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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.
//
// # 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:
//
// 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;