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    //! A simple big-integer type for slow path algorithms.
    //!
    //! This includes minimal stackvector for use in big-integer arithmetic.

    #![doc(hidden)]

    #[cfg(feature = "alloc")]
    use crate::heapvec::HeapVec;
    use crate::num::Float;
    #[cfg(not(feature = "alloc"))]
    use crate::stackvec::StackVec;
    #[cfg(not(feature = "compact"))]
    use crate::table::{LARGE_POW5, LARGE_POW5_STEP};
    use core::{cmp, ops, ptr};

    /// Number of bits in a Bigint.
    ///
    /// This needs to be at least the number of bits required to store
    /// a Bigint, which is `log2(radix**digits)`.
    /// ≅ 3600 for base-10, rounded-up.
    pub const BIGINT_BITS: usize = 4000;

    /// The number of limbs for the bigint.
    pub const BIGINT_LIMBS: usize = BIGINT_BITS / LIMB_BITS;

    #[cfg(feature = "alloc")]
    pub type VecType = HeapVec;

    #[cfg(not(feature = "alloc"))]
    pub type VecType = StackVec;

    /// Storage for a big integer type.
    ///
    /// This is used for algorithms when we have a finite number of digits.
    /// Specifically, it stores all the significant digits scaled to the
    /// proper exponent, as an integral type, and then directly compares
    /// these digits.
    ///
    /// This requires us to store the number of significant bits, plus the
    /// number of exponent bits (required) since we scale everything
    /// to the same exponent.
    #[derive(Clone, PartialEq, Eq)]
    pub struct Bigint {
        /// Significant digits for the float, stored in a big integer in LE order.
        ///
        /// This is pretty much the same number of digits for any radix, since the
        ///  significant digits balances out the zeros from the exponent:
        ///     1. Decimal is 1091 digits, 767 mantissa digits + 324 exponent zeros.
        ///     2. Base 6 is 1097 digits, or 680 mantissa digits + 417 exponent zeros.
        ///     3. Base 36 is 1086 digits, or 877 mantissa digits + 209 exponent zeros.
        ///
        /// However, the number of bytes required is larger for large radixes:
        /// for decimal, we need `log2(10**1091) ≅ 3600`, while for base 36
        /// we need `log2(36**1086) ≅ 5600`. Since we use uninitialized data,
        /// we avoid a major performance hit from the large buffer size.
        pub data: VecType,
    }

    #[allow(clippy::new_without_default)]
    impl Bigint {
        /// Construct a bigint representing 0.
        #[inline(always)]
        pub fn new() -> Self {
            Self {
                data: VecType::new(),
            }
        }

        /// Construct a bigint from an integer.
        #[inline(always)]
        pub fn from_u64(value: u64) -> Self {
            Self {
                data: VecType::from_u64(value),
            }
        }

        #[inline(always)]
        pub fn hi64(&self) -> (u64, bool) {
            self.data.hi64()
        }

        /// Multiply and assign as if by exponentiation by a power.
        #[inline]
        pub fn pow(&mut self, base: u32, exp: u32) -> Option<()> {
            debug_assert!(base == 2 || base == 5 || base == 10);
            if base % 5 == 0 {
                pow(&mut self.data, exp)?;
            }
            if base % 2 == 0 {
                shl(&mut self.data, exp as usize)?;
            }
            Some(())
        }

        /// Calculate the bit-length of the big-integer.
        #[inline]
        pub fn bit_length(&self) -> u32 {
            bit_length(&self.data)
        }
    }

    impl ops::MulAssign<&Bigint> for Bigint {
        fn mul_assign(&mut self, rhs: &Bigint) {
            self.data *= &rhs.data;
        }
    }

    /// REVERSE VIEW

    /// Reverse, immutable view of a sequence.
    pub struct ReverseView<'a, T: 'a> {
        inner: &'a [T],
    }

    impl<'a, T> ops::Index<usize> for ReverseView<'a, T> {
        type Output = T;

        #[inline]
        fn index(&self, index: usize) -> &T {
            let len = self.inner.len();
            &(*self.inner)[len - index - 1]
        }
    }

