dragon.rs.html -- source
Description: Source to the Rust file `library/core/src/num/flt2dec/strategy/dragon.rs`.
Keywords: rust, rustlang, rust-lang

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    //! Almost direct (but slightly optimized) Rust translation of Figure 3 of "Printing
    //! Floating-Point Numbers Quickly and Accurately"[^1].
    //!
    //! [^1]: Burger, R. G. and Dybvig, R. K. 1996. Printing floating-point numbers
    //!   quickly and accurately. SIGPLAN Not. 31, 5 (May. 1996), 108-116.

    use crate::cmp::Ordering;
    use crate::mem::MaybeUninit;

    use crate::num::bignum::Big32x40 as Big;
    use crate::num::bignum::Digit32 as Digit;
    use crate::num::flt2dec::estimator::estimate_scaling_factor;
    use crate::num::flt2dec::{round_up, Decoded, MAX_SIG_DIGITS};

    static POW10: [Digit; 10] =
        [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000];
    static TWOPOW10: [Digit; 10] =
        [2, 20, 200, 2000, 20000, 200000, 2000000, 20000000, 200000000, 2000000000];

    // precalculated arrays of `Digit`s for 10^(2^n)
    static POW10TO16: [Digit; 2] = [0x6fc10000, 0x2386f2];
    static POW10TO32: [Digit; 4] = [0, 0x85acef81, 0x2d6d415b, 0x4ee];
    static POW10TO64: [Digit; 7] = [0, 0, 0xbf6a1f01, 0x6e38ed64, 0xdaa797ed, 0xe93ff9f4, 0x184f03];
    static POW10TO128: [Digit; 14] = [
        0, 0, 0, 0, 0x2e953e01, 0x3df9909, 0xf1538fd, 0x2374e42f, 0xd3cff5ec, 0xc404dc08, 0xbccdb0da,
        0xa6337f19, 0xe91f2603, 0x24e,
    ];
    static POW10TO256: [Digit; 27] = [
        0, 0, 0, 0, 0, 0, 0, 0, 0x982e7c01, 0xbed3875b, 0xd8d99f72, 0x12152f87, 0x6bde50c6, 0xcf4a6e70,
        0xd595d80f, 0x26b2716e, 0xadc666b0, 0x1d153624, 0x3c42d35a, 0x63ff540e, 0xcc5573c0, 0x65f9ef17,
        0x55bc28f2, 0x80dcc7f7, 0xf46eeddc, 0x5fdcefce, 0x553f7,
    ];

    #[doc(hidden)]
    pub fn mul_pow10(x: &mut Big, n: usize) -> &mut Big {
        debug_assert!(n < 512);
        if n & 7 != 0 {
            x.mul_small(POW10[n & 7]);
        }
        if n & 8 != 0 {
            x.mul_small(POW10[8]);
        }
        if n & 16 != 0 {
            x.mul_digits(&POW10TO16);
        }
        if n & 32 != 0 {
            x.mul_digits(&POW10TO32);
        }
        if n & 64 != 0 {
            x.mul_digits(&POW10TO64);
        }
        if n & 128 != 0 {
            x.mul_digits(&POW10TO128);
        }
        if n & 256 != 0 {
            x.mul_digits(&POW10TO256);
        }
        x
    }

    fn div_2pow10(x: &mut Big, mut n: usize) -> &mut Big {
        let largest = POW10.len() - 1;
        while n > largest {
            x.div_rem_small(POW10[largest]);
            n -= largest;
        }
        x.div_rem_small(TWOPOW10[n]);
        x
    }

    // only usable when `x < 16 * scale`; `scaleN` should be `scale.mul_small(N)`
    fn div_rem_upto_16<'a>(
        x: &'a mut Big,
        scale: &Big,
        scale2: &Big,
        scale4: &Big,
        scale8: &Big,
    ) -> (u8, &'a mut Big) {
        let mut d = 0;
        if *x >= *scale8 {
            x.sub(scale8);
            d += 8;
        }
        if *x >= *scale4 {
            x.sub(scale4);
            d += 4;
        }
        if *x >= *scale2 {
            x.sub(scale2);
            d += 2;
        }
        if *x >= *scale {
            x.sub(scale);
            d += 1;
        }
        debug_assert!(*x < *scale);
        (d, x)
    }

