      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
    374
    375
    376
    377
    378
    379
    380
    381
    382
    383
    384
    385
    386
    387
    388
    389
    390
    391
    392
    393
    394
    395
    396
    397
    398
    399
    400
    401
    402
    403
    404
    405
    406
    407
    408
    409
    410
    411
    412
    413
    414
    415
    416
    417
    418
    419
    420
    421
    422
    423
    424
    425
    426
    427
    428
    429
    430
    431
    432
    433
    434
    435
    436
    437
    438
    439
    440
    441
    442
    443
    444
    445
    446
    447
    448
    449
    450
    451
    452
    453
    454
    455
    456
    457
    458
    459
    460
    461
    462
    463
    464
    465
    466
    467
    468
    469
    470
    471
    472
    473
    474
    475
    476
    477
    478
    479
    480
    481
    482
    483
    484
    485
    486
    487
    488
    489
    490
    491
    492
    493
    494
    495
    496
    497
    498
    499
    500
    501
    502
    503
    504
    505
    506
    507
    508
    509
    510
    511
    512
    513
    514
    515
    516
    517
    518
    519
    520
    521
    522
    523
    524
    525
    526
    527
    528
    529
    530
    531
    532
    533
    534
    535
    536
    537
    538
    539
    540
    541
    542
    543
    544
    545
    546
    547
    548
    549
    550
    551
    552
    553
    554
    555
    556
    557
    558
    559
    560
    561
    562
    563
    564
    565
    566
    567
    568
    569
    570
    571
    572
    573
    574
    575
    576
    577
    578
    579
    580
    581
    582
    583
    584
    585
    586
    587
    588
    589
    590
    591
    592
    593
    594
    595
    596
    597
    598
    599
    600
    601
    602
    603
    604
    605
    606
    607
    608
    609
    610
    611
    612
    613
    614
    615
    616
    617
    618
    619
    620
    621
    622
    623
    624
    625
    626
    627
    628
    629
    630
    631
    632
    633
    634
    635
    636
    637
    638
    639
    640
    641
    642
    643
    644
    645
    646
    647
    648
    649
    650
    651
    652
    653
    654
    655
    656
    657
    658
    659
    660
    661
    662
    663
    664
    665
    666
    667
    668
    669
    670
    671
    672
    673
    674
    675
    676
    677
    678
    679
    680
    681
    682
    683
    684
    685
    686
    687
    688
    689
    690
    691
    692
    693
    694
    695
    696
    697
    698
    699
    700
    701
    702
    703
    704
    705
    706
    707
    708
    709
    710
    711
    712
    713
    714
    715
    716
    717
    718
    719
    720
    721
    722
    723
    724
    725
    726
    727
    728
    729
    730
    731
    732
    733
    734
    735
    736
    737
    738
    739
    740
    741
    742
    743
    744
    745
    746
    747
    748
    749
    750
    751
    752
    753
    754
    755
    756
    757
    758
    759
    760
    761
    762
    763
    764
    765
    766
    767
    768
    769
    770
    771
    772
    773
    774
    775
    776
    777
    778
    779
    780
    781
    782
    783
    784
    785
    786
    787
    788
    789
    790
    791
    792
    793

    /*!

    Floating-point number to decimal conversion routines.

    # Problem statement

    We are given the floating-point number `v = f * 2^e` with an integer `f`,
    and its bounds `minus` and `plus` such that any number between `v - minus` and
    `v + plus` will be rounded to `v`. For the simplicity we assume that
    this range is exclusive. Then we would like to get the unique decimal
    representation `V = 0.d[0..n-1] * 10^k` such that:

    - `d[0]` is non-zero.

    - It's correctly rounded when parsed back: `v - minus < V < v + plus`.
      Furthermore it is shortest such one, i.e., there is no representation
      with less than `n` digits that is correctly rounded.

    - It's closest to the original value: `abs(V - v) <= 10^(k-n) / 2`. Note that
      there might be two representations satisfying this uniqueness requirement,
      in which case some tie-breaking mechanism is used.

    We will call this mode of operation as to the *shortest* mode. This mode is used
    when there is no additional constraint, and can be thought as a "natural" mode
    as it matches the ordinary intuition (it at least prints `0.1f32` as "0.1").

