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27 
28 #include "../include/v8stdint.h"
29 #include "checks.h"
30 #include "utils.h"
31 
32 #include "fast-dtoa.h"
33 
34 #include "cached-powers.h"
35 #include "diy-fp.h"
36 #include "double.h"
37 
38 namespace v8 {
39 namespace internal {
40 
41 // The minimal and maximal target exponent define the range of w's binary
42 // exponent, where 'w' is the result of multiplying the input by a cached power
43 // of ten.
44 //
45 // A different range might be chosen on a different platform, to optimize digit
46 // generation, but a smaller range requires more powers of ten to be cached.
47 static const int kMinimalTargetExponent = -60;
48 static const int kMaximalTargetExponent = -32;
49 
50 
51 // Adjusts the last digit of the generated number, and screens out generated
52 // solutions that may be inaccurate. A solution may be inaccurate if it is
53 // outside the safe interval, or if we ctannot prove that it is closer to the
54 // input than a neighboring representation of the same length.
55 //
56 // Input: * buffer containing the digits of too_high / 10^kappa
57 // * the buffer's length
58 // * distance_too_high_w == (too_high - w).f() * unit
59 // * unsafe_interval == (too_high - too_low).f() * unit
60 // * rest = (too_high - buffer * 10^kappa).f() * unit
61 // * ten_kappa = 10^kappa * unit
62 // * unit = the common multiplier
63 // Output: returns true if the buffer is guaranteed to contain the closest
64 // representable number to the input.
65 // Modifies the generated digits in the buffer to approach (round towards) w.
66 static bool RoundWeed(Vector<char> buffer,
67  int length,
68  uint64_t distance_too_high_w,
69  uint64_t unsafe_interval,
70  uint64_t rest,
71  uint64_t ten_kappa,
72  uint64_t unit) {
73  uint64_t small_distance = distance_too_high_w - unit;
74  uint64_t big_distance = distance_too_high_w + unit;
75  // Let w_low = too_high - big_distance, and
76  // w_high = too_high - small_distance.
77  // Note: w_low < w < w_high
78  //
79  // The real w (* unit) must lie somewhere inside the interval
80  // ]w_low; w_high[ (often written as "(w_low; w_high)")
81 
82  // Basically the buffer currently contains a number in the unsafe interval
83  // ]too_low; too_high[ with too_low < w < too_high
84  //
85  // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
86  // ^v 1 unit ^ ^ ^ ^
87  // boundary_high --------------------- . . . .
88  // ^v 1 unit . . . .
89  // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
90  // . . ^ . .
91  // . big_distance . . .
92  // . . . . rest
93  // small_distance . . . .
94  // v . . . .
95  // w_high - - - - - - - - - - - - - - - - - - . . . .
96  // ^v 1 unit . . . .
97  // w ---------------------------------------- . . . .
98  // ^v 1 unit v . . .
99  // w_low - - - - - - - - - - - - - - - - - - - - - . . .
100  // . . v
101  // buffer --------------------------------------------------+-------+--------
102  // . .
103  // safe_interval .
104  // v .
105  // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
106  // ^v 1 unit .
107  // boundary_low ------------------------- unsafe_interval
108  // ^v 1 unit v
109  // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
110  //
111  //
112  // Note that the value of buffer could lie anywhere inside the range too_low
113  // to too_high.
114  //
115  // boundary_low, boundary_high and w are approximations of the real boundaries
116  // and v (the input number). They are guaranteed to be precise up to one unit.
117  // In fact the error is guaranteed to be strictly less than one unit.
118  //
119  // Anything that lies outside the unsafe interval is guaranteed not to round
120  // to v when read again.
121  // Anything that lies inside the safe interval is guaranteed to round to v
122  // when read again.
123  // If the number inside the buffer lies inside the unsafe interval but not
124  // inside the safe interval then we simply do not know and bail out (returning
125  // false).
