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1 : // Copyright 2010 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
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10 : // disclaimer in the documentation and/or other materials provided
11 : // with the distribution.
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15 : //
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22 : // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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26 : // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 :
28 : #include <stdarg.h>
29 : #include <limits.h>
30 :
31 : #include "strtod.h"
32 : #include "bignum.h"
33 : #include "cached-powers.h"
34 : #include "ieee.h"
35 :
36 : namespace double_conversion {
37 :
38 : // 2^53 = 9007199254740992.
39 : // Any integer with at most 15 decimal digits will hence fit into a double
40 : // (which has a 53bit significand) without loss of precision.
41 : static const int kMaxExactDoubleIntegerDecimalDigits = 15;
42 : // 2^64 = 18446744073709551616 > 10^19
43 : static const int kMaxUint64DecimalDigits = 19;
44 :
45 : // Max double: 1.7976931348623157 x 10^308
46 : // Min non-zero double: 4.9406564584124654 x 10^-324
47 : // Any x >= 10^309 is interpreted as +infinity.
48 : // Any x <= 10^-324 is interpreted as 0.
49 : // Note that 2.5e-324 (despite being smaller than the min double) will be read
50 : // as non-zero (equal to the min non-zero double).
51 : static const int kMaxDecimalPower = 309;
52 : static const int kMinDecimalPower = -324;
53 :
54 : // 2^64 = 18446744073709551616
55 : static const uint64_t kMaxUint64 = UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF);
56 :
57 :
58 : static const double exact_powers_of_ten[] = {
59 : 1.0, // 10^0
60 : 10.0,
61 : 100.0,
62 : 1000.0,
63 : 10000.0,
64 : 100000.0,
65 : 1000000.0,
66 : 10000000.0,
67 : 100000000.0,
68 : 1000000000.0,
69 : 10000000000.0, // 10^10
70 : 100000000000.0,
71 : 1000000000000.0,
72 : 10000000000000.0,
73 : 100000000000000.0,
74 : 1000000000000000.0,
75 : 10000000000000000.0,
76 : 100000000000000000.0,
77 : 1000000000000000000.0,
78 : 10000000000000000000.0,
79 : 100000000000000000000.0, // 10^20
80 : 1000000000000000000000.0,
81 : // 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
82 : 10000000000000000000000.0
83 : };
84 : static const int kExactPowersOfTenSize = ARRAY_SIZE(exact_powers_of_ten);
85 :
86 : // Maximum number of significant digits in the decimal representation.
87 : // In fact the value is 772 (see conversions.cc), but to give us some margin
88 : // we round up to 780.
89 : static const int kMaxSignificantDecimalDigits = 780;
90 :
91 0 : static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
92 0 : for (int i = 0; i < buffer.length(); i++) {
93 0 : if (buffer[i] != '0') {
94 0 : return buffer.SubVector(i, buffer.length());
95 : }
96 : }
97 0 : return Vector<const char>(buffer.start(), 0);
98 : }
99 :
100 :
101 0 : static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
102 0 : for (int i = buffer.length() - 1; i >= 0; --i) {
103 0 : if (buffer[i] != '0') {
104 0 : return buffer.SubVector(0, i + 1);
105 : }
106 : }
107 0 : return Vector<const char>(buffer.start(), 0);
108 : }
109 :
110 :
111 0 : static void CutToMaxSignificantDigits(Vector<const char> buffer,
112 : int exponent,
113 : char* significant_buffer,
114 : int* significant_exponent) {
115 0 : for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
116 0 : significant_buffer[i] = buffer[i];
117 : }
118 : // The input buffer has been trimmed. Therefore the last digit must be
119 : // different from '0'.
120 0 : ASSERT(buffer[buffer.length() - 1] != '0');
121 : // Set the last digit to be non-zero. This is sufficient to guarantee
122 : // correct rounding.
123 0 : significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
124 0 : *significant_exponent =
125 0 : exponent + (buffer.length() - kMaxSignificantDecimalDigits);
126 0 : }
127 :
128 :
129 : // Trims the buffer and cuts it to at most kMaxSignificantDecimalDigits.
130 : // If possible the input-buffer is reused, but if the buffer needs to be
131 : // modified (due to cutting), then the input needs to be copied into the
132 : // buffer_copy_space.
