1 // Copyright 2011 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 <stdint.h> 6 7 #include <cmath> 8 9 #include "src/base/logging.h" 10 #include "src/utils.h" 11 12 #include "src/double.h" 13 #include "src/fixed-dtoa.h" 14 15 namespace v8 { 16 namespace internal { 17 18 // Represents a 128bit type. This class should be replaced by a native type on 19 // platforms that support 128bit integers. 20 class UInt128 { 21 public: 22 UInt128() : high_bits_(0), low_bits_(0) { } 23 UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { } 24 25 void Multiply(uint32_t multiplicand) { 26 uint64_t accumulator; 27 28 accumulator = (low_bits_ & kMask32) * multiplicand; 29 uint32_t part = static_cast<uint32_t>(accumulator & kMask32); 30 accumulator >>= 32; 31 accumulator = accumulator + (low_bits_ >> 32) * multiplicand; 32 low_bits_ = (accumulator << 32) + part; 33 accumulator >>= 32; 34 accumulator = accumulator + (high_bits_ & kMask32) * multiplicand; 35 part = static_cast<uint32_t>(accumulator & kMask32); 36 accumulator >>= 32; 37 accumulator = accumulator + (high_bits_ >> 32) * multiplicand; 38 high_bits_ = (accumulator << 32) + part; 39 DCHECK((accumulator >> 32) == 0); 40 } 41 42 void Shift(int shift_amount) { 43 DCHECK(-64 <= shift_amount && shift_amount <= 64); 44 if (shift_amount == 0) { 45 return; 46 } else if (shift_amount == -64) { 47 high_bits_ = low_bits_; 48 low_bits_ = 0; 49 } else if (shift_amount == 64) { 50 low_bits_ = high_bits_; 51 high_bits_ = 0; 52 } else if (shift_amount <= 0) { 53 high_bits_ <<= -shift_amount; 54 high_bits_ += low_bits_ >> (64 + shift_amount); 55 low_bits_ <<= -shift_amount; 56 } else { 57 low_bits_ >>= shift_amount; 58 low_bits_ += high_bits_ << (64 - shift_amount); 59 high_bits_ >>= shift_amount; 60 } 61 } 62 63 // Modifies *this to *this MOD (2^power). 64 // Returns *this DIV (2^power). 65 int DivModPowerOf2(int power) { 66 if (power >= 64) { 67 int result = static_cast<int>(high_bits_ >> (power - 64)); 68 high_bits_ -= static_cast<uint64_t>(result) << (power - 64); 69 return result; 70 } else { 71 uint64_t part_low = low_bits_ >> power; 72 uint64_t part_high = high_bits_ << (64 - power); 73 int result = static_cast<int>(part_low + part_high); 74 high_bits_ = 0; 75 low_bits_ -= part_low << power; 76 return result; 77 } 78 } 79 80 bool IsZero() const { 81 return high_bits_ == 0 && low_bits_ == 0; 82 } 83 84 int BitAt(int position) { 85 if (position >= 64) { 86 return static_cast<int>(high_bits_ >> (position - 64)) & 1; 87 } else { 88 return static_cast<int>(low_bits_ >> position) & 1; 89 } 90 } 91 92 private: 93 static const uint64_t kMask32 = 0xFFFFFFFF; 94 // Value == (high_bits_ << 64) + low_bits_ 95 uint64_t high_bits_; 96 uint64_t low_bits_; 97 }; 98 99 100 static const int kDoubleSignificandSize = 53; // Includes the hidden bit. 101 102 103 static void FillDigits32FixedLength(uint32_t number, int requested_length, 104 Vector<char> buffer, int* length) { 105 for (int i = requested_length - 1; i >= 0; --i) { 106 buffer[(*length) + i] = '0' + number % 10; 107 number /= 10; 108 } 109 *length += requested_length; 110 } 111 112 113 static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) { 114 int number_length = 0; 115 // We fill the digits in reverse order and exchange them afterwards. 116 while (number != 0) { 117 int digit = number % 10; 118 number /= 10; 119 buffer[(*length) + number_length] = '0' + digit; 120 number_length++; 121 } 122 // Exchange the digits. 123 int i = *length; 124 int j = *length + number_length - 1; 125 while (i < j) { 126 char tmp = buffer[i]; 127 buffer[i] = buffer[j]; 128 buffer[j] = tmp; 129 i++; 130 j--; 131 } 132 *length += number_length; 133 } 134 135 136 static void FillDigits64FixedLength(uint64_t number, int requested_length, 137 Vector<char> buffer, int* length) { 138 const uint32_t kTen7 = 10000000; 139 // For efficiency cut the number into 3 uint32_t parts, and print those. 140 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 141 number /= kTen7; 142 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 143 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 144 145 FillDigits32FixedLength(part0, 3, buffer, length); 146 FillDigits32FixedLength(part1, 7, buffer, length); 147 FillDigits32FixedLength(part2, 7, buffer, length); 148 } 149 150 151 static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) { 152 const uint32_t kTen7 = 10000000; 153 // For efficiency cut the number into 3 uint32_t parts, and print those. 