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27
28#include "config.h"
29
30#include <wtf/dtoa/fast-dtoa.h>
31
32#include <wtf/dtoa/cached-powers.h>
33#include <wtf/dtoa/diy-fp.h>
34#include <wtf/dtoa/ieee.h>
35
36namespace WTF {
37namespace double_conversion {
38
39// The minimal and maximal target exponent define the range of w's binary
40// exponent, where 'w' is the result of multiplying the input by a cached power
41// of ten.
42//
43// A different range might be chosen on a different platform, to optimize digit
44// generation, but a smaller range requires more powers of ten to be cached.
45static const int kMinimalTargetExponent = -60;
46static const int kMaximalTargetExponent = -32;
47
48
49// Adjusts the last digit of the generated number, and screens out generated
50// solutions that may be inaccurate. A solution may be inaccurate if it is
51// outside the safe interval, or if we cannot prove that it is closer to the
52// input than a neighboring representation of the same length.
53//
54// Input: * buffer containing the digits of too_high / 10^kappa
55// * the buffer's length
56// * distance_too_high_w == (too_high - w).f() * unit
57// * unsafe_interval == (too_high - too_low).f() * unit
58// * rest = (too_high - buffer * 10^kappa).f() * unit
59// * ten_kappa = 10^kappa * unit
60// * unit = the common multiplier
61// Output: returns true if the buffer is guaranteed to contain the closest
62// representable number to the input.
63// Modifies the generated digits in the buffer to approach (round towards) w.
64static bool RoundWeed(BufferReference<char> buffer,
65 int length,
66 uint64_t distance_too_high_w,
67 uint64_t unsafe_interval,
68 uint64_t rest,
69 uint64_t ten_kappa,
70 uint64_t unit) {
71 uint64_t small_distance = distance_too_high_w - unit;
72 uint64_t big_distance = distance_too_high_w + unit;
73 // Let w_low = too_high - big_distance, and
74 // w_high = too_high - small_distance.
75 // Note: w_low < w < w_high
76 //
77 // The real w (* unit) must lie somewhere inside the interval
78 // ]w_low; w_high[ (often written as "(w_low; w_high)")
79
80 // Basically the buffer currently contains a number in the unsafe interval
81 // ]too_low; too_high[ with too_low < w < too_high
82 //
83 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
84 // ^v 1 unit ^ ^ ^ ^
85 // boundary_high --------------------- . . . .
86 // ^v 1 unit . . . .
87 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
88 // . . ^ . .
89 // . big_distance . . .
90 // . . . . rest
91 // small_distance . . . .
92 // v . . . .
93 // w_high - - - - - - - - - - - - - - - - - - . . . .
94 // ^v 1 unit . . . .
95 // w ---------------------------------------- . . . .
96 // ^v 1 unit v . . .
97 // w_low - - - - - - - - - - - - - - - - - - - - - . . .
98 // . . v
99 // buffer --------------------------------------------------+-------+--------
100 // . .
101 // safe_interval .
102 // v .
103 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
104 // ^v 1 unit .
105 // boundary_low ------------------------- unsafe_interval
106 // ^v 1 unit v
107 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108 //
109 //
110 // Note that the value of buffer could lie anywhere inside the range too_low
111 // to too_high.
112 //
113 // boundary_low, boundary_high and w are approximations of the real boundaries
114 // and v (the input number). They are guaranteed to be precise up to one unit.
115 // In fact the error is guaranteed to be strictly less than one unit.
116 //
117 // Anything that lies outside the unsafe interval is guaranteed not to round
118 // to v when read again.
119 // Anything that lies inside the safe interval is guaranteed to round to v
120 // when read again.
121 // If the number inside the buffer lies inside the unsafe interval but not
122 // inside the safe interval then we simply do not know and bail out (returning
123 // false).
124 //
125 // Similarly we have to take into account the imprecision of 'w' when finding
126 // the closest representation of 'w'. If we have two potential
127 // representations, and one is closer to both w_low and w_high, then we know
128 // it is closer to the actual value v.
129 //
130 // By generating the digits of too_high we got the largest (closest to
131 // too_high) buffer that is still in the unsafe interval. In the case where
132 // w_high < buffer < too_high we try to decrement the buffer.
