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      1 // Copyright 2011 the V8 project authors. All rights reserved.
      2 // Use of this source code is governed by a BSD-style license that can be
      3 // found in the LICENSE file.
      4 
      5 #include <stdint.h>
      6 #include "src/base/logging.h"
      7 #include "src/utils.h"
      8 
      9 #include "src/fast-dtoa.h"
     10 
     11 #include "src/cached-powers.h"
     12 #include "src/diy-fp.h"
     13 #include "src/double.h"
     14 
     15 namespace v8 {
     16 namespace internal {
     17 
     18 // The minimal and maximal target exponent define the range of w's binary
     19 // exponent, where 'w' is the result of multiplying the input by a cached power
     20 // of ten.
     21 //
     22 // A different range might be chosen on a different platform, to optimize digit
     23 // generation, but a smaller range requires more powers of ten to be cached.
     24 static const int kMinimalTargetExponent = -60;
     25 static const int kMaximalTargetExponent = -32;
     26 
     27 
     28 // Adjusts the last digit of the generated number, and screens out generated
     29 // solutions that may be inaccurate. A solution may be inaccurate if it is
     30 // outside the safe interval, or if we ctannot prove that it is closer to the
     31 // input than a neighboring representation of the same length.
     32 //
     33 // Input: * buffer containing the digits of too_high / 10^kappa
     34 //        * the buffer's length
     35 //        * distance_too_high_w == (too_high - w).f() * unit
     36 //        * unsafe_interval == (too_high - too_low).f() * unit
     37 //        * rest = (too_high - buffer * 10^kappa).f() * unit
     38 //        * ten_kappa = 10^kappa * unit
     39 //        * unit = the common multiplier
     40 // Output: returns true if the buffer is guaranteed to contain the closest
     41 //    representable number to the input.
     42 //  Modifies the generated digits in the buffer to approach (round towards) w.
     43 static bool RoundWeed(Vector<char> buffer,
     44                       int length,
     45                       uint64_t distance_too_high_w,
     46                       uint64_t unsafe_interval,
     47                       uint64_t rest,
     48                       uint64_t ten_kappa,
     49                       uint64_t unit) {
     50   uint64_t small_distance = distance_too_high_w - unit;
     51   uint64_t big_distance = distance_too_high_w + unit;
     52   // Let w_low  = too_high - big_distance, and
     53   //     w_high = too_high - small_distance.
     54   // Note: w_low < w < w_high
     55   //
     56   // The real w (* unit) must lie somewhere inside the interval
     57   // ]w_low; w_high[ (often written as "(w_low; w_high)")
     58 
     59   // Basically the buffer currently contains a number in the unsafe interval
     60   // ]too_low; too_high[ with too_low < w < too_high
     61   //
     62   //  too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     63   //                     ^v 1 unit            ^      ^                 ^      ^
     64   //  boundary_high ---------------------     .      .                 .      .
     65   //                     ^v 1 unit            .      .                 .      .
     66   //   - - - - - - - - - - - - - - - - - - -  +  - - + - - - - - -     .      .
     67   //                                          .      .         ^       .      .
     68   //                                          .  big_distance  .       .      .
     69   //                                          .      .         .       .    rest
     70   //                              small_distance     .         .       .      .
     71   //                                          v      .         .       .      .
     72   //  w_high - - - - - - - - - - - - - - - - - -     .         .       .      .
     73   //                     ^v 1 unit                   .         .       .      .
     74   //  w ----------------------------------------     .         .       .      .
     75   //                     ^v 1 unit                   v         .       .      .
     76   //  w_low  - - - - - - - - - - - - - - - - - - - - -         .       .      .
     77   //                                                           .       .      v
     78   //  buffer --------------------------------------------------+-------+--------
     79   //                                                           .       .
     80   //                                                  safe_interval    .
     81   //                                                           v       .
     82   //   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -     .
     83   //                     ^v 1 unit                                     .
     84   //  boundary_low -------------------------                     unsafe_interval
     85   //                     ^v 1 unit                                     v
     86   //  too_low  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     87   //
     88   //
     89   // Note that the value of buffer could lie anywhere inside the range too_low
     90   // to too_high.