    /// Create a reverse view of the vector for indexing.
    #[inline]
    pub fn rview(x: &[Limb]) -> ReverseView<Limb> {
        ReverseView {
            inner: x,
        }
    }

    // COMPARE
    // -------

    /// Compare `x` to `y`, in little-endian order.
    #[inline]
    pub fn compare(x: &[Limb], y: &[Limb]) -> cmp::Ordering {
        match x.len().cmp(&y.len()) {
            cmp::Ordering::Equal => {
                let iter = x.iter().rev().zip(y.iter().rev());
                for (&xi, yi) in iter {
                    match xi.cmp(yi) {
                        cmp::Ordering::Equal => (),
                        ord => return ord,
                    }
                }
                // Equal case.
                cmp::Ordering::Equal
            },
            ord => ord,
        }
    }

    // NORMALIZE
    // ---------

    /// Normalize the integer, so any leading zero values are removed.
    #[inline]
    pub fn normalize(x: &mut VecType) {
        // We don't care if this wraps: the index is bounds-checked.
        while let Some(&value) = x.get(x.len().wrapping_sub(1)) {
            if value == 0 {
                unsafe { x.set_len(x.len() - 1) };
            } else {
                break;
            }
        }
    }

    /// Get if the big integer is normalized.
    #[inline]
    #[allow(clippy::match_like_matches_macro)]
    pub fn is_normalized(x: &[Limb]) -> bool {
        // We don't care if this wraps: the index is bounds-checked.
        match x.get(x.len().wrapping_sub(1)) {
            Some(&0) => false,
            _ => true,
        }
    }

    // FROM
    // ----

    /// Create StackVec from u64 value.
    #[inline(always)]
    #[allow(clippy::branches_sharing_code)]
    pub fn from_u64(x: u64) -> VecType {
        let mut vec = VecType::new();
        debug_assert!(vec.capacity() >= 2);
        if LIMB_BITS == 32 {
            vec.try_push(x as Limb).unwrap();
            vec.try_push((x >> 32) as Limb).unwrap();
        } else {
            vec.try_push(x as Limb).unwrap();
        }
        vec.normalize();
        vec
    }

    // HI
    // --

    /// Check if any of the remaining bits are non-zero.
    ///
    /// # Safety
    ///
    /// Safe as long as `rindex <= x.len()`.
    #[inline]
    pub fn nonzero(x: &[Limb], rindex: usize) -> bool {
        debug_assert!(rindex <= x.len());

        let len = x.len();
        let slc = &x[..len - rindex];
        slc.iter().rev().any(|&x| x != 0)
    }

    // These return the high X bits and if the bits were truncated.

    /// Shift 32-bit integer to high 64-bits.
    #[inline]
    pub fn u32_to_hi64_1(r0: u32) -> (u64, bool) {
        u64_to_hi64_1(r0 as u64)
    }

    /// Shift 2 32-bit integers to high 64-bits.
    #[inline]
    pub fn u32_to_hi64_2(r0: u32, r1: u32) -> (u64, bool) {
        let r0 = (r0 as u64) << 32;
        let r1 = r1 as u64;
        u64_to_hi64_1(r0 | r1)
    }

    /// Shift 3 32-bit integers to high 64-bits.
    #[inline]
    pub fn u32_to_hi64_3(r0: u32, r1: u32, r2: u32) -> (u64, bool) {
        let r0 = r0 as u64;
        let r1 = (r1 as u64) << 32;
        let r2 = r2 as u64;
        u64_to_hi64_2(r0, r1 | r2)
    }

    /// Shift 64-bit integer to high 64-bits.
    #[inline]
    pub fn u64_to_hi64_1(r0: u64) -> (u64, bool) {
        let ls = r0.leading_zeros();
        (r0 << ls, false)
    }

    /// Shift 2 64-bit integers to high 64-bits.
    #[inline]
    pub fn u64_to_hi64_2(r0: u64, r1: u64) -> (u64, bool) {
        let ls = r0.leading_zeros();
        let rs = 64 - ls;
        let v = match ls {
            0 => r0,
            _ => (r0 << ls) | (r1 >> rs),
        };
        let n = r1 << ls != 0;
        (v, n)
    }

