dashu_float/

root.rs

use dashu_base::{
    Approximation, CubicRoot, EstimatedLog2, Sign, SquareRoot, SquareRootRem, UnsignedAbs,
};
use dashu_int::{IBig, UBig};

use crate::{
    error::{assert_limited_precision, panic_root_zeroth, FpError, FpResult},
    fbig::FBig,
    repr::{Context, Repr, Word},
    round::{ErrorBounds, Round, Rounded},
    utils::{shl_digits, split_digits_ref},
};

/// Take the value of a [`Rounded`] result, recording in `exact` whether it was computed exactly.
///
/// Mirrors MPFR's `exact` flag: an all-exact operation chain yields the exact true value, which a
/// Ziv closure can report with radius 0 — `ziv` then accepts it without the containment test,
/// which otherwise can't certify an exactly-representable result (it sits on a one-sided preimage
/// boundary under directed rounding).
fn value_tracking_exact<T>(r: Rounded<T>, exact: &mut bool) -> T {
    if !matches!(r, Approximation::Exact(_)) {
        *exact = false;
    }
    r.value()
}

impl<R: Round, const B: Word> SquareRoot for FBig<R, B> {
    type Output = Self;
    #[inline]
    fn sqrt(&self) -> Self {
        self.context.unwrap_fp(self.context.sqrt(self.repr()))
    }
}

impl<R: Round, const B: Word> CubicRoot for FBig<R, B> {
    type Output = Self;
    #[inline]
    fn cbrt(&self) -> Self {
        self.context.unwrap_fp(self.context.cbrt(self.repr()))
    }
}

impl<R: Round, const B: Word> FBig<R, B> {
    /// Calculate the square root of the floating point number.
    ///
    /// # Panics
    ///
    /// Panics if the precision is unlimited.
    #[inline]
    pub fn sqrt(&self) -> Self {
        self.context.unwrap_fp(self.context.sqrt(&self.repr))
    }

    /// Calculate the nth root of the floating point number.
    ///
    /// When `n` is large the computation can be expensive — the significand is
    /// padded to `n · precision` digits before the integer root is taken, and
    /// the integer Newton iteration works with numbers of that size. For large
    /// `n` consider [`powf`][`FBig::powf`] with a rational exponent `1 / n`
    /// as a faster approximate alternative.
    ///
    /// # Examples
    ///
    /// ```
    /// # use core::str::FromStr;
    /// # use dashu_base::ParseError;
    /// # use dashu_float::DBig;
    /// let a = DBig::from_str("16")?;
    /// assert_eq!(a.nth_root(4), DBig::from_str("2")?);
    /// # Ok::<(), ParseError>(())
    /// ```
    ///
    /// # Panics
    ///
    /// Panics if `n` is zero, or if `n` is even and the number is negative.
    #[inline]
    pub fn nth_root(&self, n: usize) -> Self {
        self.context
            .unwrap_fp(self.context.nth_root(n, self.repr()))
    }
}

impl<R: Round> Context<R> {
    /// Calculate the square root of the floating point number.
    ///
    /// # Examples
    ///
    /// ```
    /// # use core::str::FromStr;
    /// # use dashu_base::ParseError;
    /// # use dashu_float::DBig;
    /// use dashu_base::Approximation::*;
    /// use dashu_float::{Context, round::{mode::HalfAway, Rounding::*}};
    ///
    /// let context = Context::<HalfAway>::new(2);
    /// let a = DBig::from_str("1.23")?;
    /// assert_eq!(context.sqrt(&a.repr()), Ok(Inexact(DBig::from_str("1.1")?, NoOp)));
    /// # Ok::<(), ParseError>(())
    /// ```
    ///
    /// # Panics
    ///
    /// Panics if the precision is unlimited.
    pub fn sqrt<const B: Word>(&self, x: &Repr<B>) -> FpResult<FBig<R, B>> {
        if x.is_infinite() {
            return Err(FpError::InfiniteInput);
        }
        if x.significand.is_zero() {
            // sqrt(+0) = +0, sqrt(-0) = -0 (preserve the sign of zero). Exact, so handle
            // it before the limited-precision assertion: a precision-0 (unlimited) value
            // such as the one from `try_from(0.0)` must still compute sqrt(0) exactly.
            return Ok(Approximation::Exact(FBig::new(x.clone(), *self)));
        }
        assert_limited_precision(self.precision);
        if x.sign() == Sign::Negative {
            return Err(FpError::OutOfDomain);
        }