126  //
127  // Similarly we have to take into account the imprecision of 'w' when finding
128  // the closest representation of 'w'. If we have two potential
129  // representations, and one is closer to both w_low and w_high, then we know
130  // it is closer to the actual value v.
131  //
132  // By generating the digits of too_high we got the largest (closest to
133  // too_high) buffer that is still in the unsafe interval. In the case where
134  // w_high < buffer < too_high we try to decrement the buffer.
135  // This way the buffer approaches (rounds towards) w.
136  // There are 3 conditions that stop the decrementation process:
137  // 1) the buffer is already below w_high
138  // 2) decrementing the buffer would make it leave the unsafe interval
139  // 3) decrementing the buffer would yield a number below w_high and farther
140  // away than the current number. In other words:
141  // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
142  // Instead of using the buffer directly we use its distance to too_high.
143  // Conceptually rest ~= too_high - buffer
144  // We need to do the following tests in this order to avoid over- and
145  // underflows.
146  ASSERT(rest <= unsafe_interval);
147  while (rest < small_distance && // Negated condition 1
148  unsafe_interval - rest >= ten_kappa && // Negated condition 2
149  (rest + ten_kappa < small_distance || // buffer{-1} > w_high
150  small_distance - rest >= rest + ten_kappa - small_distance)) {
151  buffer[length - 1]--;
152  rest += ten_kappa;
153  }
154 
155  // We have approached w+ as much as possible. We now test if approaching w-
156  // would require changing the buffer. If yes, then we have two possible
157  // representations close to w, but we cannot decide which one is closer.
158  if (rest < big_distance &&
159  unsafe_interval - rest >= ten_kappa &&
160  (rest + ten_kappa < big_distance ||
161  big_distance - rest > rest + ten_kappa - big_distance)) {
162  return false;
163  }
164 
165  // Weeding test.
166  // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
167  // Since too_low = too_high - unsafe_interval this is equivalent to
168  // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
169  // Conceptually we have: rest ~= too_high - buffer
170  return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
171 }
172 
173 
174 // Rounds the buffer upwards if the result is closer to v by possibly adding
175 // 1 to the buffer. If the precision of the calculation is not sufficient to
176 // round correctly, return false.
177 // The rounding might shift the whole buffer in which case the kappa is
178 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
179 //
180 // If 2*rest > ten_kappa then the buffer needs to be round up.
181 // rest can have an error of +/- 1 unit. This function accounts for the
182 // imprecision and returns false, if the rounding direction cannot be
183 // unambiguously determined.
184 //
185 // Precondition: rest < ten_kappa.
186 static bool RoundWeedCounted(Vector<char> buffer,
187  int length,
188  uint64_t rest,
189  uint64_t ten_kappa,
190  uint64_t unit,
191  int* kappa) {
192  ASSERT(rest < ten_kappa);
193  // The following tests are done in a specific order to avoid overflows. They
194  // will work correctly with any uint64 values of rest < ten_kappa and unit.
195  //
196  // If the unit is too big, then we don't know which way to round. For example
197  // a unit of 50 means that the real number lies within rest +/- 50. If
198  // 10^kappa == 40 then there is no way to tell which way to round.
199  if (unit >= ten_kappa) return false;
200  // Even if unit is just half the size of 10^kappa we are already completely
201  // lost. (And after the previous test we know that the expression will not
202  // over/underflow.)
203  if (ten_kappa - unit <= unit) return false;
204  // If 2 * (rest + unit) <= 10^kappa we can safely round down.
205  if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
206  return true;
207  }
208  // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
209  if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
210  // Increment the last digit recursively until we find a non '9' digit.
211  buffer[length - 1]++;
212  for (int i = length - 1; i > 0; --i) {
213  if (buffer[i] != '0' + 10) break;
214  buffer[i] = '0';
215  buffer[i - 1]++;
216  }
217  // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
218  // exception of the first digit all digits are now '0'. Simply switch the
219  // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
220  // the power (the kappa) is increased.
