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33 #include "cached-powers.h"
36 namespace double_conversion
{
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;
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;
54 // 2^64 = 18446744073709551616
55 static const uint64_t kMaxUint64
= UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF
);
58 static const double exact_powers_of_ten
[] = {
69 10000000000.0, // 10^10
77 1000000000000000000.0,
78 10000000000000000000.0,
79 100000000000000000000.0, // 10^20
80 1000000000000000000000.0,
81 // 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
82 10000000000000000000000.0
84 static const int kExactPowersOfTenSize
= ARRAY_SIZE(exact_powers_of_ten
);
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;
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());
97 return Vector
<const char>(buffer
.start(), 0);
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);
107 return Vector
<const char>(buffer
.start(), 0);
111 static void CutToMaxSignificantDigits(Vector
<const char> buffer
,
113 char* significant_buffer
,
114 int* significant_exponent
) {
115 for (int i
= 0; i
< kMaxSignificantDecimalDigits
- 1; ++i
) {
116 significant_buffer
[i
] = buffer
[i
];
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
123 significant_buffer
[kMaxSignificantDecimalDigits
- 1] = '1';
124 *significant_exponent
=
125 exponent
+ (buffer
.length() - kMaxSignificantDecimalDigits
);
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 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 ASSERT(space_size
>= kMaxSignificantDecimalDigits
);
141 CutToMaxSignificantDigits(right_trimmed
, exponent
,
142 buffer_copy_space
, updated_exponent
);
143 *trimmed
= Vector
<const char>(buffer_copy_space
,
144 kMaxSignificantDecimalDigits
);
146 *trimmed
= right_trimmed
;
147 *updated_exponent
= exponent
;
152 // Reads digits from the buffer and converts them to a uint64.
153 // Reads in as many digits as fit into a uint64.
154 // When the string starts with "1844674407370955161" no further digit is read.
155 // Since 2^64 = 18446744073709551616 it would still be possible read another
156 // digit if it was less or equal than 6, but this would complicate the code.
157 static uint64_t ReadUint64(Vector
<const char> buffer
,
158 int* number_of_read_digits
) {
161 while (i
< buffer
.length() && result
<= (kMaxUint64
/ 10 - 1)) {
162 int digit
= buffer
[i
++] - '0';
163 ASSERT(0 <= digit
&& digit
<= 9);
164 result
= 10 * result
+ digit
;
166 *number_of_read_digits
= i
;
171 // Reads a DiyFp from the buffer.
172 // The returned DiyFp is not necessarily normalized.
173 // If remaining_decimals is zero then the returned DiyFp is accurate.
174 // Otherwise it has been rounded and has error of at most 1/2 ulp.
175 static void ReadDiyFp(Vector
<const char> buffer
,
177 int* remaining_decimals
) {
179 uint64_t significand
= ReadUint64(buffer
, &read_digits
);
180 if (buffer
.length() == read_digits
) {
181 *result
= DiyFp(significand
, 0);
182 *remaining_decimals
= 0;
184 // Round the significand.
185 if (buffer
[read_digits
] >= '5') {
188 // Compute the binary exponent.
190 *result
= DiyFp(significand
, exponent
);
191 *remaining_decimals
= buffer
.length() - read_digits
;
196 static bool DoubleStrtod(Vector
<const char> trimmed
,
199 #if !defined(DOUBLE_CONVERSION_CORRECT_DOUBLE_OPERATIONS)
200 // On x86 the floating-point stack can be 64 or 80 bits wide. If it is
201 // 80 bits wide (as is the case on Linux) then double-rounding occurs and the
202 // result is not accurate.
203 // We know that Windows32 uses 64 bits and is therefore accurate.
204 // Note that the ARM simulator is compiled for 32bits. It therefore exhibits
208 if (trimmed
.length() <= kMaxExactDoubleIntegerDecimalDigits
) {
210 // The trimmed input fits into a double.