     91   //
     92   // boundary_low, boundary_high and w are approximations of the real boundaries
     93   // and v (the input number). They are guaranteed to be precise up to one unit.
     94   // In fact the error is guaranteed to be strictly less than one unit.
     95   //
     96   // Anything that lies outside the unsafe interval is guaranteed not to round
     97   // to v when read again.
     98   // Anything that lies inside the safe interval is guaranteed to round to v
     99   // when read again.
    100   // If the number inside the buffer lies inside the unsafe interval but not
    101   // inside the safe interval then we simply do not know and bail out (returning
    102   // false).
    103   //
    104   // Similarly we have to take into account the imprecision of 'w' when finding
    105   // the closest representation of 'w'. If we have two potential
    106   // representations, and one is closer to both w_low and w_high, then we know
    107   // it is closer to the actual value v.
    108   //
    109   // By generating the digits of too_high we got the largest (closest to
    110   // too_high) buffer that is still in the unsafe interval. In the case where
    111   // w_high < buffer < too_high we try to decrement the buffer.
    112   // This way the buffer approaches (rounds towards) w.
    113   // There are 3 conditions that stop the decrementation process:
    114   //   1) the buffer is already below w_high
    115   //   2) decrementing the buffer would make it leave the unsafe interval
    116   //   3) decrementing the buffer would yield a number below w_high and farther
    117   //      away than the current number. In other words:
    118   //              (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
    119   // Instead of using the buffer directly we use its distance to too_high.
    120   // Conceptually rest ~= too_high - buffer
    121   // We need to do the following tests in this order to avoid over- and
    122   // underflows.
    123   DCHECK(rest <= unsafe_interval);
    124   while (rest < small_distance &&  // Negated condition 1
    125          unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
    126          (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
    127           small_distance - rest >= rest + ten_kappa - small_distance)) {
    128     buffer[length - 1]--;
    129     rest += ten_kappa;
    130   }
    131 
    132   // We have approached w+ as much as possible. We now test if approaching w-
    133   // would require changing the buffer. If yes, then we have two possible
    134   // representations close to w, but we cannot decide which one is closer.
    135   if (rest < big_distance &&
    136       unsafe_interval - rest >= ten_kappa &&
    137       (rest + ten_kappa < big_distance ||
    138        big_distance - rest > rest + ten_kappa - big_distance)) {
    139     return false;
    140   }
    141 
    142   // Weeding test.
    143   //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
    144   //   Since too_low = too_high - unsafe_interval this is equivalent to
    145   //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
    146   //   Conceptually we have: rest ~= too_high - buffer
    147   return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
    148 }
    149 
    150 
    151 // Rounds the buffer upwards if the result is closer to v by possibly adding
    152 // 1 to the buffer. If the precision of the calculation is not sufficient to
    153 // round correctly, return false.
    154 // The rounding might shift the whole buffer in which case the kappa is
    155 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
    156 //
    157 // If 2*rest > ten_kappa then the buffer needs to be round up.
    158 // rest can have an error of +/- 1 unit. This function accounts for the
    159 // imprecision and returns false, if the rounding direction cannot be
    160 // unambiguously determined.
    161 //
    162 // Precondition: rest < ten_kappa.
    163 static bool RoundWeedCounted(Vector<char> buffer,
    164                              int length,
    165                              uint64_t rest,
    166                              uint64_t ten_kappa,
    167                              uint64_t unit,
    168                              int* kappa) {
    169   DCHECK(rest < ten_kappa);
    170   // The following tests are done in a specific order to avoid overflows. They
    171   // will work correctly with any uint64 values of rest < ten_kappa and unit.
    172   //
    173   // If the unit is too big, then we don't know which way to round. For example
    174   // a unit of 50 means that the real number lies within rest +/- 50. If
    175   // 10^kappa == 40 then there is no way to tell which way to round.
    176   if (unit >= ten_kappa) return false;
    177   // Even if unit is just half the size of 10^kappa we are already completely
    178   // lost. (And after the previous test we know that the expression will not
    179   // over/underflow.)