    /// The shortest mode implementation for Dragon.
    pub fn format_shortest<'a>(
        d: &Decoded,
        buf: &'a mut [MaybeUninit<u8>],
    ) -> (/*digits*/ &'a [u8], /*exp*/ i16) {
        // the number `v` to format is known to be:
        // - equal to `mant * 2^exp`;
        // - preceded by `(mant - 2 * minus) * 2^exp` in the original type; and
        // - followed by `(mant + 2 * plus) * 2^exp` in the original type.
        //
        // obviously, `minus` and `plus` cannot be zero. (for infinities, we use out-of-range values.)
        // also we assume that at least one digit is generated, i.e., `mant` cannot be zero too.
        //
        // this also means that any number between `low = (mant - minus) * 2^exp` and
        // `high = (mant + plus) * 2^exp` will map to this exact floating point number,
        // with bounds included when the original mantissa was even (i.e., `!mant_was_odd`).

        assert!(d.mant > 0);
        assert!(d.minus > 0);
        assert!(d.plus > 0);
        assert!(d.mant.checked_add(d.plus).is_some());
        assert!(d.mant.checked_sub(d.minus).is_some());
        assert!(buf.len() >= MAX_SIG_DIGITS);

        // `a.cmp(&b) < rounding` is `if d.inclusive {a <= b} else {a < b}`
        let rounding = if d.inclusive { Ordering::Greater } else { Ordering::Equal };

        // estimate `k_0` from original inputs satisfying `10^(k_0-1) < high <= 10^(k_0+1)`.
        // the tight bound `k` satisfying `10^(k-1) < high <= 10^k` is calculated later.
        let mut k = estimate_scaling_factor(d.mant + d.plus, d.exp);

        // convert `{mant, plus, minus} * 2^exp` into the fractional form so that:
        // - `v = mant / scale`
        // - `low = (mant - minus) / scale`
        // - `high = (mant + plus) / scale`
        let mut mant = Big::from_u64(d.mant);
        let mut minus = Big::from_u64(d.minus);
        let mut plus = Big::from_u64(d.plus);
        let mut scale = Big::from_small(1);
        if d.exp < 0 {
            scale.mul_pow2(-d.exp as usize);
        } else {
            mant.mul_pow2(d.exp as usize);
            minus.mul_pow2(d.exp as usize);
            plus.mul_pow2(d.exp as usize);
        }

        // divide `mant` by `10^k`. now `scale / 10 < mant + plus <= scale * 10`.
        if k >= 0 {
            mul_pow10(&mut scale, k as usize);
        } else {
            mul_pow10(&mut mant, -k as usize);
            mul_pow10(&mut minus, -k as usize);
            mul_pow10(&mut plus, -k as usize);
        }

        // fixup when `mant + plus > scale` (or `>=`).
        // we are not actually modifying `scale`, since we can skip the initial multiplication instead.
        // now `scale < mant + plus <= scale * 10` and we are ready to generate digits.
        //
        // note that `d[0]` *can* be zero, when `scale - plus < mant < scale`.
        // in this case rounding-up condition (`up` below) will be triggered immediately.
        if scale.cmp(mant.clone().add(&plus)) < rounding {
            // equivalent to scaling `scale` by 10
            k += 1;
        } else {
            mant.mul_small(10);
            minus.mul_small(10);
            plus.mul_small(10);
        }

        // cache `(2, 4, 8) * scale` for digit generation.
        let mut scale2 = scale.clone();
        scale2.mul_pow2(1);
        let mut scale4 = scale.clone();
        scale4.mul_pow2(2);
        let mut scale8 = scale.clone();
        scale8.mul_pow2(3);

        let mut down;
        let mut up;
        let mut i = 0;
        loop {
            // invariants, where `d[0..n-1]` are digits generated so far:
            // - `v = mant / scale * 10^(k-n-1) + d[0..n-1] * 10^(k-n)`
            // - `v - low = minus / scale * 10^(k-n-1)`
            // - `high - v = plus / scale * 10^(k-n-1)`
            // - `(mant + plus) / scale <= 10` (thus `mant / scale < 10`)
            // where `d[i..j]` is a shorthand for `d[i] * 10^(j-i) + ... + d[j-1] * 10 + d[j]`.

