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      1 // Copyright 2010 the V8 project authors. All rights reserved.
      2 // Redistribution and use in source and binary forms, with or without
      3 // modification, are permitted provided that the following conditions are
      4 // met:
      5 //
      6 //     * Redistributions of source code must retain the above copyright
      7 //       notice, this list of conditions and the following disclaimer.
      8 //     * Redistributions in binary form must reproduce the above
      9 //       copyright notice, this list of conditions and the following
     10 //       disclaimer in the documentation and/or other materials provided
     11 //       with the distribution.
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     14 //       from this software without specific prior written permission.
     15 //
     16 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
     17 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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     26 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
     27 
     28 #include "v8.h"
     29 
     30 #include "fast-dtoa.h"
     31 
     32 #include "cached-powers.h"
     33 #include "diy-fp.h"
     34 #include "double.h"
     35 
     36 namespace v8 {
     37 namespace internal {
     38 
     39 // The minimal and maximal target exponent define the range of w's binary
     40 // exponent, where 'w' is the result of multiplying the input by a cached power
     41 // of ten.
     42 //
     43 // A different range might be chosen on a different platform, to optimize digit
     44 // generation, but a smaller range requires more powers of ten to be cached.
     45 static const int kMinimalTargetExponent = -60;
     46 static const int kMaximalTargetExponent = -32;
     47 
     48 
     49 // Adjusts the last digit of the generated number, and screens out generated
     50 // solutions that may be inaccurate. A solution may be inaccurate if it is
     51 // outside the safe interval, or if we ctannot prove that it is closer to the
     52 // input than a neighboring representation of the same length.
     53 //
     54 // Input: * buffer containing the digits of too_high / 10^kappa
     55 //        * the buffer's length
     56 //        * distance_too_high_w == (too_high - w).f() * unit
     57 //        * unsafe_interval == (too_high - too_low).f() * unit
     58 //        * rest = (too_high - buffer * 10^kappa).f() * unit
     59 //        * ten_kappa = 10^kappa * unit
     60 //        * unit = the common multiplier
     61 // Output: returns true if the buffer is guaranteed to contain the closest
     62 //    representable number to the input.
     63 //  Modifies the generated digits in the buffer to approach (round towards) w.
     64 static bool RoundWeed(Vector<char> buffer,
     65                       int length,
     66                       uint64_t distance_too_high_w,
     67                       uint64_t unsafe_interval,
     68                       uint64_t rest,
     69                       uint64_t ten_kappa,
     70                       uint64_t unit) {
     71   uint64_t small_distance = distance_too_high_w - unit;
     72   uint64_t big_distance = distance_too_high_w + unit;
     73   // Let w_low  = too_high - big_distance, and
     74   //     w_high = too_high - small_distance.
     75   // Note: w_low < w < w_high
     76   //
     77   // The real w (* unit) must lie somewhere inside the interval
     78   // ]w_low; w_high[ (often written as "(w_low; w_high)")
     79 
     80   // Basically the buffer currently contains a number in the unsafe interval
     81   // ]too_low; too_high[ with too_low < w < too_high
     82   //
     83   //  too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     84   //                     ^v 1 unit            ^      ^                 ^      ^
     85   //  boundary_high ---------------------     .      .                 .      .
     86   //                     ^v 1 unit            .      .                 .      .
     87   //   - - - - - - - - - - - - - - - - - - -  +  - - + - - - - - -     .      .
     88   //                                          .      .         ^       .      .
     89   //                                          .  big_distance  .       .      .
     90   //                                          .      .         .       .    rest
     91   //                              small_distance     .         .       .      .
     92   //                                          v      .         .       .      .
     93   //  w_high - - - - - - - - - - - - - - - - - -     .         .       .      .
     94   //                     ^v 1 unit                   .         .       .      .
     95   //  w ----------------------------------------     .         .       .      .
     96   //                     ^v 1 unit                   v         .       .      .
     97   //  w_low  - - - - - - - - - - - - - - - - - - - - -         .       .      .
     98   //                                                           .       .      v
     99   //  buffer --------------------------------------------------+-------+--------
    100   //                                                           .       .