    180   if (ten_kappa - unit <= unit) return false;
    181   // If 2 * (rest + unit) <= 10^kappa we can safely round down.
    182   if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
    183     return true;
    184   }
    185   // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
    186   if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
    187     // Increment the last digit recursively until we find a non '9' digit.
    188     buffer[length - 1]++;
    189     for (int i = length - 1; i > 0; --i) {
    190       if (buffer[i] != '0' + 10) break;
    191       buffer[i] = '0';
    192       buffer[i - 1]++;
    193     }
    194     // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
    195     // exception of the first digit all digits are now '0'. Simply switch the
    196     // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
    197     // the power (the kappa) is increased.
    198     if (buffer[0] == '0' + 10) {
    199       buffer[0] = '1';
    200       (*kappa) += 1;
    201     }
    202     return true;
    203   }
    204   return false;
    205 }
    206 
    207 
    208 static const uint32_t kTen4 = 10000;
    209 static const uint32_t kTen5 = 100000;
    210 static const uint32_t kTen6 = 1000000;
    211 static const uint32_t kTen7 = 10000000;
    212 static const uint32_t kTen8 = 100000000;
    213 static const uint32_t kTen9 = 1000000000;
    214 
    215 // Returns the biggest power of ten that is less than or equal than the given
    216 // number. We furthermore receive the maximum number of bits 'number' has.
    217 // If number_bits == 0 then 0^-1 is returned
    218 // The number of bits must be <= 32.
    219 // Precondition: number < (1 << (number_bits + 1)).
    220 static void BiggestPowerTen(uint32_t number,
    221                             int number_bits,
    222                             uint32_t* power,
    223                             int* exponent) {
    224   switch (number_bits) {
    225     case 32:
    226     case 31:
    227     case 30:
    228       if (kTen9 <= number) {
    229         *power = kTen9;
    230         *exponent = 9;
    231         break;
    232       }  // else fallthrough
    233     case 29:
    234     case 28:
    235     case 27:
    236       if (kTen8 <= number) {
    237         *power = kTen8;
    238         *exponent = 8;
    239         break;
    240       }  // else fallthrough
    241     case 26:
    242     case 25:
    243     case 24:
    244       if (kTen7 <= number) {
    245         *power = kTen7;
    246         *exponent = 7;
    247         break;
    248       }  // else fallthrough
    249     case 23:
    250     case 22:
    251     case 21:
    252     case 20:
    253       if (kTen6 <= number) {
    254         *power = kTen6;
    255         *exponent = 6;
    256         break;
    257       }  // else fallthrough
    258     case 19:
    259     case 18:
    260     case 17:
    261       if (kTen5 <= number) {
    262         *power = kTen5;
    263         *exponent = 5;
    264         break;
    265       }  // else fallthrough
    266     case 16:
    267     case 15:
    268     case 14:
    269       if (kTen4 <= number) {
    270         *power = kTen4;
    271         *exponent = 4;
    272         break;
    273       }  // else fallthrough
    274     case 13:
    275     case 12:
    276     case 11:
    277     case 10:
    278       if (1000 <= number) {
    279         *power = 1000;
    280         *exponent = 3;
    281         break;
    282       }  // else fallthrough
    283     case 9:
    284     case 8:
    285     case 7:
    286       if (100 <= number) {
    287         *power = 100;
    288         *exponent = 2;
    289         break;
    290       }  // else fallthrough
    291     case 6:
    292     case 5:
    293     case 4:
    294       if (10 <= number) {
    295         *power = 10;
    296         *exponent = 1;
    297         break;
    298       }  // else fallthrough
    299     case 3:
    300     case 2:
    301     case 1:
    302       if (1 <= number) {
    303         *power = 1;
    304         *exponent = 0;
    305         break;
    306       }  // else fallthrough
    307     case 0:
    308       *power = 0;
    309       *exponent = -1;
    310       break;
    311     default:
    312       // Following assignments are here to silence compiler warnings.
    313       *power = 0;
    314       *exponent = 0;
    315       UNREACHABLE();
    316   }
    317 }
    318 
    319 
    320 // Generates the digits of input number w.