    We have two more modes of operation closely related to each other. In these modes
    we are given either the number of significant digits `n` or the last-digit
    limitation `limit` (which determines the actual `n`), and we would like to get
    the representation `V = 0.d[0..n-1] * 10^k` such that:

    - `d[0]` is non-zero, unless `n` was zero in which case only `k` is returned.

    - It's closest to the original value: `abs(V - v) <= 10^(k-n) / 2`. Again,
      there might be some tie-breaking mechanism.

    When `limit` is given but not `n`, we set `n` such that `k - n = limit`
    so that the last digit `d[n-1]` is scaled by `10^(k-n) = 10^limit`.
    If such `n` is negative, we clip it to zero so that we will only get `k`.
    We are also limited by the supplied buffer. This limitation is used to print
    the number up to given number of fractional digits without knowing
    the correct `k` beforehand.

    We will call the mode of operation requiring `n` as to the *exact* mode,
    and one requiring `limit` as to the *fixed* mode. The exact mode is a subset of
    the fixed mode: the sufficiently large last-digit limitation will eventually fill
    the supplied buffer and let the algorithm to return.

    # Implementation overview

    It is easy to get the floating point printing correct but slow (Russ Cox has
    [demonstrated](http://research.swtch.com/ftoa) how it's easy), or incorrect but
    fast (naïve division and modulo). But it is surprisingly hard to print
    floating point numbers correctly *and* efficiently.

    There are two classes of algorithms widely known to be correct.

    - The "Dragon" family of algorithm is first described by Guy L. Steele Jr. and
      Jon L. White. They rely on the fixed-size big integer for their correctness.
      A slight improvement was found later, which is posthumously described by
      Robert G. Burger and R. Kent Dybvig. David Gay's `dtoa.c` routine is
      a popular implementation of this strategy.

    - The "Grisu" family of algorithm is first described by Florian Loitsch.
      They use very cheap integer-only procedure to determine the close-to-correct
      representation which is at least guaranteed to be shortest. The variant,
      Grisu3, actively detects if the resulting representation is incorrect.

    We implement both algorithms with necessary tweaks to suit our requirements.
    In particular, published literatures are short of the actual implementation
    difficulties like how to avoid arithmetic overflows. Each implementation,
    available in `strategy::dragon` and `strategy::grisu` respectively,
    extensively describes all necessary justifications and many proofs for them.
    (It is still difficult to follow though. You have been warned.)

    Both implementations expose two public functions:

    - `format_shortest(decoded, buf)`, which always needs at least
      `MAX_SIG_DIGITS` digits of buffer. Implements the shortest mode.

    - `format_exact(decoded, buf, limit)`, which accepts as small as
      one digit of buffer. Implements exact and fixed modes.

    They try to fill the `u8` buffer with digits and returns the number of digits
    written and the exponent `k`. They are total for all finite `f32` and `f64`
    inputs (Grisu internally falls back to Dragon if necessary).

    The rendered digits are formatted into the actual string form with
    four functions:

    - `to_shortest_str` prints the shortest representation, which can be padded by
      zeroes to make *at least* given number of fractional digits.

    - `to_shortest_exp_str` prints the shortest representation, which can be
      padded by zeroes when its exponent is in the specified ranges,
      or can be printed in the exponential form such as `1.23e45`.

    - `to_exact_exp_str` prints the exact representation with given number of
      digits in the exponential form.

    - `to_exact_fixed_str` prints the fixed representation with *exactly*
      given number of fractional digits.

    They all return a slice of preallocated `Part` array, which corresponds to
    the individual part of strings: a fixed string, a part of rendered digits,
    a number of zeroes or a small (`u16`) number. The caller is expected to
    provide a large enough buffer and `Part` array, and to assemble the final
    string from resulting `Part`s itself.

    All algorithms and formatting functions are accompanied by extensive tests
    in `coretests::num::flt2dec` module. It also shows how to use individual
    functions.

