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3 // modification, are permitted provided that the following conditions are
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5 //
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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
TrimLeadingZeros(Vector<const char> buffer)91 static 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
TrimTrailingZeros(Vector<const char> buffer)101 static 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
CutToMaxSignificantDigits(Vector<const char> buffer,int exponent,char * significant_buffer,int * significant_exponent)111 static 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.
TrimAndCut(Vector<const char> buffer,int exponent,char * buffer_copy_space,int space_size,Vector<const char> * trimmed,int * updated_exponent)133 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 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.
ReadUint64(Vector<const char> buffer,int * number_of_read_digits)158 static 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.
ReadDiyFp(Vector<const char> buffer,DiyFp * result,int * remaining_decimals)176 static 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
DoubleStrtod(Vector<const char> trimmed,int exponent,double * result)197 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 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 }
247
248
249 // Returns 10^exponent as an exact DiyFp.
250 // The given exponent must be in the range [1; kDecimalExponentDistance[.
AdjustmentPowerOfTen(int exponent)251 static DiyFp AdjustmentPowerOfTen(int exponent) {
252 ASSERT(0 < exponent);
253 ASSERT(exponent < PowersOfTenCache::kDecimalExponentDistance);
254 // Simply hardcode the remaining powers for the given decimal exponent
255 // distance.
256 ASSERT(PowersOfTenCache::kDecimalExponentDistance == 8);
257 switch (exponent) {
258 case 1: return DiyFp(UINT64_2PART_C(0xa0000000, 00000000), -60);
259 case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57);
260 case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54);
261 case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50);
262 case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47);
263 case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44);
264 case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40);
265 default:
266 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.
DiyFpStrtod(Vector<const char> buffer,int exponent,double * result)274 static bool DiyFpStrtod(Vector<const char> buffer,
275 int exponent,
276 double* result) {
277 DiyFp input;
278 int remaining_decimals;
279 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 const int kDenominatorLog = 3;
286 const int kDenominator = 1 << kDenominatorLog;
287 // Move the remaining decimals into the exponent.
288 exponent += remaining_decimals;
289 uint64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
290
291 int old_e = input.e();
292 input.Normalize();
293 error <<= old_e - input.e();
294
295 ASSERT(exponent <= PowersOfTenCache::kMaxDecimalExponent);
296 if (exponent < PowersOfTenCache::kMinDecimalExponent) {
297 *result = 0.0;
298 return true;
299 }
300 DiyFp cached_power;
301 int cached_decimal_exponent;
302 PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
303 &cached_power,
304 &cached_decimal_exponent);
305
306 if (cached_decimal_exponent != exponent) {
307 int adjustment_exponent = exponent - cached_decimal_exponent;
308 DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
309 input.Multiply(adjustment_power);
310 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 error += kDenominator / 2;
317 }
318 }
319
320 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 int error_b = kDenominator / 2;
327 int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
328 int fixed_error = kDenominator / 2;
329 error += error_b + error_ab + fixed_error;
330
331 old_e = input.e();
332 input.Normalize();
333 error <<= old_e - input.e();
334
335 // See if the double's significand changes if we add/subtract the error.
336 int order_of_magnitude = DiyFp::kSignificandSize + input.e();
337 int effective_significand_size =
338 Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
339 int precision_digits_count =
340 DiyFp::kSignificandSize - effective_significand_size;
341 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 DiyFp::kSignificandSize + 1;
347 input.set_f(input.f() >> shift_amount);
348 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 error = (error >> shift_amount) + 1 + kDenominator;
352 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 ASSERT(precision_digits_count < 64);
357 uint64_t one64 = 1;
358 uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
359 uint64_t precision_bits = input.f() & precision_bits_mask;
360 uint64_t half_way = one64 << (precision_digits_count - 1);
361 precision_bits *= kDenominator;
362 half_way *= kDenominator;
363 DiyFp rounded_input(input.f() >> precision_digits_count,
364 input.e() + precision_digits_count);
365 if (precision_bits >= half_way + error) {
366 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 *result = Double(rounded_input).value();
373 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 return false;
378 } else {
379 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
CompareBufferWithDiyFp(Vector<const char> buffer,int exponent,DiyFp diy_fp)392 static int CompareBufferWithDiyFp(Vector<const char> buffer,
393 int exponent,
394 DiyFp diy_fp) {
395 ASSERT(buffer.length() + exponent <= kMaxDecimalPower + 1);
396 ASSERT(buffer.length() + exponent > kMinDecimalPower);
397 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 Bignum buffer_bignum;
404 Bignum diy_fp_bignum;
405 buffer_bignum.AssignDecimalString(buffer);
406 diy_fp_bignum.AssignUInt64(diy_fp.f());
407 if (exponent >= 0) {
408 buffer_bignum.MultiplyByPowerOfTen(exponent);
409 } else {
410 diy_fp_bignum.MultiplyByPowerOfTen(-exponent);
411 }
412 if (diy_fp.e() > 0) {
413 diy_fp_bignum.ShiftLeft(diy_fp.e());
414 } else {
415 buffer_bignum.ShiftLeft(-diy_fp.e());
416 }
417 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.
