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27
28#include <climits>
29#include <cstdarg>
30
31#include <double-conversion/bignum.h>
32#include <double-conversion/cached-powers.h>
33#include <double-conversion/ieee.h>
34#include <double-conversion/strtod.h>
35
36namespace 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.
41static const int kMaxExactDoubleIntegerDecimalDigits = 15;
42// 2^64 = 18446744073709551616 > 10^19
43static 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).
51static const int kMaxDecimalPower = 309;
52static const int kMinDecimalPower = -324;
53
54// 2^64 = 18446744073709551616
55static const uint64_t kMaxUint64 = UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF);
56
57
58static 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};
84static 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.
89static const int kMaxSignificantDecimalDigits = 780;
90
91static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
92 for (int i = 0; i < buffer.length(); i++) {
93 if (buffer[i] != '0') {
94 return buffer.SubVector(i, buffer.length());
95 }
96 }
97 return Vector<const char>(buffer.start(), 0);
98}
99
100
101static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
102 for (int i = buffer.length() - 1; i >= 0; --i) {
103 if (buffer[i] != '0') {
104 return buffer.SubVector(0, i + 1);
105 }
106 }
107 return Vector<const char>(buffer.start(), 0);
108}
109
110
111static void CutToMaxSignificantDigits(Vector<const char> buffer,
112 int exponent,
113 char* significant_buffer,
114 int* significant_exponent) {
115 for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
116 significant_buffer[i] = buffer[i];
117 }
118 // The input buffer has been trimmed. Therefore the last digit must be
119 // different from '0'.
120 ASSERT(buffer[buffer.length() - 1] != '0');
121 // Set the last digit to be non-zero. This is sufficient to guarantee
122 // correct rounding.
123 significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
124 *significant_exponent =
125 exponent + (buffer.length() - kMaxSignificantDecimalDigits);
126}
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.
133static 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 Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
137 Vector<const char> right_trimmed = TrimTrailingZeros(left_trimmed);
138 exponent += left_trimmed.length() - right_trimmed.length();
139 if (right_trimmed.length() > kMaxSignificantDecimalDigits) {
140 (void) space_size; // Mark variable as used.
141 ASSERT(space_size >= kMaxSignificantDecimalDigits);
142 CutToMaxSignificantDigits(right_trimmed, exponent,
143 buffer_copy_space, updated_exponent);
144 *trimmed = Vector<const char>(buffer_copy_space,
145 kMaxSignificantDecimalDigits);
146 } else {
147 *trimmed = right_trimmed;
148 *updated_exponent = exponent;
149 }
150}
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.
158static uint64_t ReadUint64(Vector<const char> buffer,
159 int* number_of_read_digits) {
160 uint64_t result = 0;
161 int i = 0;
162 while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
163 int digit = buffer[i++] - '0';
164 ASSERT(0 <= digit && digit <= 9);
165 result = 10 * result + digit;
166 }
167 *number_of_read_digits = i;
168 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.
176static void ReadDiyFp(Vector<const char> buffer,
177 DiyFp* result,
178 int* remaining_decimals) {
179 int read_digits;
180 uint64_t significand = ReadUint64(buffer, &read_digits);
181 if (buffer.length() == read_digits) {
182 *result = DiyFp(significand, 0);
183 *remaining_decimals = 0;
184 } else {
185 // Round the significand.
186 if (buffer[read_digits] >= '5') {
187 significand++;
188 }
189 // Compute the binary exponent.
190 int exponent = 0;
191 *result = DiyFp(significand, exponent);
192 *remaining_decimals = buffer.length() - read_digits;
193 }
194}
195
196
197static 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#else
209 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 if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
218 // 10^-exponent fits into a double.
219 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
220 ASSERT(read_digits == trimmed.length());
221 *result /= exact_powers_of_ten[-exponent];
222 return true;
223 }
224 if (0 <= exponent && exponent < kExactPowersOfTenSize) {
225 // 10^exponent fits into a double.