    101   //                                                  safe_interval    .
    102   //                                                           v       .
    103   //   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -     .
    104   //                     ^v 1 unit                                     .
    105   //  boundary_low -------------------------                     unsafe_interval
    106   //                     ^v 1 unit                                     v
    107   //  too_low  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    108   //
    109   //
    110   // Note that the value of buffer could lie anywhere inside the range too_low
    111   // to too_high.
    112   //
    113   // boundary_low, boundary_high and w are approximations of the real boundaries
    114   // and v (the input number). They are guaranteed to be precise up to one unit.
    115   // In fact the error is guaranteed to be strictly less than one unit.
    116   //
    117   // Anything that lies outside the unsafe interval is guaranteed not to round
    118   // to v when read again.
    119   // Anything that lies inside the safe interval is guaranteed to round to v
    120   // when read again.
    121   // If the number inside the buffer lies inside the unsafe interval but not
    122   // inside the safe interval then we simply do not know and bail out (returning
    123   // false).
    124   //
    125   // Similarly we have to take into account the imprecision of 'w' when finding
    126   // the closest representation of 'w'. If we have two potential
    127   // representations, and one is closer to both w_low and w_high, then we know
    128   // it is closer to the actual value v.
    129   //
    130   // By generating the digits of too_high we got the largest (closest to
    131   // too_high) buffer that is still in the unsafe interval. In the case where
    132   // w_high < buffer < too_high we try to decrement the buffer.
    133   // This way the buffer approaches (rounds towards) w.
    134   // There are 3 conditions that stop the decrementation process:
    135   //   1) the buffer is already below w_high
    136   //   2) decrementing the buffer would make it leave the unsafe interval
    137   //   3) decrementing the buffer would yield a number below w_high and farther
    138   //      away than the current number. In other words:
    139   //              (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
    140   // Instead of using the buffer directly we use its distance to too_high.
    141   // Conceptually rest ~= too_high - buffer
    142   // We need to do the following tests in this order to avoid over- and
    143   // underflows.
    144   ASSERT(rest <= unsafe_interval);
    145   while (rest < small_distance &&  // Negated condition 1
    146          unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
    147          (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
    148           small_distance - rest >= rest + ten_kappa - small_distance)) {
    149     buffer[length - 1]--;
    150     rest += ten_kappa;
    151   }
    152 
    153   // We have approached w+ as much as possible. We now test if approaching w-
    154   // would require changing the buffer. If yes, then we have two possible
    155   // representations close to w, but we cannot decide which one is closer.
    156   if (rest < big_distance &&
    157       unsafe_interval - rest >= ten_kappa &&
    158       (rest + ten_kappa < big_distance ||
    159        big_distance - rest > rest + ten_kappa - big_distance)) {
    160     return false;
    161   }
    162 
    163   // Weeding test.
    164   //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
    165   //   Since too_low = too_high - unsafe_interval this is equivalent to
    166   //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
    167   //   Conceptually we have: rest ~= too_high - buffer
    168   return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
    169 }
    170 
    171 
    172 // Rounds the buffer upwards if the result is closer to v by possibly adding
    173 // 1 to the buffer. If the precision of the calculation is not sufficient to
    174 // round correctly, return false.
    175 // The rounding might shift the whole buffer in which case the kappa is
    176 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
    177 //
    178 // If 2*rest > ten_kappa then the buffer needs to be round up.
    179 // rest can have an error of +/- 1 unit. This function accounts for the
    180 // imprecision and returns false, if the rounding direction cannot be
    181 // unambiguously determined.
    182 //
    183 // Precondition: rest < ten_kappa.
    184 static bool RoundWeedCounted(Vector<char> buffer,
    185                              int length,
    186                              uint64_t rest,
    187                              uint64_t ten_kappa,
    188                              uint64_t unit,
    189                              int* kappa) {
    190   ASSERT(rest < ten_kappa);
    191   // The following tests are done in a specific order to avoid overflows. They
    192   // will work correctly with any uint64 values of rest < ten_kappa and unit.
    193   //
    194   // If the unit is too big, then we don't know which way to round. For example
    195   // a unit of 50 means that the real number lies within rest +/- 50. If
    196   // 10^kappa == 40 then there is no way to tell which way to round.