133 // This way the buffer approaches (rounds towards) w.
134 // There are 3 conditions that stop the decrementation process:
135 // 1) the buffer is already below w_high
136 // 2) decrementing the buffer would make it leave the unsafe interval
137 // 3) decrementing the buffer would yield a number below w_high and farther
138 // away than the current number. In other words:
139 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
140 // Instead of using the buffer directly we use its distance to too_high.
141 // Conceptually rest ~= too_high - buffer
142 // We need to do the following tests in this order to avoid over- and
143 // underflows.
144 ASSERT(rest <= unsafe_interval);
145 while (rest < small_distance && // Negated condition 1
146 unsafe_interval - rest >= ten_kappa && // Negated condition 2
147 (rest + ten_kappa < small_distance || // buffer{-1} > w_high
148 small_distance - rest >= rest + ten_kappa - small_distance)) {
149 buffer[length - 1]--;
150 rest += ten_kappa;
151 }
152
153 // We have approached w+ as much as possible. We now test if approaching w-
154 // would require changing the buffer. If yes, then we have two possible
155 // representations close to w, but we cannot decide which one is closer.
156 if (rest < big_distance &&
157 unsafe_interval - rest >= ten_kappa &&
158 (rest + ten_kappa < big_distance ||
159 big_distance - rest > rest + ten_kappa - big_distance)) {
160 return false;
161 }
162
163 // Weeding test.
164 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
165 // Since too_low = too_high - unsafe_interval this is equivalent to
166 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
167 // Conceptually we have: rest ~= too_high - buffer
168 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
169}
170
171
172// Rounds the buffer upwards if the result is closer to v by possibly adding
173// 1 to the buffer. If the precision of the calculation is not sufficient to
174// round correctly, return false.
175// The rounding might shift the whole buffer in which case the kappa is
176// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
177//
178// If 2*rest > ten_kappa then the buffer needs to be round up.
179// rest can have an error of +/- 1 unit. This function accounts for the
180// imprecision and returns false, if the rounding direction cannot be
181// unambiguously determined.
182//
183// Precondition: rest < ten_kappa.
184static bool RoundWeedCounted(BufferReference<char> buffer,
185 int length,
186 uint64_t rest,
187 uint64_t ten_kappa,
188 uint64_t unit,
189 int* kappa) {
190 ASSERT(rest < ten_kappa);
191 // The following tests are done in a specific order to avoid overflows. They
192 // will work correctly with any uint64 values of rest < ten_kappa and unit.
193 //
194 // If the unit is too big, then we don't know which way to round. For example
195 // a unit of 50 means that the real number lies within rest +/- 50. If
196 // 10^kappa == 40 then there is no way to tell which way to round.
197 if (unit >= ten_kappa) return false;
198 // Even if unit is just half the size of 10^kappa we are already completely
199 // lost. (And after the previous test we know that the expression will not
200 // over/underflow.)
201 if (ten_kappa - unit <= unit) return false;
202 // If 2 * (rest + unit) <= 10^kappa we can safely round down.
203 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
204 return true;
205 }
206 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
207 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
208 // Increment the last digit recursively until we find a non '9' digit.
209 buffer[length - 1]++;
210 for (int i = length - 1; i > 0; --i) {
211 if (buffer[i] != '0' + 10) break;
212 buffer[i] = '0';
213 buffer[i - 1]++;
214 }
215 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
216 // exception of the first digit all digits are now '0'. Simply switch the
217 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
218 // the power (the kappa) is increased.
219 if (buffer[0] == '0' + 10) {
220 buffer[0] = '1';
221 (*kappa) += 1;
222 }
223 return true;
224 }
225 return false;
226}
227
228// Returns the biggest power of ten that is less than or equal to the given
229// number. We furthermore receive the maximum number of bits 'number' has.
230//
231// Returns power == 10^(exponent_plus_one-1) such that
232// power <= number < power * 10.
233// If number_bits == 0 then 0^(0-1) is returned.
234// The number of bits must be <= 32.
235// Precondition: number < (1 << (number_bits + 1)).