226 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
227 ASSERT(read_digits == trimmed.length());
228 *result *= exact_powers_of_ten[exponent];
229 return true;
230 }
231 int remaining_digits =
232 kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
233 if ((0 <= exponent) &&
234 (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 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
239 ASSERT(read_digits == trimmed.length());
240 *result *= exact_powers_of_ten[remaining_digits];
241 *result *= exact_powers_of_ten[exponent - remaining_digits];
242 return true;
243 }
244 }
245 return false;
246#endif
247}
248
249
250// Returns 10^exponent as an exact DiyFp.
251// The given exponent must be in the range [1; kDecimalExponentDistance[.
252static DiyFp AdjustmentPowerOfTen(int exponent) {
253 ASSERT(0 < exponent);
254 ASSERT(exponent < PowersOfTenCache::kDecimalExponentDistance);
255 // Simply hardcode the remaining powers for the given decimal exponent
256 // distance.
257 ASSERT(PowersOfTenCache::kDecimalExponentDistance == 8);
258 switch (exponent) {
259 case 1: return DiyFp(UINT64_2PART_C(0xa0000000, 00000000), -60);
260 case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57);
261 case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54);
262 case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50);
263 case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47);
264 case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44);
265 case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40);
266 default:
267 UNREACHABLE();
268 }
269}
270
271
272// If the function returns true then the result is the correct double.
273// Otherwise it is either the correct double or the double that is just below
274// the correct double.
275static bool DiyFpStrtod(Vector<const char> buffer,
276 int exponent,
277 double* result) {
278 DiyFp input;
279 int remaining_decimals;
280 ReadDiyFp(buffer, &input, &remaining_decimals);
281 // Since we may have dropped some digits the input is not accurate.
282 // If remaining_decimals is different than 0 than the error is at most
283 // .5 ulp (unit in the last place).
284 // We don't want to deal with fractions and therefore keep a common
285 // denominator.
286 const int kDenominatorLog = 3;
287 const int kDenominator = 1 << kDenominatorLog;
288 // Move the remaining decimals into the exponent.
289 exponent += remaining_decimals;
290 uint64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
291
292 int old_e = input.e();
293 input.Normalize();
294 error <<= old_e - input.e();
295
296 ASSERT(exponent <= PowersOfTenCache::kMaxDecimalExponent);
297 if (exponent < PowersOfTenCache::kMinDecimalExponent) {
298 *result = 0.0;
299 return true;
300 }
301 DiyFp cached_power;
302 int cached_decimal_exponent;
303 PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
304 &cached_power,
305 &cached_decimal_exponent);
306
307 if (cached_decimal_exponent != exponent) {
308 int adjustment_exponent = exponent - cached_decimal_exponent;
309 DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
310 input.Multiply(adjustment_power);
311 if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
312 // The product of input with the adjustment power fits into a 64 bit
313 // integer.
314 ASSERT(DiyFp::kSignificandSize == 64);
315 } else {
316 // The adjustment power is exact. There is hence only an error of 0.5.
317 error += kDenominator / 2;
318 }
319 }
320
321 input.Multiply(cached_power);
322 // The error introduced by a multiplication of a*b equals
323 // error_a + error_b + error_a*error_b/2^64 + 0.5
324 // Substituting a with 'input' and b with 'cached_power' we have
325 // error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
326 // error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
327 int error_b = kDenominator / 2;
328 int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
329 int fixed_error = kDenominator / 2;
330 error += error_b + error_ab + fixed_error;
331
332 old_e = input.e();
333 input.Normalize();
334 error <<= old_e - input.e();
335
336 // See if the double's significand changes if we add/subtract the error.
337 int order_of_magnitude = DiyFp::kSignificandSize + input.e();
338 int effective_significand_size =
339 Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
340 int precision_digits_count =
341 DiyFp::kSignificandSize - effective_significand_size;
342 if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
343 // This can only happen for very small denormals. In this case the
344 // half-way multiplied by the denominator exceeds the range of an uint64.
345 // Simply shift everything to the right.
346 int shift_amount = (precision_digits_count + kDenominatorLog) -
347 DiyFp::kSignificandSize + 1;
348 input.set_f(input.f() >> shift_amount);
349 input.set_e(input.e() + shift_amount);
350 // We add 1 for the lost precision of error, and kDenominator for
351 // the lost precision of input.f().