            // generate one digit: `d[n] = floor(mant / scale) < 10`.
            let (d, _) = div_rem_upto_16(&mut mant, &scale, &scale2, &scale4, &scale8);
            debug_assert!(d < 10);
            buf[i] = MaybeUninit::new(b'0' + d);
            i += 1;

            // this is a simplified description of the modified Dragon algorithm.
            // many intermediate derivations and completeness arguments are omitted for convenience.
            //
            // start with modified invariants, as we've updated `n`:
            // - `v = mant / scale * 10^(k-n) + d[0..n-1] * 10^(k-n)`
            // - `v - low = minus / scale * 10^(k-n)`
            // - `high - v = plus / scale * 10^(k-n)`
            //
            // assume that `d[0..n-1]` is the shortest representation between `low` and `high`,
            // i.e., `d[0..n-1]` satisfies both of the following but `d[0..n-2]` doesn't:
            // - `low < d[0..n-1] * 10^(k-n) < high` (bijectivity: digits round to `v`); and
            // - `abs(v / 10^(k-n) - d[0..n-1]) <= 1/2` (the last digit is correct).
            //
            // the second condition simplifies to `2 * mant <= scale`.
            // solving invariants in terms of `mant`, `low` and `high` yields
            // a simpler version of the first condition: `-plus < mant < minus`.
            // since `-plus < 0 <= mant`, we have the correct shortest representation
            // when `mant < minus` and `2 * mant <= scale`.
            // (the former becomes `mant <= minus` when the original mantissa is even.)
            //
            // when the second doesn't hold (`2 * mant > scale`), we need to increase the last digit.
            // this is enough for restoring that condition: we already know that
            // the digit generation guarantees `0 <= v / 10^(k-n) - d[0..n-1] < 1`.
            // in this case, the first condition becomes `-plus < mant - scale < minus`.
            // since `mant < scale` after the generation, we have `scale < mant + plus`.
            // (again, this becomes `scale <= mant + plus` when the original mantissa is even.)
            //
            // in short:
            // - stop and round `down` (keep digits as is) when `mant < minus` (or `<=`).
            // - stop and round `up` (increase the last digit) when `scale < mant + plus` (or `<=`).
            // - keep generating otherwise.
            down = mant.cmp(&minus) < rounding;
            up = scale.cmp(mant.clone().add(&plus)) < rounding;
            if down || up {
                break;
            } // we have the shortest representation, proceed to the rounding

            // restore the invariants.
            // this makes the algorithm always terminating: `minus` and `plus` always increases,
            // but `mant` is clipped modulo `scale` and `scale` is fixed.
            mant.mul_small(10);
            minus.mul_small(10);
            plus.mul_small(10);
        }

        // rounding up happens when
        // i) only the rounding-up condition was triggered, or
        // ii) both conditions were triggered and tie breaking prefers rounding up.
        if up && (!down || *mant.mul_pow2(1) >= scale) {
            // if rounding up changes the length, the exponent should also change.
            // it seems that this condition is very hard to satisfy (possibly impossible),
            // but we are just being safe and consistent here.
            // SAFETY: we initialized that memory above.
            if let Some(c) = round_up(unsafe { MaybeUninit::slice_assume_init_mut(&mut buf[..i]) }) {
                buf[i] = MaybeUninit::new(c);
                i += 1;
                k += 1;
            }
        }

        // SAFETY: we initialized that memory above.
        (unsafe { MaybeUninit::slice_assume_init_ref(&buf[..i]) }, k)
    }

    /// The exact and fixed mode implementation for Dragon.
    pub fn format_exact<'a>(
        d: &Decoded,
        buf: &'a mut [MaybeUninit<u8>],
        limit: i16,
    ) -> (/*digits*/ &'a [u8], /*exp*/ i16) {
        assert!(d.mant > 0);
        assert!(d.minus > 0);
        assert!(d.plus > 0);
        assert!(d.mant.checked_add(d.plus).is_some());
        assert!(d.mant.checked_sub(d.minus).is_some());

        // estimate `k_0` from original inputs satisfying `10^(k_0-1) < v <= 10^(k_0+1)`.
        let mut k = estimate_scaling_factor(d.mant, d.exp);