    */

    // while this is extensively documented, this is in principle private which is
    // only made public for testing. do not expose us.
    #![doc(hidden)]
    #![unstable(
        feature = "flt2dec",
        reason = "internal routines only exposed for testing",
        issue = "none"
    )]

    pub use self::decoder::{decode, DecodableFloat, Decoded, FullDecoded};

    use crate::mem::MaybeUninit;

    pub mod decoder;
    pub mod estimator;

    /// Digit-generation algorithms.
    pub mod strategy {
        pub mod dragon;
        pub mod grisu;
    }

    /// The minimum size of buffer necessary for the shortest mode.
    ///
    /// It is a bit non-trivial to derive, but this is one plus the maximal number of
    /// significant decimal digits from formatting algorithms with the shortest result.
    /// The exact formula is `ceil(# bits in mantissa * log_10 2 + 1)`.
    pub const MAX_SIG_DIGITS: usize = 17;

    /// When `d` contains decimal digits, increase the last digit and propagate carry.
    /// Returns a next digit when it causes the length to change.
    #[doc(hidden)]
    pub fn round_up(d: &mut [u8]) -> Option<u8> {
        match d.iter().rposition(|&c| c != b'9') {
            Some(i) => {
                // d[i+1..n] is all nines
                d[i] += 1;
                for j in i + 1..d.len() {
                    d[j] = b'0';
                }
                None
            }
            None if d.len() > 0 => {
                // 999..999 rounds to 1000..000 with an increased exponent
                d[0] = b'1';
                for j in 1..d.len() {
                    d[j] = b'0';
                }
                Some(b'0')
            }
            None => {
                // an empty buffer rounds up (a bit strange but reasonable)
                Some(b'1')
            }
        }
    }

    /// Formatted parts.
    #[derive(Copy, Clone, PartialEq, Eq, Debug)]
    pub enum Part<'a> {
        /// Given number of zero digits.
        Zero(usize),
        /// A literal number up to 5 digits.
        Num(u16),
        /// A verbatim copy of given bytes.
        Copy(&'a [u8]),
    }

    impl<'a> Part<'a> {
        /// Returns the exact byte length of given part.
        pub fn len(&self) -> usize {
            match *self {
                Part::Zero(nzeroes) => nzeroes,
                Part::Num(v) => {
                    if v < 1_000 {
                        if v < 10 {
                            1
                        } else if v < 100 {
                            2
                        } else {
                            3
                        }
                    } else {
                        if v < 10_000 { 4 } else { 5 }
                    }
                }
                Part::Copy(buf) => buf.len(),
            }
        }

        /// Writes a part into the supplied buffer.
        /// Returns the number of written bytes, or `None` if the buffer is not enough.
        /// (It may still leave partially written bytes in the buffer; do not rely on that.)
        pub fn write(&self, out: &mut [u8]) -> Option<usize> {
            let len = self.len();
            if out.len() >= len {
                match *self {
                    Part::Zero(nzeroes) => {
                        for c in &mut out[..nzeroes] {
                            *c = b'0';
                        }
                    }
                    Part::Num(mut v) => {
                        for c in out[..len].iter_mut().rev() {
                            *c = b'0' + (v % 10) as u8;
                            v /= 10;
                        }
                    }
                    Part::Copy(buf) => {
                        out[..buf.len()].copy_from_slice(buf);
                    }
                }
                Some(len)
            } else {
                None
            }
        }
    }

    /// Formatted result containing one or more parts.
    /// This can be written to the byte buffer or converted to the allocated string.
    #[allow(missing_debug_implementations)]
    #[derive(Clone)]
    pub struct Formatted<'a> {
        /// A byte slice representing a sign, either `""`, `"-"` or `"+"`.
        pub sign: &'static str,
        /// Formatted parts to be rendered after a sign and optional zero padding.
        pub parts: &'a [Part<'a>],
    }

    impl<'a> Formatted<'a> {
        /// Returns the exact byte length of combined formatted result.
        pub fn len(&self) -> usize {
            let mut len = self.sign.len();
            for part in self.parts {
                len += part.len();
            }
            len
        }

        /// Writes all formatted parts into the supplied buffer.
        /// Returns the number of written bytes, or `None` if the buffer is not enough.
        /// (It may still leave partially written bytes in the buffer; do not rely on that.)
        pub fn write(&self, out: &mut [u8]) -> Option<usize> {
            if out.len() < self.sign.len() {
                return None;
            }
            out[..self.sign.len()].copy_from_slice(self.sign.as_bytes());

            let mut written = self.sign.len();
            for part in self.parts {
                let len = part.write(&mut out[written..])?;
                written += len;
            }
            Some(written)
        }
    }

