Source/WTF/wtf/dtoa/fast-dtoa.cc - Issue 14238015: Move Source/WTF/wtf to Source/wtf

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Issue 14238015: Move Source/WTF/wtf to Source/wtf (Closed) Base URL: svn://svn.chromium.org/blink/trunk

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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 "config.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 WTF {

37

38 namespace double_conversion {

39

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

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

42 // of ten.

43 //

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

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

46 static const int kMinimalTargetExponent = -60;

47 static const int kMaximalTargetExponent = -32;

48

49

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

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

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

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

54 //

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

56 // * the buffer's length

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

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

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

60 // * ten_kappa = 10^kappa * unit

61 // * unit = the common multiplier

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

63 // representable number to the input.

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

65 static bool RoundWeed(Vector<char> buffer,

66 int length,

67 uint64_t distance_too_high_w,

68 uint64_t unsafe_interval,

69 uint64_t rest,

70 uint64_t ten_kappa,

71 uint64_t unit) {

72 uint64_t small_distance = distance_too_high_w - unit;

73 uint64_t big_distance = distance_too_high_w + unit;

74 // Let w_low = too_high - big_distance, and

75 // w_high = too_high - small_distance.

76 // Note: w_low < w < w_high

77 //

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

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

80

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

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

83 //

84 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

85 // ^v 1 unit ^ ^ ^ ^

86 // boundary_high --------------------- . . . .

87 // ^v 1 unit . . . .

88 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .

89 // . . ^ . .

90 // . big_distance . . .

91 // . . . . rest

92 // small_distance . . . .

93 // v . . . .

94 // w_high - - - - - - - - - - - - - - - - - - . . . .

95 // ^v 1 unit . . . .

96 // w ---------------------------------------- . . . .

97 // ^v 1 unit v . . .

98 // w_low - - - - - - - - - - - - - - - - - - - - - . . .

99 // . . v

100 // buffer --------------------------------------------------+-------+-- ------

101 // . .

102 // safe_interval .

103 // v .

104 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .

105 // ^v 1 unit .

106 // boundary_low ------------------------- unsafe_in terval

107 // ^v 1 unit v

108 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

109 //

110 //

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

112 // to too_high.

113 //

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

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

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

117 //

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

119 // to v when read again.

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

121 // when read again.

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

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

124 // false).

125 //

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

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

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

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

130 //

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

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

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

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

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

136 // 1) the buffer is already below w_high

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

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

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

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

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

142 // Conceptually rest ~= too_high - buffer

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

144 // underflows.

145 ASSERT(rest <= unsafe_interval);

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

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

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

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

150 buffer[length - 1]--;

151 rest += ten_kappa;

152 }

153

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

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

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

157 if (rest < big_distance &&

158 unsafe_interval - rest >= ten_kappa &&

159 (rest + ten_kappa < big_distance \|\|

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

161 return false;

162 }

163

164 // Weeding test.

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

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

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

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

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

170 }

171

172

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

174 // 1 to the buffer. If the precision of the calculation is not sufficient to

175 // round correctly, return false.

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

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

178 //

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

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

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

182 // unambiguously determined.

183 //

184 // Precondition: rest < ten_kappa.

185 static bool RoundWeedCounted(Vector<char> buffer,

186 int length,

187 uint64_t rest,

188 uint64_t ten_kappa,

189 uint64_t unit,

190 int* kappa) {

191 ASSERT(rest < ten_kappa);

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

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

194 //

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

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

197 // 10^kappa == 40 then there is no way to tell which way to round.

198 if (unit >= ten_kappa) return false;

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

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

201 // over/underflow.)

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

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

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

205 return true;

206 }

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

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

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

210 buffer[length - 1]++;

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

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

213 buffer[i] = '0';

214 buffer[i - 1]++;

215 }

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

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

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

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

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

221 buffer[0] = '1';

222 (*kappa) += 1;

223 }

224 return true;

225 }

226 return false;

227 }

228

229

230 static const uint32_t kTen4 = 10000;

231 static const uint32_t kTen5 = 100000;

232 static const uint32_t kTen6 = 1000000;

233 static const uint32_t kTen7 = 10000000;

234 static const uint32_t kTen8 = 100000000;

235 static const uint32_t kTen9 = 1000000000;

236

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

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

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

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

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

242 static void BiggestPowerTen(uint32_t number,

243 int number_bits,

244 uint32_t* power,

245 int* exponent) {

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

247

248 switch (number_bits) {

249 case 32:

250 case 31:

251 case 30:

252 if (kTen9 <= number) {

253 *power = kTen9;

