core/num/imp/

int_sqrt.rs

//! These functions use the [Karatsuba square root algorithm][1] to compute the
//! [integer square root](https://en.wikipedia.org/wiki/Integer_square_root)
//! for the primitive integer types.
//!
//! The signed integer functions can only handle **nonnegative** inputs, so
//! that must be checked before calling those.
//!
//! [1]: <https://web.archive.org/web/20230511212802/https://inria.hal.science/inria-00072854v1/file/RR-3805.pdf>
//! "Paul Zimmermann. Karatsuba Square Root. \[Research Report\] RR-3805,
//! INRIA. 1999, pp.8. (inria-00072854)"

/// This array stores the [integer square roots](
/// https://en.wikipedia.org/wiki/Integer_square_root) and remainders of each
/// [`u8`](prim@u8) value. For example, `U8_ISQRT_WITH_REMAINDER[17]` will be
/// `(4, 1)` because the integer square root of 17 is 4 and because 17 is 1
/// higher than 4 squared.
const U8_ISQRT_WITH_REMAINDER: [(u8, u8); 256] = {
    let mut result = [(0, 0); 256];

    let mut n: usize = 0;
    let mut isqrt_n: usize = 0;
    while n < result.len() {
        result[n] = (isqrt_n as u8, (n - isqrt_n.pow(2)) as u8);

        n += 1;
        if n == (isqrt_n + 1).pow(2) {
            isqrt_n += 1;
        }
    }

    result
};

/// Returns the [integer square root](
/// https://en.wikipedia.org/wiki/Integer_square_root) of any [`u8`](prim@u8)
/// input.
#[must_use = "this returns the result of the operation, \
              without modifying the original"]
#[inline]
pub(in crate::num) const fn u8(n: u8) -> u8 {
    U8_ISQRT_WITH_REMAINDER[n as usize].0
}

/// Generates a `u*` function that returns the [integer square root](
/// https://en.wikipedia.org/wiki/Integer_square_root) of any input of
/// a specific unsigned integer type.
macro_rules! unsigned_fn {
    ($UnsignedT:ident, $HalfBitsT:ident, $stages:ident) => {
        /// Returns the [integer square root](
        /// https://en.wikipedia.org/wiki/Integer_square_root) of any
        #[doc = concat!("[`", stringify!($UnsignedT), "`](prim@", stringify!($UnsignedT), ")")]
        /// input.
        #[must_use = "this returns the result of the operation, \
                      without modifying the original"]
        #[inline]
        pub(in crate::num) const fn $UnsignedT(mut n: $UnsignedT) -> $UnsignedT {
            if n <= <$HalfBitsT>::MAX as $UnsignedT {
                $HalfBitsT(n as $HalfBitsT) as $UnsignedT
            } else {
                // The normalization shift satisfies the Karatsuba square root
                // algorithm precondition "a₃ ≥ b/4" where a₃ is the most
                // significant quarter of `n`'s bits and b is the number of
                // values that can be represented by that quarter of the bits.
                //
                // b/4 would then be all 0s except the second most significant
                // bit (010...0) in binary. Since a₃ must be at least b/4, a₃'s
                // most significant bit or its neighbor must be a 1. Since a₃'s
                // most significant bits are `n`'s most significant bits, the
                // same applies to `n`.
                //
                // The reason to shift by an even number of bits is because an
                // even number of bits produces the square root shifted to the
                // left by half of the normalization shift:
                //
                // sqrt(n << (2 * p))
                // sqrt(2.pow(2 * p) * n)
                // sqrt(2.pow(2 * p)) * sqrt(n)
                // 2.pow(p) * sqrt(n)
                // sqrt(n) << p
                //
                // Shifting by an odd number of bits leaves an ugly sqrt(2)
                // multiplied in:
                //
                // sqrt(n << (2 * p + 1))
                // sqrt(2.pow(2 * p + 1) * n)
                // sqrt(2 * 2.pow(2 * p) * n)
                // sqrt(2) * sqrt(2.pow(2 * p)) * sqrt(n)
                // sqrt(2) * 2.pow(p) * sqrt(n)
                // sqrt(2) * (sqrt(n) << p)
                const EVEN_MAKING_BITMASK: u32 = !1;
                let normalization_shift = n.leading_zeros() & EVEN_MAKING_BITMASK;
                n <<= normalization_shift;

                let s = $stages(n);

                let denormalization_shift = normalization_shift >> 1;
                s >> denormalization_shift
            }
        }
    };
}