    /// Extract the hi bits from the buffer.
    macro_rules! hi {
        // # Safety
        //
        // Safe as long as the `stackvec.len() >= 1`.
        (@1 $self:ident, $rview:ident, $t:ident, $fn:ident) => {{
            $fn($rview[0] as $t)
        }};

        // # Safety
        //
        // Safe as long as the `stackvec.len() >= 2`.
        (@2 $self:ident, $rview:ident, $t:ident, $fn:ident) => {{
            let r0 = $rview[0] as $t;
            let r1 = $rview[1] as $t;
            $fn(r0, r1)
        }};

        // # Safety
        //
        // Safe as long as the `stackvec.len() >= 2`.
        (@nonzero2 $self:ident, $rview:ident, $t:ident, $fn:ident) => {{
            let (v, n) = hi!(@2 $self, $rview, $t, $fn);
            (v, n || nonzero($self, 2 ))
        }};

        // # Safety
        //
        // Safe as long as the `stackvec.len() >= 3`.
        (@3 $self:ident, $rview:ident, $t:ident, $fn:ident) => {{
            let r0 = $rview[0] as $t;
            let r1 = $rview[1] as $t;
            let r2 = $rview[2] as $t;
            $fn(r0, r1, r2)
        }};

        // # Safety
        //
        // Safe as long as the `stackvec.len() >= 3`.
        (@nonzero3 $self:ident, $rview:ident, $t:ident, $fn:ident) => {{
            let (v, n) = hi!(@3 $self, $rview, $t, $fn);
            (v, n || nonzero($self, 3))
        }};
    }

    /// Get the high 64 bits from the vector.
    #[inline(always)]
    pub fn hi64(x: &[Limb]) -> (u64, bool) {
        let rslc = rview(x);
        // SAFETY: the buffer must be at least length bytes long.
        match x.len() {
            0 => (0, false),
            1 if LIMB_BITS == 32 => hi!(@1 x, rslc, u32, u32_to_hi64_1),
            1 => hi!(@1 x, rslc, u64, u64_to_hi64_1),
            2 if LIMB_BITS == 32 => hi!(@2 x, rslc, u32, u32_to_hi64_2),
            2 => hi!(@2 x, rslc, u64, u64_to_hi64_2),
            _ if LIMB_BITS == 32 => hi!(@nonzero3 x, rslc, u32, u32_to_hi64_3),
            _ => hi!(@nonzero2 x, rslc, u64, u64_to_hi64_2),
        }
    }

    // POWERS
    // ------

    /// MulAssign by a power of 5.
    ///
    /// Theoretically...
    ///
    /// Use an exponentiation by squaring method, since it reduces the time
    /// complexity of the multiplication to ~`O(log(n))` for the squaring,
    /// and `O(n*m)` for the result. Since `m` is typically a lower-order
    /// factor, this significantly reduces the number of multiplications
    /// we need to do. Iteratively multiplying by small powers follows
    /// the nth triangular number series, which scales as `O(p^2)`, but
    /// where `p` is `n+m`. In short, it scales very poorly.
    ///
    /// Practically....
    ///
    /// Exponentiation by Squaring:
    ///     running 2 tests
    ///     test bigcomp_f32_lexical ... bench:       1,018 ns/iter (+/- 78)
    ///     test bigcomp_f64_lexical ... bench:       3,639 ns/iter (+/- 1,007)
    ///
    /// Exponentiation by Iterative Small Powers:
    ///     running 2 tests
    ///     test bigcomp_f32_lexical ... bench:         518 ns/iter (+/- 31)
    ///     test bigcomp_f64_lexical ... bench:         583 ns/iter (+/- 47)
    ///
    /// Exponentiation by Iterative Large Powers (of 2):
    ///     running 2 tests
    ///     test bigcomp_f32_lexical ... bench:         671 ns/iter (+/- 31)
    ///     test bigcomp_f64_lexical ... bench:       1,394 ns/iter (+/- 47)
    ///
    /// The following benchmarks were run on `1 * 5^300`, using native `pow`,
    /// a version with only small powers, and one with pre-computed powers
    /// of `5^(3 * max_exp)`, rather than `5^(5 * max_exp)`.
    ///
    /// However, using large powers is crucial for good performance for higher
    /// powers.
    ///     pow/default             time:   [426.20 ns 427.96 ns 429.89 ns]
    ///     pow/small               time:   [2.9270 us 2.9411 us 2.9565 us]
    ///     pow/large:3             time:   [838.51 ns 842.21 ns 846.27 ns]
    ///
    /// Even using worst-case scenarios, exponentiation by squaring is
    /// significantly slower for our workloads. Just multiply by small powers,
    /// in simple cases, and use precalculated large powers in other cases.
    ///
    /// Furthermore, using sufficiently big large powers is also crucial for
    /// performance. This is a tradeoff of binary size and performance, and
    /// using a single value at ~`5^(5 * max_exp)` seems optimal.
    pub fn pow(x: &mut VecType, mut exp: u32) -> Option<()> {
        // Minimize the number of iterations for large exponents: just
        // do a few steps with a large powers.
        #[cfg(not(feature = "compact"))]
        {
            while exp >= LARGE_POW5_STEP {
                large_mul(x, &LARGE_POW5)?;
                exp -= LARGE_POW5_STEP;
            }
        }

