third_party/WebKit/Source/wtf/dtoa/fast-dtoa.cc - Issue 2764243002: Move files in wtf/ to platform/wtf/ (Part 9).

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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.

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 "fast-dtoa.h"

29

30 #include "cached-powers.h"

31 #include "diy-fp.h"

32 #include "double.h"

33

34 namespace WTF {

35

36 namespace double_conversion {

37

38 // The minimal and maximal target exponent define the range of w's binary

39 // exponent, where 'w' is the result of multiplying the input by a cached po wer

40 // of ten.

41 //

42 // A different range might be chosen on a different platform, to optimize di git

43 // generation, but a smaller range requires more powers of ten to be cached.

44 static const int kMinimalTargetExponent = -60;

45 static const int kMaximalTargetExponent = -32;

46

47

48 // Adjusts the last digit of the generated number, and screens out generated

49 // solutions that may be inaccurate. A solution may be inaccurate if it is

50 // outside the safe interval, or if we cannot prove that it is closer to the

51 // input than a neighboring representation of the same length.

52 //

53 // Input: * buffer containing the digits of too_high / 10^kappa

54 // * the buffer's length

55 // * distance_too_high_w == (too_high - w).f() * unit

56 // * unsafe_interval == (too_high - too_low).f() * unit

57 // * rest = (too_high - buffer * 10^kappa).f() * unit

58 // * ten_kappa = 10^kappa * unit

59 // * unit = the common multiplier

60 // Output: returns true if the buffer is guaranteed to contain the closest

61 // representable number to the input.

62 // Modifies the generated digits in the buffer to approach (round towards) w.

63 static bool RoundWeed(Vector<char> buffer,

64 int length,

65 uint64_t distance_too_high_w,

66 uint64_t unsafe_interval,

67 uint64_t rest,

68 uint64_t ten_kappa,

69 uint64_t unit) {

70 uint64_t small_distance = distance_too_high_w - unit;

71 uint64_t big_distance = distance_too_high_w + unit;

72 // Let w_low = too_high - big_distance, and

73 // w_high = too_high - small_distance.

74 // Note: w_low < w < w_high

75 //

76 // The real w (* unit) must lie somewhere inside the interval

77 // ]w_low; w_high[ (often written as "(w_low; w_high)")

78

79 // Basically the buffer currently contains a number in the unsafe interv al

80 // ]too_low; too_high[ with too_low < w < too_high

81 //

82 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

83 // ^v 1 unit ^ ^ ^ ^

84 // boundary_high --------------------- . . . .

85 // ^v 1 unit . . . .

86 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .

87 // . . ^ . .

88 // . big_distance . . .

89 // . . . . rest

90 // small_distance . . . .

91 // v . . . .

92 // w_high - - - - - - - - - - - - - - - - - - . . . .

93 // ^v 1 unit . . . .

94 // w ---------------------------------------- . . . .

95 // ^v 1 unit v . . .

96 // w_low - - - - - - - - - - - - - - - - - - - - - . . .

97 // . . v

98 // buffer --------------------------------------------------+-------+-- ------

99 // . .

100 // safe_interval .

101 // v .

102 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .

103 // ^v 1 unit .

104 // boundary_low ------------------------- unsafe_in terval

105 // ^v 1 unit v

106 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

107 //

108 //

109 // Note that the value of buffer could lie anywhere inside the range too _low

110 // to too_high.

111 //

112 // boundary_low, boundary_high and w are approximations of the real boun daries

113 // and v (the input number). They are guaranteed to be precise up to one unit.

114 // In fact the error is guaranteed to be strictly less than one unit.

115 //

116 // Anything that lies outside the unsafe interval is guaranteed not to r ound

117 // to v when read again.

118 // Anything that lies inside the safe interval is guaranteed to round to v

119 // when read again.

120 // If the number inside the buffer lies inside the unsafe interval but n ot

121 // inside the safe interval then we simply do not know and bail out (ret urning

122 // false).