        // `v = mant / scale`.
        let mut mant = Big::from_u64(d.mant);
        let mut scale = Big::from_small(1);
        if d.exp < 0 {
            scale.mul_pow2(-d.exp as usize);
        } else {
            mant.mul_pow2(d.exp as usize);
        }

        // divide `mant` by `10^k`. now `scale / 10 < mant <= scale * 10`.
        if k >= 0 {
            mul_pow10(&mut scale, k as usize);
        } else {
            mul_pow10(&mut mant, -k as usize);
        }

        // fixup when `mant + plus >= scale`, where `plus / scale = 10^-buf.len() / 2`.
        // in order to keep the fixed-size bignum, we actually use `mant + floor(plus) >= scale`.
        // we are not actually modifying `scale`, since we can skip the initial multiplication instead.
        // again with the shortest algorithm, `d[0]` can be zero but will be eventually rounded up.
        if *div_2pow10(&mut scale.clone(), buf.len()).add(&mant) >= scale {
            // equivalent to scaling `scale` by 10
            k += 1;
        } else {
            mant.mul_small(10);
        }

        // if we are working with the last-digit limitation, we need to shorten the buffer
        // before the actual rendering in order to avoid double rounding.
        // note that we have to enlarge the buffer again when rounding up happens!
        let mut len = if k < limit {
            // oops, we cannot even produce *one* digit.
            // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
            // we return an empty buffer, with an exception of the later rounding-up case
            // which occurs when `k == limit` and has to produce exactly one digit.
            0
        } else if ((k as i32 - limit as i32) as usize) < buf.len() {
            (k - limit) as usize
        } else {
            buf.len()
        };

        if len > 0 {
            // cache `(2, 4, 8) * scale` for digit generation.
            // (this can be expensive, so do not calculate them when the buffer is empty.)
            let mut scale2 = scale.clone();
            scale2.mul_pow2(1);
            let mut scale4 = scale.clone();
            scale4.mul_pow2(2);
            let mut scale8 = scale.clone();
            scale8.mul_pow2(3);

            for i in 0..len {
                if mant.is_zero() {
                    // following digits are all zeroes, we stop here
                    // do *not* try to perform rounding! rather, fill remaining digits.
                    for c in &mut buf[i..len] {
                        *c = MaybeUninit::new(b'0');
                    }
                    // SAFETY: we initialized that memory above.
                    return (unsafe { MaybeUninit::slice_assume_init_ref(&buf[..len]) }, k);
                }

                let mut d = 0;
                if mant >= scale8 {
                    mant.sub(&scale8);
                    d += 8;
                }
                if mant >= scale4 {
                    mant.sub(&scale4);
                    d += 4;
                }
                if mant >= scale2 {
                    mant.sub(&scale2);
                    d += 2;
                }
                if mant >= scale {
                    mant.sub(&scale);
                    d += 1;
                }
                debug_assert!(mant < scale);
                debug_assert!(d < 10);
                buf[i] = MaybeUninit::new(b'0' + d);
                mant.mul_small(10);
            }
        }

        // rounding up if we stop in the middle of digits
        // if the following digits are exactly 5000..., check the prior digit and try to
        // round to even (i.e., avoid rounding up when the prior digit is even).
        let order = mant.cmp(scale.mul_small(5));
        if order == Ordering::Greater
            || (order == Ordering::Equal
                // SAFETY: `buf[len-1]` is initialized.
                && (len == 0 || unsafe { buf[len - 1].assume_init() } & 1 == 1))
        {
            // if rounding up changes the length, the exponent should also change.
            // but we've been requested a fixed number of digits, so do not alter the buffer...
            // SAFETY: we initialized that memory above.
            if let Some(c) = round_up(unsafe { MaybeUninit::slice_assume_init_mut(&mut buf[..len]) }) {
                // ...unless we've been requested the fixed precision instead.
                // we also need to check that, if the original buffer was empty,
                // the additional digit can only be added when `k == limit` (edge case).
                k += 1;
                if k > limit && len < buf.len() {
                    buf[len] = MaybeUninit::new(c);
                    len += 1;
                }
            }
        }

        // SAFETY: we initialized that memory above.
        (unsafe { MaybeUninit::slice_assume_init_ref(&buf[..len]) }, k)
    }