1 // Copyright 2012 the V8 project authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style license that can be
3 // found in the LICENSE file.
4
5 #include "src/strtod.h"
6
7 #include <stdarg.h>
8 #include <cmath>
9
10 #include "src/bignum.h"
11 #include "src/cached-powers.h"
12 #include "src/double.h"
13 #include "src/globals.h"
14 #include "src/utils.h"
15
16 namespace v8 {
17 namespace internal {
18
19 // 2^53 = 9007199254740992.
20 // Any integer with at most 15 decimal digits will hence fit into a double
21 // (which has a 53bit significand) without loss of precision.
22 static const int kMaxExactDoubleIntegerDecimalDigits = 15;
23 // 2^64 = 18446744073709551616 > 10^19
24 static const int kMaxUint64DecimalDigits = 19;
25
26 // Max double: 1.7976931348623157 x 10^308
27 // Min non-zero double: 4.9406564584124654 x 10^-324
28 // Any x >= 10^309 is interpreted as +infinity.
29 // Any x <= 10^-324 is interpreted as 0.
30 // Note that 2.5e-324 (despite being smaller than the min double) will be read
31 // as non-zero (equal to the min non-zero double).
32 static const int kMaxDecimalPower = 309;
33 static const int kMinDecimalPower = -324;
34
35 // 2^64 = 18446744073709551616
36 static const uint64_t kMaxUint64 = V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF);
37
38
39 static const double exact_powers_of_ten[] = {
40 1.0, // 10^0
41 10.0,
42 100.0,
43 1000.0,
44 10000.0,
45 100000.0,
46 1000000.0,
47 10000000.0,
48 100000000.0,
49 1000000000.0,
50 10000000000.0, // 10^10
51 100000000000.0,
52 1000000000000.0,
53 10000000000000.0,
54 100000000000000.0,
55 1000000000000000.0,
56 10000000000000000.0,
57 100000000000000000.0,
58 1000000000000000000.0,
59 10000000000000000000.0,
60 100000000000000000000.0, // 10^20
61 1000000000000000000000.0,
62 // 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
63 10000000000000000000000.0
64 };
65 static const int kExactPowersOfTenSize = arraysize(exact_powers_of_ten);
66
67 // Maximum number of significant digits in the decimal representation.
68 // In fact the value is 772 (see conversions.cc), but to give us some margin
69 // we round up to 780.
70 static const int kMaxSignificantDecimalDigits = 780;
71
TrimLeadingZeros(Vector<const char> buffer)72 static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
73 for (int i = 0; i < buffer.length(); i++) {
74 if (buffer[i] != '0') {
75 return buffer.SubVector(i, buffer.length());
76 }
77 }
78 return Vector<const char>(buffer.start(), 0);
79 }
80
81
TrimTrailingZeros(Vector<const char> buffer)82 static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
83 for (int i = buffer.length() - 1; i >= 0; --i) {
84 if (buffer[i] != '0') {
85 return buffer.SubVector(0, i + 1);
86 }
87 }
88 return Vector<const char>(buffer.start(), 0);
89 }
90
91
TrimToMaxSignificantDigits(Vector<const char> buffer,int exponent,char * significant_buffer,int * significant_exponent)92 static void TrimToMaxSignificantDigits(Vector<const char> buffer,
93 int exponent,
94 char* significant_buffer,
95 int* significant_exponent) {
96 for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
97 significant_buffer[i] = buffer[i];
98 }
99 // The input buffer has been trimmed. Therefore the last digit must be
100 // different from '0'.
101 DCHECK(buffer[buffer.length() - 1] != '0');
102 // Set the last digit to be non-zero. This is sufficient to guarantee
103 // correct rounding.
104 significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
105 *significant_exponent =
106 exponent + (buffer.length() - kMaxSignificantDecimalDigits);
107 }
108
109
110 // Reads digits from the buffer and converts them to a uint64.
111 // Reads in as many digits as fit into a uint64.
112 // When the string starts with "1844674407370955161" no further digit is read.