        // adjust the signifcand so that the exponent is even
        let digits = x.digits() as isize;
        let shift = self.precision as isize * 2 - (digits & 1) + (x.exponent & 1) - digits;
        let (signif, low, low_digits) = if shift > 0 {
            (shl_digits::<B>(&x.significand, shift as usize), IBig::ZERO, 0)
        } else {
            let shift = (-shift) as usize;
            let (hi, lo) = split_digits_ref::<B>(&x.significand, shift);
            (hi, lo, shift)
        };

        let (root, rem) = signif.unsigned_abs().sqrt_rem();
        let root = Sign::Positive * root;
        let exp = (x.exponent - shift) / 2;

        let res = if rem.is_zero() {
            Approximation::Exact(root)
        } else {
            let adjust = R::round_low_part(&root, Sign::Positive, || {
                (Sign::Positive * rem)
                    .cmp(&root)
                    .then_with(|| (low * 4u8).cmp(&Repr::<B>::BASE.pow(low_digits).into()))
            });
            Approximation::Inexact(root + adjust, adjust)
        };
        Ok(res
            .map(|signif| Repr::new(signif, exp))
            .and_then(|v| self.repr_round(v))
            .map(|v| FBig::new(v, *self)))
    }

    /// Calculate the cubic root of the floating point number.
    ///
    /// # Examples
    ///
    /// ```
    /// # use core::str::FromStr;
    /// # use dashu_base::ParseError;
    /// # use dashu_float::DBig;
    /// use dashu_base::Approximation::*;
    /// use dashu_float::{Context, round::{mode::HalfAway, Rounding::*}};
    ///
    /// let context = Context::<HalfAway>::new(2);
    /// let a = DBig::from_str("8")?;
    /// assert_eq!(context.cbrt(&a.repr()), Ok(Exact(DBig::from_str("2")?)));
    /// # Ok::<(), ParseError>(())
    /// ```
    ///
    /// # Panics
    ///
    /// Panics if the precision is unlimited.
    #[inline]
    pub fn cbrt<const B: Word>(&self, x: &Repr<B>) -> FpResult<FBig<R, B>> {
        self.nth_root(3, x)
    }

    /// Calculate the nth root of the floating point number.
    ///
    /// # Examples
    ///
    /// ```
    /// # use core::str::FromStr;
    /// # use dashu_base::ParseError;
    /// # use dashu_float::DBig;
    /// use dashu_base::Approximation::*;
    /// use dashu_float::{Context, round::{mode::HalfAway, Rounding::*}};
    ///
    /// let context = Context::<HalfAway>::new(2);
    /// let a = DBig::from_str("27")?;
    /// assert_eq!(context.nth_root(3, &a.repr()), Ok(Exact(DBig::from_str("3")?)));
    /// # Ok::<(), ParseError>(())
    /// ```
    ///
    /// # Panics
    ///
    /// Panics if `n` is zero, if the precision is unlimited, or if `n` is even and `x` is negative.
    pub fn nth_root<const B: Word>(&self, n: usize, x: &Repr<B>) -> FpResult<FBig<R, B>> {
        if x.is_infinite() {
            return Err(FpError::InfiniteInput);
        }
        assert_limited_precision(self.precision);
        if n == 0 {
            panic_root_zeroth()
        }
        debug_assert!(n < isize::MAX as usize);
        let sign = x.sign();
        if sign == Sign::Negative && n % 2 == 0 {
            return Err(FpError::OutOfDomain);
        }
        if n == 1 {
            return Ok(self.repr_round_ref(x).map(|v| FBig::new(v, *self)));
        }
        if x.significand.is_zero() {
            // UBig::ZERO.nth_root(n) erroneously returns ONE, so short-circuit here.
            // An even root of -0 already errored above, so reaching here the sign is
            // preserved: odd root of ±0 is ±0.
            return Ok(Approximation::Exact(FBig::new(x.clone(), *self)));
        }

        // operate on the magnitude so that shifting/splitting keep a clean sign;
        // the original sign is re-applied to the result at the end.
        let xmag: IBig = if sign == Sign::Negative {
            -&x.significand
        } else {
            x.significand.clone()
        };

        // adjust the significand so that the exponent is divisible by n and the
        // significand carries at least n*precision digits (required for rounding)
        let digits = x.digits() as isize;
        let r = (x.exponent + digits).rem_euclid(n as isize);
        let shift = n as isize * self.precision as isize - digits + r;
        let (signif, low, low_digits) = if shift > 0 {
            (shl_digits::<B>(&xmag, shift as usize), IBig::ZERO, 0)
        } else {
            let shift = (-shift) as usize;
            let (hi, lo) = split_digits_ref::<B>(&xmag, shift);
            (hi, lo, shift)
        };

        let mag: UBig = signif.unsigned_abs();
        let root: UBig = mag.nth_root(n);
        let rem: UBig = &mag - root.clone().pow(n);
        let exp = (x.exponent - shift) / n as isize;

