Line data Source code
1 : // Copyright 2011 the V8 project authors. All rights reserved.
2 : // Use of this source code is governed by a BSD-style license that can be
3 : // found in the LICENSE file.
4 :
5 : #include <stdint.h>
6 : #include "src/base/logging.h"
7 : #include "src/utils.h"
8 :
9 : #include "src/fast-dtoa.h"
10 :
11 : #include "src/cached-powers.h"
12 : #include "src/diy-fp.h"
13 : #include "src/double.h"
14 :
15 : namespace v8 {
16 : namespace internal {
17 :
18 : // The minimal and maximal target exponent define the range of w's binary
19 : // exponent, where 'w' is the result of multiplying the input by a cached power
20 : // of ten.
21 : //
22 : // A different range might be chosen on a different platform, to optimize digit
23 : // generation, but a smaller range requires more powers of ten to be cached.
24 : static const int kMinimalTargetExponent = -60;
25 : static const int kMaximalTargetExponent = -32;
26 :
27 :
28 : // Adjusts the last digit of the generated number, and screens out generated
29 : // solutions that may be inaccurate. A solution may be inaccurate if it is
30 : // outside the safe interval, or if we ctannot prove that it is closer to the
31 : // input than a neighboring representation of the same length.
32 : //
33 : // Input: * buffer containing the digits of too_high / 10^kappa
34 : // * the buffer's length
35 : // * distance_too_high_w == (too_high - w).f() * unit
36 : // * unsafe_interval == (too_high - too_low).f() * unit
37 : // * rest = (too_high - buffer * 10^kappa).f() * unit
38 : // * ten_kappa = 10^kappa * unit
39 : // * unit = the common multiplier
40 : // Output: returns true if the buffer is guaranteed to contain the closest
41 : // representable number to the input.
42 : // Modifies the generated digits in the buffer to approach (round towards) w.
43 2133138 : static bool RoundWeed(Vector<char> buffer,
44 : int length,
45 : uint64_t distance_too_high_w,
46 : uint64_t unsafe_interval,
47 : uint64_t rest,
48 : uint64_t ten_kappa,
49 : uint64_t unit) {
50 2133138 : uint64_t small_distance = distance_too_high_w - unit;
51 2133138 : uint64_t big_distance = distance_too_high_w + unit;
52 : // Let w_low = too_high - big_distance, and
53 : // w_high = too_high - small_distance.
54 : // Note: w_low < w < w_high
55 : //
56 : // The real w (* unit) must lie somewhere inside the interval
57 : // ]w_low; w_high[ (often written as "(w_low; w_high)")
58 :
59 : // Basically the buffer currently contains a number in the unsafe interval
60 : // ]too_low; too_high[ with too_low < w < too_high
61 : //
62 : // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
63 : // ^v 1 unit ^ ^ ^ ^
64 : // boundary_high --------------------- . . . .
65 : // ^v 1 unit . . . .
66 : // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
67 : // . . ^ . .
68 : // . big_distance . . .
69 : // . . . . rest
70 : // small_distance . . . .
71 : // v . . . .
72 : // w_high - - - - - - - - - - - - - - - - - - . . . .
73 : // ^v 1 unit . . . .
74 : // w ---------------------------------------- . . . .
75 : // ^v 1 unit v . . .
76 : // w_low - - - - - - - - - - - - - - - - - - - - - . . .
77 : // . . v
78 : // buffer --------------------------------------------------+-------+--------
79 : // . .
80 : // safe_interval .
81 : // v .
82 : // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
83 : // ^v 1 unit .
84 : // boundary_low ------------------------- unsafe_interval
85 : // ^v 1 unit v
86 : // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
87 : //
88 : //
89 : // Note that the value of buffer could lie anywhere inside the range too_low
90 : // to too_high.
91 : //
92 : // boundary_low, boundary_high and w are approximations of the real boundaries
93 : // and v (the input number). They are guaranteed to be precise up to one unit.
94 : // In fact the error is guaranteed to be strictly less than one unit.
95 : //
96 : // Anything that lies outside the unsafe interval is guaranteed not to round
97 : // to v when read again.
98 : // Anything that lies inside the safe interval is guaranteed to round to v
99 : // when read again.
100 : // If the number inside the buffer lies inside the unsafe interval but not
101 : // inside the safe interval then we simply do not know and bail out (returning
102 : // false).
