dashu_float/

root.rs

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

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

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

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

impl<R: Round, const B: Word> FBig<R, B> {
    /// 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.nth_root(n, self.repr()).value()
    }
}

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()), 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>) -> Rounded<FBig<R, B>> {
        assert_finite(x);
        assert_limited_precision(self.precision);
        if x.sign() == Sign::Negative {
            panic_root_negative()
        }

        // 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)
        };
        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()), 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>) -> Rounded<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()), 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>) -> Rounded<FBig<R, B>> {
        assert_finite(x);
        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 {
            panic_root_negative()
        }
        if n == 1 {
            return 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
            return Approximation::Exact(FBig::new(Repr::zero(), *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)
        };
        res.map(|signif| Repr::new(signif, exp))
            .and_then(|v| self.repr_round(v))
            .map(|v| FBig::new(v, *self))
    }
}