    321 // w is a floating-point number (DiyFp), consisting of a significand and an
    322 // exponent. Its exponent is bounded by kMinimalTargetExponent and
    323 // kMaximalTargetExponent.
    324 //       Hence -60 <= w.e() <= -32.
    325 //
    326 // Returns false if it fails, in which case the generated digits in the buffer
    327 // should not be used.
    328 // Preconditions:
    329 //  * low, w and high are correct up to 1 ulp (unit in the last place). That
    330 //    is, their error must be less than a unit of their last digits.
    331 //  * low.e() == w.e() == high.e()
    332 //  * low < w < high, and taking into account their error: low~ <= high~
    333 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
    334 // Postconditions: returns false if procedure fails.
    335 //   otherwise:
    336 //     * buffer is not null-terminated, but len contains the number of digits.
    337 //     * buffer contains the shortest possible decimal digit-sequence
    338 //       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
    339 //       correct values of low and high (without their error).
    340 //     * if more than one decimal representation gives the minimal number of
    341 //       decimal digits then the one closest to W (where W is the correct value
    342 //       of w) is chosen.
    343 // Remark: this procedure takes into account the imprecision of its input
    344 //   numbers. If the precision is not enough to guarantee all the postconditions
    345 //   then false is returned. This usually happens rarely (~0.5%).
    346 //
    347 // Say, for the sake of example, that
    348 //   w.e() == -48, and w.f() == 0x1234567890abcdef
    349 // w's value can be computed by w.f() * 2^w.e()
    350 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
    351 //  -> w's integral part is 0x1234
    352 //  w's fractional part is therefore 0x567890abcdef.
    353 // Printing w's integral part is easy (simply print 0x1234 in decimal).
    354 // In order to print its fraction we repeatedly multiply the fraction by 10 and
    355 // get each digit. Example the first digit after the point would be computed by
    356 //   (0x567890abcdef * 10) >> 48. -> 3
    357 // The whole thing becomes slightly more complicated because we want to stop
    358 // once we have enough digits. That is, once the digits inside the buffer
    359 // represent 'w' we can stop. Everything inside the interval low - high
    360 // represents w. However we have to pay attention to low, high and w's
    361 // imprecision.
    362 static bool DigitGen(DiyFp low,
    363                      DiyFp w,
    364                      DiyFp high,
    365                      Vector<char> buffer,
    366                      int* length,
    367                      int* kappa) {
    368   DCHECK(low.e() == w.e() && w.e() == high.e());
    369   DCHECK(low.f() + 1 <= high.f() - 1);
    370   DCHECK(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
    371   // low, w and high are imprecise, but by less than one ulp (unit in the last
    372   // place).
    373   // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
    374   // the new numbers are outside of the interval we want the final
    375   // representation to lie in.
    376   // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
    377   // numbers that are certain to lie in the interval. We will use this fact
    378   // later on.
    379   // We will now start by generating the digits within the uncertain
    380   // interval. Later we will weed out representations that lie outside the safe
    381   // interval and thus _might_ lie outside the correct interval.
    382   uint64_t unit = 1;
    383   DiyFp too_low = DiyFp(low.f() - unit, low.e());
    384   DiyFp too_high = DiyFp(high.f() + unit, high.e());
    385   // too_low and too_high are guaranteed to lie outside the interval we want the
    386   // generated number in.
    387   DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
    388   // We now cut the input number into two parts: the integral digits and the
    389   // fractionals. We will not write any decimal separator though, but adapt
    390   // kappa instead.
    391   // Reminder: we are currently computing the digits (stored inside the buffer)
    392   // such that:   too_low < buffer * 10^kappa < too_high
    393   // We use too_high for the digit_generation and stop as soon as possible.
    394   // If we stop early we effectively round down.
    395   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
    396   // Division by one is a shift.
    397   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
    398   // Modulo by one is an and.