103 : //
104 : // Similarly we have to take into account the imprecision of 'w' when finding
105 : // the closest representation of 'w'. If we have two potential
106 : // representations, and one is closer to both w_low and w_high, then we know
107 : // it is closer to the actual value v.
108 : //
109 : // By generating the digits of too_high we got the largest (closest to
110 : // too_high) buffer that is still in the unsafe interval. In the case where
111 : // w_high < buffer < too_high we try to decrement the buffer.
112 : // This way the buffer approaches (rounds towards) w.
113 : // There are 3 conditions that stop the decrementation process:
114 : // 1) the buffer is already below w_high
115 : // 2) decrementing the buffer would make it leave the unsafe interval
116 : // 3) decrementing the buffer would yield a number below w_high and farther
117 : // away than the current number. In other words:
118 : // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
119 : // Instead of using the buffer directly we use its distance to too_high.
120 : // Conceptually rest ~= too_high - buffer
121 : // We need to do the following tests in this order to avoid over- and
122 : // underflows.
123 : DCHECK(rest <= unsafe_interval);
124 2132908 : while (rest < small_distance && // Negated condition 1
125 4779444 : unsafe_interval - rest >= ten_kappa && // Negated condition 2
126 2387172 : (rest + ten_kappa < small_distance || // buffer{-1} > w_high
127 851697 : small_distance - rest >= rest + ten_kappa - small_distance)) {
128 2221662 : buffer[length - 1]--;
129 : rest += ten_kappa;
130 : }
131 :
132 : // We have approached w+ as much as possible. We now test if approaching w-
133 : // would require changing the buffer. If yes, then we have two possible
134 : // representations close to w, but we cannot decide which one is closer.
135 3294864 : if (rest < big_distance &&
136 1590933 : unsafe_interval - rest >= ten_kappa &&
137 858414 : (rest + ten_kappa < big_distance ||
138 429207 : big_distance - rest > rest + ten_kappa - big_distance)) {
139 : return false;
140 : }
141 :
142 : // Weeding test.
143 : // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
144 : // Since too_low = too_high - unsafe_interval this is equivalent to
145 : // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
146 : // Conceptually we have: rest ~= too_high - buffer
147 2128033 : return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
148 : }
149 :
150 :
151 : // Rounds the buffer upwards if the result is closer to v by possibly adding
152 : // 1 to the buffer. If the precision of the calculation is not sufficient to
153 : // round correctly, return false.
154 : // The rounding might shift the whole buffer in which case the kappa is
155 : // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
156 : //
157 : // If 2*rest > ten_kappa then the buffer needs to be round up.
158 : // rest can have an error of +/- 1 unit. This function accounts for the
159 : // imprecision and returns false, if the rounding direction cannot be
160 : // unambiguously determined.
161 : //
162 : // Precondition: rest < ten_kappa.
163 936211 : static bool RoundWeedCounted(Vector<char> buffer,
164 : int length,
165 : uint64_t rest,
166 : uint64_t ten_kappa,
167 : uint64_t unit,
168 : int* kappa) {
169 : DCHECK(rest < ten_kappa);
170 : // The following tests are done in a specific order to avoid overflows. They
171 : // will work correctly with any uint64 values of rest < ten_kappa and unit.
172 : //
173 : // If the unit is too big, then we don't know which way to round. For example
174 : // a unit of 50 means that the real number lies within rest +/- 50. If
175 : // 10^kappa == 40 then there is no way to tell which way to round.
176 936211 : if (unit >= ten_kappa) return false;
177 : // Even if unit is just half the size of 10^kappa we are already completely
178 : // lost. (And after the previous test we know that the expression will not
179 : // over/underflow.)
180 907799 : if (ten_kappa - unit <= unit) return false;
181 : // If 2 * (rest + unit) <= 10^kappa we can safely round down.
182 893949 : if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
183 : return true;
184 : }
185 : // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
186 452104 : if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
187 : // Increment the last digit recursively until we find a non '9' digit.
188 864124 : buffer[length - 1]++;
189 478865 : for (int i = length - 1; i > 0; --i) {
190 911476 : if (buffer[i] != '0' + 10) break;
191 46803 : buffer[i] = '0';
192 93606 : buffer[i - 1]++;
193 : }
194 : // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
195 : // exception of the first digit all digits are now '0'. Simply switch the
196 : // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
197 : // the power (the kappa) is increased.