236
237// Inspired by the method for finding an integer log base 10 from here:
238// http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
239static unsigned int const kSmallPowersOfTen[] =
240 {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
241 1000000000};
242
243static void BiggestPowerTen(uint32_t number,
244 int number_bits,
245 uint32_t* power,
246 int* exponent_plus_one) {
247 ASSERT(number < (1u << (number_bits + 1)));
248 // 1233/4096 is approximately 1/lg(10).
249 int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12);
250 // We increment to skip over the first entry in the kPowersOf10 table.
251 // Note: kPowersOf10[i] == 10^(i-1).
252 exponent_plus_one_guess++;
253 // We don't have any guarantees that 2^number_bits <= number.
254 if (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
255 exponent_plus_one_guess--;
256 }
257 *power = kSmallPowersOfTen[exponent_plus_one_guess];
258 *exponent_plus_one = exponent_plus_one_guess;
259}
260
261// Generates the digits of input number w.
262// w is a floating-point number (DiyFp), consisting of a significand and an
263// exponent. Its exponent is bounded by kMinimalTargetExponent and
264// kMaximalTargetExponent.
265// Hence -60 <= w.e() <= -32.
266//
267// Returns false if it fails, in which case the generated digits in the buffer
268// should not be used.
269// Preconditions:
270// * low, w and high are correct up to 1 ulp (unit in the last place). That
271// is, their error must be less than a unit of their last digits.
272// * low.e() == w.e() == high.e()
273// * low < w < high, and taking into account their error: low~ <= high~
274// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
275// Postconditions: returns false if procedure fails.
276// otherwise:
277// * buffer is not null-terminated, but len contains the number of digits.
278// * buffer contains the shortest possible decimal digit-sequence
279// such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
280// correct values of low and high (without their error).
281// * if more than one decimal representation gives the minimal number of
282// decimal digits then the one closest to W (where W is the correct value
283// of w) is chosen.
284// Remark: this procedure takes into account the imprecision of its input
285// numbers. If the precision is not enough to guarantee all the postconditions
286// then false is returned. This usually happens rarely (~0.5%).
287//
288// Say, for the sake of example, that
289// w.e() == -48, and w.f() == 0x1234567890abcdef
290// w's value can be computed by w.f() * 2^w.e()
291// We can obtain w's integral digits by simply shifting w.f() by -w.e().
292// -> w's integral part is 0x1234
293// w's fractional part is therefore 0x567890abcdef.
294// Printing w's integral part is easy (simply print 0x1234 in decimal).
295// In order to print its fraction we repeatedly multiply the fraction by 10 and
296// get each digit. Example the first digit after the point would be computed by
297// (0x567890abcdef * 10) >> 48. -> 3
298// The whole thing becomes slightly more complicated because we want to stop
299// once we have enough digits. That is, once the digits inside the buffer
300// represent 'w' we can stop. Everything inside the interval low - high
301// represents w. However we have to pay attention to low, high and w's
302// imprecision.
303static bool DigitGen(DiyFp low,
304 DiyFp w,
305 DiyFp high,
306 BufferReference<char> buffer,
307 int* length,
308 int* kappa) {
309 ASSERT(low.e() == w.e() && w.e() == high.e());
310 ASSERT(low.f() + 1 <= high.f() - 1);
311 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
312 // low, w and high are imprecise, but by less than one ulp (unit in the last
313 // place).
314 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
315 // the new numbers are outside of the interval we want the final
316 // representation to lie in.
317 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
318 // numbers that are certain to lie in the interval. We will use this fact
319 // later on.
320 // We will now start by generating the digits within the uncertain
321 // interval. Later we will weed out representations that lie outside the safe
322 // interval and thus _might_ lie outside the correct interval.
323 uint64_t unit = 1;
324 DiyFp too_low = DiyFp(low.f() - unit, low.e());
325 DiyFp too_high = DiyFp(high.f() + unit, high.e());
326 // too_low and too_high are guaranteed to lie outside the interval we want the
327 // generated number in.
328 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
329 // We now cut the input number into two parts: the integral digits and the
330 // fractionals. We will not write any decimal separator though, but adapt
331 // kappa instead.
332 // Reminder: we are currently computing the digits (stored inside the buffer)
333 // such that: too_low < buffer * 10^kappa < too_high
334 // We use too_high for the digit_generation and stop as soon as possible.
