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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;
generated by cgit v1.2.3 (git 2.39.1) at 2024-12-27 01:36:12 +0000