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