    197   if (unit >= ten_kappa) return false;
    198   // Even if unit is just half the size of 10^kappa we are already completely
    199   // lost. (And after the previous test we know that the expression will not
    200   // over/underflow.)
    201   if (ten_kappa - unit <= unit) return false;
    202   // If 2 * (rest + unit) <= 10^kappa we can safely round down.
    203   if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
    204     return true;
    205   }
    206   // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
    207   if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
    208     // Increment the last digit recursively until we find a non '9' digit.
    209     buffer[length - 1]++;
    210     for (int i = length - 1; i > 0; --i) {
    211       if (buffer[i] != '0' + 10) break;
    212       buffer[i] = '0';
    213       buffer[i - 1]++;
    214     }
    215     // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
    216     // exception of the first digit all digits are now '0'. Simply switch the
    217     // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
    218     // the power (the kappa) is increased.
    219     if (buffer[0] == '0' + 10) {
    220       buffer[0] = '1';
    221       (*kappa) += 1;
    222     }
    223     return true;
    224   }
    225   return false;
    226 }
    227 
    228 
    229 static const uint32_t kTen4 = 10000;
    230 static const uint32_t kTen5 = 100000;
    231 static const uint32_t kTen6 = 1000000;
    232 static const uint32_t kTen7 = 10000000;
    233 static const uint32_t kTen8 = 100000000;
    234 static const uint32_t kTen9 = 1000000000;
    235 
    236 // Returns the biggest power of ten that is less than or equal than the given
    237 // number. We furthermore receive the maximum number of bits 'number' has.
    238 // If number_bits == 0 then 0^-1 is returned
    239 // The number of bits must be <= 32.
    240 // Precondition: number < (1 << (number_bits + 1)).
    241 static void BiggestPowerTen(uint32_t number,
    242                             int number_bits,
    243                             uint32_t* power,
    244                             int* exponent) {
    245   switch (number_bits) {
    246     case 32:
    247     case 31:
    248     case 30:
    249       if (kTen9 <= number) {
    250         *power = kTen9;
    251         *exponent = 9;
    252         break;
    253       }  // else fallthrough
    254     case 29:
    255     case 28:
    256     case 27:
    257       if (kTen8 <= number) {
    258         *power = kTen8;
    259         *exponent = 8;
    260         break;
    261       }  // else fallthrough
    262     case 26:
    263     case 25:
    264     case 24:
    265       if (kTen7 <= number) {
    266         *power = kTen7;
    267         *exponent = 7;
    268         break;
    269       }  // else fallthrough
    270     case 23:
    271     case 22:
    272     case 21:
    273     case 20:
    274       if (kTen6 <= number) {
    275         *power = kTen6;
    276         *exponent = 6;
    277         break;
    278       }  // else fallthrough
    279     case 19:
    280     case 18:
    281     case 17:
    282       if (kTen5 <= number) {
    283         *power = kTen5;
    284         *exponent = 5;
    285         break;
    286       }  // else fallthrough
    287     case 16:
    288     case 15:
    289     case 14:
    290       if (kTen4 <= number) {
    291         *power = kTen4;
    292         *exponent = 4;
    293         break;
    294       }  // else fallthrough
    295     case 13:
    296     case 12:
    297     case 11:
    298     case 10:
    299       if (1000 <= number) {
    300         *power = 1000;
    301         *exponent = 3;
    302         break;
    303       }  // else fallthrough
    304     case 9:
    305     case 8:
    306     case 7:
    307       if (100 <= number) {
    308         *power = 100;
    309         *exponent = 2;
    310         break;
    311       }  // else fallthrough
    312     case 6:
    313     case 5:
    314     case 4:
    315       if (10 <= number) {
    316         *power = 10;
    317         *exponent = 1;
    318         break;
    319       }  // else fallthrough
    320     case 3:
    321     case 2:
    322     case 1:
    323       if (1 <= number) {
    324         *power = 1;
    325         *exponent = 0;
    326         break;
    327       }  // else fallthrough
    328     case 0:
    329       *power = 0;
    330       *exponent = -1;
    331       break;
    332     default:
    333       // Following assignments are here to silence compiler warnings.