198 432062 : if (buffer[0] == '0' + 10) {
199 1137 : buffer[0] = '1';
200 1137 : (*kappa) += 1;
201 : }
202 : return true;
203 : }
204 : return false;
205 : }
206 :
207 :
208 : static const uint32_t kTen4 = 10000;
209 : static const uint32_t kTen5 = 100000;
210 : static const uint32_t kTen6 = 1000000;
211 : static const uint32_t kTen7 = 10000000;
212 : static const uint32_t kTen8 = 100000000;
213 : static const uint32_t kTen9 = 1000000000;
214 :
215 : // Returns the biggest power of ten that is less than or equal than the given
216 : // number. We furthermore receive the maximum number of bits 'number' has.
217 : // If number_bits == 0 then 0^-1 is returned
218 : // The number of bits must be <= 32.
219 : // Precondition: number < (1 << (number_bits + 1)).
220 3138803 : static void BiggestPowerTen(uint32_t number,
221 : int number_bits,
222 : uint32_t* power,
223 : int* exponent) {
224 3138803 : switch (number_bits) {
225 : case 32:
226 : case 31:
227 : case 30:
228 47051 : if (kTen9 <= number) {
229 0 : *power = kTen9;
230 0 : *exponent = 9;
231 0 : break;
232 : }
233 : V8_FALLTHROUGH;
234 : case 29:
235 : case 28:
236 : case 27:
237 304437 : if (kTen8 <= number) {
238 218732 : *power = kTen8;
239 218732 : *exponent = 8;
240 218732 : break;
241 : }
242 : V8_FALLTHROUGH;
243 : case 26:
244 : case 25:
245 : case 24:
246 365127 : if (kTen7 <= number) {
247 273486 : *power = kTen7;
248 273486 : *exponent = 7;
249 273486 : break;
250 : }
251 : V8_FALLTHROUGH;
252 : case 23:
253 : case 22:
254 : case 21:
255 : case 20:
256 1032697 : if (kTen6 <= number) {
257 473628 : *power = kTen6;
258 473628 : *exponent = 6;
259 473628 : break;
260 : }
261 : V8_FALLTHROUGH;
262 : case 19:
263 : case 18:
264 : case 17:
265 1056647 : if (kTen5 <= number) {
266 906561 : *power = kTen5;
267 906561 : *exponent = 5;
268 906561 : break;
269 : }
270 : V8_FALLTHROUGH;
271 : case 16:
272 : case 15:
273 : case 14:
274 430296 : if (kTen4 <= number) {
275 360882 : *power = kTen4;
276 360882 : *exponent = 4;
277 360882 : break;
278 : }
279 : V8_FALLTHROUGH;
280 : case 13:
281 : case 12:
282 : case 11:
283 : case 10:
284 411073 : if (1000 <= number) {
285 291779 : *power = 1000;
286 291779 : *exponent = 3;
287 291779 : break;
288 : }
289 : V8_FALLTHROUGH;
290 : case 9:
291 : case 8:
292 : case 7:
293 366010 : if (100 <= number) {
294 274705 : *power = 100;
295 274705 : *exponent = 2;
296 274705 : break;
297 : }
298 : V8_FALLTHROUGH;
299 : case 6:
300 : case 5:
301 : case 4:
302 339030 : if (10 <= number) {
303 271702 : *power = 10;
304 271702 : *exponent = 1;
305 271702 : break;
306 : }
307 : V8_FALLTHROUGH;
308 : case 3:
309 : case 2:
310 : case 1:
311 67328 : if (1 <= number) {
312 67328 : *power = 1;
313 67328 : *exponent = 0;
314 67328 : break;
315 : }
316 : V8_FALLTHROUGH;
317 : case 0:
318 0 : *power = 0;
319 0 : *exponent = -1;
320 0 : break;
321 : default:
322 : // Following assignments are here to silence compiler warnings.
323 0 : *power = 0;
324 0 : *exponent = 0;
325 0 : UNREACHABLE();
326 : }
327 3138803 : }
328 :
329 : // Generates the digits of input number w.
330 : // w is a floating-point number (DiyFp), consisting of a significand and an
331 : // exponent. Its exponent is bounded by kMinimalTargetExponent and
332 : // kMaximalTargetExponent.
333 : // Hence -60 <= w.e() <= -32.
334 : //
335 : // Returns false if it fails, in which case the generated digits in the buffer
336 : // should not be used.