123 //

124 // Similarly we have to take into account the imprecision of 'w' when fi nding

125 // the closest representation of 'w'. If we have two potential

126 // representations, and one is closer to both w_low and w_high, then we know

127 // it is closer to the actual value v.

128 //

129 // By generating the digits of too_high we got the largest (closest to

130 // too_high) buffer that is still in the unsafe interval. In the case wh ere

131 // w_high < buffer < too_high we try to decrement the buffer.

132 // This way the buffer approaches (rounds towards) w.

133 // There are 3 conditions that stop the decrementation process:

134 // 1) the buffer is already below w_high

135 // 2) decrementing the buffer would make it leave the unsafe interval

136 // 3) decrementing the buffer would yield a number below w_high and fa rther

137 // away than the current number. In other words:

138 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high

139 // Instead of using the buffer directly we use its distance to too_high.

140 // Conceptually rest ~= too_high - buffer

141 // We need to do the following tests in this order to avoid over- and

142 // underflows.

143 ASSERT(rest <= unsafe_interval);

144 while (rest < small_distance && // Negated condition 1

145 unsafe_interval - rest >= ten_kappa && // Negated condition 2

146 (rest + ten_kappa < small_distance \|\| // buffer{-1} > w_high

147 small_distance - rest >= rest + ten_kappa - small_distance)) {

148 buffer[length - 1]--;

149 rest += ten_kappa;

150 }

151

152 // We have approached w+ as much as possible. We now test if approaching w-

153 // would require changing the buffer. If yes, then we have two possible

154 // representations close to w, but we cannot decide which one is closer.

155 if (rest < big_distance &&

156 unsafe_interval - rest >= ten_kappa &&

157 (rest + ten_kappa < big_distance \|\|

158 big_distance - rest > rest + ten_kappa - big_distance)) {

159 return false;

160 }

161

162 // Weeding test.

163 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]

164 // Since too_low = too_high - unsafe_interval this is equivalent to

165 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]

166 // Conceptually we have: rest ~= too_high - buffer

167 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);

168 }

169

170

171 // Rounds the buffer upwards if the result is closer to v by possibly adding

172 // 1 to the buffer. If the precision of the calculation is not sufficient to

173 // round correctly, return false.

174 // The rounding might shift the whole buffer in which case the kappa is

175 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.

176 //

177 // If 2*rest > ten_kappa then the buffer needs to be round up.

178 // rest can have an error of +/- 1 unit. This function accounts for the

179 // imprecision and returns false, if the rounding direction cannot be

180 // unambiguously determined.

181 //

182 // Precondition: rest < ten_kappa.

183 static bool RoundWeedCounted(Vector<char> buffer,

184 int length,

185 uint64_t rest,

186 uint64_t ten_kappa,

187 uint64_t unit,

188 int* kappa) {

189 ASSERT(rest < ten_kappa);

190 // The following tests are done in a specific order to avoid overflows. They

191 // will work correctly with any uint64 values of rest < ten_kappa and un it.

192 //

193 // If the unit is too big, then we don't know which way to round. For ex ample

194 // a unit of 50 means that the real number lies within rest +/- 50. If

195 // 10^kappa == 40 then there is no way to tell which way to round.

196 if (unit >= ten_kappa) return false;

197 // Even if unit is just half the size of 10^kappa we are already complet ely

198 // lost. (And after the previous test we know that the expression will n ot

199 // over/underflow.)

200 if (ten_kappa - unit <= unit) return false;

201 // If 2 * (rest + unit) <= 10^kappa we can safely round down.

202 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {

203 return true;

204 }

205 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.

206 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {

207 // Increment the last digit recursively until we find a non '9' digi t.

208 buffer[length - 1]++;

209 for (int i = length - 1; i > 0; --i) {

210 if (buffer[i] != '0' + 10) break;

211 buffer[i] = '0';

212 buffer[i - 1]++;

213 }

214 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the

215 // exception of the first digit all digits are now '0'. Simply switc h the

216 // first digit to '1' and adjust the kappa. Example: "99" becomes "1 0" and

217 // the power (the kappa) is increased.