    334       *power = 0;
    335       *exponent = 0;
    336       UNREACHABLE();
    337   }
    338 }
    339 
    340 
    341 // Generates the digits of input number w.
    342 // w is a floating-point number (DiyFp), consisting of a significand and an
    343 // exponent. Its exponent is bounded by kMinimalTargetExponent and
    344 // kMaximalTargetExponent.
    345 //       Hence -60 <= w.e() <= -32.
    346 //
    347 // Returns false if it fails, in which case the generated digits in the buffer
    348 // should not be used.
    349 // Preconditions:
    350 //  * low, w and high are correct up to 1 ulp (unit in the last place). That
    351 //    is, their error must be less than a unit of their last digits.
    352 //  * low.e() == w.e() == high.e()
    353 //  * low < w < high, and taking into account their error: low~ <= high~
    354 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
    355 // Postconditions: returns false if procedure fails.
    356 //   otherwise:
    357 //     * buffer is not null-terminated, but len contains the number of digits.
    358 //     * buffer contains the shortest possible decimal digit-sequence
    359 //       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
    360 //       correct values of low and high (without their error).
    361 //     * if more than one decimal representation gives the minimal number of
    362 //       decimal digits then the one closest to W (where W is the correct value
    363 //       of w) is chosen.
    364 // Remark: this procedure takes into account the imprecision of its input
    365 //   numbers. If the precision is not enough to guarantee all the postconditions
    366 //   then false is returned. This usually happens rarely (~0.5%).
    367 //
    368 // Say, for the sake of example, that
    369 //   w.e() == -48, and w.f() == 0x1234567890abcdef
    370 // w's value can be computed by w.f() * 2^w.e()
    371 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
    372 //  -> w's integral part is 0x1234
    373 //  w's fractional part is therefore 0x567890abcdef.
    374 // Printing w's integral part is easy (simply print 0x1234 in decimal).
    375 // In order to print its fraction we repeatedly multiply the fraction by 10 and
    376 // get each digit. Example the first digit after the point would be computed by
    377 //   (0x567890abcdef * 10) >> 48. -> 3
    378 // The whole thing becomes slightly more complicated because we want to stop
    379 // once we have enough digits. That is, once the digits inside the buffer
    380 // represent 'w' we can stop. Everything inside the interval low - high
    381 // represents w. However we have to pay attention to low, high and w's
    382 // imprecision.
    383 static bool DigitGen(DiyFp low,
    384                      DiyFp w,
    385                      DiyFp high,
    386                      Vector<char> buffer,
    387                      int* length,
    388                      int* kappa) {
    389   ASSERT(low.e() == w.e() && w.e() == high.e());
    390   ASSERT(low.f() + 1 <= high.f() - 1);
    391   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
    392   // low, w and high are imprecise, but by less than one ulp (unit in the last
    393   // place).
    394   // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
    395   // the new numbers are outside of the interval we want the final
    396   // representation to lie in.
    397   // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
    398   // numbers that are certain to lie in the interval. We will use this fact
    399   // later on.
    400   // We will now start by generating the digits within the uncertain
    401   // interval. Later we will weed out representations that lie outside the safe
    402   // interval and thus _might_ lie outside the correct interval.
    403   uint64_t unit = 1;
    404   DiyFp too_low = DiyFp(low.f() - unit, low.e());
    405   DiyFp too_high = DiyFp(high.f() + unit, high.e());
    406   // too_low and too_high are guaranteed to lie outside the interval we want the
    407   // generated number in.
    408   DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
    409   // We now cut the input number into two parts: the integral digits and the
    410   // fractionals. We will not write any decimal separator though, but adapt
    411   // kappa instead.
    412   // Reminder: we are currently computing the digits (stored inside the buffer)
    413   // such that:   too_low < buffer * 10^kappa < too_high
    414   // We use too_high for the digit_generation and stop as soon as possible.
    415   // If we stop early we effectively round down.
    416   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
    417   // Division by one is a shift.
    418   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
    419   // Modulo by one is an and.
