bignum.rs.html -- source
Description: Source to the Rust file `src/libcore/num/bignum.rs`.
Keywords: rust, rustlang, rust-lang

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    //! Custom arbitrary-precision number (bignum) implementation.
    //!
    //! This is designed to avoid the heap allocation at expense of stack memory.
    //! The most used bignum type, `Big32x40`, is limited by 32 × 40 = 1,280 bits
    //! and will take at most 160 bytes of stack memory. This is more than enough
    //! for round-tripping all possible finite `f64` values.
    //!
    //! In principle it is possible to have multiple bignum types for different
    //! inputs, but we don't do so to avoid the code bloat. Each bignum is still
    //! tracked for the actual usages, so it normally doesn't matter.

    // This module is only for dec2flt and flt2dec, and only public because of coretests.
    // It is not intended to ever be stabilized.
    #![doc(hidden)]
    #![unstable(
        feature = "core_private_bignum",
        reason = "internal routines only exposed for testing",
        issue = "none"
    )]
    #![macro_use]

    use crate::intrinsics;
    use crate::mem;

    /// Arithmetic operations required by bignums.
    pub trait FullOps: Sized {
        /// Returns `(carry', v')` such that `carry' * 2^W + v' = self + other + carry`,
        /// where `W` is the number of bits in `Self`.
        fn full_add(self, other: Self, carry: bool) -> (bool /* carry */, Self);

        /// Returns `(carry', v')` such that `carry' * 2^W + v' = self * other + carry`,
        /// where `W` is the number of bits in `Self`.
        fn full_mul(self, other: Self, carry: Self) -> (Self /* carry */, Self);

        /// Returns `(carry', v')` such that `carry' * 2^W + v' = self * other + other2 + carry`,
        /// where `W` is the number of bits in `Self`.
        fn full_mul_add(self, other: Self, other2: Self, carry: Self) -> (Self /* carry */, Self);

        /// Returns `(quo, rem)` such that `borrow * 2^W + self = quo * other + rem`
        /// and `0 <= rem < other`, where `W` is the number of bits in `Self`.
        fn full_div_rem(self, other: Self, borrow: Self)
        -> (Self /* quotient */, Self /* remainder */);
    }

    macro_rules! impl_full_ops {
        ($($ty:ty: add($addfn:path), mul/div($bigty:ident);)*) => (
            $(
                impl FullOps for $ty {
                    fn full_add(self, other: $ty, carry: bool) -> (bool, $ty) {
                        // This cannot overflow; the output is between `0` and `2 * 2^nbits - 1`.
                        // FIXME: will LLVM optimize this into ADC or similar?
                        let (v, carry1) = intrinsics::add_with_overflow(self, other);
                        let (v, carry2) = intrinsics::add_with_overflow(v, if carry {1} else {0});
                        (carry1 || carry2, v)
                    }

                    fn full_mul(self, other: $ty, carry: $ty) -> ($ty, $ty) {
                        // This cannot overflow;
                        // the output is between `0` and `2^nbits * (2^nbits - 1)`.
                        // FIXME: will LLVM optimize this into ADC or similar?
                        let nbits = mem::size_of::<$ty>() * 8;
                        let v = (self as $bigty) * (other as $bigty) + (carry as $bigty);
                        ((v >> nbits) as $ty, v as $ty)
                    }

                    fn full_mul_add(self, other: $ty, other2: $ty, carry: $ty) -> ($ty, $ty) {
                        // This cannot overflow;
                        // the output is between `0` and `2^nbits * (2^nbits - 1)`.
                        let nbits = mem::size_of::<$ty>() * 8;
                        let v = (self as $bigty) * (other as $bigty) + (other2 as $bigty) +
                                (carry as $bigty);
                        ((v >> nbits) as $ty, v as $ty)
                    }

                    fn full_div_rem(self, other: $ty, borrow: $ty) -> ($ty, $ty) {
                        debug_assert!(borrow < other);
                        // This cannot overflow; the output is between `0` and `other * (2^nbits - 1)`.
                        let nbits = mem::size_of::<$ty>() * 8;
                        let lhs = ((borrow as $bigty) << nbits) | (self as $bigty);
                        let rhs = other as $bigty;
                        ((lhs / rhs) as $ty, (lhs % rhs) as $ty)
                    }
                }
            )*
        )
    }

    impl_full_ops! {
        u8:  add(intrinsics::u8_add_with_overflow),  mul/div(u16);
        u16: add(intrinsics::u16_add_with_overflow), mul/div(u32);
        u32: add(intrinsics::u32_add_with_overflow), mul/div(u64);
        // See RFC #521 for enabling this.
        // u64: add(intrinsics::u64_add_with_overflow), mul/div(u128);
    }