254 *exponent = 9;

255 break;

256 } // else fallthrough

257 case 29:

258 case 28:

259 case 27:

260 if (kTen8 <= number) {

261 *power = kTen8;

262 *exponent = 8;

263 break;

264 } // else fallthrough

265 case 26:

266 case 25:

267 case 24:

268 if (kTen7 <= number) {

269 *power = kTen7;

270 *exponent = 7;

271 break;

272 } // else fallthrough

273 case 23:

274 case 22:

275 case 21:

276 case 20:

277 if (kTen6 <= number) {

278 *power = kTen6;

279 *exponent = 6;

280 break;

281 } // else fallthrough

282 case 19:

283 case 18:

284 case 17:

285 if (kTen5 <= number) {

286 *power = kTen5;

287 *exponent = 5;

288 break;

289 } // else fallthrough

290 case 16:

291 case 15:

292 case 14:

293 if (kTen4 <= number) {

294 *power = kTen4;

295 *exponent = 4;

296 break;

297 } // else fallthrough

298 case 13:

299 case 12:

300 case 11:

301 case 10:

302 if (1000 <= number) {

303 *power = 1000;

304 *exponent = 3;

305 break;

306 } // else fallthrough

307 case 9:

308 case 8:

309 case 7:

310 if (100 <= number) {

311 *power = 100;

312 *exponent = 2;

313 break;

314 } // else fallthrough

315 case 6:

316 case 5:

317 case 4:

318 if (10 <= number) {

319 *power = 10;

320 *exponent = 1;

321 break;

322 } // else fallthrough

323 case 3:

324 case 2:

325 case 1:

326 if (1 <= number) {

327 *power = 1;

328 *exponent = 0;

329 break;

330 } // else fallthrough

331 case 0:

332 *power = 0;

333 *exponent = -1;

334 break;

335 default:

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

337 *power = 0;

338 *exponent = 0;

339 UNREACHABLE();

340 }

341 }

342

343

344 // Generates the digits of input number w.

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

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

347 // kMaximalTargetExponent.

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

349 //

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

351 // should not be used.

352 // Preconditions:

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

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

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

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

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

358 // Postconditions: returns false if procedure fails.

359 // otherwise:

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

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

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

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

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

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

366 // of w) is chosen.

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

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

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

370 //

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

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

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

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

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

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

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

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

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

380 // (0x567890abcdef * 10) >> 48. -> 3

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

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

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

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

385 // imprecision.

386 static bool DigitGen(DiyFp low,

387 DiyFp w,

388 DiyFp high,

389 Vector<char> buffer,

390 int* length,

391 int* kappa) {

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

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

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

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

396 // place).

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

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

399 // representation to lie in.

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

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

402 // later on.

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

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

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

406 uint64_t unit = 1;

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

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

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

410 // generated number in.

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

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

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

414 // kappa instead.

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

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

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

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

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

420 // Division by one is a shift.

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

422 // Modulo by one is an and.

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

424 uint32_t divisor;

425 int divisor_exponent;

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

427 &divisor, &divisor_exponent);

428 *kappa = divisor_exponent + 1;

429 *length = 0;

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

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

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

433 // that is smaller than integrals.

434 while (*kappa > 0) {

435 int digit = integrals / divisor;

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

437 (*length)++;

438 integrals %= divisor;

439 (*kappa)--;

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

441 // invariant thus holds again.

442 uint64_t rest =

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

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

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

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

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

448 // that lies within the unsafe interval.

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

450 unsafe_interval.f(), rest,

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

452 }

453 divisor /= 10;

454 }

455

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

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

458 // 10 and divide by one. We just need to pay attention to multiply assoc iated

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

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

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

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

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

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

465 while (true) {

466 fractionals *= 10;

467 unit *= 10;

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

469 // Integer division by one.

470 int digit = static_cast<int>(fractionals >> -one.e());

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

472 (*length)++;

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

474 (*kappa)--;

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

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

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

478 }

479 }

480 }

481

482

483

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

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

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

487 // kMaximalTargetExponent.

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

489 //

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

491 // should not be used.

492 // Preconditions:

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

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

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

496 //

497 // Postconditions: returns false if procedure fails.

498 // otherwise:

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

500 // digits.

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

502 // requested_digits digits.

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

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

505 // * kappa is such that

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

507 //

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

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

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

511 // increases with higher requested_digits.

512 static bool DigitGenCounted(DiyFp w,

513 int requested_digits,

514 Vector<char> buffer,

515 int* length,

516 int* kappa) {

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

518 ASSERT(kMinimalTargetExponent >= -60);

519 ASSERT(kMaximalTargetExponent <= -32);

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

521 // also scale its error.