    399   uint64_t fractionals = too_high.f() & (one.f() - 1);
    400   uint32_t divisor;
    401   int divisor_exponent;
    402   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
    403                   &divisor, &divisor_exponent);
    404   *kappa = divisor_exponent + 1;
    405   *length = 0;
    406   // Loop invariant: buffer = too_high / 10^kappa  (integer division)
    407   // The invariant holds for the first iteration: kappa has been initialized
    408   // with the divisor exponent + 1. And the divisor is the biggest power of ten
    409   // that is smaller than integrals.
    410   while (*kappa > 0) {
    411     int digit = integrals / divisor;
    412     buffer[*length] = '0' + digit;
    413     (*length)++;
    414     integrals %= divisor;
    415     (*kappa)--;
    416     // Note that kappa now equals the exponent of the divisor and that the
    417     // invariant thus holds again.
    418     uint64_t rest =
    419         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    420     // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
    421     // Reminder: unsafe_interval.e() == one.e()
    422     if (rest < unsafe_interval.f()) {
    423       // Rounding down (by not emitting the remaining digits) yields a number
    424       // that lies within the unsafe interval.
    425       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
    426                        unsafe_interval.f(), rest,
    427                        static_cast<uint64_t>(divisor) << -one.e(), unit);
    428     }
    429     divisor /= 10;
    430   }
    431 
    432   // The integrals have been generated. We are at the point of the decimal
    433   // separator. In the following loop we simply multiply the remaining digits by
    434   // 10 and divide by one. We just need to pay attention to multiply associated
    435   // data (like the interval or 'unit'), too.
    436   // Note that the multiplication by 10 does not overflow, because w.e >= -60
    437   // and thus one.e >= -60.
    438   DCHECK(one.e() >= -60);
    439   DCHECK(fractionals < one.f());
    440   DCHECK(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
    441   while (true) {
    442     fractionals *= 10;
    443     unit *= 10;
    444     unsafe_interval.set_f(unsafe_interval.f() * 10);
    445     // Integer division by one.
    446     int digit = static_cast<int>(fractionals >> -one.e());
    447     buffer[*length] = '0' + digit;
    448     (*length)++;
    449     fractionals &= one.f() - 1;  // Modulo by one.
    450     (*kappa)--;
    451     if (fractionals < unsafe_interval.f()) {
    452       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
    453                        unsafe_interval.f(), fractionals, one.f(), unit);
    454     }
    455   }
    456 }
    457 
    458 
    459 
    460 // Generates (at most) requested_digits of input number w.
    461 // w is a floating-point number (DiyFp), consisting of a significand and an
    462 // exponent. Its exponent is bounded by kMinimalTargetExponent and
    463 // kMaximalTargetExponent.
    464 //       Hence -60 <= w.e() <= -32.
    465 //
    466 // Returns false if it fails, in which case the generated digits in the buffer
    467 // should not be used.
    468 // Preconditions:
    469 //  * w is correct up to 1 ulp (unit in the last place). That
    470 //    is, its error must be strictly less than a unit of its last digit.
    471 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
    472 //
    473 // Postconditions: returns false if procedure fails.
    474 //   otherwise:
    475 //     * buffer is not null-terminated, but length contains the number of
    476 //       digits.
    477 //     * the representation in buffer is the most precise representation of
    478 //       requested_digits digits.
    479 //     * buffer contains at most requested_digits digits of w. If there are less
    480 //       than requested_digits digits then some trailing '0's have been removed.
    481 //     * kappa is such that
    482 //            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
    483 //
    484 // Remark: This procedure takes into account the imprecision of its input
    485 //   numbers. If the precision is not enough to guarantee all the postconditions
    486 //   then false is returned. This usually happens rarely, but the failure-rate
    487 //   increases with higher requested_digits.
    488 static bool DigitGenCounted(DiyFp w,
    489                             int requested_digits,
    490                             Vector<char> buffer,
    491                             int* length,
    492                             int* kappa) {
    493   DCHECK(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
    494   DCHECK(kMinimalTargetExponent >= -60);
    495   DCHECK(kMaximalTargetExponent <= -32);
    496   // w is assumed to have an error less than 1 unit. Whenever w is scaled we
    497   // also scale its error.
    498   uint64_t w_error = 1;
    499   // We cut the input number into two parts: the integral digits and the
    500   // fractional digits. We don't emit any decimal separator, but adapt kappa
    501   // instead. Example: instead of writing "1.2" we put "12" into the buffer and
    502   // increase kappa by 1.