113 // Since 2^64 = 18446744073709551616 it would still be possible read another
114 // 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)115 static uint64_t ReadUint64(Vector<const char> buffer,
116 int* number_of_read_digits) {
117 uint64_t result = 0;
118 int i = 0;
119 while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
120 int digit = buffer[i++] - '0';
121 DCHECK(0 <= digit && digit <= 9);
122 result = 10 * result + digit;
123 }
124 *number_of_read_digits = i;
125 return result;
126 }
127
128
129 // Reads a DiyFp from the buffer.
130 // The returned DiyFp is not necessarily normalized.
131 // If remaining_decimals is zero then the returned DiyFp is accurate.
132 // Otherwise it has been rounded and has error of at most 1/2 ulp.
ReadDiyFp(Vector<const char> buffer,DiyFp * result,int * remaining_decimals)133 static void ReadDiyFp(Vector<const char> buffer,
134 DiyFp* result,
135 int* remaining_decimals) {
136 int read_digits;
137 uint64_t significand = ReadUint64(buffer, &read_digits);
138 if (buffer.length() == read_digits) {
139 *result = DiyFp(significand, 0);
140 *remaining_decimals = 0;
141 } else {
142 // Round the significand.
143 if (buffer[read_digits] >= '5') {
144 significand++;
145 }
146 // Compute the binary exponent.
147 int exponent = 0;
148 *result = DiyFp(significand, exponent);
149 *remaining_decimals = buffer.length() - read_digits;
150 }
151 }
152
153
DoubleStrtod(Vector<const char> trimmed,int exponent,double * result)154 static bool DoubleStrtod(Vector<const char> trimmed,
155 int exponent,
156 double* result) {
157 #if (V8_TARGET_ARCH_IA32 || V8_TARGET_ARCH_X87 || defined(USE_SIMULATOR)) && \
158 !defined(_MSC_VER)
159 // On x86 the floating-point stack can be 64 or 80 bits wide. If it is
160 // 80 bits wide (as is the case on Linux) then double-rounding occurs and the
161 // result is not accurate.
162 // We know that Windows32 with MSVC, unlike with MinGW32, uses 64 bits and is
163 // therefore accurate.
164 // Note that the ARM and MIPS simulators are compiled for 32bits. They
165 // therefore exhibit the same problem.
166 return false;
167 #endif
168 if (trimmed.length() <= kMaxExactDoubleIntegerDecimalDigits) {
169 int read_digits;
170 // The trimmed input fits into a double.
171 // If the 10^exponent (resp. 10^-exponent) fits into a double too then we
172 // can compute the result-double simply by multiplying (resp. dividing) the
173 // two numbers.
174 // This is possible because IEEE guarantees that floating-point operations
175 // return the best possible approximation.
176 if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
177 // 10^-exponent fits into a double.
178 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
179 DCHECK(read_digits == trimmed.length());
180 *result /= exact_powers_of_ten[-exponent];
181 return true;
182 }
183 if (0 <= exponent && exponent < kExactPowersOfTenSize) {
184 // 10^exponent fits into a double.
185 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
186 DCHECK(read_digits == trimmed.length());
187 *result *= exact_powers_of_ten[exponent];
188 return true;
189 }
190 int remaining_digits =
191 kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
192 if ((0 <= exponent) &&
193 (exponent - remaining_digits < kExactPowersOfTenSize)) {
194 // The trimmed string was short and we can multiply it with
195 // 10^remaining_digits. As a result the remaining exponent now fits
196 // into a double too.
197 *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
198 DCHECK(read_digits == trimmed.length());
199 *result *= exact_powers_of_ten[remaining_digits];
200 *result *= exact_powers_of_ten[exponent - remaining_digits];
201 return true;
202 }
203 }
204 return false;
205 }
206
207
208 // Returns 10^exponent as an exact DiyFp.
209 // The given exponent must be in the range [1; kDecimalExponentDistance[.
AdjustmentPowerOfTen(int exponent)210 static DiyFp AdjustmentPowerOfTen(int exponent) {
211 DCHECK(0 < exponent);
212 DCHECK(exponent < PowersOfTenCache::kDecimalExponentDistance);
213 // Simply hardcode the remaining powers for the given decimal exponent
214 // distance.