133 0 : static void TrimAndCut(Vector<const char> buffer, int exponent,
134 : char* buffer_copy_space, int space_size,
135 : Vector<const char>* trimmed, int* updated_exponent) {
136 0 : Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
137 0 : Vector<const char> right_trimmed = TrimTrailingZeros(left_trimmed);
138 0 : exponent += left_trimmed.length() - right_trimmed.length();
139 0 : if (right_trimmed.length() > kMaxSignificantDecimalDigits) {
140 : (void) space_size; // Mark variable as used.
141 0 : ASSERT(space_size >= kMaxSignificantDecimalDigits);
142 : CutToMaxSignificantDigits(right_trimmed, exponent,
143 0 : buffer_copy_space, updated_exponent);
144 0 : *trimmed = Vector<const char>(buffer_copy_space,
145 0 : kMaxSignificantDecimalDigits);
146 : } else {
147 0 : *trimmed = right_trimmed;
148 0 : *updated_exponent = exponent;
149 : }
150 0 : }
151 :
152 :
153 : // Reads digits from the buffer and converts them to a uint64.
154 : // Reads in as many digits as fit into a uint64.
155 : // When the string starts with "1844674407370955161" no further digit is read.
156 : // Since 2^64 = 18446744073709551616 it would still be possible read another
157 : // digit if it was less or equal than 6, but this would complicate the code.
158 0 : static uint64_t ReadUint64(Vector<const char> buffer,
159 : int* number_of_read_digits) {
160 0 : uint64_t result = 0;
161 0 : int i = 0;
162 0 : while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
163 0 : int digit = buffer[i++] - '0';
164 0 : ASSERT(0 <= digit && digit <= 9);
165 0 : result = 10 * result + digit;
166 : }
167 0 : *number_of_read_digits = i;
168 0 : return result;
169 : }
170 :
171 :
172 : // Reads a DiyFp from the buffer.
173 : // The returned DiyFp is not necessarily normalized.
174 : // If remaining_decimals is zero then the returned DiyFp is accurate.
175 : // Otherwise it has been rounded and has error of at most 1/2 ulp.
176 0 : static void ReadDiyFp(Vector<const char> buffer,
177 : DiyFp* result,
178 : int* remaining_decimals) {
179 : int read_digits;
180 0 : uint64_t significand = ReadUint64(buffer, &read_digits);
181 0 : if (buffer.length() == read_digits) {
182 0 : *result = DiyFp(significand, 0);
183 0 : *remaining_decimals = 0;
184 : } else {
185 : // Round the significand.
186 0 : if (buffer[read_digits] >= '5') {
187 0 : significand++;
188 : }
189 : // Compute the binary exponent.
190 0 : int exponent = 0;
191 0 : *result = DiyFp(significand, exponent);
192 0 : *remaining_decimals = buffer.length() - read_digits;
193 : }
194 0 : }
195 :
196 :
197 0 : static bool DoubleStrtod(Vector<const char> trimmed,
198 : int exponent,
199 : double* result) {
200 : #if !defined(DOUBLE_CONVERSION_CORRECT_DOUBLE_OPERATIONS)
201 : // On x86 the floating-point stack can be 64 or 80 bits wide. If it is
202 : // 80 bits wide (as is the case on Linux) then double-rounding occurs and the
203 : // result is not accurate.
204 : // We know that Windows32 uses 64 bits and is therefore accurate.
205 : // Note that the ARM simulator is compiled for 32bits. It therefore exhibits
206 : // the same problem.
207 : return false;
208 : #endif
209 0 : if (trimmed.length() <= kMaxExactDoubleIntegerDecimalDigits) {
210 : int read_digits;
211 : // The trimmed input fits into a double.
212 : // If the 10^exponent (resp. 10^-exponent) fits into a double too then we
213 : // can compute the result-double simply by multiplying (resp. dividing) the
214 : // two numbers.
215 : // This is possible because IEEE guarantees that floating-point operations
216 : // return the best possible approximation.
217 0 : if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
218 : // 10^-exponent fits into a double.
219 0 : *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
220 0 : ASSERT(read_digits == trimmed.length());
221 0 : *result /= exact_powers_of_ten[-exponent];
222 0 : return true;
223 : }
224 0 : if (0 <= exponent && exponent < kExactPowersOfTenSize) {
225 : // 10^exponent fits into a double.
226 0 : *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
227 0 : ASSERT(read_digits == trimmed.length());
228 0 : *result *= exact_powers_of_ten[exponent];
229 0 : return true;
230 : }
231 : int remaining_digits =
232 0 : kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
233 0 : if ((0 <= exponent) &&
234 0 : (exponent - remaining_digits < kExactPowersOfTenSize)) {
235 : // The trimmed string was short and we can multiply it with
236 : // 10^remaining_digits. As a result the remaining exponent now fits
237 : // into a double too.