221  if (buffer[0] == '0' + 10) {
222  buffer[0] = '1';
223  (*kappa) += 1;
224  }
225  return true;
226  }
227  return false;
228 }
229 
230 
231 static const uint32_t kTen4 = 10000;
232 static const uint32_t kTen5 = 100000;
233 static const uint32_t kTen6 = 1000000;
234 static const uint32_t kTen7 = 10000000;
235 static const uint32_t kTen8 = 100000000;
236 static const uint32_t kTen9 = 1000000000;
237 
238 // Returns the biggest power of ten that is less than or equal than the given
239 // number. We furthermore receive the maximum number of bits 'number' has.
240 // If number_bits == 0 then 0^-1 is returned
241 // The number of bits must be <= 32.
242 // Precondition: number < (1 << (number_bits + 1)).
243 static void BiggestPowerTen(uint32_t number,
244  int number_bits,
245  uint32_t* power,
246  int* exponent) {
247  switch (number_bits) {
248  case 32:
249  case 31:
250  case 30:
251  if (kTen9 <= number) {
252  *power = kTen9;
253  *exponent = 9;
254  break;
255  } // else fallthrough
256  case 29:
257  case 28:
258  case 27:
259  if (kTen8 <= number) {
260  *power = kTen8;
261  *exponent = 8;
262  break;
263  } // else fallthrough
264  case 26:
265  case 25:
266  case 24:
267  if (kTen7 <= number) {
268  *power = kTen7;
269  *exponent = 7;
270  break;
271  } // else fallthrough
272  case 23:
273  case 22:
274  case 21:
275  case 20:
276  if (kTen6 <= number) {
277  *power = kTen6;
278  *exponent = 6;
279  break;
280  } // else fallthrough
281  case 19:
282  case 18:
283  case 17:
284  if (kTen5 <= number) {
285  *power = kTen5;
286  *exponent = 5;
287  break;
288  } // else fallthrough
289  case 16:
290  case 15:
291  case 14:
292  if (kTen4 <= number) {
293  *power = kTen4;
294  *exponent = 4;
295  break;
296  } // else fallthrough
297  case 13:
298  case 12:
299  case 11:
300  case 10:
301  if (1000 <= number) {
302  *power = 1000;
303  *exponent = 3;
304  break;
305  } // else fallthrough
306  case 9:
307  case 8:
308  case 7:
309  if (100 <= number) {
310  *power = 100;
311  *exponent = 2;
312  break;
313  } // else fallthrough
314  case 6:
315  case 5:
316  case 4:
317  if (10 <= number) {
318  *power = 10;
319  *exponent = 1;
320  break;
321  } // else fallthrough
322  case 3:
323  case 2:
324  case 1:
325  if (1 <= number) {
326  *power = 1;
327  *exponent = 0;
328  break;
329  } // else fallthrough
330  case 0:
331  *power = 0;
332  *exponent = -1;
333  break;
334  default:
335  // Following assignments are here to silence compiler warnings.
336  *power = 0;
337  *exponent = 0;
338  UNREACHABLE();
339  }
340 }
341 
342 
343 // Generates the digits of input number w.
344 // w is a floating-point number (DiyFp), consisting of a significand and an
345 // exponent. Its exponent is bounded by kMinimalTargetExponent and
346 // kMaximalTargetExponent.
347 // Hence -60 <= w.e() <= -32.
348 //
349 // Returns false if it fails, in which case the generated digits in the buffer
350 // should not be used.
351 // Preconditions:
352 // * low, w and high are correct up to 1 ulp (unit in the last place). That
353 // is, their error must be less than a unit of their last digits.
354 // * low.e() == w.e() == high.e()
355 // * low < w < high, and taking into account their error: low~ <= high~
356 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
357 // Postconditions: returns false if procedure fails.