154 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 155 number /= kTen7; 156 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 157 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 158 159 if (part0 != 0) { 160 FillDigits32(part0, buffer, length); 161 FillDigits32FixedLength(part1, 7, buffer, length); 162 FillDigits32FixedLength(part2, 7, buffer, length); 163 } else if (part1 != 0) { 164 FillDigits32(part1, buffer, length); 165 FillDigits32FixedLength(part2, 7, buffer, length); 166 } else { 167 FillDigits32(part2, buffer, length); 168 } 169 } 170 171 172 static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) { 173 // An empty buffer represents 0. 174 if (*length == 0) { 175 buffer[0] = '1'; 176 *decimal_point = 1; 177 *length = 1; 178 return; 179 } 180 // Round the last digit until we either have a digit that was not '9' or until 181 // we reached the first digit. 182 buffer[(*length) - 1]++; 183 for (int i = (*length) - 1; i > 0; --i) { 184 if (buffer[i] != '0' + 10) { 185 return; 186 } 187 buffer[i] = '0'; 188 buffer[i - 1]++; 189 } 190 // If the first digit is now '0' + 10, we would need to set it to '0' and add 191 // a '1' in front. However we reach the first digit only if all following 192 // digits had been '9' before rounding up. Now all trailing digits are '0' and 193 // we simply switch the first digit to '1' and update the decimal-point 194 // (indicating that the point is now one digit to the right). 195 if (buffer[0] == '0' + 10) { 196 buffer[0] = '1'; 197 (*decimal_point)++; 198 } 199 } 200 201 202 // The given fractionals number represents a fixed-point number with binary 203 // point at bit (-exponent). 204 // Preconditions: 205 // -128 <= exponent <= 0. 206 // 0 <= fractionals * 2^exponent < 1 207 // The buffer holds the result. 208 // The function will round its result. During the rounding-process digits not 209 // generated by this function might be updated, and the decimal-point variable 210 // might be updated. If this function generates the digits 99 and the buffer 211 // already contained "199" (thus yielding a buffer of "19999") then a 212 // rounding-up will change the contents of the buffer to "20000". 213 static void FillFractionals(uint64_t fractionals, int exponent, 214 int fractional_count, Vector<char> buffer, 215 int* length, int* decimal_point) { 216 DCHECK(-128 <= exponent && exponent <= 0); 217 // 'fractionals' is a fixed-point number, with binary point at bit 218 // (-exponent). Inside the function the non-converted remainder of fractionals 219 // is a fixed-point number, with binary point at bit 'point'. 220 if (-exponent <= 64) { 221 // One 64 bit number is sufficient. 222 DCHECK(fractionals >> 56 == 0); 223 int point = -exponent; 224 for (int i = 0; i < fractional_count; ++i) { 225 if (fractionals == 0) break; 226 // Instead of multiplying by 10 we multiply by 5 and adjust the point 227 // location. This way the fractionals variable will not overflow. 228 // Invariant at the beginning of the loop: fractionals < 2^point. 229 // Initially we have: point <= 64 and fractionals < 2^56 230 // After each iteration the point is decremented by one. 231 // Note that 5^3 = 125 < 128 = 2^7. 232 // Therefore three iterations of this loop will not overflow fractionals 233 // (even without the subtraction at the end of the loop body). At this 234 // time point will satisfy point <= 61 and therefore fractionals < 2^point 235 // and any further multiplication of fractionals by 5 will not overflow. 236 fractionals *= 5; 237 point--; 238 int digit = static_cast<int>(fractionals >> point); 239 buffer[*length] = '0' + digit; 240 (*length)++; 241 fractionals -= static_cast<uint64_t>(digit) << point; 242 } 243 // If the first bit after the point is set we have to round up. 244 if (((fractionals >> (point - 1)) & 1) == 1) { 245 RoundUp(buffer, length, decimal_point); 246 } 247 } else { // We need 128 bits. 248 DCHECK(64 < -exponent && -exponent <= 128); 249 UInt128 fractionals128 = UInt128(fractionals, 0); 250 fractionals128.Shift(-exponent - 64); 251 int point = 128; 252 for (int i = 0; i < fractional_count; ++i) { 253 if (fractionals128.IsZero()) break; 254 // As before: instead of multiplying by 10 we multiply by 5 and adjust the 255 // point location. 256 // This multiplication will not overflow for the same reasons as before. 