238 0 : *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
239 0 : ASSERT(read_digits == trimmed.length());
240 0 : *result *= exact_powers_of_ten[remaining_digits];
241 0 : *result *= exact_powers_of_ten[exponent - remaining_digits];
242 0 : return true;
243 : }
244 : }
245 0 : return false;
246 : }
247 :
248 :
249 : // Returns 10^exponent as an exact DiyFp.
250 : // The given exponent must be in the range [1; kDecimalExponentDistance[.
251 0 : static DiyFp AdjustmentPowerOfTen(int exponent) {
252 0 : ASSERT(0 < exponent);
253 0 : ASSERT(exponent < PowersOfTenCache::kDecimalExponentDistance);
254 : // Simply hardcode the remaining powers for the given decimal exponent
255 : // distance.
256 0 : ASSERT(PowersOfTenCache::kDecimalExponentDistance == 8);
257 0 : switch (exponent) {
258 0 : case 1: return DiyFp(UINT64_2PART_C(0xa0000000, 00000000), -60);
259 0 : case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57);
260 0 : case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54);
261 0 : case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50);
262 0 : case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47);
263 0 : case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44);
264 0 : case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40);
265 : default:
266 0 : UNREACHABLE();
267 : }
268 : }
269 :
270 :
271 : // If the function returns true then the result is the correct double.
272 : // Otherwise it is either the correct double or the double that is just below
273 : // the correct double.
274 0 : static bool DiyFpStrtod(Vector<const char> buffer,
275 : int exponent,
276 : double* result) {
277 0 : DiyFp input;
278 : int remaining_decimals;
279 0 : ReadDiyFp(buffer, &input, &remaining_decimals);
280 : // Since we may have dropped some digits the input is not accurate.
281 : // If remaining_decimals is different than 0 than the error is at most
282 : // .5 ulp (unit in the last place).
283 : // We don't want to deal with fractions and therefore keep a common
284 : // denominator.
285 0 : const int kDenominatorLog = 3;
286 0 : const int kDenominator = 1 << kDenominatorLog;
287 : // Move the remaining decimals into the exponent.
288 0 : exponent += remaining_decimals;
289 0 : uint64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
290 :
291 0 : int old_e = input.e();
292 0 : input.Normalize();
293 0 : error <<= old_e - input.e();
294 :
295 0 : ASSERT(exponent <= PowersOfTenCache::kMaxDecimalExponent);
296 0 : if (exponent < PowersOfTenCache::kMinDecimalExponent) {
297 0 : *result = 0.0;
298 0 : return true;
299 : }
300 0 : DiyFp cached_power;
301 : int cached_decimal_exponent;
302 : PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
303 : &cached_power,
304 0 : &cached_decimal_exponent);
305 :
306 0 : if (cached_decimal_exponent != exponent) {
307 0 : int adjustment_exponent = exponent - cached_decimal_exponent;
308 0 : DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
309 0 : input.Multiply(adjustment_power);
310 0 : if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
311 : // The product of input with the adjustment power fits into a 64 bit
312 : // integer.
313 : ASSERT(DiyFp::kSignificandSize == 64);
314 : } else {
315 : // The adjustment power is exact. There is hence only an error of 0.5.
316 0 : error += kDenominator / 2;
317 : }
318 : }
319 :
320 0 : input.Multiply(cached_power);
321 : // The error introduced by a multiplication of a*b equals
322 : // error_a + error_b + error_a*error_b/2^64 + 0.5
323 : // Substituting a with 'input' and b with 'cached_power' we have
324 : // error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
325 : // error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
326 0 : int error_b = kDenominator / 2;
327 0 : int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
328 0 : int fixed_error = kDenominator / 2;
329 0 : error += error_b + error_ab + fixed_error;
330 :
331 0 : old_e = input.e();
332 0 : input.Normalize();
333 0 : error <<= old_e - input.e();
334 :
335 : // See if the double's significand changes if we add/subtract the error.
336 0 : int order_of_magnitude = DiyFp::kSignificandSize + input.e();
337 : int effective_significand_size =
338 0 : Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
339 : int precision_digits_count =
340 0 : DiyFp::kSignificandSize - effective_significand_size;
341 0 : if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
342 : // This can only happen for very small denormals. In this case the
343 : // half-way multiplied by the denominator exceeds the range of an uint64.
344 : // Simply shift everything to the right.