335 // If we stop early we effectively round down.
336 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
337 // Division by one is a shift.
338 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
339 // Modulo by one is an and.
340 uint64_t fractionals = too_high.f() & (one.f() - 1);
341 uint32_t divisor;
342 int divisor_exponent_plus_one;
343 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
344 &divisor, &divisor_exponent_plus_one);
345 *kappa = divisor_exponent_plus_one;
346 *length = 0;
347 // Loop invariant: buffer = too_high / 10^kappa (integer division)
348 // The invariant holds for the first iteration: kappa has been initialized
349 // with the divisor exponent + 1. And the divisor is the biggest power of ten
350 // that is smaller than integrals.
351 while (*kappa > 0) {
352 int digit = integrals / divisor;
353 ASSERT(digit <= 9);
354 buffer[*length] = static_cast<char>('0' + digit);
355 (*length)++;
356 integrals %= divisor;
357 (*kappa)--;
358 // Note that kappa now equals the exponent of the divisor and that the
359 // invariant thus holds again.
360 uint64_t rest =
361 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
362 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
363 // Reminder: unsafe_interval.e() == one.e()
364 if (rest < unsafe_interval.f()) {
365 // Rounding down (by not emitting the remaining digits) yields a number
366 // that lies within the unsafe interval.
367 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
368 unsafe_interval.f(), rest,
369 static_cast<uint64_t>(divisor) << -one.e(), unit);
370 }
371 divisor /= 10;
372 }
373
374 // The integrals have been generated. We are at the point of the decimal
375 // separator. In the following loop we simply multiply the remaining digits by
376 // 10 and divide by one. We just need to pay attention to multiply associated
377 // data (like the interval or 'unit'), too.
378 // Note that the multiplication by 10 does not overflow, because w.e >= -60
379 // and thus one.e >= -60.
380 ASSERT(one.e() >= -60);
381 ASSERT(fractionals < one.f());
382 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
383 for (;;) {
384 fractionals *= 10;
385 unit *= 10;
386 unsafe_interval.set_f(unsafe_interval.f() * 10);
387 // Integer division by one.
388 int digit = static_cast<int>(fractionals >> -one.e());
389 ASSERT(digit <= 9);
390 buffer[*length] = static_cast<char>('0' + digit);
391 (*length)++;
392 fractionals &= one.f() - 1; // Modulo by one.
393 (*kappa)--;
394 if (fractionals < unsafe_interval.f()) {
395 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
396 unsafe_interval.f(), fractionals, one.f(), unit);
397 }
398 }
399}
400
401
402
403// Generates (at most) requested_digits digits of input number w.
404// w is a floating-point number (DiyFp), consisting of a significand and an
405// exponent. Its exponent is bounded by kMinimalTargetExponent and
406// kMaximalTargetExponent.
407// Hence -60 <= w.e() <= -32.
408//
409// Returns false if it fails, in which case the generated digits in the buffer
410// should not be used.
411// Preconditions:
412// * w is correct up to 1 ulp (unit in the last place). That
413// is, its error must be strictly less than a unit of its last digit.
414// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
415//
416// Postconditions: returns false if procedure fails.
417// otherwise:
418// * buffer is not null-terminated, but length contains the number of
419// digits.
420// * the representation in buffer is the most precise representation of
421// requested_digits digits.
422// * buffer contains at most requested_digits digits of w. If there are less
423// than requested_digits digits then some trailing '0's have been removed.
424// * kappa is such that
425// w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
426//
427// Remark: This procedure takes into account the imprecision of its input
428// numbers. If the precision is not enough to guarantee all the postconditions
429// then false is returned. This usually happens rarely, but the failure-rate
430// increases with higher requested_digits.
431static bool DigitGenCounted(DiyFp w,
432 int requested_digits,
433 BufferReference<char> buffer,
434 int* length,
435 int* kappa) {
436 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
437 ASSERT(kMinimalTargetExponent >= -60);
438 ASSERT(kMaximalTargetExponent <= -32);
439 // w is assumed to have an error less than 1 unit. Whenever w is scaled we
440 // also scale its error.