/// Generates the first stage of the computation after normalization.
///
/// # Safety
///
/// `$n` must be nonzero.
macro_rules! first_stage {
    ($original_bits:literal, $n:ident) => {{
        debug_assert!($n != 0, "`$n` is  zero in `first_stage!`.");

        const N_SHIFT: u32 = $original_bits - 8;
        let n = $n >> N_SHIFT;

        let (s, r) = U8_ISQRT_WITH_REMAINDER[n as usize];

        // Inform the optimizer that `s` is nonzero. This will allow it to
        // avoid generating code to handle division-by-zero panics in the next
        // stage.
        //
        // SAFETY: If the original `$n` is zero, the top of the `unsigned_fn`
        // macro recurses instead of continuing to this point, so the original
        // `$n` wasn't a 0 if we've reached here.
        //
        // Then the `unsigned_fn` macro normalizes `$n` so that at least one of
        // its two most-significant bits is a 1.
        //
        // Then this stage puts the eight most-significant bits of `$n` into
        // `n`. This means that `n` here has at least one 1 bit in its two
        // most-significant bits, making `n` nonzero.
        //
        // `U8_ISQRT_WITH_REMAINDER[n as usize]` will give a nonzero `s` when
        // given a nonzero `n`.
        unsafe { crate::hint::assert_unchecked(s != 0) };
        (s, r)
    }};
}

/// Generates a middle stage of the computation.
///
/// # Safety
///
/// `$s` must be nonzero.
macro_rules! middle_stage {
    ($original_bits:literal, $ty:ty, $n:ident, $s:ident, $r:ident) => {{
        debug_assert!($s != 0, "`$s` is  zero in `middle_stage!`.");

        const N_SHIFT: u32 = $original_bits - <$ty>::BITS;
        let n = ($n >> N_SHIFT) as $ty;

        const HALF_BITS: u32 = <$ty>::BITS >> 1;
        const QUARTER_BITS: u32 = <$ty>::BITS >> 2;
        const LOWER_HALF_1_BITS: $ty = (1 << HALF_BITS) - 1;
        const LOWEST_QUARTER_1_BITS: $ty = (1 << QUARTER_BITS) - 1;

        let lo = n & LOWER_HALF_1_BITS;
        let numerator = (($r as $ty) << QUARTER_BITS) | (lo >> QUARTER_BITS);
        let denominator = ($s as $ty) << 1;
        let q = numerator / denominator;
        let u = numerator % denominator;

        let mut s = ($s << QUARTER_BITS) as $ty + q;
        let (mut r, overflow) =
            ((u << QUARTER_BITS) | (lo & LOWEST_QUARTER_1_BITS)).overflowing_sub(q * q);
        if overflow {
            r = r.wrapping_add(2 * s - 1);
            s -= 1;
        }

        // Inform the optimizer that `s` is nonzero. This will allow it to
        // avoid generating code to handle division-by-zero panics in the next
        // stage.
        //
        // SAFETY: If the original `$n` is zero, the top of the `unsigned_fn`
        // macro recurses instead of continuing to this point, so the original
        // `$n` wasn't a 0 if we've reached here.
        //
        // Then the `unsigned_fn` macro normalizes `$n` so that at least one of
        // its two most-significant bits is a 1.
        //
        // Then these stages take as many of the most-significant bits of `$n`
        // as will fit in this stage's type. For example, the stage that
        // handles `u32` deals with the 32 most-significant bits of `$n`. This
        // means that each stage has at least one 1 bit in `n`'s two
        // most-significant bits, making `n` nonzero.
        //
        // Then this stage will produce the correct integer square root for
        // that `n` value. Since `n` is nonzero, `s` will also be nonzero.
        unsafe { crate::hint::assert_unchecked(s != 0) };
        (s, r)
    }};
}

