAST_ZIV_RETRIES.with(|c| c.set(usize::MAX));
        let r = ctx.ziv(4, |_| (F::from(7u8), F::ZERO));
        assert!(matches!(r, Exact(_)));
        assert_eq!(LAST_ZIV_RETRIES.with(|c| c.get()), usize::MAX);
    }

    // ziv_pair accepts an exact pair (both radii 0) on the first attempt.
    #[test]
    fn ziv_pair_accepts_exact_first_attempt() {
        let ctx: Context<mode::HalfEven> = Context::new(10);
        LAST_ZIV_RETRIES.with(|c| c.set(usize::MAX));
        let (r1, r2) = ctx.ziv_pair(4, |_| ((F::ONE, F::ZERO), (F::from(2u8), F::ZERO)));
        assert!(matches!(r1, Exact(_)));
        assert!(matches!(r2, Exact(_)));
        assert_eq!(LAST_ZIV_RETRIES.with(|c| c.get()), 0);
    }

    // ziv_pair retries while *either* value's interval straddles a boundary; here the second value
    // carries the shrinking radius, so the pair must retry together.
    #[test]
    fn ziv_pair_retries_until_both_contained() {
        let ctx: Context<mode::HalfEven> = Context::new(4);
        let (r1, r2) = ctx.ziv_pair(2, |guard| {
            let radius = F::ONE >> guard as isize;
            ((F::ONE, F::ZERO), (F::ONE, radius))
        });
        let _ = (r1.value(), r2.value());
        assert!(LAST_ZIV_RETRIES.with(|c| c.get()) >= 1);
    }

    // The guard-digit heuristic should let exp/ln converge in at most one retry for typical
    // inputs. A single retry on a near-tie is by design (Ziv certifies correctness; the guard only
    // controls the first-attempt hit rate). This catches gross guard mis-sizing (many retries),
    // not the occasional near-tie retry.
    #[test]
    fn ziv_few_retries_for_typical_inputs() {
        let cases = [
            F::try_from(0.5f64).unwrap(),
            F::try_from(1.5f64).unwrap(),
            F::try_from(2.0f64).unwrap(),
            F::try_from(10.0f64).unwrap(),
            F::try_from(1000.0f64).unwrap(),
            F::try_from(1e-6f64).unwrap(),
        ];
        const MAX_RETRIES: usize = 1;
        for p in [10usize, 24, 53, 100, 200] {
            for x in &cases {
                let x = x.clone().with_precision(p).value();
                LAST_ZIV_RETRIES.with(|c| c.set(usize::MAX));
                let _ = x.ln();
                let ln_retries = LAST_ZIV_RETRIES.with(|c| c.get());
                assert!(
                    ln_retries <= MAX_RETRIES,
                    "ln({x}) at p={p} took {ln_retries} retries (expected <= {MAX_RETRIES})"
                );
                LAST_ZIV_RETRIES.with(|c| c.set(usize::MAX));
                let _ = x.exp();
                let exp_retries = LAST_ZIV_RETRIES.with(|c| c.get());
                assert!(
                    exp_retries <= MAX_RETRIES,
                    "exp({x}) at p={p} took {exp_retries} retries (expected <= {MAX_RETRIES})"
                );
            }
        }
    }
}