        // Now use our pre-computed small powers iteratively.
        // This is calculated as `⌊log(2^BITS - 1, 5)⌋`.
        let small_step = if LIMB_BITS == 32 {
            13
        } else {
            27
        };
        let max_native = (5 as Limb).pow(small_step);
        while exp >= small_step {
            small_mul(x, max_native)?;
            exp -= small_step;
        }
        if exp != 0 {
            // SAFETY: safe, since `exp < small_step`.
            let small_power = unsafe { f64::int_pow_fast_path(exp as usize, 5) };
            small_mul(x, small_power as Limb)?;
        }
        Some(())
    }

    // SCALAR
    // ------

    /// Add two small integers and return the resulting value and if overflow happens.
    #[inline(always)]
    pub fn scalar_add(x: Limb, y: Limb) -> (Limb, bool) {
        x.overflowing_add(y)
    }

    /// Multiply two small integers (with carry) (and return the overflow contribution).
    ///
    /// Returns the (low, high) components.
    #[inline(always)]
    pub fn scalar_mul(x: Limb, y: Limb, carry: Limb) -> (Limb, Limb) {
        // Cannot overflow, as long as wide is 2x as wide. This is because
        // the following is always true:
        // `Wide::MAX - (Narrow::MAX * Narrow::MAX) >= Narrow::MAX`
        let z: Wide = (x as Wide) * (y as Wide) + (carry as Wide);
        (z as Limb, (z >> LIMB_BITS) as Limb)
    }

    // SMALL
    // -----

    /// Add small integer to bigint starting from offset.
    #[inline]
    pub fn small_add_from(x: &mut VecType, y: Limb, start: usize) -> Option<()> {
        let mut index = start;
        let mut carry = y;
        while carry != 0 && index < x.len() {
            let result = scalar_add(x[index], carry);
            x[index] = result.0;
            carry = result.1 as Limb;
            index += 1;
        }
        // If we carried past all the elements, add to the end of the buffer.
        if carry != 0 {
            x.try_push(carry)?;
        }
        Some(())
    }

    /// Add small integer to bigint.
    #[inline(always)]
    pub fn small_add(x: &mut VecType, y: Limb) -> Option<()> {
        small_add_from(x, y, 0)
    }

    /// Multiply bigint by small integer.
    #[inline]
    pub fn small_mul(x: &mut VecType, y: Limb) -> Option<()> {
        let mut carry = 0;
        for xi in x.iter_mut() {
            let result = scalar_mul(*xi, y, carry);
            *xi = result.0;
            carry = result.1;
        }
        // If we carried past all the elements, add to the end of the buffer.
        if carry != 0 {
            x.try_push(carry)?;
        }
        Some(())
    }

    // LARGE
    // -----

    /// Add bigint to bigint starting from offset.
    pub fn large_add_from(x: &mut VecType, y: &[Limb], start: usize) -> Option<()> {
        // The effective x buffer is from `xstart..x.len()`, so we need to treat
        // that as the current range. If the effective y buffer is longer, need
        // to resize to that, + the start index.
        if y.len() > x.len().saturating_sub(start) {
            // Ensure we panic if we can't extend the buffer.
            // This avoids any unsafe behavior afterwards.
            x.try_resize(y.len() + start, 0)?;
        }