352 error = (error >> shift_amount) + 1 + kDenominator;
353 precision_digits_count -= shift_amount;
354 }
355 // We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
356 ASSERT(DiyFp::kSignificandSize == 64);
357 ASSERT(precision_digits_count < 64);
358 uint64_t one64 = 1;
359 uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
360 uint64_t precision_bits = input.f() & precision_bits_mask;
361 uint64_t half_way = one64 << (precision_digits_count - 1);
362 precision_bits *= kDenominator;
363 half_way *= kDenominator;
364 DiyFp rounded_input(input.f() >> precision_digits_count,
365 input.e() + precision_digits_count);
366 if (precision_bits >= half_way + error) {
367 rounded_input.set_f(rounded_input.f() + 1);
368 }
369 // If the last_bits are too close to the half-way case than we are too
370 // inaccurate and round down. In this case we return false so that we can
371 // fall back to a more precise algorithm.
372
373 *result = Double(rounded_input).value();
374 if (half_way - error < precision_bits && precision_bits < half_way + error) {
375 // Too imprecise. The caller will have to fall back to a slower version.
376 // However the returned number is guaranteed to be either the correct
377 // double, or the next-lower double.
378 return false;
379 } else {
380 return true;
381 }
382}
383
384
385// Returns
386// - -1 if buffer*10^exponent < diy_fp.
387// - 0 if buffer*10^exponent == diy_fp.
388// - +1 if buffer*10^exponent > diy_fp.
389// Preconditions:
390// buffer.length() + exponent <= kMaxDecimalPower + 1
391// buffer.length() + exponent > kMinDecimalPower
392// buffer.length() <= kMaxDecimalSignificantDigits
393static int CompareBufferWithDiyFp(Vector<const char> buffer,
394 int exponent,
395 DiyFp diy_fp) {
396 ASSERT(buffer.length() + exponent <= kMaxDecimalPower + 1);
397 ASSERT(buffer.length() + exponent > kMinDecimalPower);
398 ASSERT(buffer.length() <= kMaxSignificantDecimalDigits);
399 // Make sure that the Bignum will be able to hold all our numbers.
400 // Our Bignum implementation has a separate field for exponents. Shifts will
401 // consume at most one bigit (< 64 bits).
402 // ln(10) == 3.3219...
403 ASSERT(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
404 Bignum buffer_bignum;
405 Bignum diy_fp_bignum;
406 buffer_bignum.AssignDecimalString(buffer);
407 diy_fp_bignum.AssignUInt64(diy_fp.f());
408 if (exponent >= 0) {
409 buffer_bignum.MultiplyByPowerOfTen(exponent);
410 } else {
411 diy_fp_bignum.MultiplyByPowerOfTen(-exponent);
412 }
413 if (diy_fp.e() > 0) {
414 diy_fp_bignum.ShiftLeft(diy_fp.e());
415 } else {
416 buffer_bignum.ShiftLeft(-diy_fp.e());
417 }
418 return Bignum::Compare(buffer_bignum, diy_fp_bignum);
419}
420
421
422// Returns true if the guess is the correct double.
423// Returns false, when guess is either correct or the next-lower double.
424static bool ComputeGuess(Vector<const char> trimmed, int exponent,
425 double* guess) {
426 if (trimmed.length() == 0) {
427 *guess = 0.0;
428 return true;
429 }
430 if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) {
431 *guess = Double::Infinity();
432 return true;
433 }
434 if (exponent + trimmed.length() <= kMinDecimalPower) {
435 *guess = 0.0;
436 return true;
437 }
438
439 if (DoubleStrtod(trimmed, exponent, guess) ||
440 DiyFpStrtod(trimmed, exponent, guess)) {
441 return true;
442 }
443 if (*guess == Double::Infinity()) {
444 return true;
445 }
446 return false;
447}
448
449double Strtod(Vector<const char> buffer, int exponent) {
450 char copy_buffer[kMaxSignificantDecimalDigits];
451 Vector<const char> trimmed;
452 int updated_exponent;
453 TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
454 &trimmed, &updated_exponent);
455 exponent = updated_exponent;
456
457 double guess;
458 bool is_correct = ComputeGuess(trimmed, exponent, &guess);
459 if (is_correct) return guess;
460
461 DiyFp upper_boundary = Double(guess).UpperBoundary();
462 int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
463 if (comparison < 0) {
464 return guess;
465 } else if (comparison > 0) {
466 return Double(guess).NextDouble();
467 } else if ((Double(guess).Significand() & 1) == 0) {
468 // Round towards even.