215 DCHECK(PowersOfTenCache::kDecimalExponentDistance == 8);
216 switch (exponent) {
217 case 1: return DiyFp(V8_2PART_UINT64_C(0xa0000000, 00000000), -60);
218 case 2: return DiyFp(V8_2PART_UINT64_C(0xc8000000, 00000000), -57);
219 case 3: return DiyFp(V8_2PART_UINT64_C(0xfa000000, 00000000), -54);
220 case 4: return DiyFp(V8_2PART_UINT64_C(0x9c400000, 00000000), -50);
221 case 5: return DiyFp(V8_2PART_UINT64_C(0xc3500000, 00000000), -47);
222 case 6: return DiyFp(V8_2PART_UINT64_C(0xf4240000, 00000000), -44);
223 case 7: return DiyFp(V8_2PART_UINT64_C(0x98968000, 00000000), -40);
224 default:
225 UNREACHABLE();
226 return DiyFp(0, 0);
227 }
228 }
229
230
231 // If the function returns true then the result is the correct double.
232 // Otherwise it is either the correct double or the double that is just below
233 // the correct double.
DiyFpStrtod(Vector<const char> buffer,int exponent,double * result)234 static bool DiyFpStrtod(Vector<const char> buffer,
235 int exponent,
236 double* result) {
237 DiyFp input;
238 int remaining_decimals;
239 ReadDiyFp(buffer, &input, &remaining_decimals);
240 // Since we may have dropped some digits the input is not accurate.
241 // If remaining_decimals is different than 0 than the error is at most
242 // .5 ulp (unit in the last place).
243 // We don't want to deal with fractions and therefore keep a common
244 // denominator.
245 const int kDenominatorLog = 3;
246 const int kDenominator = 1 << kDenominatorLog;
247 // Move the remaining decimals into the exponent.
248 exponent += remaining_decimals;
249 int64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
250
251 int old_e = input.e();
252 input.Normalize();
253 error <<= old_e - input.e();
254
255 DCHECK(exponent <= PowersOfTenCache::kMaxDecimalExponent);
256 if (exponent < PowersOfTenCache::kMinDecimalExponent) {
257 *result = 0.0;
258 return true;
259 }
260 DiyFp cached_power;
261 int cached_decimal_exponent;
262 PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
263 &cached_power,
264 &cached_decimal_exponent);
265
266 if (cached_decimal_exponent != exponent) {
267 int adjustment_exponent = exponent - cached_decimal_exponent;
268 DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
269 input.Multiply(adjustment_power);
270 if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
271 // The product of input with the adjustment power fits into a 64 bit
272 // integer.
273 DCHECK(DiyFp::kSignificandSize == 64);
274 } else {
275 // The adjustment power is exact. There is hence only an error of 0.5.
276 error += kDenominator / 2;
277 }
278 }
279
280 input.Multiply(cached_power);
281 // The error introduced by a multiplication of a*b equals
282 // error_a + error_b + error_a*error_b/2^64 + 0.5
283 // Substituting a with 'input' and b with 'cached_power' we have
284 // error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
285 // error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
286 int error_b = kDenominator / 2;
287 int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
288 int fixed_error = kDenominator / 2;
289 error += error_b + error_ab + fixed_error;
290
291 old_e = input.e();
292 input.Normalize();
293 error <<= old_e - input.e();
294
295 // See if the double's significand changes if we add/subtract the error.
296 int order_of_magnitude = DiyFp::kSignificandSize + input.e();
297 int effective_significand_size =
298 Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
299 int precision_digits_count =
300 DiyFp::kSignificandSize - effective_significand_size;
301 if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
302 // This can only happen for very small denormals. In this case the
303 // half-way multiplied by the denominator exceeds the range of an uint64.
304 // Simply shift everything to the right.
305 int shift_amount = (precision_digits_count + kDenominatorLog) -
306 DiyFp::kSignificandSize + 1;
307 input.set_f(input.f() >> shift_amount);
308 input.set_e(input.e() + shift_amount);
309 // We add 1 for the lost precision of error, and kDenominator for
310 // the lost precision of input.f().
