LCOV - code coverage report
Current view: directory - js/src/v8-dtoa - fast-dtoa.cc (source / functions) Found Hit Coverage
Test: app.info Lines: 123 114 92.7 %
Date: 2012-06-02 Functions: 5 5 100.0 %

       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.
      12                 : //     * Neither the name of Google Inc. nor the names of its
      13                 : //       contributors may be used to endorse or promote products derived
      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
      18                 : // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
      19                 : // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
      20                 : // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
      21                 : // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
      22                 : // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
      23                 : // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
      24                 : // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
      25                 : // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
      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 minimal_target_exponent = -60;
      46                 : static const int maximal_target_exponent = -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          257335 : 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          257335 :   uint64_t small_distance = distance_too_high_w - unit;
      72          257335 :   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_low[ (often written as "(w_low; w_low)")
      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 rounding
     126                 :   // the buffer. If we have two potential representations we need to make sure
     127                 :   // that the chosen one is closer to w_low and w_high since v can be anywhere
     128                 :   // between them.
     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          581554 :   while (rest < small_distance &&  // Negated condition 1
     143                 :          unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
     144                 :          (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
     145                 :           small_distance - rest >= rest + ten_kappa - small_distance)) {
     146           66884 :     buffer[length - 1]--;
     147           66884 :     rest += ten_kappa;
     148                 :   }
     149                 : 
     150                 :   // We have approached w+ as much as possible. We now test if approaching w-
     151                 :   // would require changing the buffer. If yes, then we have two possible
     152                 :   // representations close to w, but we cannot decide which one is closer.
     153          257335 :   if (rest < big_distance &&
     154                 :       unsafe_interval - rest >= ten_kappa &&
     155                 :       (rest + ten_kappa < big_distance ||
     156                 :        big_distance - rest > rest + ten_kappa - big_distance)) {
     157             315 :     return false;
     158                 :   }
     159                 : 
     160                 :   // Weeding test.
     161                 :   //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
     162                 :   //   Since too_low = too_high - unsafe_interval this is equivalent to
     163                 :   //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
     164                 :   //   Conceptually we have: rest ~= too_high - buffer
     165          257020 :   return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
     166                 : }
     167                 : 
     168                 : 
     169                 : 
     170                 : static const uint32_t kTen4 = 10000;
     171                 : static const uint32_t kTen5 = 100000;
     172                 : static const uint32_t kTen6 = 1000000;
     173                 : static const uint32_t kTen7 = 10000000;
     174                 : static const uint32_t kTen8 = 100000000;
     175                 : static const uint32_t kTen9 = 1000000000;
     176                 : 
     177                 : // Returns the biggest power of ten that is less than or equal than the given
     178                 : // number. We furthermore receive the maximum number of bits 'number' has.
     179                 : // If number_bits == 0 then 0^-1 is returned
     180                 : // The number of bits must be <= 32.
     181                 : // Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)).
     182          257335 : static void BiggestPowerTen(uint32_t number,
     183                 :                             int number_bits,
     184                 :                             uint32_t* power,
     185                 :                             int* exponent) {
     186          257335 :   switch (number_bits) {
     187                 :     case 32:
     188                 :     case 31:
     189                 :     case 30:
     190             540 :       if (kTen9 <= number) {
     191               0 :         *power = kTen9;
     192               0 :         *exponent = 9;
     193               0 :         break;
     194                 :       }  // else fallthrough
     195                 :     case 29:
     196                 :     case 28:
     197                 :     case 27:
     198           34702 :       if (kTen8 <= number) {
     199           34150 :         *power = kTen8;
     200           34150 :         *exponent = 8;
     201           34150 :         break;
     202                 :       }  // else fallthrough
     203                 :     case 26:
     204                 :     case 25:
     205                 :     case 24:
     206             976 :       if (kTen7 <= number) {
     207             924 :         *power = kTen7;
     208             924 :         *exponent = 7;
     209             924 :         break;
     210                 :       }  // else fallthrough
     211                 :     case 23:
     212                 :     case 22:
     213                 :     case 21:
     214                 :     case 20:
     215            9417 :       if (kTen6 <= number) {
     216             426 :         *power = kTen6;