337 : // Preconditions:
338 : // * low, w and high are correct up to 1 ulp (unit in the last place). That
339 : // is, their error must be less than a unit of their last digits.
340 : // * low.e() == w.e() == high.e()
341 : // * low < w < high, and taking into account their error: low~ <= high~
342 : // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
343 : // Postconditions: returns false if procedure fails.
344 : // otherwise:
345 : // * buffer is not null-terminated, but len contains the number of digits.
346 : // * buffer contains the shortest possible decimal digit-sequence
347 : // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
348 : // correct values of low and high (without their error).
349 : // * if more than one decimal representation gives the minimal number of
350 : // decimal digits then the one closest to W (where W is the correct value
351 : // of w) is chosen.
352 : // Remark: this procedure takes into account the imprecision of its input
353 : // numbers. If the precision is not enough to guarantee all the postconditions
354 : // then false is returned. This usually happens rarely (~0.5%).
355 : //
356 : // Say, for the sake of example, that
357 : // w.e() == -48, and w.f() == 0x1234567890ABCDEF
358 : // w's value can be computed by w.f() * 2^w.e()
359 : // We can obtain w's integral digits by simply shifting w.f() by -w.e().
360 : // -> w's integral part is 0x1234
361 : // w's fractional part is therefore 0x567890ABCDEF.
362 : // Printing w's integral part is easy (simply print 0x1234 in decimal).
363 : // In order to print its fraction we repeatedly multiply the fraction by 10 and
364 : // get each digit. Example the first digit after the point would be computed by
365 : // (0x567890ABCDEF * 10) >> 48. -> 3
366 : // The whole thing becomes slightly more complicated because we want to stop
367 : // once we have enough digits. That is, once the digits inside the buffer
368 : // represent 'w' we can stop. Everything inside the interval low - high
369 : // represents w. However we have to pay attention to low, high and w's
370 : // imprecision.
371 2133139 : static bool DigitGen(DiyFp low,
372 : DiyFp w,
373 : DiyFp high,
374 : Vector<char> buffer,
375 : int* length,
376 : int* kappa) {
377 : DCHECK(low.e() == w.e() && w.e() == high.e());
378 : DCHECK(low.f() + 1 <= high.f() - 1);
379 : DCHECK(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
380 : // low, w and high are imprecise, but by less than one ulp (unit in the last
381 : // place).
382 : // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
383 : // the new numbers are outside of the interval we want the final
384 : // representation to lie in.
385 : // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
386 : // numbers that are certain to lie in the interval. We will use this fact
387 : // later on.
388 : // We will now start by generating the digits within the uncertain
389 : // interval. Later we will weed out representations that lie outside the safe
390 : // interval and thus _might_ lie outside the correct interval.
391 : uint64_t unit = 1;
392 2133139 : DiyFp too_low = DiyFp(low.f() - unit, low.e());
393 2133139 : DiyFp too_high = DiyFp(high.f() + unit, high.e());
394 : // too_low and too_high are guaranteed to lie outside the interval we want the
395 : // generated number in.
396 : DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
397 : // We now cut the input number into two parts: the integral digits and the
398 : // fractionals. We will not write any decimal separator though, but adapt
399 : // kappa instead.
400 : // Reminder: we are currently computing the digits (stored inside the buffer)
401 : // such that: too_low < buffer * 10^kappa < too_high
402 : // We use too_high for the digit_generation and stop as soon as possible.
403 : // If we stop early we effectively round down.
404 2133139 : DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
405 : // Division by one is a shift.
406 2133139 : uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
407 : // Modulo by one is an and.
408 2133139 : uint64_t fractionals = too_high.f() & (one.f() - 1);
409 : uint32_t divisor;
410 : int divisor_exponent;
411 2133139 : BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
412 2133139 : &divisor, &divisor_exponent);
413 2133138 : *kappa = divisor_exponent + 1;
414 2133138 : *length = 0;
415 : // Loop invariant: buffer = too_high / 10^kappa (integer division)
416 : // The invariant holds for the first iteration: kappa has been initialized
417 : // with the divisor exponent + 1. And the divisor is the biggest power of ten
418 : // that is smaller than integrals.
419 13892970 : while (*kappa > 0) {
420 11834387 : int digit = integrals / divisor;
421 23668774 : buffer[*length] = '0' + digit;
422 11834387 : (*length)++;
423 11834387 : integrals %= divisor;
424 11834387 : (*kappa)--;
425 : // Note that kappa now equals the exponent of the divisor and that the
426 : // invariant thus holds again.