218 if (buffer[0] == '0' + 10) {

219 buffer[0] = '1';

220 (*kappa) += 1;

221 }

222 return true;

223 }

224 return false;

225 }

226

227

228 static const uint32_t kTen4 = 10000;

229 static const uint32_t kTen5 = 100000;

230 static const uint32_t kTen6 = 1000000;

231 static const uint32_t kTen7 = 10000000;

232 static const uint32_t kTen8 = 100000000;

233 static const uint32_t kTen9 = 1000000000;

234

235 // Returns the biggest power of ten that is less than or equal to the given

236 // number. We furthermore receive the maximum number of bits 'number' has.

237 // If number_bits == 0 then 0^-1 is returned

238 // The number of bits must be <= 32.

239 // Precondition: number < (1 << (number_bits + 1)).

240 static void BiggestPowerTen(uint32_t number,

241 int number_bits,

242 uint32_t* power,

243 int* exponent) {

244 ASSERT(number < (uint32_t)(1 << (number_bits + 1)));

245

246 switch (number_bits) {

247 case 32:

248 case 31:

249 case 30:

250 if (kTen9 <= number) {

251 *power = kTen9;

252 *exponent = 9;

253 break;

254 } // else fallthrough

255 case 29:

256 case 28:

257 case 27:

258 if (kTen8 <= number) {

259 *power = kTen8;

260 *exponent = 8;

261 break;

262 } // else fallthrough

263 case 26:

264 case 25:

265 case 24:

266 if (kTen7 <= number) {

267 *power = kTen7;

268 *exponent = 7;

269 break;

270 } // else fallthrough

271 case 23:

272 case 22:

273 case 21:

274 case 20:

275 if (kTen6 <= number) {

276 *power = kTen6;

277 *exponent = 6;

278 break;

279 } // else fallthrough

280 case 19:

281 case 18:

282 case 17:

283 if (kTen5 <= number) {

284 *power = kTen5;

285 *exponent = 5;

286 break;

287 } // else fallthrough

288 case 16:

289 case 15:

290 case 14:

291 if (kTen4 <= number) {

292 *power = kTen4;

293 *exponent = 4;

294 break;

295 } // else fallthrough

296 case 13:

297 case 12:

298 case 11:

299 case 10:

300 if (1000 <= number) {

301 *power = 1000;

302 *exponent = 3;

303 break;

304 } // else fallthrough

305 case 9:

306 case 8:

307 case 7:

308 if (100 <= number) {

309 *power = 100;

310 *exponent = 2;

311 break;

312 } // else fallthrough

313 case 6:

314 case 5:

315 case 4:

316 if (10 <= number) {

317 *power = 10;

318 *exponent = 1;

319 break;

320 } // else fallthrough

321 case 3:

322 case 2:

323 case 1:

324 if (1 <= number) {

325 *power = 1;

326 *exponent = 0;

327 break;

328 } // else fallthrough

329 case 0:

330 *power = 0;

331 *exponent = -1;

332 break;

333 default:

334 // Following assignments are here to silence compiler warnings.

335 *power = 0;

336 *exponent = 0;

337 UNREACHABLE();

338 }

339 }

340

341

342 // Generates the digits of input number w.

343 // w is a floating-point number (DiyFp), consisting of a significand and an

344 // exponent. Its exponent is bounded by kMinimalTargetExponent and

345 // kMaximalTargetExponent.

346 // Hence -60 <= w.e() <= -32.

347 //

348 // Returns false if it fails, in which case the generated digits in the buff er

349 // should not be used.

350 // Preconditions:

351 // * low, w and high are correct up to 1 ulp (unit in the last place). That

352 // is, their error must be less than a unit of their last digits.