     217             426 :         *exponent = 6;
     218             426 :         break;
     219                 :       }  // else fallthrough
     220                 :     case 19:
     221                 :     case 18:
     222                 :     case 17:
     223           59992 :       if (kTen5 <= number) {
     224           59698 :         *power = kTen5;
     225           59698 :         *exponent = 5;
     226           59698 :         break;
     227                 :       }  // else fallthrough
     228                 :     case 16:
     229                 :     case 15:
     230                 :     case 14:
     231           46516 :       if (kTen4 <= number) {
     232             675 :         *power = kTen4;
     233             675 :         *exponent = 4;
     234             675 :         break;
     235                 :       }  // else fallthrough
     236                 :     case 13:
     237                 :     case 12:
     238                 :     case 11:
     239                 :     case 10:
     240           92035 :       if (1000 <= number) {
     241           82309 :         *power = 1000;
     242           82309 :         *exponent = 3;
     243           82309 :         break;
     244                 :       }  // else fallthrough
     245                 :     case 9:
     246                 :     case 8:
     247                 :     case 7:
     248           71302 :       if (100 <= number) {
     249           61427 :         *power = 100;
     250           61427 :         *exponent = 2;
     251           61427 :         break;
     252                 :       }  // else fallthrough
     253                 :     case 6:
     254                 :     case 5:
     255                 :     case 4:
     256           17726 :       if (10 <= number) {
     257           16923 :         *power = 10;
     258           16923 :         *exponent = 1;
     259           16923 :         break;
     260                 :       }  // else fallthrough
     261                 :     case 3:
     262                 :     case 2:
     263                 :     case 1:
     264             803 :       if (1 <= number) {
     265             803 :         *power = 1;
     266             803 :         *exponent = 0;
     267             803 :         break;
     268                 :       }  // else fallthrough
     269                 :     case 0:
     270               0 :       *power = 0;
     271               0 :       *exponent = -1;
     272               0 :       break;
     273                 :     default:
     274                 :       // Following assignments are here to silence compiler warnings.
     275               0 :       *power = 0;
     276               0 :       *exponent = 0;
     277               0 :       UNREACHABLE();
     278                 :   }
     279          257335 : }
     280                 : 
     281                 : 
     282                 : // Generates the digits of input number w.
     283                 : // w is a floating-point number (DiyFp), consisting of a significand and an
     284                 : // exponent. Its exponent is bounded by minimal_target_exponent and
     285                 : // maximal_target_exponent.
     286                 : //       Hence -60 <= w.e() <= -32.
     287                 : //
     288                 : // Returns false if it fails, in which case the generated digits in the buffer
     289                 : // should not be used.
     290                 : // Preconditions:
     291                 : //  * low, w and high are correct up to 1 ulp (unit in the last place). That
     292                 : //    is, their error must be less that a unit of their last digits.
     293                 : //  * low.e() == w.e() == high.e()
     294                 : //  * low < w < high, and taking into account their error: low~ <= high~
     295                 : //  * minimal_target_exponent <= w.e() <= maximal_target_exponent
     296                 : // Postconditions: returns false if procedure fails.
     297                 : //   otherwise:
     298                 : //     * buffer is not null-terminated, but len contains the number of digits.
     299                 : //     * buffer contains the shortest possible decimal digit-sequence
     300                 : //       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
     301                 : //       correct values of low and high (without their error).
     302                 : //     * if more than one decimal representation gives the minimal number of
     303                 : //       decimal digits then the one closest to W (where W is the correct value
     304                 : //       of w) is chosen.
     305                 : // Remark: this procedure takes into account the imprecision of its input
     306                 : //   numbers. If the precision is not enough to guarantee all the postconditions
     307                 : //   then false is returned. This usually happens rarely (~0.5%).
     308                 : //
     309                 : // Say, for the sake of example, that
     310                 : //   w.e() == -48, and w.f() == 0x1234567890abcdef
     311                 : // w's value can be computed by w.f() * 2^w.e()
     312                 : // We can obtain w's integral digits by simply shifting w.f() by -w.e().
     313                 : //  -> w's integral part is 0x1234
     314                 : //  w's fractional part is therefore 0x567890abcdef.
     315                 : // Printing w's integral part is easy (simply print 0x1234 in decimal).
     316                 : // In order to print its fraction we repeatedly multiply the fraction by 10 and
     317                 : // get each digit. Example the first digit after the point would be computed by
     318                 : //   (0x567890abcdef * 10) >> 48. -> 3
     319                 : // The whole thing becomes slightly more complicated because we want to stop
     320                 : // once we have enough digits. That is, once the digits inside the buffer
     321                 : // represent 'w' we can stop. Everything inside the interval low - high
     322                 : // represents w. However we have to pay attention to low, high and w's
     323                 : // imprecision.