427 : uint64_t rest =
428 11834387 : (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
429 : // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
430 : // Reminder: unsafe_interval.e() == one.e()
431 11834387 : if (rest < unsafe_interval.f()) {
432 : // Rounding down (by not emitting the remaining digits) yields a number
433 : // that lies within the unsafe interval.
434 149110 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
435 : unsafe_interval.f(), rest,
436 74555 : static_cast<uint64_t>(divisor) << -one.e(), unit);
437 : }
438 11759832 : divisor /= 10;
439 : }
440 :
441 : // The integrals have been generated. We are at the point of the decimal
442 : // separator. In the following loop we simply multiply the remaining digits by
443 : // 10 and divide by one. We just need to pay attention to multiply associated
444 : // data (like the interval or 'unit'), too.
445 : // Note that the multiplication by 10 does not overflow, because w.e >= -60
446 : // and thus one.e >= -60.
447 : DCHECK_GE(one.e(), -60);
448 : DCHECK(fractionals < one.f());
449 : DCHECK(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
450 : while (true) {
451 21280685 : fractionals *= 10;
452 21280685 : unit *= 10;
453 21280685 : unsafe_interval.set_f(unsafe_interval.f() * 10);
454 : // Integer division by one.
455 21280685 : int digit = static_cast<int>(fractionals >> -one.e());
456 42561370 : buffer[*length] = '0' + digit;
457 21280685 : (*length)++;
458 21280685 : fractionals &= one.f() - 1; // Modulo by one.
459 21280685 : (*kappa)--;
460 21280685 : if (fractionals < unsafe_interval.f()) {
461 2058583 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
462 2058583 : unsafe_interval.f(), fractionals, one.f(), unit);
463 : }
464 : }
465 : }
466 :
467 :
468 :
469 : // Generates (at most) requested_digits of input number w.
470 : // w is a floating-point number (DiyFp), consisting of a significand and an
471 : // exponent. Its exponent is bounded by kMinimalTargetExponent and
472 : // kMaximalTargetExponent.
473 : // Hence -60 <= w.e() <= -32.
474 : //
475 : // Returns false if it fails, in which case the generated digits in the buffer
476 : // should not be used.
477 : // Preconditions:
478 : // * w is correct up to 1 ulp (unit in the last place). That
479 : // is, its error must be strictly less than a unit of its last digit.
480 : // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
481 : //
482 : // Postconditions: returns false if procedure fails.
483 : // otherwise:
484 : // * buffer is not null-terminated, but length contains the number of
485 : // digits.
486 : // * the representation in buffer is the most precise representation of
487 : // requested_digits digits.
488 : // * buffer contains at most requested_digits digits of w. If there are less
489 : // than requested_digits digits then some trailing '0's have been removed.
490 : // * kappa is such that
491 : // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
492 : //
493 : // Remark: This procedure takes into account the imprecision of its input
494 : // numbers. If the precision is not enough to guarantee all the postconditions
495 : // then false is returned. This usually happens rarely, but the failure-rate
496 : // increases with higher requested_digits.
497 1005665 : static bool DigitGenCounted(DiyFp w,
498 : int requested_digits,
499 : Vector<char> buffer,
500 : int* length,
501 : int* kappa) {
502 : DCHECK(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
503 : DCHECK_GE(kMinimalTargetExponent, -60);
504 : DCHECK_LE(kMaximalTargetExponent, -32);
505 : // w is assumed to have an error less than 1 unit. Whenever w is scaled we
506 : // also scale its error.
507 : uint64_t w_error = 1;
508 : // We cut the input number into two parts: the integral digits and the
509 : // fractional digits. We don't emit any decimal separator, but adapt kappa
510 : // instead. Example: instead of writing "1.2" we put "12" into the buffer and
511 : // increase kappa by 1.
512 1005665 : DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
513 : // Division by one is a shift.
514 1005665 : uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
515 : // Modulo by one is an and.
516 1005665 : uint64_t fractionals = w.f() & (one.f() - 1);
517 : uint32_t divisor;
518 : int divisor_exponent;
519 1005665 : BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
520 1005665 : &divisor, &divisor_exponent);
521 1005665 : *kappa = divisor_exponent + 1;
522 1005665 : *length = 0;
523 :
524 : // Loop invariant: buffer = w / 10^kappa (integer division)
525 : // The invariant holds for the first iteration: kappa has been initialized
526 : // with the divisor exponent + 1. And the divisor is the biggest power of ten
527 : // that is smaller than 'integrals'.