353 // * low.e() == w.e() == high.e()

354 // * low < w < high, and taking into account their error: low~ <= high~

355 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent

356 // Postconditions: returns false if procedure fails.

357 // otherwise:

358 // * buffer is not null-terminated, but len contains the number of digit s.

359 // * buffer contains the shortest possible decimal digit-sequence

360 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are th e

361 // correct values of low and high (without their error).

362 // * if more than one decimal representation gives the minimal number of

363 // decimal digits then the one closest to W (where W is the correct va lue

364 // of w) is chosen.

365 // Remark: this procedure takes into account the imprecision of its input

366 // numbers. If the precision is not enough to guarantee all the postcondit ions

367 // then false is returned. This usually happens rarely (~0.5%).

368 //

369 // Say, for the sake of example, that

370 // w.e() == -48, and w.f() == 0x1234567890abcdef

371 // w's value can be computed by w.f() * 2^w.e()

372 // We can obtain w's integral digits by simply shifting w.f() by -w.e().

373 // -> w's integral part is 0x1234

374 // w's fractional part is therefore 0x567890abcdef.

375 // Printing w's integral part is easy (simply print 0x1234 in decimal).

376 // In order to print its fraction we repeatedly multiply the fraction by 10 and

377 // get each digit. Example the first digit after the point would be computed by

378 // (0x567890abcdef * 10) >> 48. -> 3

379 // The whole thing becomes slightly more complicated because we want to stop

380 // once we have enough digits. That is, once the digits inside the buffer

381 // represent 'w' we can stop. Everything inside the interval low - high

382 // represents w. However we have to pay attention to low, high and w's

383 // imprecision.

384 static bool DigitGen(DiyFp low,

385 DiyFp w,

386 DiyFp high,

387 Vector<char> buffer,

388 int* length,

389 int* kappa) {

390 ASSERT(low.e() == w.e() && w.e() == high.e());

391 ASSERT(low.f() + 1 <= high.f() - 1);

392 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponen t);

393 // low, w and high are imprecise, but by less than one ulp (unit in the last

394 // place).

395 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain t hat

396 // the new numbers are outside of the interval we want the final

397 // representation to lie in.

398 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would y ield

399 // numbers that are certain to lie in the interval. We will use this fac t

400 // later on.

401 // We will now start by generating the digits within the uncertain

402 // interval. Later we will weed out representations that lie outside the safe

403 // interval and thus _might_ lie outside the correct interval.

404 uint64_t unit = 1;

405 DiyFp too_low = DiyFp(low.f() - unit, low.e());

406 DiyFp too_high = DiyFp(high.f() + unit, high.e());

407 // too_low and too_high are guaranteed to lie outside the interval we wa nt the

408 // generated number in.

409 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);

410 // We now cut the input number into two parts: the integral digits and t he

411 // fractionals. We will not write any decimal separator though, but adap t

412 // kappa instead.

413 // Reminder: we are currently computing the digits (stored inside the bu ffer)

414 // such that: too_low < buffer * 10^kappa < too_high

415 // We use too_high for the digit_generation and stop as soon as possible .

416 // If we stop early we effectively round down.

417 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());

418 // Division by one is a shift.

419 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());

420 // Modulo by one is an and.

421 uint64_t fractionals = too_high.f() & (one.f() - 1);

422 uint32_t divisor;

423 int divisor_exponent;

424 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),

425 &divisor, &divisor_exponent);

426 *kappa = divisor_exponent + 1;

427 *length = 0;

428 // Loop invariant: buffer = too_high / 10^kappa (integer division)

429 // The invariant holds for the first iteration: kappa has been initializ ed

430 // with the divisor exponent + 1. And the divisor is the biggest power o f ten

431 // that is smaller than integrals.

432 while (*kappa > 0) {

433 char digit = static_cast<char>(integrals / divisor);

434 buffer[*length] = '0' + digit;

435 (*length)++;

436 integrals %= divisor;

437 (*kappa)--;

438 // Note that kappa now equals the exponent of the divisor and that t he

439 // invariant thus holds again.