        let result_sign = if sign == Sign::Negative {
            Sign::Negative
        } else {
            Sign::Positive
        };
        let signed_root: IBig = result_sign * root.clone();

        let res = if rem.is_zero() && low.is_zero() {
            Approximation::Exact(signed_root)
        } else {
            let adjust = R::round_low_part(&signed_root, result_sign, || {
                // The true value is (mag + low / BASE^low_digits)^(1/n) and
                // root = floor(mag^(1/n)); its fractional part is compared to 1/2.
                // frac < 1/2  <=>  2^n * full < (2*root + 1)^n * BASE^low_digits,
                // where full = mag * BASE^low_digits + low (the full significand).
                let base_pow = Repr::<B>::BASE.pow(low_digits);
                let full = &mag * &base_pow + low.unsigned_abs();
                let lhs = full << n;
                let rhs = ((root.clone() << 1) + UBig::from_word(1)).pow(n) * base_pow;
                lhs.cmp(&rhs)
            });
            Approximation::Inexact(signed_root.clone() + adjust, adjust)
        };
        Ok(res
            .map(|signif| Repr::new(signif, exp))
            .and_then(|v| self.repr_round(v))
            .map(|v| FBig::new(v, *self)))
    }
}

impl<R: ErrorBounds> Context<R> {
    /// Compute `sqrt(a² + b²)` without spurious overflow/underflow.
    ///
    /// This is the overflow-safe scaled sum-of-squares: the larger-magnitude operand is never
    /// squared. Writing `m = max(|a|, |b|)` and `r = min(|a|,|b|) / m` (so `|r| ≤ 1`), the result is
    /// `m · sqrt(1 + r²)`, where `1 + r² ∈ [1, 2]` cannot overflow. The result is correctly rounded
    /// via a Ziv retry loop (`hypot(±inf, ·) = +inf`, `hypot(0, 0) = +0`).
    ///
    /// This is a field-arithmetic-class op (no constant cache), like `sqrt`/`atan2`.
    ///
    /// # Panics
    ///
    /// Panics if the precision is unlimited.
    pub fn hypot<const B: Word>(&self, a: &Repr<B>, b: &Repr<B>) -> FpResult<FBig<R, B>> {
        if a.is_infinite() || b.is_infinite() {
            return Ok(Approximation::Exact(FBig::new(Repr::infinity(), *self)));
        }
        assert_limited_precision(self.precision);
        if a.significand.is_zero() && b.significand.is_zero() {
            return Ok(Approximation::Exact(FBig::new(Repr::zero(), *self)));
        }

        // magnitudes, ordered large >= small (both finite, not both zero here)
        let a_mag = if a.sign() == Sign::Negative {
            -a.clone()
        } else {
            a.clone()
        };
        let b_mag = if b.sign() == Sign::Negative {
            -b.clone()
        } else {
            b.clone()
        };
        let (large, small) = if a_mag.cmp(&b_mag).is_ge() {
            (a_mag, b_mag)
        } else {
            (b_mag, a_mag)
        };

        if small.significand.is_zero() {
            // hypot(x, 0) = |x|; `large` is already a magnitude.
            return Ok(self.repr_round_ref(&large).map(|v| FBig::new(v, *self)));
        }

        // The result is `sqrt(large² + small²)`, i.e. ∈ [large, large·√2]. It overflows only when
        // `large` is so large that the result reaches the infinity sentinel exponent — unreachable
        // for real inputs, but pre-checked here so the Ziv closure can use infallible `FBig`
        // arithmetic.
        if large.exponent >= isize::MAX - 1 {
            return Err(FpError::Overflow(Sign::Positive));
        }

        let initial_guard = crate::utils::ceil_usize(self.precision.log2_est()) + 10;
        Ok(self.ziv(initial_guard, |guard| {
            let gctx = Context::<R>::new(self.precision + guard);
            // result = sqrt(large² + small²), with both operands scaled down by `k` base-B digits
            // before squaring (so `large²` can't overflow the exponent) and the root scaled back:
            // sqrt(L² + S²) · B^k = sqrt(large² + small²) for L = large·B⁻ᵏ, S = small·B⁻ᵏ. No
            // division — so for integer inputs every step is exact (MPFR's `exact` flag), and an
            // all-exact chain yields the exact true value. Report radius 0 then, which `ziv`
            // accepts without the containment test (it can't certify an exactly-representable result
            // under directed rounding — e.g. hypot(3,4)=5, hypot(5,12)=13 — which sits on a
            // one-sided preimage boundary).
            let k = (large.exponent as i128 - (isize::MAX as i128 - 2) / 2).max(0) as isize;
            let mut exact = true;
            let large_f =
                FBig::new(value_tracking_exact(gctx.repr_round_ref(&large), &mut exact), gctx);
            let small_f =
                FBig::new(value_tracking_exact(gctx.repr_round_ref(&small), &mut exact), gctx);
            let l_sq = value_tracking_exact(gctx.sqr((large_f >> k).repr()).unwrap(), &mut exact);
            let s_sq = value_tracking_exact(gctx.sqr((small_f >> k).repr()).unwrap(), &mut exact);
            let sum = value_tracking_exact(gctx.add(l_sq.repr(), s_sq.repr()).unwrap(), &mut exact);
            let root = value_tracking_exact(gctx.sqrt(sum.repr()).unwrap(), &mut exact);
            let result = root << k; // exact exponent shift — scales back, doesn't affect `exact`
            let radius = if exact {
                FBig::<R, B>::ZERO
            } else {
                result.ulp() * 8
            };
            (result, radius)
        }))
    }
}