358 // otherwise:
359 // * buffer is not null-terminated, but len contains the number of digits.
360 // * buffer contains the shortest possible decimal digit-sequence
361 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
362 // correct values of low and high (without their error).
363 // * if more than one decimal representation gives the minimal number of
364 // decimal digits then the one closest to W (where W is the correct value
365 // of w) is chosen.
366 // Remark: this procedure takes into account the imprecision of its input
367 // numbers. If the precision is not enough to guarantee all the postconditions
368 // then false is returned. This usually happens rarely (~0.5%).
369 //
370 // Say, for the sake of example, that
371 // w.e() == -48, and w.f() == 0x1234567890abcdef
372 // w's value can be computed by w.f() * 2^w.e()
373 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
374 // -> w's integral part is 0x1234
375 // w's fractional part is therefore 0x567890abcdef.
376 // Printing w's integral part is easy (simply print 0x1234 in decimal).
377 // In order to print its fraction we repeatedly multiply the fraction by 10 and
378 // get each digit. Example the first digit after the point would be computed by
379 // (0x567890abcdef * 10) >> 48. -> 3
380 // The whole thing becomes slightly more complicated because we want to stop
381 // once we have enough digits. That is, once the digits inside the buffer
382 // represent 'w' we can stop. Everything inside the interval low - high
383 // represents w. However we have to pay attention to low, high and w's
384 // imprecision.
385 static bool DigitGen(DiyFp low,
386  DiyFp w,
387  DiyFp high,
388  Vector<char> buffer,
389  int* length,
390  int* kappa) {
391  ASSERT(low.e() == w.e() && w.e() == high.e());
392  ASSERT(low.f() + 1 <= high.f() - 1);
393  ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
394  // low, w and high are imprecise, but by less than one ulp (unit in the last
395  // place).
396  // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
397  // the new numbers are outside of the interval we want the final
398  // representation to lie in.
399  // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
400  // numbers that are certain to lie in the interval. We will use this fact
401  // later on.
402  // We will now start by generating the digits within the uncertain
403  // interval. Later we will weed out representations that lie outside the safe
404  // interval and thus _might_ lie outside the correct interval.
405  uint64_t unit = 1;
406  DiyFp too_low = DiyFp(low.f() - unit, low.e());
407  DiyFp too_high = DiyFp(high.f() + unit, high.e());
408  // too_low and too_high are guaranteed to lie outside the interval we want the
409  // generated number in.
410  DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
411  // We now cut the input number into two parts: the integral digits and the
412  // fractionals. We will not write any decimal separator though, but adapt
413  // kappa instead.
414  // Reminder: we are currently computing the digits (stored inside the buffer)
415  // such that: too_low < buffer * 10^kappa < too_high
416  // We use too_high for the digit_generation and stop as soon as possible.
417  // If we stop early we effectively round down.
418  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
419  // Division by one is a shift.
420  uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
421  // Modulo by one is an and.
422  uint64_t fractionals = too_high.f() & (one.f() - 1);
423  uint32_t divisor;
424  int divisor_exponent;
425  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
426  &divisor, &divisor_exponent);
427  *kappa = divisor_exponent + 1;
428  *length = 0;
429  // Loop invariant: buffer = too_high / 10^kappa (integer division)
430  // The invariant holds for the first iteration: kappa has been initialized
431  // with the divisor exponent + 1. And the divisor is the biggest power of ten
432  // that is smaller than integrals.
433  while (*kappa > 0) {
434  int digit = integrals / divisor;
435  buffer[*length] = '0' + digit;
436  (*length)++;
437  integrals %= divisor;
438  (*kappa)--;
439  // Note that kappa now equals the exponent of the divisor and that the
440  // invariant thus holds again.
441  uint64_t rest =
442  (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
443  // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
444  // Reminder: unsafe_interval.e() == one.e()
445  if (rest < unsafe_interval.f()) {
446  // Rounding down (by not emitting the remaining digits) yields a number
447  // that lies within the unsafe interval.