440 uint64_t rest =

441 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;

442 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())

443 // Reminder: unsafe_interval.e() == one.e()

444 if (rest < unsafe_interval.f()) {

445 // Rounding down (by not emitting the remaining digits) yields a number

446 // that lies within the unsafe interval.

447 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),

448 unsafe_interval.f(), rest,

449 static_cast<uint64_t>(divisor) << -one.e(), uni t);

450 }

451 divisor /= 10;

452 }

453

454 // The integrals have been generated. We are at the point of the decimal

455 // separator. In the following loop we simply multiply the remaining dig its by

456 // 10 and divide by one. We just need to pay attention to multiply assoc iated

457 // data (like the interval or 'unit'), too.

458 // Note that the multiplication by 10 does not overflow, because w.e >= -60

459 // and thus one.e >= -60.

460 ASSERT(one.e() >= -60);

461 ASSERT(fractionals < one.f());

462 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());

463 while (true) {

464 fractionals *= 10;

465 unit *= 10;

466 unsafe_interval.set_f(unsafe_interval.f() * 10);

467 // Integer division by one.

468 char digit = static_cast<char>(fractionals >> -one.e());

469 buffer[*length] = '0' + digit;

470 (*length)++;

471 fractionals &= one.f() - 1; // Modulo by one.

472 (*kappa)--;

473 if (fractionals < unsafe_interval.f()) {

474 return RoundWeed(buffer, length, DiyFp::Minus(too_high, w).f() unit,

475 unsafe_interval.f(), fractionals, one.f(), unit );

476 }

477 }

478 }

479

480

481

482 // Generates (at most) requested_digits digits of input number w.

483 // w is a floating-point number (DiyFp), consisting of a significand and an

484 // exponent. Its exponent is bounded by kMinimalTargetExponent and

485 // kMaximalTargetExponent.

486 // Hence -60 <= w.e() <= -32.

487 //

488 // Returns false if it fails, in which case the generated digits in the buff er

489 // should not be used.

490 // Preconditions:

491 // * w is correct up to 1 ulp (unit in the last place). That

492 // is, its error must be strictly less than a unit of its last digit.

493 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent

494 //

495 // Postconditions: returns false if procedure fails.

496 // otherwise:

497 // * buffer is not null-terminated, but length contains the number of

498 // digits.

499 // * the representation in buffer is the most precise representation of

500 // requested_digits digits.

501 // * buffer contains at most requested_digits digits of w. If there are less

502 // than requested_digits digits then some trailing '0's have been remo ved.

503 // * kappa is such that

504 // w = buffer * 10^kappa + eps with \|eps\| < 10^kappa / 2.

505 //

506 // Remark: This procedure takes into account the imprecision of its input

507 // numbers. If the precision is not enough to guarantee all the postcondit ions

508 // then false is returned. This usually happens rarely, but the failure-ra te

509 // increases with higher requested_digits.

510 static bool DigitGenCounted(DiyFp w,

511 int requested_digits,

512 Vector<char> buffer,

513 int* length,

514 int* kappa) {

515 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponen t);

516 ASSERT(kMinimalTargetExponent >= -60);

517 ASSERT(kMaximalTargetExponent <= -32);

518 // w is assumed to have an error less than 1 unit. Whenever w is scaled we

519 // also scale its error.

520 uint64_t w_error = 1;

521 // We cut the input number into two parts: the integral digits and the

522 // fractional digits. We don't emit any decimal separator, but adapt kap pa

523 // instead. Example: instead of writing "1.2" we put "12" into the buffe r and

524 // increase kappa by 1.

525 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());

526 // Division by one is a shift.

527 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());

528 // Modulo by one is an and.