345 : int shift_amount = (precision_digits_count + kDenominatorLog) -
346 0 : DiyFp::kSignificandSize + 1;
347 0 : input.set_f(input.f() >> shift_amount);
348 0 : input.set_e(input.e() + shift_amount);
349 : // We add 1 for the lost precision of error, and kDenominator for
350 : // the lost precision of input.f().
351 0 : error = (error >> shift_amount) + 1 + kDenominator;
352 0 : precision_digits_count -= shift_amount;
353 : }
354 : // We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
355 : ASSERT(DiyFp::kSignificandSize == 64);
356 0 : ASSERT(precision_digits_count < 64);
357 0 : uint64_t one64 = 1;
358 0 : uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
359 0 : uint64_t precision_bits = input.f() & precision_bits_mask;
360 0 : uint64_t half_way = one64 << (precision_digits_count - 1);
361 0 : precision_bits *= kDenominator;
362 0 : half_way *= kDenominator;
363 0 : DiyFp rounded_input(input.f() >> precision_digits_count,
364 0 : input.e() + precision_digits_count);
365 0 : if (precision_bits >= half_way + error) {
366 0 : rounded_input.set_f(rounded_input.f() + 1);
367 : }
368 : // If the last_bits are too close to the half-way case than we are too
369 : // inaccurate and round down. In this case we return false so that we can
370 : // fall back to a more precise algorithm.
371 :
372 0 : *result = Double(rounded_input).value();
373 0 : if (half_way - error < precision_bits && precision_bits < half_way + error) {
374 : // Too imprecise. The caller will have to fall back to a slower version.
375 : // However the returned number is guaranteed to be either the correct
376 : // double, or the next-lower double.
377 0 : return false;
378 : } else {
379 0 : return true;
380 : }
381 : }
382 :
383 :
384 : // Returns
385 : // - -1 if buffer*10^exponent < diy_fp.
386 : // - 0 if buffer*10^exponent == diy_fp.
387 : // - +1 if buffer*10^exponent > diy_fp.
388 : // Preconditions:
389 : // buffer.length() + exponent <= kMaxDecimalPower + 1
390 : // buffer.length() + exponent > kMinDecimalPower
391 : // buffer.length() <= kMaxDecimalSignificantDigits
392 0 : static int CompareBufferWithDiyFp(Vector<const char> buffer,
393 : int exponent,
394 : DiyFp diy_fp) {
395 0 : ASSERT(buffer.length() + exponent <= kMaxDecimalPower + 1);
396 0 : ASSERT(buffer.length() + exponent > kMinDecimalPower);
397 0 : ASSERT(buffer.length() <= kMaxSignificantDecimalDigits);
398 : // Make sure that the Bignum will be able to hold all our numbers.
399 : // Our Bignum implementation has a separate field for exponents. Shifts will
400 : // consume at most one bigit (< 64 bits).
401 : // ln(10) == 3.3219...
402 : ASSERT(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
403 0 : Bignum buffer_bignum;
404 0 : Bignum diy_fp_bignum;
405 0 : buffer_bignum.AssignDecimalString(buffer);
406 0 : diy_fp_bignum.AssignUInt64(diy_fp.f());
407 0 : if (exponent >= 0) {
408 0 : buffer_bignum.MultiplyByPowerOfTen(exponent);
409 : } else {
410 0 : diy_fp_bignum.MultiplyByPowerOfTen(-exponent);
411 : }
412 0 : if (diy_fp.e() > 0) {
413 0 : diy_fp_bignum.ShiftLeft(diy_fp.e());
414 : } else {
415 0 : buffer_bignum.ShiftLeft(-diy_fp.e());
416 : }
417 0 : return Bignum::Compare(buffer_bignum, diy_fp_bignum);
418 : }
419 :
420 :
421 : // Returns true if the guess is the correct double.
422 : // Returns false, when guess is either correct or the next-lower double.
