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