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1 : // Copyright 2012 the V8 project authors. All rights reserved.
2 : // Redistribution and use in source and binary forms, with or without
3 : // modification, are permitted provided that the following conditions are
4 : // met:
5 : //
6 : // * Redistributions of source code must retain the above copyright
7 : // notice, this list of conditions and the following disclaimer.
8 : // * Redistributions in binary form must reproduce the above
9 : // copyright notice, this list of conditions and the following
10 : // disclaimer in the documentation and/or other materials provided
11 : // with the distribution.
12 : // * Neither the name of Google Inc. nor the names of its
13 : // contributors may be used to endorse or promote products derived
14 : // from this software without specific prior written permission.
15 : //
16 : // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
17 : // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
18 : // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
19 : // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
20 : // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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22 : // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 : // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 : // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 : // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
26 : // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 :
28 : #include "fast-dtoa.h"
29 :
30 : #include "cached-powers.h"
31 : #include "diy-fp.h"
32 : #include "ieee.h"
33 :
34 : namespace 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.
42 : static const int kMinimalTargetExponent = -60;
43 : static 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.
61 8 : static 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 8 : uint64_t small_distance = distance_too_high_w - unit;
69 8 : 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 8 : ASSERT(rest <= unsafe_interval);
142 13 : while (rest < small_distance && // Negated condition 1
143 20 : unsafe_interval - rest >= ten_kappa && // Negated condition 2
144 10 : (rest + ten_kappa < small_distance || // buffer{-1} > w_high
145 4 : small_distance - rest >= rest + ten_kappa - small_distance)) {
146 6 : buffer[length - 1]--;
147 6 : 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 12 : if (rest < big_distance &&
154 6 : unsafe_interval - rest >= ten_kappa &&
155 4 : (rest + ten_kappa < big_distance ||
156 2 : big_distance - rest > rest + ten_kappa - big_distance)) {
157 0 : 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 8 : 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.
181 0 : static 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 0 : 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 0 : 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 0 : if (ten_kappa - unit <= unit) return false;
199 : // If 2 * (rest + unit) <= 10^kappa we can safely round down.
200 0 : if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
201 0 : return true;
202 : }
203 : // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
204 0 : if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
205 : // Increment the last digit recursively until we find a non '9' digit.
206 0 : buffer[length - 1]++;
207 0 : for (int i = length - 1; i > 0; --i) {
208 0 : if (buffer[i] != '0' + 10) break;
209 0 : buffer[i] = '0';
210 0 : 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 0 : if (buffer[0] == '0' + 10) {
217 0 : buffer[0] = '1';
218 0 : (*kappa) += 1;
219 : }
220 0 : return true;
221 : }
222 0 : 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
236 : static unsigned int const kSmallPowersOfTen[] =
237 : {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
238 : 1000000000};
239 :
240 8 : static void BiggestPowerTen(uint32_t number,
241 : int number_bits,
242 : uint32_t* power,
243 : int* exponent_plus_one) {
244 8 : ASSERT(number < (1u << (number_bits + 1)));
245 : // 1233/4096 is approximately 1/lg(10).
246 8 : 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 8 : exponent_plus_one_guess++;
250 : // We don't have any guarantees that 2^number_bits <= number.
251 8 : if (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
252 5 : exponent_plus_one_guess--;
253 : }
254 8 : *power = kSmallPowersOfTen[exponent_plus_one_guess];
255 8 : *exponent_plus_one = exponent_plus_one_guess;
256 8 : }
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.
300 8 : static bool DigitGen(DiyFp low,
301 : DiyFp w,
302 : DiyFp high,
303 : Vector<char> buffer,
304 : int* length,
305 : int* kappa) {
306 8 : ASSERT(low.e() == w.e() && w.e() == high.e());
307 8 : ASSERT(low.f() + 1 <= high.f() - 1);
308 8 : 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 8 : uint64_t unit = 1;
321 8 : DiyFp too_low = DiyFp(low.f() - unit, low.e());
322 8 : 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 8 : 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 8 : DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
334 : // Division by one is a shift.
335 8 : uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
336 : // Modulo by one is an and.
337 8 : uint64_t fractionals = too_high.f() & (one.f() - 1);
338 : uint32_t divisor;
339 : int divisor_exponent_plus_one;
340 8 : BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
341 8 : &divisor, &divisor_exponent_plus_one);
342 8 : *kappa = divisor_exponent_plus_one;
343 8 : *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 92 : while (*kappa > 0) {
349 44 : int digit = integrals / divisor;
350 44 : ASSERT(digit <= 9);
351 44 : buffer[*length] = static_cast<char>('0' + digit);
352 44 : (*length)++;
353 44 : integrals %= divisor;
354 44 : (*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 44 : (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 44 : 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 4 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
365 : unsafe_interval.f(), rest,
366 4 : static_cast<uint64_t>(divisor) << -one.e(), unit);
367 : }
368 42 : 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 6 : ASSERT(one.e() >= -60);
378 6 : ASSERT(fractionals < one.f());
379 6 : ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
380 : for (;;) {
381 61 : fractionals *= 10;
382 61 : unit *= 10;
383 61 : unsafe_interval.set_f(unsafe_interval.f() * 10);
384 : // Integer division by one.
385 61 : int digit = static_cast<int>(fractionals >> -one.e());
386 61 : ASSERT(digit <= 9);
387 61 : buffer[*length] = static_cast<char>('0' + digit);
388 61 : (*length)++;
389 61 : fractionals &= one.f() - 1; // Modulo by one.