        // Iteratively add elements from y to x.
        let mut carry = false;
        for (index, &yi) in y.iter().enumerate() {
            // We panicked in `try_resize` if this wasn't true.
            let xi = x.get_mut(start + index).unwrap();

            // Only one op of the two ops can overflow, since we added at max
            // Limb::max_value() + Limb::max_value(). Add the previous carry,
            // and store the current carry for the next.
            let result = scalar_add(*xi, yi);
            *xi = result.0;
            let mut tmp = result.1;
            if carry {
                let result = scalar_add(*xi, 1);
                *xi = result.0;
                tmp |= result.1;
            }
            carry = tmp;
        }

        // Handle overflow.
        if carry {
            small_add_from(x, 1, y.len() + start)?;
        }
        Some(())
    }

    /// Add bigint to bigint.
    #[inline(always)]
    pub fn large_add(x: &mut VecType, y: &[Limb]) -> Option<()> {
        large_add_from(x, y, 0)
    }

    /// Grade-school multiplication algorithm.
    ///
    /// Slow, naive algorithm, using limb-bit bases and just shifting left for
    /// each iteration. This could be optimized with numerous other algorithms,
    /// but it's extremely simple, and works in O(n*m) time, which is fine
    /// by me. Each iteration, of which there are `m` iterations, requires
    /// `n` multiplications, and `n` additions, or grade-school multiplication.
    ///
    /// Don't use Karatsuba multiplication, since out implementation seems to
    /// be slower asymptotically, which is likely just due to the small sizes
    /// we deal with here. For example, running on the following data:
    ///
    /// ```text
    /// const SMALL_X: &[u32] = &[
    ///     766857581, 3588187092, 1583923090, 2204542082, 1564708913, 2695310100, 3676050286,
    ///     1022770393, 468044626, 446028186
    /// ];
    /// const SMALL_Y: &[u32] = &[
    ///     3945492125, 3250752032, 1282554898, 1708742809, 1131807209, 3171663979, 1353276095,
    ///     1678845844, 2373924447, 3640713171
    /// ];
    /// const LARGE_X: &[u32] = &[
    ///     3647536243, 2836434412, 2154401029, 1297917894, 137240595, 790694805, 2260404854,
    ///     3872698172, 690585094, 99641546, 3510774932, 1672049983, 2313458559, 2017623719,
    ///     638180197, 1140936565, 1787190494, 1797420655, 14113450, 2350476485, 3052941684,
    ///     1993594787, 2901001571, 4156930025, 1248016552, 848099908, 2660577483, 4030871206,
    ///     692169593, 2835966319, 1781364505, 4266390061, 1813581655, 4210899844, 2137005290,
    ///     2346701569, 3715571980, 3386325356, 1251725092, 2267270902, 474686922, 2712200426,
    ///     197581715, 3087636290, 1379224439, 1258285015, 3230794403, 2759309199, 1494932094,
    ///     326310242
    /// ];
    /// const LARGE_Y: &[u32] = &[
    ///     1574249566, 868970575, 76716509, 3198027972, 1541766986, 1095120699, 3891610505,
    ///     2322545818, 1677345138, 865101357, 2650232883, 2831881215, 3985005565, 2294283760,
    ///     3468161605, 393539559, 3665153349, 1494067812, 106699483, 2596454134, 797235106,
    ///     705031740, 1209732933, 2732145769, 4122429072, 141002534, 790195010, 4014829800,
    ///     1303930792, 3649568494, 308065964, 1233648836, 2807326116, 79326486, 1262500691,
    ///     621809229, 2258109428, 3819258501, 171115668, 1139491184, 2979680603, 1333372297,
    ///     1657496603, 2790845317, 4090236532, 4220374789, 601876604, 1828177209, 2372228171,
    ///     2247372529
    /// ];
    /// ```
    ///
    /// We get the following results:

    /// ```text
    /// mul/small:long          time:   [220.23 ns 221.47 ns 222.81 ns]
    /// Found 4 outliers among 100 measurements (4.00%)
    ///   2 (2.00%) high mild
    ///   2 (2.00%) high severe
    /// mul/small:karatsuba     time:   [233.88 ns 234.63 ns 235.44 ns]
    /// Found 11 outliers among 100 measurements (11.00%)
    ///   8 (8.00%) high mild
    ///   3 (3.00%) high severe
    /// mul/large:long          time:   [1.9365 us 1.9455 us 1.9558 us]
    /// Found 12 outliers among 100 measurements (12.00%)
    ///   7 (7.00%) high mild
    ///   5 (5.00%) high severe
    /// mul/large:karatsuba     time:   [4.4250 us 4.4515 us 4.4812 us]
    /// ```
    ///
    /// In short, Karatsuba multiplication is never worthwhile for out use-case.
    pub fn long_mul(x: &[Limb], y: &[Limb]) -> Option<VecType> {
        // Using the immutable value, multiply by all the scalars in y, using
        // the algorithm defined above. Use a single buffer to avoid
        // frequent reallocations. Handle the first case to avoid a redundant
        // addition, since we know y.len() >= 1.
        let mut z = VecType::try_from(x)?;
        if !y.is_empty() {
            let y0 = y[0];
            small_mul(&mut z, y0)?;

            for (index, &yi) in y.iter().enumerate().skip(1) {
                if yi != 0 {
                    let mut zi = VecType::try_from(x)?;
                    small_mul(&mut zi, yi)?;
                    large_add_from(&mut z, &zi, index)?;
                }
            }
        }

        z.normalize();
        Some(z)
    }

    /// Multiply bigint by bigint using grade-school multiplication algorithm.
    #[inline(always)]
    pub fn large_mul(x: &mut VecType, y: &[Limb]) -> Option<()> {
        // Karatsuba multiplication never makes sense, so just use grade school
        // multiplication.
        if y.len() == 1 {
            // SAFETY: safe since `y.len() == 1`.
            small_mul(x, y[0])?;
        } else {
            *x = long_mul(y, x)?;
        }
        Some(())
    }

    // SHIFT
    // -----

    /// Shift-left `n` bits inside a buffer.
    #[inline]
    pub fn shl_bits(x: &mut VecType, n: usize) -> Option<()> {
        debug_assert!(n != 0);

        // Internally, for each item, we shift left by n, and add the previous
        // right shifted limb-bits.
        // For example, we transform (for u8) shifted left 2, to:
        //      b10100100 b01000010
        //      b10 b10010001 b00001000
        debug_assert!(n < LIMB_BITS);
        let rshift = LIMB_BITS - n;
        let lshift = n;
        let mut prev: Limb = 0;
        for xi in x.iter_mut() {
            let tmp = *xi;
            *xi <<= lshift;
            *xi |= prev >> rshift;
            prev = tmp;
        }

        // Always push the carry, even if it creates a non-normal result.
        let carry = prev >> rshift;
        if carry != 0 {
            x.try_push(carry)?;
        }

        Some(())
    }

    /// Shift-left `n` limbs inside a buffer.
    #[inline]
    pub fn shl_limbs(x: &mut VecType, n: usize) -> Option<()> {
        debug_assert!(n != 0);
        if n + x.len() > x.capacity() {
            None
        } else if !x.is_empty() {
            let len = n + x.len();
            // SAFE: since x is not empty, and `x.len() + n <= x.capacity()`.
            unsafe {
                // Move the elements.
                let src = x.as_ptr();
                let dst = x.as_mut_ptr().add(n);
                ptr::copy(src, dst, x.len());
                // Write our 0s.
                ptr::write_bytes(x.as_mut_ptr(), 0, n);
                x.set_len(len);
            }
            Some(())
        } else {
            Some(())
        }
    }

    /// Shift-left buffer by n bits.
    #[inline]
    pub fn shl(x: &mut VecType, n: usize) -> Option<()> {
        let rem = n % LIMB_BITS;
        let div = n / LIMB_BITS;
        if rem != 0 {
            shl_bits(x, rem)?;
        }
        if div != 0 {
            shl_limbs(x, div)?;
        }
        Some(())
    }