    /// Formats given decimal digits `0.<...buf...> * 10^exp` into the decimal form
    /// with at least given number of fractional digits. The result is stored to
    /// the supplied parts array and a slice of written parts is returned.
    ///
    /// `frac_digits` can be less than the number of actual fractional digits in `buf`;
    /// it will be ignored and full digits will be printed. It is only used to print
    /// additional zeroes after rendered digits. Thus `frac_digits` of 0 means that
    /// it will only print given digits and nothing else.
    fn digits_to_dec_str<'a>(
        buf: &'a [u8],
        exp: i16,
        frac_digits: usize,
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> &'a [Part<'a>] {
        assert!(!buf.is_empty());
        assert!(buf[0] > b'0');
        assert!(parts.len() >= 4);

        // if there is the restriction on the last digit position, `buf` is assumed to be
        // left-padded with the virtual zeroes. the number of virtual zeroes, `nzeroes`,
        // equals to `max(0, exp + frac_digits - buf.len())`, so that the position of
        // the last digit `exp - buf.len() - nzeroes` is no more than `-frac_digits`:
        //
        //                       |<-virtual->|
        //       |<---- buf ---->|  zeroes   |     exp
        //    0. 1 2 3 4 5 6 7 8 9 _ _ _ _ _ _ x 10
        //    |                  |           |
        // 10^exp    10^(exp-buf.len())   10^(exp-buf.len()-nzeroes)
        //
        // `nzeroes` is individually calculated for each case in order to avoid overflow.

        if exp <= 0 {
            // the decimal point is before rendered digits: [0.][000...000][1234][____]
            let minus_exp = -(exp as i32) as usize;
            parts[0] = MaybeUninit::new(Part::Copy(b"0."));
            parts[1] = MaybeUninit::new(Part::Zero(minus_exp));
            parts[2] = MaybeUninit::new(Part::Copy(buf));
            if frac_digits > buf.len() && frac_digits - buf.len() > minus_exp {
                parts[3] = MaybeUninit::new(Part::Zero((frac_digits - buf.len()) - minus_exp));
                // SAFETY: we just initialized the elements `..4`.
                unsafe { MaybeUninit::slice_assume_init_ref(&parts[..4]) }
            } else {
                // SAFETY: we just initialized the elements `..3`.
                unsafe { MaybeUninit::slice_assume_init_ref(&parts[..3]) }
            }
        } else {
            let exp = exp as usize;
            if exp < buf.len() {
                // the decimal point is inside rendered digits: [12][.][34][____]
                parts[0] = MaybeUninit::new(Part::Copy(&buf[..exp]));
                parts[1] = MaybeUninit::new(Part::Copy(b"."));
                parts[2] = MaybeUninit::new(Part::Copy(&buf[exp..]));
                if frac_digits > buf.len() - exp {
                    parts[3] = MaybeUninit::new(Part::Zero(frac_digits - (buf.len() - exp)));
                    // SAFETY: we just initialized the elements `..4`.
                    unsafe { MaybeUninit::slice_assume_init_ref(&parts[..4]) }
                } else {
                    // SAFETY: we just initialized the elements `..3`.
                    unsafe { MaybeUninit::slice_assume_init_ref(&parts[..3]) }
                }
            } else {
                // the decimal point is after rendered digits: [1234][____0000] or [1234][__][.][__].
                parts[0] = MaybeUninit::new(Part::Copy(buf));
                parts[1] = MaybeUninit::new(Part::Zero(exp - buf.len()));
                if frac_digits > 0 {
                    parts[2] = MaybeUninit::new(Part::Copy(b"."));
                    parts[3] = MaybeUninit::new(Part::Zero(frac_digits));
                    // SAFETY: we just initialized the elements `..4`.
                    unsafe { MaybeUninit::slice_assume_init_ref(&parts[..4]) }
                } else {
                    // SAFETY: we just initialized the elements `..2`.
                    unsafe { MaybeUninit::slice_assume_init_ref(&parts[..2]) }
                }
            }
        }
    }