ComputeGuess(Vector<const char> trimmed,int exponent,double * guess)423 static bool ComputeGuess(Vector<const char> trimmed, int exponent,
424 double* guess) {
425 if (trimmed.length() == 0) {
426 *guess = 0.0;
427 return true;
428 }
429 if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) {
430 *guess = Double::Infinity();
431 return true;
432 }
433 if (exponent + trimmed.length() <= kMinDecimalPower) {
434 *guess = 0.0;
435 return true;
436 }
437
438 if (DoubleStrtod(trimmed, exponent, guess) ||
439 DiyFpStrtod(trimmed, exponent, guess)) {
440 return true;
441 }
442 if (*guess == Double::Infinity()) {
443 return true;
444 }
445 return false;
446 }
447
Strtod(Vector<const char> buffer,int exponent)448 double Strtod(Vector<const char> buffer, int exponent) {
449 char copy_buffer[kMaxSignificantDecimalDigits];
450 Vector<const char> trimmed;
451 int updated_exponent;
452 TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
453 &trimmed, &updated_exponent);
454 exponent = updated_exponent;
455
456 double guess;
457 bool is_correct = ComputeGuess(trimmed, exponent, &guess);
458 if (is_correct) return guess;
459
460 DiyFp upper_boundary = Double(guess).UpperBoundary();
461 int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
462 if (comparison < 0) {
463 return guess;
464 } else if (comparison > 0) {
465 return Double(guess).NextDouble();
466 } else if ((Double(guess).Significand() & 1) == 0) {
467 // Round towards even.
468 return guess;
469 } else {
470 return Double(guess).NextDouble();
471 }
472 }
473
Strtof(Vector<const char> buffer,int exponent)474 float Strtof(Vector<const char> buffer, int exponent) {
475 char copy_buffer[kMaxSignificantDecimalDigits];
476 Vector<const char> trimmed;
477 int updated_exponent;
478 TrimAndCut(buffer, exponent, copy_buffer, kMaxSignificantDecimalDigits,
479 &trimmed, &updated_exponent);
480 exponent = updated_exponent;
481
482 double double_guess;
483 bool is_correct = ComputeGuess(trimmed, exponent, &double_guess);
484
485 float float_guess = static_cast<float>(double_guess);
486 if (float_guess == double_guess) {
487 // This shortcut triggers for integer values.
488 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 double double_next = Double(double_guess).NextDouble();
506 double double_previous = Double(double_guess).PreviousDouble();
507
508 float f1 = static_cast<float>(double_previous);
509 float f2 = float_guess;
510 float f3 = static_cast<float>(double_next);
511 float f4;
512 if (is_correct) {
513 f4 = f3;
514 } else {
515 double double_next2 = Double(double_next).NextDouble();
516 f4 = static_cast<float>(double_next2);
517 }
518 (void) f2; // Mark variable as used.
519 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 if (f1 == f4) {
524 return float_guess;
525 }
526
527 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 float guess = f1;
534 float next = f4;
535 DiyFp upper_boundary;
536 if (guess == 0.0f) {
537 float min_float = 1e-45f;
538 upper_boundary = Double(static_cast<double>(min_float) / 2).AsDiyFp();
539 } else {
540 upper_boundary = Single(guess).UpperBoundary();
541 }
542 int comparison = CompareBufferWithDiyFp(trimmed, exponent, upper_boundary);
543 if (comparison < 0) {
544 return guess;
545 } else if (comparison > 0) {
546 return next;
547 } else if ((Single(guess).Significand() & 1) == 0) {
548 // Round towards even.
549 return guess;
550 } else {
551 return next;
552 }
553 }
554
555 } // namespace double_conversion
556