469 return guess;
470 } else {
471 return Double(guess).NextDouble();
472 }
473}
474
475static float SanitizedDoubletof(double d) {
476 ASSERT(d >= 0.0);
477 // ASAN has a sanitize check that disallows casting doubles to floats if
478 // they are too big.
479 // https://clang.llvm.org/docs/UndefinedBehaviorSanitizer.html#available-checks
480 // The behavior should be covered by IEEE 754, but some projects use this
481 // flag, so work around it.
482 float max_finite = 3.4028234663852885981170418348451692544e+38;
483 // The half-way point between the max-finite and infinity value.
484 // Since infinity has an even significand everything equal or greater than
485 // this value should become infinity.
486 double half_max_finite_infinity =
487 3.40282356779733661637539395458142568448e+38;
488 if (d >= max_finite) {
489 if (d >= half_max_finite_infinity) {
490 return Single::Infinity();
491 } else {
492 return max_finite;
493 }
494 } else {
495 return static_cast<float>(d);
496 }
497}
498
499float Strtof(Vector<const char> buffer, int exponent) {
500 char copy_buffer[kMaxSignificantDecimalDigits];
501 Vector<const char> trimmed;
502 int updated_exponent;
503 TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
504 &trimmed, &updated_exponent);
505 exponent = updated_exponent;
506
507 double double_guess;
508 bool is_correct = ComputeGuess(trimmed, exponent, &double_guess);
509
510 float float_guess = SanitizedDoubletof(double_guess);
511 if (float_guess == double_guess) {
512 // This shortcut triggers for integer values.
513 return float_guess;
514 }
515
516 // We must catch double-rounding. Say the double has been rounded up, and is
517 // now a boundary of a float, and rounds up again. This is why we have to
518 // look at previous too.
519 // Example (in decimal numbers):
520 // input: 12349
521 // high-precision (4 digits): 1235
522 // low-precision (3 digits):
523 // when read from input: 123
524 // when rounded from high precision: 124.
525 // To do this we simply look at the neigbors of the correct result and see
526 // if they would round to the same float. If the guess is not correct we have
527 // to look at four values (since two different doubles could be the correct
528 // double).
529
530 double double_next = Double(double_guess).NextDouble();
531 double double_previous = Double(double_guess).PreviousDouble();
532
533 float f1 = SanitizedDoubletof(double_previous);
534 float f2 = float_guess;
535 float f3 = SanitizedDoubletof(double_next);
536 float f4;
537 if (is_correct) {
538 f4 = f3;
539 } else {
540 double double_next2 = Double(double_next).NextDouble();
541 f4 = SanitizedDoubletof(double_next2);
542 }
543 (void) f2; // Mark variable as used.
544 ASSERT(f1 <= f2 && f2 <= f3 && f3 <= f4);
545
546 // If the guess doesn't lie near a single-precision boundary we can simply
547 // return its float-value.
548 if (f1 == f4) {
549 return float_guess;
550 }
551
552 ASSERT((f1 != f2 && f2 == f3 && f3 == f4) ||
553 (f1 == f2 && f2 != f3 && f3 == f4) ||
554 (f1 == f2 && f2 == f3 && f3 != f4));
555
556 // guess and next are the two possible candidates (in the same way that
557 // double_guess was the lower candidate for a double-precision guess).
558 float guess = f1;
559 float next = f4;
560 DiyFp upper_boundary;
561 if (guess == 0.0f) {
562 float min_float = 1e-45f;
563 upper_boundary = Double(static_cast<double>(min_float) / 2).AsDiyFp();
564 } else {
565 upper_boundary = Single(guess).UpperBoundary();
566 }
567 int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
568 if (comparison < 0) {
569 return guess;
570 } else if (comparison > 0) {
571 return next;
572 } else if ((Single(guess).Significand() & 1) == 0) {
573 // Round towards even.
574 return guess;
575 } else {
576 return next;
577 }
578}
579
580} // namespace double_conversion
581