    503   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
    504   // Division by one is a shift.
    505   uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
    506   // Modulo by one is an and.
    507   uint64_t fractionals = w.f() & (one.f() - 1);
    508   uint32_t divisor;
    509   int divisor_exponent;
    510   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
    511                   &divisor, &divisor_exponent);
    512   *kappa = divisor_exponent + 1;
    513   *length = 0;
    514 
    515   // Loop invariant: buffer = w / 10^kappa  (integer division)
    516   // The invariant holds for the first iteration: kappa has been initialized
    517   // with the divisor exponent + 1. And the divisor is the biggest power of ten
    518   // that is smaller than 'integrals'.
    519   while (*kappa > 0) {
    520     int digit = integrals / divisor;
    521     buffer[*length] = '0' + digit;
    522     (*length)++;
    523     requested_digits--;
    524     integrals %= divisor;
    525     (*kappa)--;
    526     // Note that kappa now equals the exponent of the divisor and that the
    527     // invariant thus holds again.
    528     if (requested_digits == 0) break;
    529     divisor /= 10;
    530   }
    531 
    532   if (requested_digits == 0) {
    533     uint64_t rest =
    534         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    535     return RoundWeedCounted(buffer, *length, rest,
    536                             static_cast<uint64_t>(divisor) << -one.e(), w_error,
    537                             kappa);
    538   }
    539 
    540   // The integrals have been generated. We are at the point of the decimal
    541   // separator. In the following loop we simply multiply the remaining digits by
    542   // 10 and divide by one. We just need to pay attention to multiply associated
    543   // data (the 'unit'), too.
    544   // Note that the multiplication by 10 does not overflow, because w.e >= -60
    545   // and thus one.e >= -60.
    546   DCHECK(one.e() >= -60);
    547   DCHECK(fractionals < one.f());
    548   DCHECK(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
    549   while (requested_digits > 0 && fractionals > w_error) {
    550     fractionals *= 10;
    551     w_error *= 10;
    552     // Integer division by one.
    553     int digit = static_cast<int>(fractionals >> -one.e());
    554     buffer[*length] = '0' + digit;
    555     (*length)++;
    556     requested_digits--;
    557     fractionals &= one.f() - 1;  // Modulo by one.
    558     (*kappa)--;
    559   }
    560   if (requested_digits != 0) return false;
    561   return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
    562                           kappa);
    563 }
    564 
    565 
    566 // Provides a decimal representation of v.
    567 // Returns true if it succeeds, otherwise the result cannot be trusted.
    568 // There will be *length digits inside the buffer (not null-terminated).
    569 // If the function returns true then
    570 //        v == (double) (buffer * 10^decimal_exponent).
    571 // The digits in the buffer are the shortest representation possible: no
    572 // 0.09999999999999999 instead of 0.1. The shorter representation will even be
    573 // chosen even if the longer one would be closer to v.
    574 // The last digit will be closest to the actual v. That is, even if several
    575 // digits might correctly yield 'v' when read again, the closest will be
    576 // computed.
    577 static bool Grisu3(double v,
    578                    Vector<char> buffer,
    579                    int* length,
    580                    int* decimal_exponent) {
    581   DiyFp w = Double(v).AsNormalizedDiyFp();
    582   // boundary_minus and boundary_plus are the boundaries between v and its
    583   // closest floating-point neighbors. Any number strictly between
    584   // boundary_minus and boundary_plus will round to v when convert to a double.
    585   // Grisu3 will never output representations that lie exactly on a boundary.
    586   DiyFp boundary_minus, boundary_plus;
    587   Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
    588   DCHECK(boundary_plus.e() == w.e());
    589   DiyFp ten_mk;  // Cached power of ten: 10^-k
    590   int mk;        // -k
    591   int ten_mk_minimal_binary_exponent =
    592      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    593   int ten_mk_maximal_binary_exponent =
    594      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    595   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
    596       ten_mk_minimal_binary_exponent,
    597       ten_mk_maximal_binary_exponent,
    598       &ten_mk, &mk);
    599   DCHECK((kMinimalTargetExponent <= w.e() + ten_mk.e() +
    600           DiyFp::kSignificandSize) &&
    601          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
    602           DiyFp::kSignificandSize));
    603   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
    604   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
    605 
    606   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
    607   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
    608   // off by a small amount.