448  return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
449  unsafe_interval.f(), rest,
450  static_cast<uint64_t>(divisor) << -one.e(), unit);
451  }
452  divisor /= 10;
453  }
454 
455  // The integrals have been generated. We are at the point of the decimal
456  // separator. In the following loop we simply multiply the remaining digits by
457  // 10 and divide by one. We just need to pay attention to multiply associated
458  // data (like the interval or 'unit'), too.
459  // Note that the multiplication by 10 does not overflow, because w.e >= -60
460  // and thus one.e >= -60.
461  ASSERT(one.e() >= -60);
462  ASSERT(fractionals < one.f());
463  ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
464  while (true) {
465  fractionals *= 10;
466  unit *= 10;
467  unsafe_interval.set_f(unsafe_interval.f() * 10);
468  // Integer division by one.
469  int digit = static_cast<int>(fractionals >> -one.e());
470  buffer[*length] = '0' + digit;
471  (*length)++;
472  fractionals &= one.f() - 1; // Modulo by one.
473  (*kappa)--;
474  if (fractionals < unsafe_interval.f()) {
475  return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
476  unsafe_interval.f(), fractionals, one.f(), unit);
477  }
478  }
479 }
480 
481 
482 
483 // Generates (at most) requested_digits of input number w.
484 // w is a floating-point number (DiyFp), consisting of a significand and an
485 // exponent. Its exponent is bounded by kMinimalTargetExponent and
486 // kMaximalTargetExponent.
487 // Hence -60 <= w.e() <= -32.
488 //
489 // Returns false if it fails, in which case the generated digits in the buffer
490 // should not be used.
491 // Preconditions:
492 // * w is correct up to 1 ulp (unit in the last place). That
493 // is, its error must be strictly less than a unit of its last digit.
494 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
495 //
496 // Postconditions: returns false if procedure fails.
497 // otherwise:
498 // * buffer is not null-terminated, but length contains the number of
499 // digits.
500 // * the representation in buffer is the most precise representation of
501 // requested_digits digits.
502 // * buffer contains at most requested_digits digits of w. If there are less
503 // than requested_digits digits then some trailing '0's have been removed.
504 // * kappa is such that
505 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
506 //
507 // Remark: This procedure takes into account the imprecision of its input
508 // numbers. If the precision is not enough to guarantee all the postconditions
509 // then false is returned. This usually happens rarely, but the failure-rate
510 // increases with higher requested_digits.
511 static bool DigitGenCounted(DiyFp w,
512  int requested_digits,
513  Vector<char> buffer,
514  int* length,
515  int* kappa) {
516  ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
517  ASSERT(kMinimalTargetExponent >= -60);
518  ASSERT(kMaximalTargetExponent <= -32);
519  // w is assumed to have an error less than 1 unit. Whenever w is scaled we
520  // also scale its error.
521  uint64_t w_error = 1;
522  // We cut the input number into two parts: the integral digits and the
523  // fractional digits. We don't emit any decimal separator, but adapt kappa
524  // instead. Example: instead of writing "1.2" we put "12" into the buffer and
525  // increase kappa by 1.
526  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
527  // Division by one is a shift.
528  uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
529  // Modulo by one is an and.
530  uint64_t fractionals = w.f() & (one.f() - 1);
531  uint32_t divisor;
532  int divisor_exponent;
533  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
534  &divisor, &divisor_exponent);
535  *kappa = divisor_exponent + 1;
536  *length = 0;
537 
538  // Loop invariant: buffer = w / 10^kappa (integer division)
539  // The invariant holds for the first iteration: kappa has been initialized
540  // with the divisor exponent + 1. And the divisor is the biggest power of ten
541  // that is smaller than 'integrals'.