    /// Formats the given decimal digits `0.<...buf...> * 10^exp` into the exponential
    /// form with at least the given number of significant digits. When `upper` is `true`,
    /// the exponent will be prefixed by `E`; otherwise that's `e`. The result is
    /// stored to the supplied parts array and a slice of written parts is returned.
    ///
    /// `min_digits` can be less than the number of actual significant digits in `buf`;
    /// it will be ignored and full digits will be printed. It is only used to print
    /// additional zeroes after rendered digits. Thus, `min_digits == 0` means that
    /// it will only print the given digits and nothing else.
    fn digits_to_exp_str<'a>(
        buf: &'a [u8],
        exp: i16,
        min_ndigits: usize,
        upper: bool,
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> &'a [Part<'a>] {
        assert!(!buf.is_empty());
        assert!(buf[0] > b'0');
        assert!(parts.len() >= 6);

        let mut n = 0;

        parts[n] = MaybeUninit::new(Part::Copy(&buf[..1]));
        n += 1;

        if buf.len() > 1 || min_ndigits > 1 {
            parts[n] = MaybeUninit::new(Part::Copy(b"."));
            parts[n + 1] = MaybeUninit::new(Part::Copy(&buf[1..]));
            n += 2;
            if min_ndigits > buf.len() {
                parts[n] = MaybeUninit::new(Part::Zero(min_ndigits - buf.len()));
                n += 1;
            }
        }

        // 0.1234 x 10^exp = 1.234 x 10^(exp-1)
        let exp = exp as i32 - 1; // avoid underflow when exp is i16::MIN
        if exp < 0 {
            parts[n] = MaybeUninit::new(Part::Copy(if upper { b"E-" } else { b"e-" }));
            parts[n + 1] = MaybeUninit::new(Part::Num(-exp as u16));
        } else {
            parts[n] = MaybeUninit::new(Part::Copy(if upper { b"E" } else { b"e" }));
            parts[n + 1] = MaybeUninit::new(Part::Num(exp as u16));
        }
        // SAFETY: we just initialized the elements `..n + 2`.
        unsafe { MaybeUninit::slice_assume_init_ref(&parts[..n + 2]) }
    }

    /// Sign formatting options.
    #[derive(Copy, Clone, PartialEq, Eq, Debug)]
    pub enum Sign {
        /// Prints `-` only for the negative non-zero values.
        Minus, // -inf -1  0  0  1  inf nan
        /// Prints `-` only for any negative values (including the negative zero).
        MinusRaw, // -inf -1 -0  0  1  inf nan
        /// Prints `-` for the negative non-zero values, or `+` otherwise.
        MinusPlus, // -inf -1 +0 +0 +1 +inf nan
        /// Prints `-` for any negative values (including the negative zero), or `+` otherwise.
        MinusPlusRaw, // -inf -1 -0 +0 +1 +inf nan
    }

    /// Returns the static byte string corresponding to the sign to be formatted.
    /// It can be either `""`, `"+"` or `"-"`.
    fn determine_sign(sign: Sign, decoded: &FullDecoded, negative: bool) -> &'static str {
        match (*decoded, sign) {
            (FullDecoded::Nan, _) => "",
            (FullDecoded::Zero, Sign::Minus) => "",
            (FullDecoded::Zero, Sign::MinusRaw) => {
                if negative {
                    "-"
                } else {
                    ""
                }
            }
            (FullDecoded::Zero, Sign::MinusPlus) => "+",
            (FullDecoded::Zero, Sign::MinusPlusRaw) => {
                if negative {
                    "-"
                } else {
                    "+"
                }
            }
            (_, Sign::Minus | Sign::MinusRaw) => {
                if negative {
                    "-"
                } else {
                    ""
                }
            }
            (_, Sign::MinusPlus | Sign::MinusPlusRaw) => {
                if negative {
                    "-"
                } else {
                    "+"
                }
            }
        }
    }

    /// Formats the given floating point number into the decimal form with at least
    /// given number of fractional digits. The result is stored to the supplied parts
    /// array while utilizing given byte buffer as a scratch. `upper` is currently
    /// unused but left for the future decision to change the case of non-finite values,
    /// i.e., `inf` and `nan`. The first part to be rendered is always a `Part::Sign`
    /// (which can be an empty string if no sign is rendered).
    ///
    /// `format_shortest` should be the underlying digit-generation function.
    /// It should return the part of the buffer that it initialized.
    /// You probably would want `strategy::grisu::format_shortest` for this.
    ///
    /// `frac_digits` can be less than the number of actual fractional digits in `v`;
    /// it will be ignored and full digits will be printed. It is only used to print
    /// additional zeroes after rendered digits. Thus `frac_digits` of 0 means that
    /// it will only print given digits and nothing else.
    ///
    /// The byte buffer should be at least `MAX_SIG_DIGITS` bytes long.
    /// There should be at least 4 parts available, due to the worst case like
    /// `[+][0.][0000][2][0000]` with `frac_digits = 10`.
    pub fn to_shortest_str<'a, T, F>(
        mut format_shortest: F,
        v: T,
        sign: Sign,
        frac_digits: usize,
        buf: &'a mut [MaybeUninit<u8>],
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> Formatted<'a>
    where
        T: DecodableFloat,
        F: FnMut(&Decoded, &'a mut [MaybeUninit<u8>]) -> (&'a [u8], i16),
    {
        assert!(parts.len() >= 4);
        assert!(buf.len() >= MAX_SIG_DIGITS);