    /// Table of powers of 5 representable in digits. Specifically, the largest {u8, u16, u32} value
    /// that's a power of five, plus the corresponding exponent. Used in `mul_pow5`.
    const SMALL_POW5: [(u64, usize); 3] = [(125, 3), (15625, 6), (1_220_703_125, 13)];

    macro_rules! define_bignum {
        ($name:ident: type=$ty:ty, n=$n:expr) => {
            /// Stack-allocated arbitrary-precision (up to certain limit) integer.
            ///
            /// This is backed by a fixed-size array of given type ("digit").
            /// While the array is not very large (normally some hundred bytes),
            /// copying it recklessly may result in the performance hit.
            /// Thus this is intentionally not `Copy`.
            ///
            /// All operations available to bignums panic in the case of overflows.
            /// The caller is responsible to use large enough bignum types.
            pub struct $name {
                /// One plus the offset to the maximum "digit" in use.
                /// This does not decrease, so be aware of the computation order.
                /// `base[size..]` should be zero.
                size: usize,
                /// Digits. `[a, b, c, ...]` represents `a + b*2^W + c*2^(2W) + ...`
                /// where `W` is the number of bits in the digit type.
                base: [$ty; $n],
            }

            impl $name {
                /// Makes a bignum from one digit.
                pub fn from_small(v: $ty) -> $name {
                    let mut base = [0; $n];
                    base[0] = v;
                    $name { size: 1, base: base }
                }

                /// Makes a bignum from `u64` value.
                pub fn from_u64(mut v: u64) -> $name {
                    use crate::mem;

                    let mut base = [0; $n];
                    let mut sz = 0;
                    while v > 0 {
                        base[sz] = v as $ty;
                        v >>= mem::size_of::<$ty>() * 8;
                        sz += 1;
                    }
                    $name { size: sz, base: base }
                }

                /// Returns the internal digits as a slice `[a, b, c, ...]` such that the numeric
                /// value is `a + b * 2^W + c * 2^(2W) + ...` where `W` is the number of bits in
                /// the digit type.
                pub fn digits(&self) -> &[$ty] {
                    &self.base[..self.size]
                }

                /// Returns the `i`-th bit where bit 0 is the least significant one.
                /// In other words, the bit with weight `2^i`.
                pub fn get_bit(&self, i: usize) -> u8 {
                    use crate::mem;

                    let digitbits = mem::size_of::<$ty>() * 8;
                    let d = i / digitbits;
                    let b = i % digitbits;
                    ((self.base[d] >> b) & 1) as u8
                }

                /// Returns `true` if the bignum is zero.
                pub fn is_zero(&self) -> bool {
                    self.digits().iter().all(|&v| v == 0)
                }

                /// Returns the number of bits necessary to represent this value. Note that zero
                /// is considered to need 0 bits.
                pub fn bit_length(&self) -> usize {
                    use crate::mem;

                    // Skip over the most significant digits which are zero.
                    let digits = self.digits();
                    let zeros = digits.iter().rev().take_while(|&&x| x == 0).count();
                    let end = digits.len() - zeros;
                    let nonzero = &digits[..end];

                    if nonzero.is_empty() {
                        // There are no non-zero digits, i.e., the number is zero.
                        return 0;
                    }
                    // This could be optimized with leading_zeros() and bit shifts, but that's
                    // probably not worth the hassle.
                    let digitbits = mem::size_of::<$ty>() * 8;
                    let mut i = nonzero.len() * digitbits - 1;
                    while self.get_bit(i) == 0 {
                        i -= 1;
                    }
                    i + 1
                }

                /// Adds `other` to itself and returns its own mutable reference.
                pub fn add<'a>(&'a mut self, other: &$name) -> &'a mut $name {
                    use crate::cmp;
                    use crate::num::bignum::FullOps;

                    let mut sz = cmp::max(self.size, other.size);
                    let mut carry = false;
                    for (a, b) in self.base[..sz].iter_mut().zip(&other.base[..sz]) {
                        let (c, v) = (*a).full_add(*b, carry);
                        *a = v;
                        carry = c;
                    }
                    if carry {
                        self.base[sz] = 1;
                        sz += 1;
                    }
                    self.size = sz;
                    self
                }

                pub fn add_small(&mut self, other: $ty) -> &mut $name {
                    use crate::num::bignum::FullOps;

                    let (mut carry, v) = self.base[0].full_add(other, false);
                    self.base[0] = v;
                    let mut i = 1;
                    while carry {
                        let (c, v) = self.base[i].full_add(0, carry);
                        self.base[i] = v;
                        carry = c;
                        i += 1;
                    }
                    if i > self.size {
                        self.size = i;
                    }
                    self
                }