    609   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
    610   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
    611   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
    612   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
    613   DCHECK(scaled_w.e() ==
    614          boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
    615   // In theory it would be possible to avoid some recomputations by computing
    616   // the difference between w and boundary_minus/plus (a power of 2) and to
    617   // compute scaled_boundary_minus/plus by subtracting/adding from
    618   // scaled_w. However the code becomes much less readable and the speed
    619   // enhancements are not terriffic.
    620   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
    621   DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);
    622 
    623   // DigitGen will generate the digits of scaled_w. Therefore we have
    624   // v == (double) (scaled_w * 10^-mk).
    625   // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
    626   // integer than it will be updated. For instance if scaled_w == 1.23 then
    627   // the buffer will be filled with "123" und the decimal_exponent will be
    628   // decreased by 2.
    629   int kappa;
    630   bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
    631                          buffer, length, &kappa);
    632   *decimal_exponent = -mk + kappa;
    633   return result;
    634 }
    635 
    636 
    637 // The "counted" version of grisu3 (see above) only generates requested_digits
    638 // number of digits. This version does not generate the shortest representation,
    639 // and with enough requested digits 0.1 will at some point print as 0.9999999...
    640 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
    641 // therefore the rounding strategy for halfway cases is irrelevant.
    642 static bool Grisu3Counted(double v,
    643                           int requested_digits,
    644                           Vector<char> buffer,
    645                           int* length,
    646                           int* decimal_exponent) {
    647   DiyFp w = Double(v).AsNormalizedDiyFp();
    648   DiyFp ten_mk;  // Cached power of ten: 10^-k
    649   int mk;        // -k
    650   int ten_mk_minimal_binary_exponent =
    651      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    652   int ten_mk_maximal_binary_exponent =
    653      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    654   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
    655       ten_mk_minimal_binary_exponent,
    656       ten_mk_maximal_binary_exponent,
    657       &ten_mk, &mk);
    658   DCHECK((kMinimalTargetExponent <= w.e() + ten_mk.e() +
    659           DiyFp::kSignificandSize) &&
    660          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
    661           DiyFp::kSignificandSize));
    662   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
    663   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
    664 
    665   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
    666   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
    667   // off by a small amount.
    668   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
    669   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
    670   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
    671   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
    672 
    673   // We now have (double) (scaled_w * 10^-mk).
    674   // DigitGen will generate the first requested_digits digits of scaled_w and
    675   // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
    676   // will not always be exactly the same since DigitGenCounted only produces a
    677   // limited number of digits.)
    678   int kappa;
    679   bool result = DigitGenCounted(scaled_w, requested_digits,
    680                                 buffer, length, &kappa);
    681   *decimal_exponent = -mk + kappa;
    682   return result;
    683 }
    684 
    685 
    686 bool FastDtoa(double v,
    687               FastDtoaMode mode,
    688               int requested_digits,
    689               Vector<char> buffer,
    690               int* length,
    691               int* decimal_point) {
    692   DCHECK(v > 0);
    693   DCHECK(!Double(v).IsSpecial());
    694 
    695   bool result = false;
    696   int decimal_exponent = 0;
    697   switch (mode) {
    698     case FAST_DTOA_SHORTEST:
    699       result = Grisu3(v, buffer, length, &decimal_exponent);
    700       break;
    701     case FAST_DTOA_PRECISION:
    702       result = Grisu3Counted(v, requested_digits,
    703                              buffer, length, &decimal_exponent);
    704       break;
    705     default:
    706       UNREACHABLE();
    707   }
    708   if (result) {
    709     *decimal_point = *length + decimal_exponent;
    710     buffer[*length] = '\0';
    711   }
    712   return result;
    713 }
    714 
    715 }  // namespace internal
    716 }  // namespace v8
    717