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

      1
      2
      3
      4
      5
      6
      7
      8
      9
     10
     11
     12
     13
     14
     15
     16
     17
     18
     19
     20
     21
     22
     23
     24
     25
     26
     27
     28
     29
     30
     31
     32
     33
     34
     35
     36
     37
     38
     39
     40
     41
     42
     43
     44
     45
     46
     47
     48
     49
     50
     51
     52
     53
     54
     55
     56
     57
     58
     59
     60
     61
     62
     63
     64
     65
     66
     67
     68
     69
     70
     71
     72
     73
     74
     75
     76
     77
     78
     79
     80
     81
     82
     83
     84
     85
     86
     87
     88
     89
     90
     91
     92
     93
     94
     95
     96
     97
     98
     99
    100
    101
    102
    103
    104
    105
    106
    107
    108
    109
    110
    111
    112
    113
    114
    115
    116
    117
    118
    119
    120
    121
    122
    123
    124
    125
    126
    127
    128
    129
    130
    131
    132
    133
    134
    135
    136
    137
    138
    139
    140
    141
    142
    143
    144
    145
    146
    147
    148
    149
    150
    151
    152
    153
    154
    155
    156
    157
    158
    159
    160
    161
    162
    163
    164
    165
    166
    167
    168
    169
    170
    171
    172
    173
    174
    175
    176
    177
    178
    179
    180
    181
    182
    183
    184
    185
    186
    187
    188
    189
    190
    191
    192
    193
    194
    195
    196
    197
    198
    199
    200
    201
    202
    203
    204
    205
    206
    207
    208
    209
    210
    211
    212
    213
    214
    215
    216
    217
    218
    219
    220
    221
    222
    223
    224
    225
    226
    227
    228
    229
    230
    231
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    261
    262
    263
    264
    265
    266
    267
    268
    269
    270
    271
    272
    273
    274
    275
    276
    277
    278
    279
    280
    281
    282
    283
    284
    285
    286
    287
    288
    289
    290
    291
    292
    293
    294
    295
    296
    297
    298
    299
    300
    301
    302
    303
    304
    305
    306
    307
    308
    309
    310
    311
    312
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    324
    325
    326
    327
    328
    329
    330
    331
    332
    333
    334
    335
    336
    337
    338
    339
    340
    341
    342
    343
    344
    345
    346
    347
    348
    349
    350
    351
    352
    353
    354
    355
    356
    357
    358
    359
    360
    361
    362
    363
    364
    365
    366
    367
    368
    369
    370
    371
    372
    373

    //! 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::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(d: &Decoded, buf: &mut [u8]) -> (/*#digits*/ usize, /*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] = 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.
            if let Some(c) = round_up(buf, i) {
                buf[i] = c;
                i += 1;
                k += 1;
            }
        }

        (i, k)
    }

    /// The exact and fixed mode implementation for Dragon.
    pub fn format_exact(d: &Decoded, buf: &mut [u8], limit: i16) -> (/*#digits*/ usize, /*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 = b'0';
                    }
                    return (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] = 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 && (len == 0 || buf[len - 1] & 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...
            if let Some(c) = round_up(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] = c;
                    len += 1;
                }
            }
        }

        (len, k)
    }