        let (negative, full_decoded) = decode(v);
        let sign = determine_sign(sign, &full_decoded, negative);
        match full_decoded {
            FullDecoded::Nan => {
                parts[0] = MaybeUninit::new(Part::Copy(b"NaN"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Infinite => {
                parts[0] = MaybeUninit::new(Part::Copy(b"inf"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Zero => {
                if frac_digits > 0 {
                    // [0.][0000]
                    parts[0] = MaybeUninit::new(Part::Copy(b"0."));
                    parts[1] = MaybeUninit::new(Part::Zero(frac_digits));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..2`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..2]) },
                    }
                } else {
                    parts[0] = MaybeUninit::new(Part::Copy(b"0"));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..1`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) },
                    }
                }
            }
            FullDecoded::Finite(ref decoded) => {
                let (buf, exp) = format_shortest(decoded, buf);
                Formatted { sign, parts: digits_to_dec_str(buf, exp, frac_digits, parts) }
            }
        }
    }

    /// Formats the given floating point number into the decimal form or
    /// the exponential form, depending on the resulting exponent. The result is
    /// stored to the supplied parts array while utilizing given byte buffer
    /// as a scratch. `upper` is used to determine the case of non-finite values
    /// (`inf` and `nan`) or the case of the exponent prefix (`e` or `E`).
    /// The first part to be rendered is always a `Part::Sign` (which can be
    /// an empty string if no sign is rendered).
    ///
    /// `format_shortest` should be the underlying digit-generation function.
    /// It should return the part of the buffer that it initialized.
    /// You probably would want `strategy::grisu::format_shortest` for this.
    ///
    /// The `dec_bounds` is a tuple `(lo, hi)` such that the number is formatted
    /// as decimal only when `10^lo <= V < 10^hi`. Note that this is the *apparent* `V`
    /// instead of the actual `v`! Thus any printed exponent in the exponential form
    /// cannot be in this range, avoiding any confusion.
    ///
    /// The byte buffer should be at least `MAX_SIG_DIGITS` bytes long.
    /// There should be at least 6 parts available, due to the worst case like
    /// `[+][1][.][2345][e][-][6]`.
    pub fn to_shortest_exp_str<'a, T, F>(
        mut format_shortest: F,
        v: T,
        sign: Sign,
        dec_bounds: (i16, i16),
        upper: bool,
        buf: &'a mut [MaybeUninit<u8>],
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> Formatted<'a>
    where
        T: DecodableFloat,
        F: FnMut(&Decoded, &'a mut [MaybeUninit<u8>]) -> (&'a [u8], i16),
    {
        assert!(parts.len() >= 6);
        assert!(buf.len() >= MAX_SIG_DIGITS);
        assert!(dec_bounds.0 <= dec_bounds.1);

        let (negative, full_decoded) = decode(v);
        let sign = determine_sign(sign, &full_decoded, negative);
        match full_decoded {
            FullDecoded::Nan => {
                parts[0] = MaybeUninit::new(Part::Copy(b"NaN"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Infinite => {
                parts[0] = MaybeUninit::new(Part::Copy(b"inf"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Zero => {
                parts[0] = if dec_bounds.0 <= 0 && 0 < dec_bounds.1 {
                    MaybeUninit::new(Part::Copy(b"0"))
                } else {
                    MaybeUninit::new(Part::Copy(if upper { b"0E0" } else { b"0e0" }))
                };
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Finite(ref decoded) => {
                let (buf, exp) = format_shortest(decoded, buf);
                let vis_exp = exp as i32 - 1;
                let parts = if dec_bounds.0 as i32 <= vis_exp && vis_exp < dec_bounds.1 as i32 {
                    digits_to_dec_str(buf, exp, 0, parts)
                } else {
                    digits_to_exp_str(buf, exp, 0, upper, parts)
                };
                Formatted { sign, parts }
            }
        }
    }