    420   uint64_t fractionals = too_high.f() & (one.f() - 1);
    421   uint32_t divisor;
    422   int divisor_exponent;
    423   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
    424                   &divisor, &divisor_exponent);
    425   *kappa = divisor_exponent + 1;
    426   *length = 0;
    427   // Loop invariant: buffer = too_high / 10^kappa  (integer division)
    428   // The invariant holds for the first iteration: kappa has been initialized
    429   // with the divisor exponent + 1. And the divisor is the biggest power of ten
    430   // that is smaller than integrals.
    431   while (*kappa > 0) {
    432     int digit = integrals / divisor;
    433     buffer[*length] = '0' + digit;
    434     (*length)++;
    435     integrals %= divisor;
    436     (*kappa)--;
    437     // Note that kappa now equals the exponent of the divisor and that the
    438     // invariant thus holds again.
    439     uint64_t rest =
    440         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    441     // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
    442     // Reminder: unsafe_interval.e() == one.e()
    443     if (rest < unsafe_interval.f()) {
    444       // Rounding down (by not emitting the remaining digits) yields a number
    445       // that lies within the unsafe interval.
    446       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
    447                        unsafe_interval.f(), rest,
    448                        static_cast<uint64_t>(divisor) << -one.e(), unit);
    449     }
    450     divisor /= 10;
    451   }
    452 
    453   // The integrals have been generated. We are at the point of the decimal
    454   // separator. In the following loop we simply multiply the remaining digits by
    455   // 10 and divide by one. We just need to pay attention to multiply associated
    456   // data (like the interval or 'unit'), too.
    457   // Note that the multiplication by 10 does not overflow, because w.e >= -60
    458   // and thus one.e >= -60.
    459   ASSERT(one.e() >= -60);
    460   ASSERT(fractionals < one.f());
    461   ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
    462   while (true) {
    463     fractionals *= 10;
    464     unit *= 10;
    465     unsafe_interval.set_f(unsafe_interval.f() * 10);
    466     // Integer division by one.
    467     int digit = static_cast<int>(fractionals >> -one.e());
    468     buffer[*length] = '0' + digit;
    469     (*length)++;
    470     fractionals &= one.f() - 1;  // Modulo by one.
    471     (*kappa)--;
    472     if (fractionals < unsafe_interval.f()) {
    473       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
    474                        unsafe_interval.f(), fractionals, one.f(), unit);
    475     }
    476   }
    477 }
    478 
    479 
    480 
    481 // Generates (at most) requested_digits of input number w.
    482 // w is a floating-point number (DiyFp), consisting of a significand and an
    483 // exponent. Its exponent is bounded by kMinimalTargetExponent and
    484 // kMaximalTargetExponent.
    485 //       Hence -60 <= w.e() <= -32.
    486 //
    487 // Returns false if it fails, in which case the generated digits in the buffer
    488 // should not be used.
    489 // Preconditions:
    490 //  * w is correct up to 1 ulp (unit in the last place). That
    491 //    is, its error must be strictly less than a unit of its last digit.
    492 //  * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
    493 //
    494 // Postconditions: returns false if procedure fails.
    495 //   otherwise:
    496 //     * buffer is not null-terminated, but length contains the number of
    497 //       digits.
    498 //     * the representation in buffer is the most precise representation of
    499 //       requested_digits digits.
    500 //     * buffer contains at most requested_digits digits of w. If there are less
    501 //       than requested_digits digits then some trailing '0's have been removed.
    502 //     * kappa is such that
    503 //            w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
    504 //
    505 // Remark: This procedure takes into account the imprecision of its input
    506 //   numbers. If the precision is not enough to guarantee all the postconditions
    507 //   then false is returned. This usually happens rarely, but the failure-rate
    508 //   increases with higher requested_digits.
    509 static bool DigitGenCounted(DiyFp w,
    510                             int requested_digits,
    511                             Vector<char> buffer,
    512                             int* length,
    513                             int* kappa) {
    514   ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
    515   ASSERT(kMinimalTargetExponent >= -60);
    516   ASSERT(kMaximalTargetExponent <= -32);
    517   // w is assumed to have an error less than 1 unit. Whenever w is scaled we
    518   // also scale its error.