211 // If the 10^exponent (resp. 10^-exponent) fits into a double too then we
212 // can compute the result-double simply by multiplying (resp. dividing) the
214 // This is possible because IEEE guarantees that floating-point operations
215 // return the best possible approximation.
216 if (exponent
< 0 && -exponent
< kExactPowersOfTenSize
) {
217 // 10^-exponent fits into a double.
218 *result
= static_cast<double>(ReadUint64(trimmed
, &read_digits
));
219 ASSERT(read_digits
== trimmed
.length());
220 *result
/= exact_powers_of_ten
[-exponent
];
223 if (0 <= exponent
&& exponent
< kExactPowersOfTenSize
) {
224 // 10^exponent fits into a double.
225 *result
= static_cast<double>(ReadUint64(trimmed
, &read_digits
));
226 ASSERT(read_digits
== trimmed
.length());
227 *result
*= exact_powers_of_ten
[exponent
];
230 int remaining_digits
=
231 kMaxExactDoubleIntegerDecimalDigits
- trimmed
.length();
232 if ((0 <= exponent
) &&
233 (exponent
- remaining_digits
< kExactPowersOfTenSize
)) {
234 // The trimmed string was short and we can multiply it with
235 // 10^remaining_digits. As a result the remaining exponent now fits
236 // into a double too.
237 *result
= static_cast<double>(ReadUint64(trimmed
, &read_digits
));
238 ASSERT(read_digits
== trimmed
.length());
239 *result
*= exact_powers_of_ten
[remaining_digits
];
240 *result
*= exact_powers_of_ten
[exponent
- remaining_digits
];
248 // Returns 10^exponent as an exact DiyFp.
249 // The given exponent must be in the range [1; kDecimalExponentDistance[.
250 static DiyFp
AdjustmentPowerOfTen(int exponent
) {
251 ASSERT(0 < exponent
);
252 ASSERT(exponent
< PowersOfTenCache::kDecimalExponentDistance
);
253 // Simply hardcode the remaining powers for the given decimal exponent
255 ASSERT(PowersOfTenCache::kDecimalExponentDistance
== 8);
257 case 1: return DiyFp(UINT64_2PART_C(0xa0000000, 00000000), -60);
258 case 2: return DiyFp(UINT64_2PART_C(0xc8000000, 00000000), -57);
259 case 3: return DiyFp(UINT64_2PART_C(0xfa000000, 00000000), -54);
260 case 4: return DiyFp(UINT64_2PART_C(0x9c400000, 00000000), -50);
261 case 5: return DiyFp(UINT64_2PART_C(0xc3500000, 00000000), -47);
262 case 6: return DiyFp(UINT64_2PART_C(0xf4240000, 00000000), -44);
263 case 7: return DiyFp(UINT64_2PART_C(0x98968000, 00000000), -40);
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 static bool DiyFpStrtod(Vector
<const char> buffer
,
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
285 const int kDenominatorLog
= 3;
286 const int kDenominator
= 1 << kDenominatorLog
;
287 // Move the remaining decimals into the exponent.
288 exponent
+= remaining_decimals
;
289 int error
= (remaining_decimals
== 0 ? 0 : kDenominator
/ 2);
291 int old_e
= input
.e();
293 error
<<= old_e
- input
.e();
295 ASSERT(exponent
<= PowersOfTenCache::kMaxDecimalExponent
);
296 if (exponent
< PowersOfTenCache::kMinDecimalExponent
) {
301 int cached_decimal_exponent
;
302 PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent
,
304 &cached_decimal_exponent
);
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
313 ASSERT(DiyFp::kSignificandSize
== 64);
315 // The adjustment power is exact. There is hence only an error of 0.5.
316 error
+= kDenominator
/ 2;
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
;
333 error
<<= old_e
- input
.e();
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
;
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);
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);
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.
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.
385 // - -1 if buffer*10^exponent < diy_fp.