441 uint64_t w_error = 1;
442 // We cut the input number into two parts: the integral digits and the
443 // fractional digits. We don't emit any decimal separator, but adapt kappa
444 // instead. Example: instead of writing "1.2" we put "12" into the buffer and
445 // increase kappa by 1.
446 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
447 // Division by one is a shift.
448 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
449 // Modulo by one is an and.
450 uint64_t fractionals = w.f() & (one.f() - 1);
451 uint32_t divisor;
452 int divisor_exponent_plus_one;
453 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
454 &divisor, &divisor_exponent_plus_one);
455 *kappa = divisor_exponent_plus_one;
456 *length = 0;
457
458 // Loop invariant: buffer = w / 10^kappa (integer division)
459 // The invariant holds for the first iteration: kappa has been initialized
460 // with the divisor exponent + 1. And the divisor is the biggest power of ten
461 // that is smaller than 'integrals'.
462 while (*kappa > 0) {
463 int digit = integrals / divisor;
464 ASSERT(digit <= 9);
465 buffer[*length] = static_cast<char>('0' + digit);
466 (*length)++;
467 requested_digits--;
468 integrals %= divisor;
469 (*kappa)--;
470 // Note that kappa now equals the exponent of the divisor and that the
471 // invariant thus holds again.
472 if (requested_digits == 0) break;
473 divisor /= 10;
474 }
475
476 if (requested_digits == 0) {
477 uint64_t rest =
478 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
479 return RoundWeedCounted(buffer, *length, rest,
480 static_cast<uint64_t>(divisor) << -one.e(), w_error,
481 kappa);
482 }
483
484 // The integrals have been generated. We are at the point of the decimal
485 // separator. In the following loop we simply multiply the remaining digits by
486 // 10 and divide by one. We just need to pay attention to multiply associated
487 // data (the 'unit'), too.
488 // Note that the multiplication by 10 does not overflow, because w.e >= -60
489 // and thus one.e >= -60.
490 ASSERT(one.e() >= -60);
491 ASSERT(fractionals < one.f());
492 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
493 while (requested_digits > 0 && fractionals > w_error) {
494 fractionals *= 10;
495 w_error *= 10;
496 // Integer division by one.
497 int digit = static_cast<int>(fractionals >> -one.e());
498 ASSERT(digit <= 9);
499 buffer[*length] = static_cast<char>('0' + digit);
500 (*length)++;
501 requested_digits--;
502 fractionals &= one.f() - 1; // Modulo by one.
503 (*kappa)--;
504 }
505 if (requested_digits != 0) return false;
506 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
507 kappa);
508}
509
510
511// Provides a decimal representation of v.
512// Returns true if it succeeds, otherwise the result cannot be trusted.
513// There will be *length digits inside the buffer (not null-terminated).
514// If the function returns true then
515// v == (double) (buffer * 10^decimal_exponent).
516// The digits in the buffer are the shortest representation possible: no
517// 0.09999999999999999 instead of 0.1. The shorter representation will even be
518// chosen even if the longer one would be closer to v.
519// The last digit will be closest to the actual v. That is, even if several
520// digits might correctly yield 'v' when read again, the closest will be
521// computed.
522static bool Grisu3(double v,
523 FastDtoaMode mode,
524 BufferReference<char> buffer,
525 int* length,
526 int* decimal_exponent) {
527 DiyFp w = Double(v).AsNormalizedDiyFp();
528 // boundary_minus and boundary_plus are the boundaries between v and its
529 // closest floating-point neighbors. Any number strictly between
530 // boundary_minus and boundary_plus will round to v when convert to a double.
531 // Grisu3 will never output representations that lie exactly on a boundary.
532 DiyFp boundary_minus, boundary_plus;
533 if (mode == FAST_DTOA_SHORTEST) {
534 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
535 } else {
536 ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
537 float single_v = static_cast<float>(v);
538 Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
539 }
540 ASSERT(boundary_plus.e() == w.e());
541 DiyFp ten_mk; // Cached power of ten: 10^-k
542 int mk; // -k
543 int ten_mk_minimal_binary_exponent =
544 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
545 int ten_mk_maximal_binary_exponent =
546 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
547 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
548 ten_mk_minimal_binary_exponent,
549 ten_mk_maximal_binary_exponent,
550 &ten_mk, &mk);
551 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
552 DiyFp::kSignificandSize) &&
553 (kMaximalTargetExponent >= w.e() + ten_mk.e() +
554 DiyFp::kSignificandSize));
555 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
556 // 64 bit significand and ten_mk is thus only precise up to 64 bits.