528 5386015 : while (*kappa > 0) {
529 4634081 : int digit = integrals / divisor;
530 9268162 : buffer[*length] = '0' + digit;
531 4634081 : (*length)++;
532 4634081 : requested_digits--;
533 4634081 : integrals %= divisor;
534 4634081 : (*kappa)--;
535 : // Note that kappa now equals the exponent of the divisor and that the
536 : // invariant thus holds again.
537 4634081 : if (requested_digits == 0) break;
538 4380350 : divisor /= 10;
539 : }
540 :
541 1005665 : if (requested_digits == 0) {
542 : uint64_t rest =
543 253731 : (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
544 253731 : return RoundWeedCounted(buffer, *length, rest,
545 : static_cast<uint64_t>(divisor) << -one.e(), w_error,
546 253731 : kappa);
547 : }
548 :
549 : // The integrals have been generated. We are at the point of the decimal
550 : // separator. In the following loop we simply multiply the remaining digits by
551 : // 10 and divide by one. We just need to pay attention to multiply associated
552 : // data (the 'unit'), too.
553 : // Note that the multiplication by 10 does not overflow, because w.e >= -60
554 : // and thus one.e >= -60.
555 : DCHECK_GE(one.e(), -60);
556 : DCHECK(fractionals < one.f());
557 : DCHECK(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
558 7063305 : while (requested_digits > 0 && fractionals > w_error) {
559 6311371 : fractionals *= 10;
560 6311371 : w_error *= 10;
561 : // Integer division by one.
562 6311371 : int digit = static_cast<int>(fractionals >> -one.e());
563 12622742 : buffer[*length] = '0' + digit;
564 6311371 : (*length)++;
565 6311371 : requested_digits--;
566 6311371 : fractionals &= one.f() - 1; // Modulo by one.
567 6311371 : (*kappa)--;
568 : }
569 751934 : if (requested_digits != 0) return false;
570 682480 : return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
571 682480 : kappa);
572 : }
573 :
574 :
575 : // Provides a decimal representation of v.
576 : // Returns true if it succeeds, otherwise the result cannot be trusted.
577 : // There will be *length digits inside the buffer (not null-terminated).
578 : // If the function returns true then
579 : // v == (double) (buffer * 10^decimal_exponent).
580 : // The digits in the buffer are the shortest representation possible: no
581 : // 0.09999999999999999 instead of 0.1. The shorter representation will even be
582 : // chosen even if the longer one would be closer to v.
583 : // The last digit will be closest to the actual v. That is, even if several
584 : // digits might correctly yield 'v' when read again, the closest will be
585 : // computed.
586 2133139 : static bool Grisu3(double v,
587 : Vector<char> buffer,
588 : int* length,
589 : int* decimal_exponent) {
590 2133139 : DiyFp w = Double(v).AsNormalizedDiyFp();
591 : // boundary_minus and boundary_plus are the boundaries between v and its
592 : // closest floating-point neighbors. Any number strictly between
593 : // boundary_minus and boundary_plus will round to v when convert to a double.
594 : // Grisu3 will never output representations that lie exactly on a boundary.
595 : DiyFp boundary_minus, boundary_plus;
596 2133138 : Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
597 : DCHECK(boundary_plus.e() == w.e());
598 : DiyFp ten_mk; // Cached power of ten: 10^-k
599 : int mk; // -k
600 : int ten_mk_minimal_binary_exponent =
601 2133138 : kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
602 : int ten_mk_maximal_binary_exponent =
603 2133138 : kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
604 : PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
605 : ten_mk_minimal_binary_exponent,
606 : ten_mk_maximal_binary_exponent,
607 2133138 : &ten_mk, &mk);
608 : DCHECK((kMinimalTargetExponent <= w.e() + ten_mk.e() +
609 : DiyFp::kSignificandSize) &&
610 : (kMaximalTargetExponent >= w.e() + ten_mk.e() +
611 : DiyFp::kSignificandSize));
612 : // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
613 : // 64 bit significand and ten_mk is thus only precise up to 64 bits.
614 :
615 : // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
616 : // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
617 : // off by a small amount.