390 61 : (*kappa)--;
391 61 : if (fractionals < unsafe_interval.f()) {
392 12 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
393 6 : unsafe_interval.f(), fractionals, one.f(), unit);
394 : }
395 55 : }
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.
428 0 : static bool DigitGenCounted(DiyFp w,
429 : int requested_digits,
430 : Vector<char> buffer,
431 : int* length,
432 : int* kappa) {
433 0 : 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 0 : 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 0 : DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
444 : // Division by one is a shift.
445 0 : uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
446 : // Modulo by one is an and.
447 0 : uint64_t fractionals = w.f() & (one.f() - 1);
448 : uint32_t divisor;
449 : int divisor_exponent_plus_one;
450 0 : BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
451 0 : &divisor, &divisor_exponent_plus_one);
452 0 : *kappa = divisor_exponent_plus_one;
453 0 : *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 0 : while (*kappa > 0) {
460 0 : int digit = integrals / divisor;
461 0 : ASSERT(digit <= 9);
462 0 : buffer[*length] = static_cast<char>('0' + digit);
463 0 : (*length)++;
464 0 : requested_digits--;
465 0 : integrals %= divisor;
466 0 : (*kappa)--;
467 : // Note that kappa now equals the exponent of the divisor and that the
468 : // invariant thus holds again.
469 0 : if (requested_digits == 0) break;
470 0 : divisor /= 10;
471 : }
472 :
473 0 : if (requested_digits == 0) {
474 : uint64_t rest =
475 0 : (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
476 0 : return RoundWeedCounted(buffer, *length, rest,
477 0 : static_cast<uint64_t>(divisor) << -one.e(), w_error,
478 0 : 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 0 : ASSERT(one.e() >= -60);
488 0 : ASSERT(fractionals < one.f());
489 0 : ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
490 0 : while (requested_digits > 0 && fractionals > w_error) {
491 0 : fractionals *= 10;
492 0 : w_error *= 10;
493 : // Integer division by one.
494 0 : int digit = static_cast<int>(fractionals >> -one.e());
495 0 : ASSERT(digit <= 9);
496 0 : buffer[*length] = static_cast<char>('0' + digit);
497 0 : (*length)++;
498 0 : requested_digits--;
499 0 : fractionals &= one.f() - 1; // Modulo by one.
500 0 : (*kappa)--;
501 : }
502 0 : if (requested_digits != 0) return false;
503 0 : return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
504 0 : 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.
519 8 : static bool Grisu3(double v,
520 : FastDtoaMode mode,
521 : Vector<char> buffer,
522 : int* length,
523 : int* decimal_exponent) {
524 8 : 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 8 : DiyFp boundary_minus, boundary_plus;
530 8 : if (mode == FAST_DTOA_SHORTEST) {
531 8 : Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
532 : } else {
533 0 : ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE);
534 0 : float single_v = static_cast<float>(v);
535 0 : Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
536 : }
537 8 : ASSERT(boundary_plus.e() == w.e());
538 8 : DiyFp ten_mk; // Cached power of ten: 10^-k
539 : int mk; // -k
540 : int ten_mk_minimal_binary_exponent =
541 8 : kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
542 : int ten_mk_maximal_binary_exponent =
543 8 : kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
544 : PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
545 : ten_mk_minimal_binary_exponent,
546 : ten_mk_maximal_binary_exponent,
547 8 : &ten_mk, &mk);
548 8 : 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 8 : DiyFp scaled_w = DiyFp::Times(w, ten_mk);
562 8 : 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 8 : DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
570 8 : 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 8 : buffer, length, &kappa);
581 8 : *decimal_exponent = -mk + kappa;
582 8 : 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.
591 0 : static bool Grisu3Counted(double v,
592 : int requested_digits,
593 : Vector<char> buffer,
594 : int* length,
595 : int* decimal_exponent) {
596 0 : DiyFp w = Double(v).AsNormalizedDiyFp();
597 0 : DiyFp ten_mk; // Cached power of ten: 10^-k
598 : int mk; // -k
599 : int ten_mk_minimal_binary_exponent =
600 0 : kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
601 : int ten_mk_maximal_binary_exponent =
602 0 : kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
603 : PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
604 : ten_mk_minimal_binary_exponent,
605 : ten_mk_maximal_binary_exponent,
606 0 : &ten_mk, &mk);
607 0 : 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 0 : 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 0 : buffer, length, &kappa);
630 0 : *decimal_exponent = -mk + kappa;
631 0 : return result;
632 : }
633 :
634 :
635 8 : bool FastDtoa(double v,
636 : FastDtoaMode mode,
637 : int requested_digits,
638 : Vector<char> buffer,
639 : int* length,
640 : int* decimal_point) {
641 8 : ASSERT(v > 0);
642 8 : ASSERT(!Double(v).IsSpecial());
643 :
644 8 : bool result = false;
645 8 : int decimal_exponent = 0;
646 8 : switch (mode) {
647 : case FAST_DTOA_SHORTEST:
648 : case FAST_DTOA_SHORTEST_SINGLE:
649 8 : result = Grisu3(v, mode, buffer, length, &decimal_exponent);
650 8 : break;
651 : case FAST_DTOA_PRECISION:
652 : result = Grisu3Counted(v, requested_digits,
653 0 : buffer, length, &decimal_exponent);
654 0 : break;
655 : default:
656 0 : UNREACHABLE();
657 : }
658 8 : if (result) {
659 8 : *decimal_point = *length + decimal_exponent;
660 8 : buffer[*length] = '\0';
661 : }
662 8 : return result;
663 : }
664 :
665 : } // namespace double_conversion
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