423 0 : static bool ComputeGuess(Vector<const char> trimmed, int exponent,
424 : double* guess) {
425 0 : if (trimmed.length() == 0) {
426 0 : *guess = 0.0;
427 0 : return true;
428 : }
429 0 : if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) {
430 0 : *guess = Double::Infinity();
431 0 : return true;
432 : }
433 0 : if (exponent + trimmed.length() <= kMinDecimalPower) {
434 0 : *guess = 0.0;
435 0 : return true;
436 : }
437 :
438 0 : if (DoubleStrtod(trimmed, exponent, guess) ||
439 0 : DiyFpStrtod(trimmed, exponent, guess)) {
440 0 : return true;
441 : }
442 0 : if (*guess == Double::Infinity()) {
443 0 : return true;
444 : }
445 0 : return false;
446 : }
447 :
448 0 : double Strtod(Vector<const char> buffer, int exponent) {
449 : char copy_buffer[kMaxSignificantDecimalDigits];
450 0 : Vector<const char> trimmed;
451 : int updated_exponent;
452 : TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
453 0 : &trimmed, &updated_exponent);
454 0 : exponent = updated_exponent;
455 :
456 : double guess;
457 0 : bool is_correct = ComputeGuess(trimmed, exponent, &guess);
458 0 : if (is_correct) return guess;
459 :
460 0 : DiyFp upper_boundary = Double(guess).UpperBoundary();
461 0 : int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
462 0 : if (comparison < 0) {
463 0 : return guess;
464 0 : } else if (comparison > 0) {
465 0 : return Double(guess).NextDouble();
466 0 : } else if ((Double(guess).Significand() & 1) == 0) {
467 : // Round towards even.
468 0 : return guess;
469 : } else {
470 0 : return Double(guess).NextDouble();
471 : }
472 : }
473 :
474 0 : float Strtof(Vector<const char> buffer, int exponent) {
475 : char copy_buffer[kMaxSignificantDecimalDigits];
476 0 : Vector<const char> trimmed;
477 : int updated_exponent;
478 : TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
479 0 : &trimmed, &updated_exponent);
480 0 : exponent = updated_exponent;
481 :
482 : double double_guess;
483 0 : bool is_correct = ComputeGuess(trimmed, exponent, &double_guess);
484 :
485 0 : float float_guess = static_cast<float>(double_guess);
486 0 : if (float_guess == double_guess) {
487 : // This shortcut triggers for integer values.
488 0 : return float_guess;
489 : }
490 :
491 : // We must catch double-rounding. Say the double has been rounded up, and is
492 : // now a boundary of a float, and rounds up again. This is why we have to
493 : // look at previous too.
494 : // Example (in decimal numbers):
495 : // input: 12349
496 : // high-precision (4 digits): 1235
497 : // low-precision (3 digits):
498 : // when read from input: 123
499 : // when rounded from high precision: 124.
500 : // To do this we simply look at the neigbors of the correct result and see
501 : // if they would round to the same float. If the guess is not correct we have
502 : // to look at four values (since two different doubles could be the correct
503 : // double).
504 :
505 0 : double double_next = Double(double_guess).NextDouble();
506 0 : double double_previous = Double(double_guess).PreviousDouble();
507 :
508 0 : float f1 = static_cast<float>(double_previous);
509 0 : float f2 = float_guess;
510 0 : float f3 = static_cast<float>(double_next);
511 : float f4;
512 0 : if (is_correct) {
513 0 : f4 = f3;
514 : } else {
515 0 : double double_next2 = Double(double_next).NextDouble();
516 0 : f4 = static_cast<float>(double_next2);
517 : }
518 : (void) f2; // Mark variable as used.
519 0 : ASSERT(f1 <= f2 && f2 <= f3 && f3 <= f4);
520 :
521 : // If the guess doesn't lie near a single-precision boundary we can simply
522 : // return its float-value.
523 0 : if (f1 == f4) {
524 0 : return float_guess;
525 : }
526 :
527 0 : ASSERT((f1 != f2 && f2 == f3 && f3 == f4) ||
528 : (f1 == f2 && f2 != f3 && f3 == f4) ||
529 : (f1 == f2 && f2 == f3 && f3 != f4));
530 :
531 : // guess and next are the two possible canditates (in the same way that
532 : // double_guess was the lower candidate for a double-precision guess).
533 0 : float guess = f1;
534 0 : float next = f4;
535 0 : DiyFp upper_boundary;
536 0 : if (guess == 0.0f) {
537 0 : float min_float = 1e-45f;
538 0 : upper_boundary = Double(static_cast<double>(min_float) / 2).AsDiyFp();
539 : } else {
540 0 : upper_boundary = Single(guess).UpperBoundary();
541 : }
542 0 : int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
543 0 : if (comparison < 0) {
544 0 : return guess;
545 0 : } else if (comparison > 0) {
546 0 : return next;
547 0 : } else if ((Single(guess).Significand() & 1) == 0) {
548 : // Round towards even.
549 0 : return guess;
550 : } else {
551 0 : return next;
552 : }
553 : }
554 :
555 : } // namespace double_conversion
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