618 : // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
619 : // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
620 : // (f-1) * 2^e < w*10^k < (f+1) * 2^e
621 : DiyFp scaled_w = DiyFp::Times(w, ten_mk);
622 : DCHECK(scaled_w.e() ==
623 : boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
624 : // In theory it would be possible to avoid some recomputations by computing
625 : // the difference between w and boundary_minus/plus (a power of 2) and to
626 : // compute scaled_boundary_minus/plus by subtracting/adding from
627 : // scaled_w. However the code becomes much less readable and the speed
628 : // enhancements are not terriffic.
629 : DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
630 : DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
631 :
632 : // DigitGen will generate the digits of scaled_w. Therefore we have
633 : // v == (double) (scaled_w * 10^-mk).
634 : // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
635 : // integer than it will be updated. For instance if scaled_w == 1.23 then
636 : // the buffer will be filled with "123" und the decimal_exponent will be
637 : // decreased by 2.
638 : int kappa;
639 : bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
640 2133139 : buffer, length, &kappa);
641 2133138 : *decimal_exponent = -mk + kappa;
642 2133138 : return result;
643 : }
644 :
645 :
646 : // The "counted" version of grisu3 (see above) only generates requested_digits
647 : // number of digits. This version does not generate the shortest representation,
648 : // and with enough requested digits 0.1 will at some point print as 0.9999999...
649 : // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
650 : // therefore the rounding strategy for halfway cases is irrelevant.
651 1005665 : static bool Grisu3Counted(double v,
652 : int requested_digits,
653 : Vector<char> buffer,
654 : int* length,
655 : int* decimal_exponent) {
656 1005665 : DiyFp w = Double(v).AsNormalizedDiyFp();
657 : DiyFp ten_mk; // Cached power of ten: 10^-k
658 : int mk; // -k
659 : int ten_mk_minimal_binary_exponent =
660 1005665 : kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
661 : int ten_mk_maximal_binary_exponent =
662 1005665 : kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
663 : PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
664 : ten_mk_minimal_binary_exponent,
665 : ten_mk_maximal_binary_exponent,
666 1005665 : &ten_mk, &mk);
667 : DCHECK((kMinimalTargetExponent <= w.e() + ten_mk.e() +
668 : DiyFp::kSignificandSize) &&
669 : (kMaximalTargetExponent >= w.e() + ten_mk.e() +
670 : DiyFp::kSignificandSize));
671 : // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
672 : // 64 bit significand and ten_mk is thus only precise up to 64 bits.
673 :
674 : // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
675 : // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
676 : // off by a small amount.
677 : // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
678 : // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
679 : // (f-1) * 2^e < w*10^k < (f+1) * 2^e
680 : DiyFp scaled_w = DiyFp::Times(w, ten_mk);
681 :
682 : // We now have (double) (scaled_w * 10^-mk).
683 : // DigitGen will generate the first requested_digits digits of scaled_w and
684 : // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
685 : // will not always be exactly the same since DigitGenCounted only produces a
686 : // limited number of digits.)
687 : int kappa;
688 : bool result = DigitGenCounted(scaled_w, requested_digits,
689 1005665 : buffer, length, &kappa);
690 1005665 : *decimal_exponent = -mk + kappa;
691 1005665 : return result;
692 : }
693 :
694 :
695 3138804 : bool FastDtoa(double v,
696 : FastDtoaMode mode,
697 : int requested_digits,
698 : Vector<char> buffer,
699 : int* length,
700 : int* decimal_point) {
701 : DCHECK_GT(v, 0);
702 : DCHECK(!Double(v).IsSpecial());
703 :
704 : bool result = false;
705 3138804 : int decimal_exponent = 0;
706 3138804 : switch (mode) {
707 : case FAST_DTOA_SHORTEST:
708 2133139 : result = Grisu3(v, buffer, length, &decimal_exponent);
709 2133137 : break;
710 : case FAST_DTOA_PRECISION:
711 : result = Grisu3Counted(v, requested_digits,
712 1005665 : buffer, length, &decimal_exponent);
713 1005665 : break;
714 : default:
715 0 : UNREACHABLE();
716 : }
717 3138802 : if (result) {
718 2997391 : *decimal_point = *length + decimal_exponent;
719 5994782 : buffer[*length] = '\0';
720 : }
721 3138802 : return result;
722 : }
723 :
724 : } // namespace internal
725 122004 : } // namespace v8
|