    /// Get number of leading zero bits in the storage.
    #[inline]
    pub fn leading_zeros(x: &[Limb]) -> u32 {
        let length = x.len();
        // wrapping_sub is fine, since it'll just return None.
        if let Some(&value) = x.get(length.wrapping_sub(1)) {
            value.leading_zeros()
        } else {
            0
        }
    }

    /// Calculate the bit-length of the big-integer.
    #[inline]
    pub fn bit_length(x: &[Limb]) -> u32 {
        let nlz = leading_zeros(x);
        LIMB_BITS as u32 * x.len() as u32 - nlz
    }

    // LIMB
    // ----

    //  Type for a single limb of the big integer.
    //
    //  A limb is analogous to a digit in base10, except, it stores 32-bit
    //  or 64-bit numbers instead. We want types where 64-bit multiplication
    //  is well-supported by the architecture, rather than emulated in 3
    //  instructions. The quickest way to check this support is using a
    //  cross-compiler for numerous architectures, along with the following
    //  source file and command:
    //
    //  Compile with `gcc main.c -c -S -O3 -masm=intel`
    //
    //  And the source code is:
    //  ```text
    //  #include <stdint.h>
    //
    //  struct i128 {
    //      uint64_t hi;
    //      uint64_t lo;
    //  };
    //
    //  // Type your code here, or load an example.
    //  struct i128 square(uint64_t x, uint64_t y) {
    //      __int128 prod = (__int128)x * (__int128)y;
    //      struct i128 z;
    //      z.hi = (uint64_t)(prod >> 64);
    //      z.lo = (uint64_t)prod;
    //      return z;
    //  }
    //  ```
    //
    //  If the result contains `call __multi3`, then the multiplication
    //  is emulated by the compiler. Otherwise, it's natively supported.
    //
    //  This should be all-known 64-bit platforms supported by Rust.
    //      https://forge.rust-lang.org/platform-support.html
    //
    //  # Supported
    //
    //  Platforms where native 128-bit multiplication is explicitly supported:
    //      - x86_64 (Supported via `MUL`).
    //      - mips64 (Supported via `DMULTU`, which `HI` and `LO` can be read-from).
    //      - s390x (Supported via `MLGR`).
    //
    //  # Efficient
    //
    //  Platforms where native 64-bit multiplication is supported and
    //  you can extract hi-lo for 64-bit multiplications.
    //      - aarch64 (Requires `UMULH` and `MUL` to capture high and low bits).
    //      - powerpc64 (Requires `MULHDU` and `MULLD` to capture high and low bits).
    //      - riscv64 (Requires `MUL` and `MULH` to capture high and low bits).
    //
    //  # Unsupported
    //
    //  Platforms where native 128-bit multiplication is not supported,
    //  requiring software emulation.
    //      sparc64 (`UMUL` only supports double-word arguments).
    //      sparcv9 (Same as sparc64).
    //
    //  These tests are run via `xcross`, my own library for C cross-compiling,
    //  which supports numerous targets (far in excess of Rust's tier 1 support,
    //  or rust-embedded/cross's list). xcross may be found here:
    //      https://github.com/Alexhuszagh/xcross
    //
    //  To compile for the given target, run:
    //      `xcross gcc main.c -c -S -O3 --target $target`
    //
    //  All 32-bit architectures inherently do not have support. That means
    //  we can essentially look for 64-bit architectures that are not SPARC.

    #[cfg(all(target_pointer_width = "64", not(target_arch = "sparc")))]
    pub type Limb = u64;
    #[cfg(all(target_pointer_width = "64", not(target_arch = "sparc")))]
    pub type Wide = u128;
    #[cfg(all(target_pointer_width = "64", not(target_arch = "sparc")))]
    pub const LIMB_BITS: usize = 64;

    #[cfg(not(all(target_pointer_width = "64", not(target_arch = "sparc"))))]
    pub type Limb = u32;
    #[cfg(not(all(target_pointer_width = "64", not(target_arch = "sparc"))))]
    pub type Wide = u64;
    #[cfg(not(all(target_pointer_width = "64", not(target_arch = "sparc"))))]
    pub const LIMB_BITS: usize = 32;