impl<R: ErrorBounds, const B: Word> FBig<R, B> {
    /// Compute `sqrt(self² + other²)` without spurious overflow/underflow.
    ///
    /// The result precision is `max(self.precision(), other.precision())`. See
    /// [`Context::hypot`] for the overflow-safety strategy.
    ///
    /// # Examples
    ///
    /// ```
    /// # use core::str::FromStr;
    /// # use dashu_base::ParseError;
    /// # use dashu_float::DBig;
    /// let a = DBig::from_str("3")?;
    /// let b = DBig::from_str("4")?;
    /// assert_eq!(a.hypot(&b), DBig::from_str("5")?);
    /// # Ok::<(), ParseError>(())
    /// ```
    ///
    /// # Panics
    ///
    /// Panics if the precision is unlimited.
    #[inline]
    pub fn hypot(&self, other: &Self) -> Self {
        let context = Context::max(self.context, other.context);
        context.unwrap_fp(context.hypot(&self.repr, &other.repr))
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::round::mode;

    #[test]
    #[should_panic]
    fn test_fbig_sqrt_negative_panics() {
        // sqrt(-1) is out of domain; the FBig layer panics.
        let neg_one = FBig::<mode::HalfEven>::try_from(-1.0f64).unwrap();
        let _ = neg_one.sqrt();
    }

    #[test]
    fn test_hypot_pythagorean() {
        let ctx = Context::<mode::HalfEven>::new(53);
        let mk = |v: i32| Repr::<2>::new(v.into(), 0);
        // hypot(3, 4) = 5
        let r = ctx.hypot(&mk(3), &mk(4)).unwrap().value();
        assert_eq!(r.repr().significand(), &5.into());
        // hypot(5, 0) = 5
        let r = ctx.hypot(&mk(5), &mk(0)).unwrap().value();
        assert_eq!(r.repr().significand(), &5.into());
        // hypot(0, 0) = 0
        let r = ctx.hypot(&mk(0), &mk(0)).unwrap().value();
        assert!(r.repr().is_pos_zero());
        // hypot(inf, x) = +inf
        let r = ctx.hypot(&Repr::infinity(), &mk(3)).unwrap().value();
        assert!(r.repr().is_infinite());
        assert_eq!(r.repr().sign(), Sign::Positive);
    }

    fn check_hypot_exact_triples<R: ErrorBounds>(ctx: Context<R>) {
        let mk = |v: i32| Repr::<2>::new(v.into(), 0);
        // Pythagorean triples: the result is exactly representable, so under a directed mode it
        // sits on a one-sided preimage boundary. The closure must terminate (radius 0 from the
        // all-exact `sqrt(large²+small²)` chain) rather than infinite-retry.
        for (a, b, h) in [(3, 4, 5), (5, 12, 13), (8, 15, 17)] {
            let r = ctx.hypot(&mk(a), &mk(b)).unwrap().value();
            assert_eq!(r.repr().significand(), &h.into(), "hypot({a}, {b})");
        }
    }

    #[test]
    fn test_hypot_exact_under_directed_rounding() {
        check_hypot_exact_triples(Context::<mode::Down>::new(53));
        check_hypot_exact_triples(Context::<mode::Up>::new(53));
        check_hypot_exact_triples(Context::<mode::Zero>::new(53));
    }

    #[test]
    fn test_hypot_no_spurious_overflow() {
        // a value whose square would collide with the +inf sentinel exponent, but whose
        // hypot is itself representable: hypot(a, 0) = |a| must not overflow via a².
        let ctx = Context::<mode::HalfEven>::new(53);
        // exponent near isize::MAX/2 so that a² would overflow, but |a| is fine
        let a = Repr::<2>::new(IBig::from(3), isize::MAX / 2);
        let r = ctx.hypot(&a, &Repr::<2>::zero()).unwrap().value();
        assert_eq!(r.repr().exponent(), isize::MAX / 2);
    }
}