/// Generates the last stage of the computation before denormalization.
///
/// # Safety
///
/// `$s` must be nonzero.
macro_rules! last_stage {
    ($ty:ty, $n:ident, $s:ident, $r:ident) => {{
        debug_assert!($s != 0, "`$s` is  zero in `last_stage!`.");

        const HALF_BITS: u32 = <$ty>::BITS >> 1;
        const QUARTER_BITS: u32 = <$ty>::BITS >> 2;
        const LOWER_HALF_1_BITS: $ty = (1 << HALF_BITS) - 1;

        let lo = $n & LOWER_HALF_1_BITS;
        let numerator = (($r as $ty) << QUARTER_BITS) | (lo >> QUARTER_BITS);
        let denominator = ($s as $ty) << 1;

        let q = numerator / denominator;
        let mut s = ($s << QUARTER_BITS) as $ty + q;
        let (s_squared, overflow) = s.overflowing_mul(s);
        if overflow || s_squared > $n {
            s -= 1;
        }
        s
    }};
}

/// Takes the normalized [`u16`](prim@u16) input and gets its normalized
/// [integer square root](https://en.wikipedia.org/wiki/Integer_square_root).
///
/// # Safety
///
/// `n` must be nonzero.
#[inline]
const fn u16_stages(n: u16) -> u16 {
    let (s, r) = first_stage!(16, n);
    last_stage!(u16, n, s, r)
}

/// Takes the normalized [`u32`](prim@u32) input and gets its normalized
/// [integer square root](https://en.wikipedia.org/wiki/Integer_square_root).
///
/// # Safety
///
/// `n` must be nonzero.
#[inline]
const fn u32_stages(n: u32) -> u32 {
    let (s, r) = first_stage!(32, n);
    let (s, r) = middle_stage!(32, u16, n, s, r);
    last_stage!(u32, n, s, r)
}

/// Takes the normalized [`u64`](prim@u64) input and gets its normalized
/// [integer square root](https://en.wikipedia.org/wiki/Integer_square_root).
///
/// # Safety
///
/// `n` must be nonzero.
#[inline]
const fn u64_stages(n: u64) -> u64 {
    let (s, r) = first_stage!(64, n);
    let (s, r) = middle_stage!(64, u16, n, s, r);
    let (s, r) = middle_stage!(64, u32, n, s, r);
    last_stage!(u64, n, s, r)
}

/// Takes the normalized [`u128`](prim@u128) input and gets its normalized
/// [integer square root](https://en.wikipedia.org/wiki/Integer_square_root).
///
/// # Safety
///
/// `n` must be nonzero.
#[inline]
const fn u128_stages(n: u128) -> u128 {
    let (s, r) = first_stage!(128, n);
    let (s, r) = middle_stage!(128, u16, n, s, r);
    let (s, r) = middle_stage!(128, u32, n, s, r);
    let (s, r) = middle_stage!(128, u64, n, s, r);
    last_stage!(u128, n, s, r)
}

unsigned_fn!(u16, u8, u16_stages);
unsigned_fn!(u32, u16, u32_stages);
unsigned_fn!(u64, u32, u64_stages);
unsigned_fn!(u128, u64, u128_stages);

/// Instantiate this panic logic once, rather than for all the isqrt methods
/// on every single primitive type.
#[cold]
#[track_caller]
pub(in crate::num) const fn panic_for_negative_argument() -> ! {
    panic!("argument of integer square root cannot be negative")
}