557
558 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
559 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
560 // off by a small amount.
561 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
562 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
563 // (f-1) * 2^e < w*10^k < (f+1) * 2^e
564 DiyFp scaled_w = DiyFp::Times(w, ten_mk);
565 ASSERT(scaled_w.e() ==
566 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
567 // In theory it would be possible to avoid some recomputations by computing
568 // the difference between w and boundary_minus/plus (a power of 2) and to
569 // compute scaled_boundary_minus/plus by subtracting/adding from
570 // scaled_w. However the code becomes much less readable and the speed
571 // enhancements are not terriffic.
572 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
573 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
574
575 // DigitGen will generate the digits of scaled_w. Therefore we have
576 // v == (double) (scaled_w * 10^-mk).
577 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
578 // integer than it will be updated. For instance if scaled_w == 1.23 then
579 // the buffer will be filled with "123" und the decimal_exponent will be
580 // decreased by 2.
581 int kappa;
582 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
583 buffer, length, &kappa);
584 *decimal_exponent = -mk + kappa;
585 return result;
586}
587
588
589// The "counted" version of grisu3 (see above) only generates requested_digits
590// number of digits. This version does not generate the shortest representation,
591// and with enough requested digits 0.1 will at some point print as 0.9999999...
592// Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
593// therefore the rounding strategy for halfway cases is irrelevant.
594static bool Grisu3Counted(double v,
595 int requested_digits,
596 BufferReference<char> buffer,
597 int* length,
598 int* decimal_exponent) {
599 DiyFp w = Double(v).AsNormalizedDiyFp();
600 DiyFp ten_mk; // Cached power of ten: 10^-k
601 int mk; // -k
602 int ten_mk_minimal_binary_exponent =
603 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
604 int ten_mk_maximal_binary_exponent =
605 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
606 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
607 ten_mk_minimal_binary_exponent,
608 ten_mk_maximal_binary_exponent,
609 &ten_mk, &mk);
610 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() +
611 DiyFp::kSignificandSize) &&
612 (kMaximalTargetExponent >= w.e() + ten_mk.e() +
613 DiyFp::kSignificandSize));
614 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
615 // 64 bit significand and ten_mk is thus only precise up to 64 bits.
616
617 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
618 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
619 // off by a small amount.
620 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
621 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
622 // (f-1) * 2^e < w*10^k < (f+1) * 2^e
623 DiyFp scaled_w = DiyFp::Times(w, ten_mk);
624
625 // We now have (double) (scaled_w * 10^-mk).
626 // DigitGen will generate the first requested_digits digits of scaled_w and
627 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
628 // will not always be exactly the same since DigitGenCounted only produces a
629 // limited number of digits.)
630 int kappa;
631 bool result = DigitGenCounted(scaled_w, requested_digits,
632 buffer, length, &kappa);
633 *decimal_exponent = -mk + kappa;
634 return result;
635}
636
637
638bool FastDtoa(double v,
639 FastDtoaMode mode,
640 int requested_digits,
641 BufferReference<char> buffer,
642 int* length,
643 int* decimal_point) {
644 ASSERT(v > 0);
645 ASSERT(!Double(v).IsSpecial());
646
647 bool result = false;
648 int decimal_exponent = 0;
649 switch (mode) {
650 case FAST_DTOA_SHORTEST:
651 case FAST_DTOA_SHORTEST_SINGLE:
652 result = Grisu3(v, mode, buffer, length, &decimal_exponent);
653 break;
654 case FAST_DTOA_PRECISION:
655 result = Grisu3Counted(v, requested_digits,
656 buffer, length, &decimal_exponent);
657 break;
658 default:
659 UNREACHABLE();
660 }
661 if (result) {
662 *decimal_point = *length + decimal_exponent;
663 buffer[*length] = '\0';
664 }
665 return result;
666}
667
668} // namespace double_conversion
669} // namespace WTF
670