LCOV - app.info - js/src/v8-dtoa/fast-dtoa.cc

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: | 2012-04-07 | 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          213007 : 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          213007 :   uint64_t small_distance = distance_too_high_w - unit;
          72          213007 :   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          491987 :   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           65973 :     buffer[length - 1]--;
         147           65973 :     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          213007 :   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             311 :     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          212696 :   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          213007 : static void BiggestPowerTen(uint32_t number,
         183                 :                             int number_bits,
         184                 :                             uint32_t* power,
         185                 :                             int* exponent) {
         186          213007 :   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            1179 :       if (kTen8 <= number) {
         199            1044 :         *power = kTen8;
         200            1044 :         *exponent = 8;
         201            1044 :         break;
         202                 :       }  // else fallthrough
         203                 :     case 26:
         204                 :     case 25:
         205                 :     case 24:
         206             261 :       if (kTen7 <= number) {
         207             252 :         *power = kTen7;
         208             252 :         *exponent = 7;
         209             252 :         break;
         210                 :       }  // else fallthrough
         211                 :     case 23:
         212                 :     case 22:
         213                 :     case 21:
         214                 :     case 20:
         215            9322 :       if (kTen6 <= number) {
         216             351 :         *power = kTen6;
         217             351 :         *exponent = 6;
         218             351 :         break;
         219                 :       }  // else fallthrough
         220                 :     case 19:
         221                 :     case 18:
         222                 :     case 17:
         223           52919 :       if (kTen5 <= number) {
         224           52652 :         *power = kTen5;
         225           52652 :         *exponent = 5;
         226           52652 :         break;
         227                 :       }  // else fallthrough
         228                 :     case 16:
         229                 :     case 15:
         230                 :     case 14:
         231           45821 :       if (kTen4 <= number) {
         232             605 :         *power = kTen4;
         233             605 :         *exponent = 4;
         234             605 :         break;
         235                 :       }  // else fallthrough
         236                 :     case 13:
         237                 :     case 12:
         238                 :     case 11:
         239                 :     case 10:
         240           88740 :       if (1000 <= number) {
         241           79119 :         *power = 1000;
         242           79119 :         *exponent = 3;
         243           79119 :         break;
         244                 :       }  // else fallthrough
         245                 :     case 9:
         246                 :     case 8:
         247                 :     case 7:
         248           71154 :       if (100 <= number) {
         249           61290 :         *power = 100;
         250           61290 :         *exponent = 2;
         251           61290 :         break;
         252                 :       }  // else fallthrough
         253                 :     case 6:
         254                 :     case 5:
         255                 :     case 4:
         256           17694 :       if (10 <= number) {
         257           16902 :         *power = 10;
         258           16902 :         *exponent = 1;
         259           16902 :         break;
         260                 :       }  // else fallthrough
         261                 :     case 3:
         262                 :     case 2:
         263                 :     case 1:
         264             792 :       if (1 <= number) {
         265             792 :         *power = 1;
         266             792 :         *exponent = 0;
         267             792 :         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          213007 : }
         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          213007 : bool DigitGen(DiyFp low,
         325                 :               DiyFp w,
         326                 :               DiyFp high,
         327                 :               Vector<char> buffer,
         328                 :               int* length,
         329                 :               int* kappa) {
         330          213007 :   ASSERT(low.e() == w.e() && w.e() == high.e());
         331          213007 :   ASSERT(low.f() + 1 <= high.f() - 1);
         332          213007 :   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          213007 :   uint64_t unit = 1;
         345          213007 :   DiyFp too_low = DiyFp(low.f() - unit, low.e());
         346          213007 :   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          213007 :   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          213007 :   DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
         358                 :   // Division by one is a shift.
         359          213007 :   uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
         360                 :   // Modulo by one is an and.
         361          213007 :   uint64_t fractionals = too_high.f() & (one.f() - 1);
         362                 :   uint32_t divider;
         363                 :   int divider_exponent;
         364          213007 :   BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
         365          213007 :                   &divider, &divider_exponent);
         366          213007 :   *kappa = divider_exponent + 1;
         367          213007 :   *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         1240293 :   while (*kappa > 0) {
         373          830650 :     int digit = integrals / divider;
         374          830650 :     buffer[*length] = '0' + digit;
         375          830650 :     (*length)++;
         376          830650 :     integrals %= divider;
         377          830650 :     (*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          830650 :         (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          830650 :     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           16371 :                        static_cast<uint64_t>(divider) << -one.e(), unit);
         390                 :     }
         391          814279 :     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         1453778 :   while (true) {
         409         1650414 :     fractionals *= 5;
         410         1650414 :     unit *= 5;
         411         1650414 :     unsafe_interval.set_f(unsafe_interval.f() * 5);
         412         1650414 :     unsafe_interval.set_e(unsafe_interval.e() + 1);  // Will be optimized out.
         413         1650414 :     one.set_f(one.f() >> 1);
         414         1650414 :     one.set_e(one.e() + 1);
         415                 :     // Integer division by one.
         416         1650414 :     int digit = static_cast<int>(fractionals >> -one.e());
         417         1650414 :     buffer[*length] = '0' + digit;
         418         1650414 :     (*length)++;
         419         1650414 :     fractionals &= one.f() - 1;  // Modulo by one.
         420         1650414 :     (*kappa)--;
         421         1650414 :     if (fractionals < unsafe_interval.f()) {
         422          196636 :       return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
         423          393272 :                        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          213007 : bool grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) {
         441          213007 :   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          213007 :   DiyFp boundary_minus, boundary_plus;
         447          213007 :   Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
         448          213007 :   ASSERT(boundary_plus.e() == w.e());
         449          213007 :   DiyFp ten_mk;  // Cached power of ten: 10^-k
         450                 :   int mk;        // -k
         451          213007 :   GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent,
         452          213007 :                  maximal_target_exponent, &mk, &ten_mk);
         453          213007 :   ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() +
         454                 :          DiyFp::kSignificandSize &&
         455                 :          maximal_target_exponent >= w.e() + ten_mk.e() +
         456          213007 :          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          213007 :   DiyFp scaled_w = DiyFp::Times(w, ten_mk);
         467          213007 :   ASSERT(scaled_w.e() ==
         468          213007 :          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          213007 :   DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
         475          213007 :   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          213007 :                          buffer, length, &kappa);
         486          213007 :   *decimal_exponent = -mk + kappa;
         487          213007 :   return result;
         488                 : }
         489                 :
         490                 :
         491          213007 : bool FastDtoa(double v,
         492                 :               Vector<char> buffer,
         493                 :               int* length,
         494                 :               int* point) {
         495          213007 :   ASSERT(v > 0);
         496          213007 :   ASSERT(!Double(v).IsSpecial());
         497                 :
         498                 :   int decimal_exponent;
         499          213007 :   bool result = grisu3(v, buffer, length, &decimal_exponent);
         500          213007 :   *point = *length + decimal_exponent;
         501          213007 :   buffer[*length] = '\0';
         502          213007 :   return result;
         503                 : }
         504                 :
         505                 : } }  // namespace v8::internal

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