542  while (*kappa > 0) {
543  int digit = integrals / divisor;
544  buffer[*length] = '0' + digit;
545  (*length)++;
546  requested_digits--;
547  integrals %= divisor;
548  (*kappa)--;
549  // Note that kappa now equals the exponent of the divisor and that the
550  // invariant thus holds again.
551  if (requested_digits == 0) break;
552  divisor /= 10;
553  }
554 
555  if (requested_digits == 0) {
556  uint64_t rest =
557  (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
558  return RoundWeedCounted(buffer, *length, rest,
559  static_cast<uint64_t>(divisor) << -one.e(), w_error,
560  kappa);
561  }
562 
563  // The integrals have been generated. We are at the point of the decimal
564  // separator. In the following loop we simply multiply the remaining digits by
565  // 10 and divide by one. We just need to pay attention to multiply associated
566  // data (the 'unit'), too.
567  // Note that the multiplication by 10 does not overflow, because w.e >= -60
568  // and thus one.e >= -60.
569  ASSERT(one.e() >= -60);
570  ASSERT(fractionals < one.f());
571  ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
572  while (requested_digits > 0 && fractionals > w_error) {
573  fractionals *= 10;
574  w_error *= 10;
575  // Integer division by one.
576  int digit = static_cast<int>(fractionals >> -one.e());
577  buffer[*length] = '0' + digit;
578  (*length)++;
579  requested_digits--;
580  fractionals &= one.f() - 1; // Modulo by one.
581  (*kappa)--;
582  }
583  if (requested_digits != 0) return false;
584  return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
585  kappa);
586 }
587 
588 
589 // Provides a decimal representation of v.
590 // Returns true if it succeeds, otherwise the result cannot be trusted.
591 // There will be *length digits inside the buffer (not null-terminated).
592 // If the function returns true then
593 // v == (double) (buffer * 10^decimal_exponent).
594 // The digits in the buffer are the shortest representation possible: no
595 // 0.09999999999999999 instead of 0.1. The shorter representation will even be
596 // chosen even if the longer one would be closer to v.
597 // The last digit will be closest to the actual v. That is, even if several
598 // digits might correctly yield 'v' when read again, the closest will be
599 // computed.
600 static bool Grisu3(double v,
601  Vector<char> buffer,
602  int* length,
603  int* decimal_exponent) {
604  DiyFp w = Double(v).AsNormalizedDiyFp();
605  // boundary_minus and boundary_plus are the boundaries between v and its
606  // closest floating-point neighbors. Any number strictly between
607  // boundary_minus and boundary_plus will round to v when convert to a double.
608  // Grisu3 will never output representations that lie exactly on a boundary.
609  DiyFp boundary_minus, boundary_plus;
610  Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
611  ASSERT(boundary_plus.e() == w.e());
612  DiyFp ten_mk; // Cached power of ten: 10^-k
613  int mk; // -k
614  int ten_mk_minimal_binary_exponent =
615  kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
616  int ten_mk_maximal_binary_exponent =
617  kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
619  ten_mk_minimal_binary_exponent,
620  ten_mk_maximal_binary_exponent,
621  &ten_mk, &mk);
622  ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
624  (kMaximalTargetExponent >= w.e() + ten_mk.e() +
626  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
627  // 64 bit significand and ten_mk is thus only precise up to 64 bits.
628 
629  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
630  // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
631  // off by a small amount.
632  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
633  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
634  // (f-1) * 2^e < w*10^k < (f+1) * 2^e
635  DiyFp scaled_w = DiyFp::Times(w, ten_mk);
636  ASSERT(scaled_w.e() ==
637  boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
638  // In theory it would be possible to avoid some recomputations by computing
639  // the difference between w and boundary_minus/plus (a power of 2) and to
640  // compute scaled_boundary_minus/plus by subtracting/adding from
641  // scaled_w. However the code becomes much less readable and the speed
642  // enhancements are not terriffic.
643  DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
644  DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
645 
646  // DigitGen will generate the digits of scaled_w. Therefore we have
647  // v == (double) (scaled_w * 10^-mk).