                /// Subtracts `other` from itself and returns its own mutable reference.
                pub fn sub<'a>(&'a mut self, other: &$name) -> &'a mut $name {
                    use crate::cmp;
                    use crate::num::bignum::FullOps;

                    let sz = cmp::max(self.size, other.size);
                    let mut noborrow = true;
                    for (a, b) in self.base[..sz].iter_mut().zip(&other.base[..sz]) {
                        let (c, v) = (*a).full_add(!*b, noborrow);
                        *a = v;
                        noborrow = c;
                    }
                    assert!(noborrow);
                    self.size = sz;
                    self
                }

                /// Multiplies itself by a digit-sized `other` and returns its own
                /// mutable reference.
                pub fn mul_small(&mut self, other: $ty) -> &mut $name {
                    use crate::num::bignum::FullOps;

                    let mut sz = self.size;
                    let mut carry = 0;
                    for a in &mut self.base[..sz] {
                        let (c, v) = (*a).full_mul(other, carry);
                        *a = v;
                        carry = c;
                    }
                    if carry > 0 {
                        self.base[sz] = carry;
                        sz += 1;
                    }
                    self.size = sz;
                    self
                }

                /// Multiplies itself by `2^bits` and returns its own mutable reference.
                pub fn mul_pow2(&mut self, bits: usize) -> &mut $name {
                    use crate::mem;

                    let digitbits = mem::size_of::<$ty>() * 8;
                    let digits = bits / digitbits;
                    let bits = bits % digitbits;

                    assert!(digits < $n);
                    debug_assert!(self.base[$n - digits..].iter().all(|&v| v == 0));
                    debug_assert!(bits == 0 || (self.base[$n - digits - 1] >> (digitbits - bits)) == 0);

                    // shift by `digits * digitbits` bits
                    for i in (0..self.size).rev() {
                        self.base[i + digits] = self.base[i];
                    }
                    for i in 0..digits {
                        self.base[i] = 0;
                    }

                    // shift by `bits` bits
                    let mut sz = self.size + digits;
                    if bits > 0 {
                        let last = sz;
                        let overflow = self.base[last - 1] >> (digitbits - bits);
                        if overflow > 0 {
                            self.base[last] = overflow;
                            sz += 1;
                        }
                        for i in (digits + 1..last).rev() {
                            self.base[i] =
                                (self.base[i] << bits) | (self.base[i - 1] >> (digitbits - bits));
                        }
                        self.base[digits] <<= bits;
                        // self.base[..digits] is zero, no need to shift
                    }

                    self.size = sz;
                    self
                }

                /// Multiplies itself by `5^e` and returns its own mutable reference.
                pub fn mul_pow5(&mut self, mut e: usize) -> &mut $name {
                    use crate::mem;
                    use crate::num::bignum::SMALL_POW5;

                    // There are exactly n trailing zeros on 2^n, and the only relevant digit sizes
                    // are consecutive powers of two, so this is well suited index for the table.
                    let table_index = mem::size_of::<$ty>().trailing_zeros() as usize;
                    let (small_power, small_e) = SMALL_POW5[table_index];
                    let small_power = small_power as $ty;

                    // Multiply with the largest single-digit power as long as possible ...
                    while e >= small_e {
                        self.mul_small(small_power);
                        e -= small_e;
                    }

                    // ... then finish off the remainder.
                    let mut rest_power = 1;
                    for _ in 0..e {
                        rest_power *= 5;
                    }
                    self.mul_small(rest_power);

                    self
                }

                /// Multiplies itself by a number described by `other[0] + other[1] * 2^W +
                /// other[2] * 2^(2W) + ...` (where `W` is the number of bits in the digit type)
                /// and returns its own mutable reference.
                pub fn mul_digits<'a>(&'a mut self, other: &[$ty]) -> &'a mut $name {
                    // the internal routine. works best when aa.len() <= bb.len().
                    fn mul_inner(ret: &mut [$ty; $n], aa: &[$ty], bb: &[$ty]) -> usize {
                        use crate::num::bignum::FullOps;