     324          257335 : bool DigitGen(DiyFp low,
     325                 :               DiyFp w,
     326                 :               DiyFp high,
     327                 :               Vector<char> buffer,
     328                 :               int* length,
     329                 :               int* kappa) {
     330          257335 :   ASSERT(low.e() == w.e() && w.e() == high.e());
     331          257335 :   ASSERT(low.f() + 1 <= high.f() - 1);
     332          257335 :   ASSERT(minimal_target_exponent <= w.e() && w.e() <= maximal_target_exponent);
     333                 :   // low, w and high are imprecise, but by less than one ulp (unit in the last
     334                 :   // place).
     335                 :   // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
     336                 :   // the new numbers are outside of the interval we want the final
     337                 :   // representation to lie in.
     338                 :   // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
     339                 :   // numbers that are certain to lie in the interval. We will use this fact
     340                 :   // later on.
     341                 :   // We will now start by generating the digits within the uncertain
     342                 :   // interval. Later we will weed out representations that lie outside the safe
     343                 :   // interval and thus _might_ lie outside the correct interval.
     344          257335 :   uint64_t unit = 1;
     345          257335 :   DiyFp too_low = DiyFp(low.f() - unit, low.e());
     346          257335 :   DiyFp too_high = DiyFp(high.f() + unit, high.e());
     347                 :   // too_low and too_high are guaranteed to lie outside the interval we want the
     348                 :   // generated number in.
     349          257335 :   DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
     350                 :   // We now cut the input number into two parts: the integral digits and the
     351                 :   // fractionals. We will not write any decimal separator though, but adapt
     352                 :   // kappa instead.
     353                 :   // Reminder: we are currently computing the digits (stored inside the buffer)
     354                 :   // such that:   too_low < buffer * 10^kappa < too_high
     355                 :   // We use too_high for the digit_generation and stop as soon as possible.
     356                 :   // If we stop early we effectively round down.
     357          257335 :   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
     358                 :   // Division by one is a shift.
     359          257335 :   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
     360                 :   // Modulo by one is an and.
     361          257335 :   uint64_t fractionals = too_high.f() & (one.f() - 1);
     362                 :   uint32_t divider;
     363                 :   int divider_exponent;
     364          257335 :   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
     365          257335 :                   &divider, &divider_exponent);
     366          257335 :   *kappa = divider_exponent + 1;
     367          257335 :   *length = 0;
     368                 :   // Loop invariant: buffer = too_high / 10^kappa  (integer division)
     369                 :   // The invariant holds for the first iteration: kappa has been initialized
     370                 :   // with the divider exponent + 1. And the divider is the biggest power of ten
     371                 :   // that is smaller than integrals.
     372         1687432 :   while (*kappa > 0) {
     373         1189631 :     int digit = integrals / divider;
     374         1189631 :     buffer[*length] = '0' + digit;
     375         1189631 :     (*length)++;
     376         1189631 :     integrals %= divider;
     377         1189631 :     (*kappa)--;
     378                 :     // Note that kappa now equals the exponent of the divider and that the
     379                 :     // invariant thus holds again.
     380                 :     uint64_t rest =
     381         1189631 :         (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
     382                 :     // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
     383                 :     // Reminder: unsafe_interval.e() == one.e()
     384         1189631 :     if (rest < unsafe_interval.f()) {
     385                 :       // Rounding down (by not emitting the remaining digits) yields a number
     386                 :       // that lies within the unsafe interval.
     387                 :       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
     388                 :                        unsafe_interval.f(), rest,
     389           16869 :                        static_cast<uint64_t>(divider) << -one.e(), unit);
     390                 :     }
     391         1172762 :     divider /= 10;
     392                 :   }
     393                 : 
     394                 :   // The integrals have been generated. We are at the point of the decimal
     395                 :   // separator. In the following loop we simply multiply the remaining digits by
     396                 :   // 10 and divide by one. We just need to pay attention to multiply associated
     397                 :   // data (like the interval or 'unit'), too.
     398                 :   // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
     399                 :   // increase its (imaginary) exponent. At the same time we decrease the
     400                 :   // divider's (one's) exponent and shift its significand.
     401                 :   // Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
     402                 :   //      fractionals.f *= 10;
     403                 :   //      fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
     404                 :   //      one.f >>= 1; one.e++;                 // value remains unchanged.
     405                 :   //      and we have again fractionals.e == one.e which allows us to divide
     406                 :   //           fractionals.f() by one.f()
     407                 :   // We simply combine the *= 10 and the >>= 1.
     408         1612840 :   while (true) {
     409         1853306 :     fractionals *= 5;
     410         1853306 :     unit *= 5;
     411         1853306 :     unsafe_interval.set_f(unsafe_interval.f() * 5);
     412         1853306 :     unsafe_interval.set_e(unsafe_interval.e() + 1);  // Will be optimized out.