    /// Returns a rather crude approximation (upper bound) for the maximum buffer size
    /// calculated from the given decoded exponent.
    ///
    /// The exact limit is:
    ///
    /// - when `exp < 0`, the maximum length is `ceil(log_10 (5^-exp * (2^64 - 1)))`.
    /// - when `exp >= 0`, the maximum length is `ceil(log_10 (2^exp * (2^64 - 1)))`.
    ///
    /// `ceil(log_10 (x^exp * (2^64 - 1)))` is less than `ceil(log_10 (2^64 - 1)) +
    /// ceil(exp * log_10 x)`, which is in turn less than `20 + (1 + exp * log_10 x)`.
    /// We use the facts that `log_10 2 < 5/16` and `log_10 5 < 12/16`, which is
    /// enough for our purposes.
    ///
    /// Why do we need this? `format_exact` functions will fill the entire buffer
    /// unless limited by the last digit restriction, but it is possible that
    /// the number of digits requested is ridiculously large (say, 30,000 digits).
    /// The vast majority of buffer will be filled with zeroes, so we don't want to
    /// allocate all the buffer beforehand. Consequently, for any given arguments,
    /// 826 bytes of buffer should be sufficient for `f64`. Compare this with
    /// the actual number for the worst case: 770 bytes (when `exp = -1074`).
    fn estimate_max_buf_len(exp: i16) -> usize {
        21 + ((if exp < 0 { -12 } else { 5 } * exp as i32) as usize >> 4)
    }

    /// Formats given floating point number into the exponential form with
    /// exactly given number of significant digits. The result is stored to
    /// the supplied parts array while utilizing given byte buffer as a scratch.
    /// `upper` is used to determine the case of the exponent prefix (`e` or `E`).
    /// The first part to be rendered is always a `Part::Sign` (which can be
    /// an empty string if no sign is rendered).
    ///
    /// `format_exact` should be the underlying digit-generation function.
    /// It should return the part of the buffer that it initialized.
    /// You probably would want `strategy::grisu::format_exact` for this.
    ///
    /// The byte buffer should be at least `ndigits` bytes long unless `ndigits` is
    /// so large that only the fixed number of digits will be ever written.
    /// (The tipping point for `f64` is about 800, so 1000 bytes should be enough.)
    /// There should be at least 6 parts available, due to the worst case like
    /// `[+][1][.][2345][e][-][6]`.
    pub fn to_exact_exp_str<'a, T, F>(
        mut format_exact: F,
        v: T,
        sign: Sign,
        ndigits: usize,
        upper: bool,
        buf: &'a mut [MaybeUninit<u8>],
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> Formatted<'a>
    where
        T: DecodableFloat,
        F: FnMut(&Decoded, &'a mut [MaybeUninit<u8>], i16) -> (&'a [u8], i16),
    {
        assert!(parts.len() >= 6);
        assert!(ndigits > 0);

        let (negative, full_decoded) = decode(v);
        let sign = determine_sign(sign, &full_decoded, negative);
        match full_decoded {
            FullDecoded::Nan => {
                parts[0] = MaybeUninit::new(Part::Copy(b"NaN"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Infinite => {
                parts[0] = MaybeUninit::new(Part::Copy(b"inf"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Zero => {
                if ndigits > 1 {
                    // [0.][0000][e0]
                    parts[0] = MaybeUninit::new(Part::Copy(b"0."));
                    parts[1] = MaybeUninit::new(Part::Zero(ndigits - 1));
                    parts[2] = MaybeUninit::new(Part::Copy(if upper { b"E0" } else { b"e0" }));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..3`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..3]) },
                    }
                } else {
                    parts[0] = MaybeUninit::new(Part::Copy(if upper { b"0E0" } else { b"0e0" }));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..1`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) },
                    }
                }
            }
            FullDecoded::Finite(ref decoded) => {
                let maxlen = estimate_max_buf_len(decoded.exp);
                assert!(buf.len() >= ndigits || buf.len() >= maxlen);

                let trunc = if ndigits < maxlen { ndigits } else { maxlen };
                let (buf, exp) = format_exact(decoded, &mut buf[..trunc], i16::MIN);
                Formatted { sign, parts: digits_to_exp_str(buf, exp, ndigits, upper, parts) }
            }
        }
    }