648  // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
649  // integer than it will be updated. For instance if scaled_w == 1.23 then
650  // the buffer will be filled with "123" und the decimal_exponent will be
651  // decreased by 2.
652  int kappa;
653  bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
654  buffer, length, &kappa);
655  *decimal_exponent = -mk + kappa;
656  return result;
657 }
658 
659 
660 // The "counted" version of grisu3 (see above) only generates requested_digits
661 // number of digits. This version does not generate the shortest representation,
662 // and with enough requested digits 0.1 will at some point print as 0.9999999...
663 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
664 // therefore the rounding strategy for halfway cases is irrelevant.
665 static bool Grisu3Counted(double v,
666  int requested_digits,
667  Vector<char> buffer,
668  int* length,
669  int* decimal_exponent) {
670  DiyFp w = Double(v).AsNormalizedDiyFp();
671  DiyFp ten_mk; // Cached power of ten: 10^-k
672  int mk; // -k
673  int ten_mk_minimal_binary_exponent =
674  kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
675  int ten_mk_maximal_binary_exponent =
676  kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
678  ten_mk_minimal_binary_exponent,
679  ten_mk_maximal_binary_exponent,
680  &ten_mk, &mk);
681  ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
683  (kMaximalTargetExponent >= w.e() + ten_mk.e() +
685  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
686  // 64 bit significand and ten_mk is thus only precise up to 64 bits.
687 
688  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
689  // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
690  // off by a small amount.
691  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
692  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
693  // (f-1) * 2^e < w*10^k < (f+1) * 2^e
694  DiyFp scaled_w = DiyFp::Times(w, ten_mk);
695 
696  // We now have (double) (scaled_w * 10^-mk).
697  // DigitGen will generate the first requested_digits digits of scaled_w and
698  // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
699  // will not always be exactly the same since DigitGenCounted only produces a
700  // limited number of digits.)
701  int kappa;
702  bool result = DigitGenCounted(scaled_w, requested_digits,
703  buffer, length, &kappa);
704  *decimal_exponent = -mk + kappa;
705  return result;
706 }
707 
708 
709 bool FastDtoa(double v,
710  FastDtoaMode mode,
711  int requested_digits,
712  Vector<char> buffer,
713  int* length,
714  int* decimal_point) {
715  ASSERT(v > 0);
716  ASSERT(!Double(v).IsSpecial());
717 
718  bool result = false;
719  int decimal_exponent = 0;
720  switch (mode) {
721  case FAST_DTOA_SHORTEST:
722  result = Grisu3(v, buffer, length, &decimal_exponent);
723  break;
724  case FAST_DTOA_PRECISION:
725  result = Grisu3Counted(v, requested_digits,
726  buffer, length, &decimal_exponent);
727  break;
728  default:
729  UNREACHABLE();
730  }
731  if (result) {
732  *decimal_point = *length + decimal_exponent;
733  buffer[*length] = '\0';
734  }
735  return result;
736 }
737 
738 } } // namespace v8::internal
static DiyFp Minus(const DiyFp &a, const DiyFp &b)
Definition: diy-fp.h:59
static const int kSignificandSize
Definition: diy-fp.h:41
uint64_t f() const
Definition: diy-fp.h:102
#define ASSERT(condition)
Definition: checks.h:270
#define UNREACHABLE()
Definition: checks.h:50
bool FastDtoa(double v, FastDtoaMode mode, int requested_digits, Vector< char > buffer, int *length, int *decimal_point)
Definition: fast-dtoa.cc:709
#define V8_2PART_UINT64_C(a, b)
Definition: globals.h:187
static DiyFp Times(const DiyFp &a, const DiyFp &b)
Definition: diy-fp.h:70
static void GetCachedPowerForBinaryExponentRange(int min_exponent, int max_exponent, DiyFp *power, int *decimal_exponent)