    519   uint64_t w_error = 1;
    520   // We cut the input number into two parts: the integral digits and the
    521   // fractional digits. We don't emit any decimal separator, but adapt kappa
    522   // instead. Example: instead of writing "1.2" we put "12" into the buffer and
    523   // increase kappa by 1.
    524   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
    525   // Division by one is a shift.
    526   uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
    527   // Modulo by one is an and.
    528   uint64_t fractionals = w.f() & (one.f() - 1);
    529   uint32_t divisor;
    530   int divisor_exponent;
    531   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
    532                   &divisor, &divisor_exponent);
    533   *kappa = divisor_exponent + 1;
    534   *length = 0;
    535 
    536   // Loop invariant: buffer = w / 10^kappa  (integer division)
    537   // The invariant holds for the first iteration: kappa has been initialized
    538   // with the divisor exponent + 1. And the divisor is the biggest power of ten
    539   // that is smaller than 'integrals'.
    540   while (*kappa > 0) {
    541     int digit = integrals / divisor;
    542     buffer[*length] = '0' + digit;
    543     (*length)++;
    544     requested_digits--;
    545     integrals %= divisor;
    546     (*kappa)--;
    547     // Note that kappa now equals the exponent of the divisor and that the
    548     // invariant thus holds again.
    549     if (requested_digits == 0) break;
    550     divisor /= 10;
    551   }
    552 
    553   if (requested_digits == 0) {
    554     uint64_t rest =
    555         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    556     return RoundWeedCounted(buffer, *length, rest,
    557                             static_cast<uint64_t>(divisor) << -one.e(), w_error,
    558                             kappa);
    559   }
    560 
    561   // The integrals have been generated. We are at the point of the decimal
    562   // separator. In the following loop we simply multiply the remaining digits by
    563   // 10 and divide by one. We just need to pay attention to multiply associated
    564   // data (the 'unit'), too.
    565   // Note that the multiplication by 10 does not overflow, because w.e >= -60
    566   // and thus one.e >= -60.
    567   ASSERT(one.e() >= -60);
    568   ASSERT(fractionals < one.f());
    569   ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
    570   while (requested_digits > 0 && fractionals > w_error) {
    571     fractionals *= 10;
    572     w_error *= 10;
    573     // Integer division by one.
    574     int digit = static_cast<int>(fractionals >> -one.e());
    575     buffer[*length] = '0' + digit;
    576     (*length)++;
    577     requested_digits--;
    578     fractionals &= one.f() - 1;  // Modulo by one.
    579     (*kappa)--;
    580   }
    581   if (requested_digits != 0) return false;
    582   return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
    583                           kappa);
    584 }
    585 
    586 
    587 // Provides a decimal representation of v.
    588 // Returns true if it succeeds, otherwise the result cannot be trusted.
    589 // There will be *length digits inside the buffer (not null-terminated).
    590 // If the function returns true then
    591 //        v == (double) (buffer * 10^decimal_exponent).
    592 // The digits in the buffer are the shortest representation possible: no
    593 // 0.09999999999999999 instead of 0.1. The shorter representation will even be
    594 // chosen even if the longer one would be closer to v.
    595 // The last digit will be closest to the actual v. That is, even if several
    596 // digits might correctly yield 'v' when read again, the closest will be
    597 // computed.
    598 static bool Grisu3(double v,
    599                    Vector<char> buffer,
    600                    int* length,
    601                    int* decimal_exponent) {
    602   DiyFp w = Double(v).AsNormalizedDiyFp();
    603   // boundary_minus and boundary_plus are the boundaries between v and its
    604   // closest floating-point neighbors. Any number strictly between
    605   // boundary_minus and boundary_plus will round to v when convert to a double.
    606   // Grisu3 will never output representations that lie exactly on a boundary.