529 uint64_t fractionals = w.f() & (one.f() - 1);

530 uint32_t divisor;

531 int divisor_exponent;

532 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),

533 &divisor, &divisor_exponent);

534 *kappa = divisor_exponent + 1;

535 *length = 0;

536

537 // Loop invariant: buffer = w / 10^kappa (integer division)

538 // The invariant holds for the first iteration: kappa has been initializ ed

539 // with the divisor exponent + 1. And the divisor is the biggest power o f ten

540 // that is smaller than 'integrals'.

541 while (*kappa > 0) {

542 char digit = static_cast<char>(integrals / divisor);

543 buffer[*length] = '0' + digit;

544 (*length)++;

545 requested_digits--;

546 integrals %= divisor;

547 (*kappa)--;

548 // Note that kappa now equals the exponent of the divisor and that t he

549 // invariant thus holds again.

550 if (requested_digits == 0) break;

551 divisor /= 10;

552 }

553

554 if (requested_digits == 0) {

555 uint64_t rest =

556 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;

557 return RoundWeedCounted(buffer, *length, rest,

558 static_cast<uint64_t>(divisor) << -one.e(), w_error,

559 kappa);

560 }

561

562 // The integrals have been generated. We are at the point of the decimal

563 // separator. In the following loop we simply multiply the remaining dig its by

564 // 10 and divide by one. We just need to pay attention to multiply assoc iated

565 // data (the 'unit'), too.

566 // Note that the multiplication by 10 does not overflow, because w.e >= -60

567 // and thus one.e >= -60.

568 ASSERT(one.e() >= -60);

569 ASSERT(fractionals < one.f());

570 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());

571 while (requested_digits > 0 && fractionals > w_error) {

572 fractionals *= 10;

573 w_error *= 10;

574 // Integer division by one.

575 char digit = static_cast<char>(fractionals >> -one.e());

576 buffer[*length] = '0' + digit;

577 (*length)++;

578 requested_digits--;

579 fractionals &= one.f() - 1; // Modulo by one.

580 (*kappa)--;

581 }

582 if (requested_digits != 0) return false;

583 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,

584 kappa);

585 }

586

587

588 // Provides a decimal representation of v.

589 // Returns true if it succeeds, otherwise the result cannot be trusted.

590 // There will be *length digits inside the buffer (not null-terminated).

591 // If the function returns true then

592 // v == (double) (buffer * 10^decimal_exponent).

593 // The digits in the buffer are the shortest representation possible: no

594 // 0.09999999999999999 instead of 0.1. The shorter representation will even be

595 // chosen even if the longer one would be closer to v.

596 // The last digit will be closest to the actual v. That is, even if several

597 // digits might correctly yield 'v' when read again, the closest will be

598 // computed.

599 static bool Grisu3(double v,

600 Vector<char> buffer,

601 int* length,

602 int* decimal_exponent) {

603 DiyFp w = Double(v).AsNormalizedDiyFp();

604 // boundary_minus and boundary_plus are the boundaries between v and its

605 // closest floating-point neighbors. Any number strictly between

606 // boundary_minus and boundary_plus will round to v when convert to a do uble.

607 // Grisu3 will never output representations that lie exactly on a bounda ry.

608 DiyFp boundary_minus, boundary_plus;

609 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);

610 ASSERT(boundary_plus.e() == w.e());

611 DiyFp ten_mk; // Cached power of ten: 10^-k

612 int mk; // -k

613 int ten_mk_minimal_binary_exponent =

614 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);

615 int ten_mk_maximal_binary_exponent =

616 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);

617 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

618 ten_mk_minimal_bi nary_exponent,

619 ten_mk_maximal_bi nary_exponent,

620 &ten_mk, &mk);

621 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +

622 DiyFp::kSignificandSize) &&

623 (kMaximalTargetExponent >= w.e() + ten_mk.e() +

624 DiyFp::kSignificandSize));

625 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only cont ains a

626 // 64 bit significand and ten_mk is thus only precise up to 64 bits.

627

628 // The DiyFp::Times procedure rounds its result, and ten_mk is approxima ted

629 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) ar e now

630 // off by a small amount.

631 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_ w.