311 error = (error >> shift_amount) + 1 + kDenominator;
312 precision_digits_count -= shift_amount;
313 }
314 // We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
315 DCHECK(DiyFp::kSignificandSize == 64);
316 DCHECK(precision_digits_count < 64);
317 uint64_t one64 = 1;
318 uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
319 uint64_t precision_bits = input.f() & precision_bits_mask;
320 uint64_t half_way = one64 << (precision_digits_count - 1);
321 precision_bits *= kDenominator;
322 half_way *= kDenominator;
323 DiyFp rounded_input(input.f() >> precision_digits_count,
324 input.e() + precision_digits_count);
325 if (precision_bits >= half_way + error) {
326 rounded_input.set_f(rounded_input.f() + 1);
327 }
328 // If the last_bits are too close to the half-way case than we are too
329 // inaccurate and round down. In this case we return false so that we can
330 // fall back to a more precise algorithm.
331
332 *result = Double(rounded_input).value();
333 if (half_way - error < precision_bits && precision_bits < half_way + error) {
334 // Too imprecise. The caller will have to fall back to a slower version.
335 // However the returned number is guaranteed to be either the correct
336 // double, or the next-lower double.
337 return false;
338 } else {
339 return true;
340 }
341 }
342
343
344 // Returns the correct double for the buffer*10^exponent.
345 // The variable guess should be a close guess that is either the correct double
346 // or its lower neighbor (the nearest double less than the correct one).
347 // Preconditions:
348 // buffer.length() + exponent <= kMaxDecimalPower + 1
349 // buffer.length() + exponent > kMinDecimalPower
350 // buffer.length() <= kMaxDecimalSignificantDigits
BignumStrtod(Vector<const char> buffer,int exponent,double guess)351 static double BignumStrtod(Vector<const char> buffer,
352 int exponent,
353 double guess) {
354 if (guess == V8_INFINITY) {
355 return guess;
356 }
357
358 DiyFp upper_boundary = Double(guess).UpperBoundary();
359
360 DCHECK(buffer.length() + exponent <= kMaxDecimalPower + 1);
361 DCHECK(buffer.length() + exponent > kMinDecimalPower);
362 DCHECK(buffer.length() <= kMaxSignificantDecimalDigits);
363 // Make sure that the Bignum will be able to hold all our numbers.
364 // Our Bignum implementation has a separate field for exponents. Shifts will
365 // consume at most one bigit (< 64 bits).
366 // ln(10) == 3.3219...
367 DCHECK(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
368 Bignum input;
369 Bignum boundary;
370 input.AssignDecimalString(buffer);
371 boundary.AssignUInt64(upper_boundary.f());
372 if (exponent >= 0) {
373 input.MultiplyByPowerOfTen(exponent);
374 } else {
375 boundary.MultiplyByPowerOfTen(-exponent);
376 }
377 if (upper_boundary.e() > 0) {
378 boundary.ShiftLeft(upper_boundary.e());
379 } else {
380 input.ShiftLeft(-upper_boundary.e());
381 }
382 int comparison = Bignum::Compare(input, boundary);
383 if (comparison < 0) {
384 return guess;
385 } else if (comparison > 0) {
386 return Double(guess).NextDouble();
387 } else if ((Double(guess).Significand() & 1) == 0) {
388 // Round towards even.
389 return guess;
390 } else {
391 return Double(guess).NextDouble();
392 }
393 }
394
395
Strtod(Vector<const char> buffer,int exponent)396 double Strtod(Vector<const char> buffer, int exponent) {
397 Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
398 Vector<const char> trimmed = TrimTrailingZeros(left_trimmed);
399 exponent += left_trimmed.length() - trimmed.length();
400 if (trimmed.length() == 0) return 0.0;
401 if (trimmed.length() > kMaxSignificantDecimalDigits) {
402 char significant_buffer[kMaxSignificantDecimalDigits];
403 int significant_exponent;
404 TrimToMaxSignificantDigits(trimmed, exponent,
405 significant_buffer, &significant_exponent);
406 return Strtod(Vector<const char>(significant_buffer,
407 kMaxSignificantDecimalDigits),
408 significant_exponent);
409 }
410 if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) return V8_INFINITY;
411 if (exponent + trimmed.length() <= kMinDecimalPower) return 0.0;
412
413 double guess;
414 if (DoubleStrtod(trimmed, exponent, &guess) ||
415 DiyFpStrtod(trimmed, exponent, &guess)) {
416 return guess;
417 }
418 return BignumStrtod(trimmed, exponent, guess);
419 }
420
421 } // namespace internal
422 } // namespace v8
423