    607   DiyFp boundary_minus, boundary_plus;
    608   Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
    609   ASSERT(boundary_plus.e() == w.e());
    610   DiyFp ten_mk;  // Cached power of ten: 10^-k
    611   int mk;        // -k
    612   int ten_mk_minimal_binary_exponent =
    613      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    614   int ten_mk_maximal_binary_exponent =
    615      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    616   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
    617       ten_mk_minimal_binary_exponent,
    618       ten_mk_maximal_binary_exponent,
    619       &ten_mk, &mk);
    620   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
    621           DiyFp::kSignificandSize) &&
    622          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
    623           DiyFp::kSignificandSize));
    624   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
    625   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
    626 
    627   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
    628   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
    629   // off by a small amount.
    630   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
    631   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
    632   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
    633   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
    634   ASSERT(scaled_w.e() ==
    635          boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
    636   // In theory it would be possible to avoid some recomputations by computing
    637   // the difference between w and boundary_minus/plus (a power of 2) and to
    638   // compute scaled_boundary_minus/plus by subtracting/adding from
    639   // scaled_w. However the code becomes much less readable and the speed
    640   // enhancements are not terriffic.
    641   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
    642   DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);
    643 
    644   // DigitGen will generate the digits of scaled_w. Therefore we have
    645   // v == (double) (scaled_w * 10^-mk).
    646   // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
    647   // integer than it will be updated. For instance if scaled_w == 1.23 then
    648   // the buffer will be filled with "123" und the decimal_exponent will be
    649   // decreased by 2.
    650   int kappa;
    651   bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
    652                          buffer, length, &kappa);
    653   *decimal_exponent = -mk + kappa;
    654   return result;
    655 }
    656 
    657 
    658 // The "counted" version of grisu3 (see above) only generates requested_digits
    659 // number of digits. This version does not generate the shortest representation,
    660 // and with enough requested digits 0.1 will at some point print as 0.9999999...
    661 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
    662 // therefore the rounding strategy for halfway cases is irrelevant.
    663 static bool Grisu3Counted(double v,
    664                           int requested_digits,
    665                           Vector<char> buffer,
    666                           int* length,
    667                           int* decimal_exponent) {
    668   DiyFp w = Double(v).AsNormalizedDiyFp();
    669   DiyFp ten_mk;  // Cached power of ten: 10^-k
    670   int mk;        // -k
    671   int ten_mk_minimal_binary_exponent =
    672      kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    673   int ten_mk_maximal_binary_exponent =
    674      kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
    675   PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
    676       ten_mk_minimal_binary_exponent,
    677       ten_mk_maximal_binary_exponent,
    678       &ten_mk, &mk);
    679   ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
    680           DiyFp::kSignificandSize) &&
    681          (kMaximalTargetExponent >= w.e() + ten_mk.e() +
    682           DiyFp::kSignificandSize));
    683   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
    684   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
    685 
    686   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
    687   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
    688   // off by a small amount.
    689   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
    690   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
    691   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
    692   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
    693 
    694   // We now have (double) (scaled_w * 10^-mk).
    695   // DigitGen will generate the first requested_digits digits of scaled_w and
    696   // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
    697   // will not always be exactly the same since DigitGenCounted only produces a
    698   // limited number of digits.)
    699   int kappa;
    700   bool result = DigitGenCounted(scaled_w, requested_digits,
    701                                 buffer, length, &kappa);
    702   *decimal_exponent = -mk + kappa;
    703   return result;
    704 }
    705 
    706 
    707 bool FastDtoa(double v,
    708               FastDtoaMode mode,
    709               int requested_digits,
    710               Vector<char> buffer,
    711               int* length,
    712               int* decimal_point) {
    713   ASSERT(v > 0);
    714   ASSERT(!Double(v).IsSpecial());
    715 
    716   bool result = false;
    717   int decimal_exponent = 0;
    718   switch (mode) {
    719     case FAST_DTOA_SHORTEST:
    720       result = Grisu3(v, buffer, length, &decimal_exponent);
    721       break;
    722     case FAST_DTOA_PRECISION:
    723       result = Grisu3Counted(v, requested_digits,
    724                              buffer, length, &decimal_exponent);
    725       break;
    726     default:
    727       UNREACHABLE();
    728   }
    729   if (result) {
    730     *decimal_point = *length + decimal_exponent;
    731     buffer[*length] = '\0';
    732   }
    733   return result;
    734 }
    735 
    736 } }  // namespace v8::internal
    737