    /// Formats given floating point number into the decimal form with exactly
    /// given number of fractional digits. The result is stored to the supplied parts
    /// array while utilizing given byte buffer as a scratch. `upper` is currently
    /// unused but left for the future decision to change the case of non-finite values,
    /// i.e., `inf` and `nan`. The first part to be rendered is always a `Part::Sign`
    /// (which can be an empty string if no sign is rendered).
    ///
    /// `format_exact` should be the underlying digit-generation function.
    /// It should return the part of the buffer that it initialized.
    /// You probably would want `strategy::grisu::format_exact` for this.
    ///
    /// The byte buffer should be enough for the output unless `frac_digits` is
    /// so large that only the fixed number of digits will be ever written.
    /// (The tipping point for `f64` is about 800, and 1000 bytes should be enough.)
    /// There should be at least 4 parts available, due to the worst case like
    /// `[+][0.][0000][2][0000]` with `frac_digits = 10`.
    pub fn to_exact_fixed_str<'a, T, F>(
        mut format_exact: F,
        v: T,
        sign: Sign,
        frac_digits: usize,
        buf: &'a mut [MaybeUninit<u8>],
        parts: &'a mut [MaybeUninit<Part<'a>>],
    ) -> Formatted<'a>
    where
        T: DecodableFloat,
        F: FnMut(&Decoded, &'a mut [MaybeUninit<u8>], i16) -> (&'a [u8], i16),
    {
        assert!(parts.len() >= 4);

        let (negative, full_decoded) = decode(v);
        let sign = determine_sign(sign, &full_decoded, negative);
        match full_decoded {
            FullDecoded::Nan => {
                parts[0] = MaybeUninit::new(Part::Copy(b"NaN"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Infinite => {
                parts[0] = MaybeUninit::new(Part::Copy(b"inf"));
                // SAFETY: we just initialized the elements `..1`.
                Formatted { sign, parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) } }
            }
            FullDecoded::Zero => {
                if frac_digits > 0 {
                    // [0.][0000]
                    parts[0] = MaybeUninit::new(Part::Copy(b"0."));
                    parts[1] = MaybeUninit::new(Part::Zero(frac_digits));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..2`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..2]) },
                    }
                } else {
                    parts[0] = MaybeUninit::new(Part::Copy(b"0"));
                    Formatted {
                        sign,
                        // SAFETY: we just initialized the elements `..1`.
                        parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) },
                    }
                }
            }
            FullDecoded::Finite(ref decoded) => {
                let maxlen = estimate_max_buf_len(decoded.exp);
                assert!(buf.len() >= maxlen);

                // it *is* possible that `frac_digits` is ridiculously large.
                // `format_exact` will end rendering digits much earlier in this case,
                // because we are strictly limited by `maxlen`.
                let limit = if frac_digits < 0x8000 { -(frac_digits as i16) } else { i16::MIN };
                let (buf, exp) = format_exact(decoded, &mut buf[..maxlen], limit);
                if exp <= limit {
                    // the restriction couldn't been met, so this should render like zero no matter
                    // `exp` was. this does not include the case that the restriction has been met
                    // only after the final rounding-up; it's a regular case with `exp = limit + 1`.
                    debug_assert_eq!(buf.len(), 0);
                    if frac_digits > 0 {
                        // [0.][0000]
                        parts[0] = MaybeUninit::new(Part::Copy(b"0."));
                        parts[1] = MaybeUninit::new(Part::Zero(frac_digits));
                        Formatted {
                            sign,
                            // SAFETY: we just initialized the elements `..2`.
                            parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..2]) },
                        }
                    } else {
                        parts[0] = MaybeUninit::new(Part::Copy(b"0"));
                        Formatted {
                            sign,
                            // SAFETY: we just initialized the elements `..1`.
                            parts: unsafe { MaybeUninit::slice_assume_init_ref(&parts[..1]) },
                        }
                    }
                } else {
                    Formatted { sign, parts: digits_to_dec_str(buf, exp, frac_digits, parts) }
                }
            }
        }
    }