632 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then

633 // (f-1) * 2^e < w10^k < (f+1) 2^e

634 DiyFp scaled_w = DiyFp::Times(w, ten_mk);

635 ASSERT(scaled_w.e() ==

636 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);

637 // In theory it would be possible to avoid some recomputations by comput ing

638 // the difference between w and boundary_minus/plus (a power of 2) and t o

639 // compute scaled_boundary_minus/plus by subtracting/adding from

640 // scaled_w. However the code becomes much less readable and the speed

641 // enhancements are not terriffic.

642 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);

643 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);

644

645 // DigitGen will generate the digits of scaled_w. Therefore we have

646 // v == (double) (scaled_w * 10^-mk).

647 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is n ot an

648 // integer than it will be updated. For instance if scaled_w == 1.23 the n

649 // the buffer will be filled with "123" und the decimal_exponent will be

650 // decreased by 2.

651 int kappa;

652 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_ plus,

653 buffer, length, &kappa);

654 *decimal_exponent = -mk + kappa;

655 return result;

656 }

657

658

659 // The "counted" version of grisu3 (see above) only generates requested_digi ts

660 // number of digits. This version does not generate the shortest representat ion,

661 // and with enough requested digits 0.1 will at some point print as 0.999999 9...

662 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and

663 // therefore the rounding strategy for halfway cases is irrelevant.

664 static bool Grisu3Counted(double v,

665 int requested_digits,

666 Vector<char> buffer,

667 int* length,

668 int* decimal_exponent) {

669 DiyFp w = Double(v).AsNormalizedDiyFp();

670 DiyFp ten_mk; // Cached power of ten: 10^-k

671 int mk; // -k

672 int ten_mk_minimal_binary_exponent =

673 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);

674 int ten_mk_maximal_binary_exponent =

675 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);

676 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

677 ten_mk_minimal_bi nary_exponent,

678 ten_mk_maximal_bi nary_exponent,

679 &ten_mk, &mk);

680 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +

681 DiyFp::kSignificandSize) &&

682 (kMaximalTargetExponent >= w.e() + ten_mk.e() +

683 DiyFp::kSignificandSize));

684 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only cont ains a

685 // 64 bit significand and ten_mk is thus only precise up to 64 bits.

686

687 // The DiyFp::Times procedure rounds its result, and ten_mk is approxima ted

688 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) ar e now

689 // off by a small amount.

690 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_ w.

691 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then

692 // (f-1) * 2^e < w10^k < (f+1) 2^e

693 DiyFp scaled_w = DiyFp::Times(w, ten_mk);

694

695 // We now have (double) (scaled_w * 10^-mk).

696 // DigitGen will generate the first requested_digits digits of scaled_w and

697 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It

698 // will not always be exactly the same since DigitGenCounted only produc es a

699 // limited number of digits.)

700 int kappa;

701 bool result = DigitGenCounted(scaled_w, requested_digits,

702 buffer, length, &kappa);

703 *decimal_exponent = -mk + kappa;

704 return result;

705 }

706

707

708 bool FastDtoa(double v,

709 FastDtoaMode mode,

710 int requested_digits,

711 Vector<char> buffer,

712 int* length,

713 int* decimal_point) {

714 ASSERT(v > 0);

715 ASSERT(!Double(v).IsSpecial());

716

717 bool result = false;

718 int decimal_exponent = 0;

719 switch (mode) {

720 case FAST_DTOA_SHORTEST:

721 result = Grisu3(v, buffer, length, &decimal_exponent);

722 break;

723 case FAST_DTOA_PRECISION:

724 result = Grisu3Counted(v, requested_digits,

725 buffer, length, &decimal_exponent);

726 break;

727 default:

728 UNREACHABLE();

729 }

730 if (result) {

731 decimal_point = length + decimal_exponent;

732 buffer[*length] = '\0';

733 }

734 return result;

735 }

736

737 } // namespace double_conversion

738

739 } // namespace WTF

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