386 // - 0 if buffer*10^exponent == diy_fp.
387 // - +1 if buffer*10^exponent > diy_fp.
389 // buffer.length() + exponent <= kMaxDecimalPower + 1
390 // buffer.length() + exponent > kMinDecimalPower
391 // buffer.length() <= kMaxDecimalSignificantDigits
392 static int CompareBufferWithDiyFp(Vector
<const char> buffer
,
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());
408 buffer_bignum
.MultiplyByPowerOfTen(exponent
);
410 diy_fp_bignum
.MultiplyByPowerOfTen(-exponent
);
412 if (diy_fp
.e() > 0) {
413 diy_fp_bignum
.ShiftLeft(diy_fp
.e());
415 buffer_bignum
.ShiftLeft(-diy_fp
.e());
417 return Bignum::Compare(buffer_bignum
, diy_fp_bignum
);
421 // Returns true if the guess is the correct double.
422 // Returns false, when guess is either correct or the next-lower double.
423 static bool ComputeGuess(Vector
<const char> trimmed
, int exponent
,
425 if (trimmed
.length() == 0) {
429 if (exponent
+ trimmed
.length() - 1 >= kMaxDecimalPower
) {
430 *guess
= Double::Infinity();
433 if (exponent
+ trimmed
.length() <= kMinDecimalPower
) {
438 if (DoubleStrtod(trimmed
, exponent
, guess
) ||
439 DiyFpStrtod(trimmed
, exponent
, guess
)) {
442 if (*guess
== Double::Infinity()) {
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
;
457 bool is_correct
= ComputeGuess(trimmed
, exponent
, &guess
);
458 if (is_correct
) return guess
;
460 DiyFp upper_boundary
= Double(guess
).UpperBoundary();
461 int comparison
= CompareBufferWithDiyFp(trimmed
, exponent
, upper_boundary
);
462 if (comparison
< 0) {
464 } else if (comparison
> 0) {
465 return Double(guess
).NextDouble();
466 } else if ((Double(guess
).Significand() & 1) == 0) {
467 // Round towards even.
470 return Double(guess
).NextDouble();
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
;
483 bool is_correct
= ComputeGuess(trimmed
, exponent
, &double_guess
);
485 float float_guess
= static_cast<float>(double_guess
);
486 if (float_guess
== double_guess
) {
487 // This shortcut triggers for integer values.
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):
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
505 double double_next
= Double(double_guess
).NextDouble();
506 double double_previous
= Double(double_guess
).PreviousDouble();
508 float f1
= static_cast<float>(double_previous
);
510 float f2
= float_guess
;
512 float f3
= static_cast<float>(double_next
);
517 double double_next2
= Double(double_next
).NextDouble();
518 f4
= static_cast<float>(double_next2
);
520 ASSERT(f1
<= f2
&& f2
<= f3
&& f3
<= f4
);
522 // If the guess doesn't lie near a single-precision boundary we can simply
523 // return its float-value.
528 ASSERT((f1
!= f2
&& f2
== f3
&& f3
== f4
) ||
529 (f1
== f2
&& f2
!= f3
&& f3
== f4
) ||
530 (f1
== f2
&& f2
== f3
&& f3
!= f4
));
532 // guess and next are the two possible canditates (in the same way that
533 // double_guess was the lower candidate for a double-precision guess).
536 DiyFp upper_boundary
;
538 float min_float
= 1e-45f
;
539 upper_boundary
= Double(static_cast<double>(min_float
) / 2).AsDiyFp();
541 upper_boundary
= Single(guess
).UpperBoundary();
543 int comparison
= CompareBufferWithDiyFp(trimmed
, exponent
, upper_boundary
);
544 if (comparison
< 0) {
546 } else if (comparison
> 0) {
548 } else if ((Single(guess
).Significand() & 1) == 0) {
549 // Round towards even.
556 } // namespace double_conversion