522 uint64_t w_error = 1;

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

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

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

526 // increase kappa by 1.

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

528 // Division by one is a shift.

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

530 // Modulo by one is an and.

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

532 uint32_t divisor;

533 int divisor_exponent;

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

535 &divisor, &divisor_exponent);

536 *kappa = divisor_exponent + 1;

537 *length = 0;

538

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

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

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

542 // that is smaller than 'integrals'.

543 while (*kappa > 0) {

544 int digit = integrals / divisor;

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

546 (*length)++;

547 requested_digits--;

548 integrals %= divisor;

549 (*kappa)--;

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

551 // invariant thus holds again.

552 if (requested_digits == 0) break;

553 divisor /= 10;

554 }

555

556 if (requested_digits == 0) {

557 uint64_t rest =

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

559 return RoundWeedCounted(buffer, *length, rest,

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

561 kappa);

562 }

563

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

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

566 // 10 and divide by one. We just need to pay attention to multiply assoc iated

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

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

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

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

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

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

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

574 fractionals *= 10;

575 w_error *= 10;

576 // Integer division by one.

577 int digit = static_cast<int>(fractionals >> -one.e());

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

579 (*length)++;

580 requested_digits--;

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

582 (*kappa)--;

583 }

584 if (requested_digits != 0) return false;

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

586 kappa);

587 }

588

589

590 // Provides a decimal representation of v.

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

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

593 // If the function returns true then

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

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

596 // 0.09999999999999999 instead of 0.1. The shorter representation will even be

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

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

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

600 // computed.

601 static bool Grisu3(double v,

602 Vector<char> buffer,

603 int* length,

604 int* decimal_exponent) {

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

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

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

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

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

610 DiyFp boundary_minus, boundary_plus;

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

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

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

614 int mk; // -k

615 int ten_mk_minimal_binary_exponent =

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

617 int ten_mk_maximal_binary_exponent =

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

619 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

620 ten_mk_minimal_bi nary_exponent,

621 ten_mk_maximal_bi nary_exponent,

622 &ten_mk, &mk);

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

624 DiyFp::kSignificandSize) &&

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

626 DiyFp::kSignificandSize));

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

628 // 64 bit significand and ten_mk is thus only precise up to 64 bits.

629

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

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

632 // off by a small amount.

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

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

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

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

637 ASSERT(scaled_w.e() ==

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

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

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

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

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

643 // enhancements are not terriffic.

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

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

646

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

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

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

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

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

652 // decreased by 2.

653 int kappa;

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

655 buffer, length, &kappa);

656 *decimal_exponent = -mk + kappa;

657 return result;

658 }

659

660

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

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

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

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

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

666 static bool Grisu3Counted(double v,

667 int requested_digits,

668 Vector<char> buffer,

669 int* length,

670 int* decimal_exponent) {

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

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

673 int mk; // -k

674 int ten_mk_minimal_binary_exponent =

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

676 int ten_mk_maximal_binary_exponent =

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

678 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(

679 ten_mk_minimal_bi nary_exponent,

680 ten_mk_maximal_bi nary_exponent,

681 &ten_mk, &mk);

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

683 DiyFp::kSignificandSize) &&

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

685 DiyFp::kSignificandSize));

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

687 // 64 bit significand and ten_mk is thus only precise up to 64 bits.

688

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

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

691 // off by a small amount.

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

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

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

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

696

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

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

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

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

701 // limited number of digits.)

702 int kappa;

703 bool result = DigitGenCounted(scaled_w, requested_digits,

704 buffer, length, &kappa);

705 *decimal_exponent = -mk + kappa;

706 return result;

707 }

708

709

710 bool FastDtoa(double v,

711 FastDtoaMode mode,

712 int requested_digits,

713 Vector<char> buffer,

714 int* length,

715 int* decimal_point) {

716 ASSERT(v > 0);

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

718

719 bool result = false;

720 int decimal_exponent = 0;

721 switch (mode) {

722 case FAST_DTOA_SHORTEST:

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

724 break;

725 case FAST_DTOA_PRECISION:

726 result = Grisu3Counted(v, requested_digits,

727 buffer, length, &decimal_exponent);

728 break;

729 default:

730 UNREACHABLE();

731 }

732 if (result) {

733 decimal_point = length + decimal_exponent;

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

735 }

736 return result;

737 }

738

739 } // namespace double_conversion

740

741 } // namespace WTF

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