                        let mut retsz = 0;
                        for (i, &a) in aa.iter().enumerate() {
                            if a == 0 {
                                continue;
                            }
                            let mut sz = bb.len();
                            let mut carry = 0;
                            for (j, &b) in bb.iter().enumerate() {
                                let (c, v) = a.full_mul_add(b, ret[i + j], carry);
                                ret[i + j] = v;
                                carry = c;
                            }
                            if carry > 0 {
                                ret[i + sz] = carry;
                                sz += 1;
                            }
                            if retsz < i + sz {
                                retsz = i + sz;
                            }
                        }
                        retsz
                    }

                    let mut ret = [0; $n];
                    let retsz = if self.size < other.len() {
                        mul_inner(&mut ret, &self.digits(), other)
                    } else {
                        mul_inner(&mut ret, other, &self.digits())
                    };
                    self.base = ret;
                    self.size = retsz;
                    self
                }

                /// Divides itself by a digit-sized `other` and returns its own
                /// mutable reference *and* the remainder.
                pub fn div_rem_small(&mut self, other: $ty) -> (&mut $name, $ty) {
                    use crate::num::bignum::FullOps;

                    assert!(other > 0);

                    let sz = self.size;
                    let mut borrow = 0;
                    for a in self.base[..sz].iter_mut().rev() {
                        let (q, r) = (*a).full_div_rem(other, borrow);
                        *a = q;
                        borrow = r;
                    }
                    (self, borrow)
                }

                /// Divide self by another bignum, overwriting `q` with the quotient and `r` with the
                /// remainder.
                pub fn div_rem(&self, d: &$name, q: &mut $name, r: &mut $name) {
                    use crate::mem;

                    // Stupid slow base-2 long division taken from
                    // https://en.wikipedia.org/wiki/Division_algorithm
                    // FIXME use a greater base ($ty) for the long division.
                    assert!(!d.is_zero());
                    let digitbits = mem::size_of::<$ty>() * 8;
                    for digit in &mut q.base[..] {
                        *digit = 0;
                    }
                    for digit in &mut r.base[..] {
                        *digit = 0;
                    }
                    r.size = d.size;
                    q.size = 1;
                    let mut q_is_zero = true;
                    let end = self.bit_length();
                    for i in (0..end).rev() {
                        r.mul_pow2(1);
                        r.base[0] |= self.get_bit(i) as $ty;
                        if &*r >= d {
                            r.sub(d);
                            // Set bit `i` of q to 1.
                            let digit_idx = i / digitbits;
                            let bit_idx = i % digitbits;
                            if q_is_zero {
                                q.size = digit_idx + 1;
                                q_is_zero = false;
                            }
                            q.base[digit_idx] |= 1 << bit_idx;
                        }
                    }
                    debug_assert!(q.base[q.size..].iter().all(|&d| d == 0));
                    debug_assert!(r.base[r.size..].iter().all(|&d| d == 0));
                }
            }

            impl crate::cmp::PartialEq for $name {
                fn eq(&self, other: &$name) -> bool {
                    self.base[..] == other.base[..]
                }
            }

            impl crate::cmp::Eq for $name {}

            impl crate::cmp::PartialOrd for $name {
                fn partial_cmp(&self, other: &$name) -> crate::option::Option<crate::cmp::Ordering> {
                    crate::option::Option::Some(self.cmp(other))
                }
            }

            impl crate::cmp::Ord for $name {
                fn cmp(&self, other: &$name) -> crate::cmp::Ordering {
                    use crate::cmp::max;
                    let sz = max(self.size, other.size);
                    let lhs = self.base[..sz].iter().cloned().rev();
                    let rhs = other.base[..sz].iter().cloned().rev();
                    lhs.cmp(rhs)
                }
            }

            impl crate::clone::Clone for $name {
                fn clone(&self) -> Self {
                    Self { size: self.size, base: self.base }
                }
            }

            impl crate::fmt::Debug for $name {
                fn fmt(&self, f: &mut crate::fmt::Formatter<'_>) -> crate::fmt::Result {
                    use crate::mem;

                    let sz = if self.size < 1 { 1 } else { self.size };
                    let digitlen = mem::size_of::<$ty>() * 2;

                    write!(f, "{:#x}", self.base[sz - 1])?;
                    for &v in self.base[..sz - 1].iter().rev() {
                        write!(f, "_{:01$x}", v, digitlen)?;
                    }
                    crate::result::Result::Ok(())
                }
            }
        };
    }

    /// The digit type for `Big32x40`.
    pub type Digit32 = u32;

    define_bignum!(Big32x40: type=Digit32, n=40);

    // this one is used for testing only.
    #[doc(hidden)]
    pub mod tests {
        define_bignum!(Big8x3: type=u8, n=3);
    }