257 fractionals128.Multiply(5); 258 point--; 259 int digit = fractionals128.DivModPowerOf2(point); 260 buffer[*length] = '0' + digit; 261 (*length)++; 262 } 263 if (fractionals128.BitAt(point - 1) == 1) { 264 RoundUp(buffer, length, decimal_point); 265 } 266 } 267 } 268 269 270 // Removes leading and trailing zeros. 271 // If leading zeros are removed then the decimal point position is adjusted. 272 static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) { 273 while (*length > 0 && buffer[(*length) - 1] == '0') { 274 (*length)--; 275 } 276 int first_non_zero = 0; 277 while (first_non_zero < *length && buffer[first_non_zero] == '0') { 278 first_non_zero++; 279 } 280 if (first_non_zero != 0) { 281 for (int i = first_non_zero; i < *length; ++i) { 282 buffer[i - first_non_zero] = buffer[i]; 283 } 284 *length -= first_non_zero; 285 *decimal_point -= first_non_zero; 286 } 287 } 288 289 290 bool FastFixedDtoa(double v, 291 int fractional_count, 292 Vector<char> buffer, 293 int* length, 294 int* decimal_point) { 295 const uint32_t kMaxUInt32 = 0xFFFFFFFF; 296 uint64_t significand = Double(v).Significand(); 297 int exponent = Double(v).Exponent(); 298 // v = significand * 2^exponent (with significand a 53bit integer). 299 // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we 300 // don't know how to compute the representation. 2^73 ~= 9.5*10^21. 301 // If necessary this limit could probably be increased, but we don't need 302 // more. 303 if (exponent > 20) return false; 304 if (fractional_count > 20) return false; 305 *length = 0; 306 // At most kDoubleSignificandSize bits of the significand are non-zero. 307 // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero 308 // bits: 0..11*..0xxx..53*..xx 309 if (exponent + kDoubleSignificandSize > 64) { 310 // The exponent must be > 11. 311 // 312 // We know that v = significand * 2^exponent. 313 // And the exponent > 11. 314 // We simplify the task by dividing v by 10^17. 315 // The quotient delivers the first digits, and the remainder fits into a 64 316 // bit number. 317 // Dividing by 10^17 is equivalent to dividing by 5^17*2^17. 318 const uint64_t kFive17 = V8_2PART_UINT64_C(0xB1, A2BC2EC5); // 5^17 319 uint64_t divisor = kFive17; 320 int divisor_power = 17; 321 uint64_t dividend = significand; 322 uint32_t quotient; 323 uint64_t remainder; 324 // Let v = f * 2^e with f == significand and e == exponent. 325 // Then need q (quotient) and r (remainder) as follows: 326 // v = q * 10^17 + r 327 // f * 2^e = q * 10^17 + r 328 // f * 2^e = q * 5^17 * 2^17 + r 329 // If e > 17 then 330 // f * 2^(e-17) = q * 5^17 + r/2^17 331 // else 332 // f = q * 5^17 * 2^(17-e) + r/2^e 333 if (exponent > divisor_power) { 334 // We only allow exponents of up to 20 and therefore (17 - e) <= 3 335 dividend <<= exponent - divisor_power; 336 quotient = static_cast<uint32_t>(dividend / divisor); 337 remainder = (dividend % divisor) << divisor_power; 338 } else { 339 divisor <<= divisor_power - exponent; 340 quotient = static_cast<uint32_t>(dividend / divisor); 341 remainder = (dividend % divisor) << exponent; 342 } 343 FillDigits32(quotient, buffer, length); 344 FillDigits64FixedLength(remainder, divisor_power, buffer, length); 345 *decimal_point = *length; 346 } else if (exponent >= 0) { 347 // 0 <= exponent <= 11 348 significand <<= exponent; 349 FillDigits64(significand, buffer, length); 350 *decimal_point = *length; 351 } else if (exponent > -kDoubleSignificandSize) { 352 // We have to cut the number. 353 uint64_t integrals = significand >> -exponent; 354 uint64_t fractionals = significand - (integrals << -exponent); 355 if (integrals > kMaxUInt32) { 356 FillDigits64(integrals, buffer, length); 357 } else { 358 FillDigits32(static_cast<uint32_t>(integrals), buffer, length); 359 } 360 *decimal_point = *length; 361 FillFractionals(fractionals, exponent, fractional_count, 362 buffer, length, decimal_point); 363 } else if (exponent < -128) { 364 // This configuration (with at most 20 digits) means that all digits must be 365 // 0. 366 DCHECK(fractional_count <= 20); 367 buffer[0] = '\0'; 368 *length = 0; 369 *decimal_point = -fractional_count; 370 } else { 371 *decimal_point = 0; 372 FillFractionals(significand, exponent, fractional_count, 373 buffer, length, decimal_point); 374 } 375 TrimZeros(buffer, length, decimal_point); 376 buffer[*length] = '\0'; 377 if ((*length) == 0) { 378 // The string is empty and the decimal_point thus has no importance. Mimick 379 // Gay's dtoa and and set it to -fractional_count. 380 *decimal_point = -fractional_count; 381 } 382 return true; 383 } 384 385 } // namespace internal 386 } // namespace v8 387