     413         1853306 :     one.set_f(one.f() >> 1);
     414         1853306 :     one.set_e(one.e() + 1);
     415                 :     // Integer division by one.
     416         1853306 :     int digit = static_cast<int>(fractionals >> -one.e());
     417         1853306 :     buffer[*length] = '0' + digit;
     418         1853306 :     (*length)++;
     419         1853306 :     fractionals &= one.f() - 1;  // Modulo by one.
     420         1853306 :     (*kappa)--;
     421         1853306 :     if (fractionals < unsafe_interval.f()) {
     422          240466 :       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
     423          480932 :                        unsafe_interval.f(), fractionals, one.f(), unit);
     424                 :     }
     425                 :   }
     426                 : }
     427                 : 
     428                 : 
     429                 : // Provides a decimal representation of v.
     430                 : // Returns true if it succeeds, otherwise the result cannot be trusted.
     431                 : // There will be *length digits inside the buffer (not null-terminated).
     432                 : // If the function returns true then
     433                 : //        v == (double) (buffer * 10^decimal_exponent).
     434                 : // The digits in the buffer are the shortest representation possible: no
     435                 : // 0.09999999999999999 instead of 0.1. The shorter representation will even be
     436                 : // chosen even if the longer one would be closer to v.
     437                 : // The last digit will be closest to the actual v. That is, even if several
     438                 : // digits might correctly yield 'v' when read again, the closest will be
     439                 : // computed.
     440          257335 : bool grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) {
     441          257335 :   DiyFp w = Double(v).AsNormalizedDiyFp();
     442                 :   // boundary_minus and boundary_plus are the boundaries between v and its
     443                 :   // closest floating-point neighbors. Any number strictly between
     444                 :   // boundary_minus and boundary_plus will round to v when convert to a double.
     445                 :   // Grisu3 will never output representations that lie exactly on a boundary.
     446          257335 :   DiyFp boundary_minus, boundary_plus;
     447          257335 :   Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
     448          257335 :   ASSERT(boundary_plus.e() == w.e());
     449          257335 :   DiyFp ten_mk;  // Cached power of ten: 10^-k
     450                 :   int mk;        // -k
     451          257335 :   GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent,
     452          257335 :                  maximal_target_exponent, &mk, &ten_mk);
     453          257335 :   ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() +
     454                 :          DiyFp::kSignificandSize &&
     455                 :          maximal_target_exponent >= w.e() + ten_mk.e() +
     456          257335 :          DiyFp::kSignificandSize);
     457                 :   // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
     458                 :   // 64 bit significand and ten_mk is thus only precise up to 64 bits.
     459                 : 
     460                 :   // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
     461                 :   // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
     462                 :   // off by a small amount.
     463                 :   // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
     464                 :   // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
     465                 :   //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
     466          257335 :   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
     467          257335 :   ASSERT(scaled_w.e() ==
     468          257335 :          boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
     469                 :   // In theory it would be possible to avoid some recomputations by computing
     470                 :   // the difference between w and boundary_minus/plus (a power of 2) and to
     471                 :   // compute scaled_boundary_minus/plus by subtracting/adding from
     472                 :   // scaled_w. However the code becomes much less readable and the speed
     473                 :   // enhancements are not terriffic.
     474          257335 :   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
     475          257335 :   DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);
     476                 : 
     477                 :   // DigitGen will generate the digits of scaled_w. Therefore we have
     478                 :   // v == (double) (scaled_w * 10^-mk).
     479                 :   // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
     480                 :   // integer than it will be updated. For instance if scaled_w == 1.23 then
     481                 :   // the buffer will be filled with "123" und the decimal_exponent will be
     482                 :   // decreased by 2.
     483                 :   int kappa;
     484                 :   bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
     485          257335 :                          buffer, length, &kappa);
     486          257335 :   *decimal_exponent = -mk + kappa;
     487          257335 :   return result;
     488                 : }
     489                 : 
     490                 : 
     491          257335 : bool FastDtoa(double v,
     492                 :               Vector<char> buffer,
     493                 :               int* length,
     494                 :               int* point) {
     495          257335 :   ASSERT(v > 0);
     496          257335 :   ASSERT(!Double(v).IsSpecial());
     497                 : 
     498                 :   int decimal_exponent;
     499          257335 :   bool result = grisu3(v, buffer, length, &decimal_exponent);
     500          257335 :   *point = *length + decimal_exponent;
     501          257335 :   buffer[*length] = '\0';
     502          257335 :   return result;
     503                 : }
     504                 : 
     505                 : } }  // namespace v8::internal

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