1 // Copyright 2010 the V8 project authors. All rights reserved. 2 // Redistribution and use in source and binary forms, with or without 3 // modification, are permitted provided that the following conditions are 4 // met: 5 // 6 // * Redistributions of source code must retain the above copyright 7 // notice, this list of conditions and the following disclaimer. 8 // * Redistributions in binary form must reproduce the above 9 // copyright notice, this list of conditions and the following 10 // disclaimer in the documentation and/or other materials provided 11 // with the distribution. 12 // * Neither the name of Google Inc. nor the names of its 13 // contributors may be used to endorse or promote products derived 14 // from this software without specific prior written permission. 15 // 16 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 17 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 18 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 19 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT 20 // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 21 // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT 22 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 23 // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 24 // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 25 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 26 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 27 28 #include "config.h" 29 30 #include <math.h> 31 32 #include "double.h" 33 #include "fixed-dtoa.h" 34 #include "wtf/UnusedParam.h" 35 36 namespace WTF { 37 38 namespace double_conversion { 39 40 // Represents a 128bit type. This class should be replaced by a native type on 41 // platforms that support 128bit integers. 42 class UInt128 { 43 public: 44 UInt128() : high_bits_(0), low_bits_(0) { } 45 UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { } 46 47 void Multiply(uint32_t multiplicand) { 48 uint64_t accumulator; 49 50 accumulator = (low_bits_ & kMask32) * multiplicand; 51 uint32_t part = static_cast<uint32_t>(accumulator & kMask32); 52 accumulator >>= 32; 53 accumulator = accumulator + (low_bits_ >> 32) * multiplicand; 54 low_bits_ = (accumulator << 32) + part; 55 accumulator >>= 32; 56 accumulator = accumulator + (high_bits_ & kMask32) * multiplicand; 57 part = static_cast<uint32_t>(accumulator & kMask32); 58 accumulator >>= 32; 59 accumulator = accumulator + (high_bits_ >> 32) * multiplicand; 60 high_bits_ = (accumulator << 32) + part; 61 ASSERT((accumulator >> 32) == 0); 62 } 63 64 void Shift(int shift_amount) { 65 ASSERT(-64 <= shift_amount && shift_amount <= 64); 66 if (shift_amount == 0) { 67 return; 68 } else if (shift_amount == -64) { 69 high_bits_ = low_bits_; 70 low_bits_ = 0; 71 } else if (shift_amount == 64) { 72 low_bits_ = high_bits_; 73 high_bits_ = 0; 74 } else if (shift_amount <= 0) { 75 high_bits_ <<= -shift_amount; 76 high_bits_ += low_bits_ >> (64 + shift_amount); 77 low_bits_ <<= -shift_amount; 78 } else { 79 low_bits_ >>= shift_amount; 80 low_bits_ += high_bits_ << (64 - shift_amount); 81 high_bits_ >>= shift_amount; 82 } 83 } 84 85 // Modifies *this to *this MOD (2^power). 86 // Returns *this DIV (2^power). 87 int DivModPowerOf2(int power) { 88 if (power >= 64) { 89 int result = static_cast<int>(high_bits_ >> (power - 64)); 90 high_bits_ -= static_cast<uint64_t>(result) << (power - 64); 91 return result; 92 } else { 93 uint64_t part_low = low_bits_ >> power; 94 uint64_t part_high = high_bits_ << (64 - power); 95 int result = static_cast<int>(part_low + part_high); 96 high_bits_ = 0; 97 low_bits_ -= part_low << power; 98 return result; 99 } 100 } 101 102 bool IsZero() const { 103 return high_bits_ == 0 && low_bits_ == 0; 104 } 105 106 int BitAt(int position) { 107 if (position >= 64) { 108 return static_cast<int>(high_bits_ >> (position - 64)) & 1; 109 } else { 110 return static_cast<int>(low_bits_ >> position) & 1; 111 } 112 } 113 114 private: 115 static const uint64_t kMask32 = 0xFFFFFFFF; 116 // Value == (high_bits_ << 64) + low_bits_ 117 uint64_t high_bits_; 118 uint64_t low_bits_; 119 }; 120 121 122 static const int kDoubleSignificandSize = 53; // Includes the hidden bit. 123 124 125 static void FillDigits32FixedLength(uint32_t number, int requested_length, 126 Vector<char> buffer, int* length) { 127 for (int i = requested_length - 1; i >= 0; --i) { 128 buffer[(*length) + i] = '0' + number % 10; 129 number /= 10; 130 } 131 *length += requested_length; 132 } 133 134 135 static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) { 136 int number_length = 0; 137 // We fill the digits in reverse order and exchange them afterwards. 138 while (number != 0) { 139 int digit = number % 10; 140 number /= 10; 141 buffer[(*length) + number_length] = '0' + digit; 142 number_length++; 143 } 144 // Exchange the digits. 145 int i = *length; 146 int j = *length + number_length - 1; 147 while (i < j) { 148 char tmp = buffer[i]; 149 buffer[i] = buffer[j]; 150 buffer[j] = tmp; 151 i++; 152 j--; 153 } 154 *length += number_length; 155 } 156 157 158 static void FillDigits64FixedLength(uint64_t number, int requested_length, 159 Vector<char> buffer, int* length) { 160 UNUSED_PARAM(requested_length); 161 const uint32_t kTen7 = 10000000; 162 // For efficiency cut the number into 3 uint32_t parts, and print those. 163 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 164 number /= kTen7; 165 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 166 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 167 168 FillDigits32FixedLength(part0, 3, buffer, length); 169 FillDigits32FixedLength(part1, 7, buffer, length); 170 FillDigits32FixedLength(part2, 7, buffer, length); 171 } 172 173 174 static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) { 175 const uint32_t kTen7 = 10000000; 176 // For efficiency cut the number into 3 uint32_t parts, and print those. 177 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 178 number /= kTen7; 179 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 180 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 181 182 if (part0 != 0) { 183 FillDigits32(part0, buffer, length); 184 FillDigits32FixedLength(part1, 7, buffer, length); 185 FillDigits32FixedLength(part2, 7, buffer, length); 186 } else if (part1 != 0) { 187 FillDigits32(part1, buffer, length); 188 FillDigits32FixedLength(part2, 7, buffer, length); 189 } else { 190 FillDigits32(part2, buffer, length); 191 } 192 } 193 194 195 static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) { 196 // An empty buffer represents 0. 197 if (*length == 0) { 198 buffer[0] = '1'; 199 *decimal_point = 1; 200 *length = 1; 201 return; 202 } 203 // Round the last digit until we either have a digit that was not '9' or until 204 // we reached the first digit. 205 buffer[(*length) - 1]++; 206 for (int i = (*length) - 1; i > 0; --i) { 207 if (buffer[i] != '0' + 10) { 208 return; 209 } 210 buffer[i] = '0'; 211 buffer[i - 1]++; 212 } 213 // If the first digit is now '0' + 10, we would need to set it to '0' and add 214 // a '1' in front. However we reach the first digit only if all following 215 // digits had been '9' before rounding up. Now all trailing digits are '0' and 216 // we simply switch the first digit to '1' and update the decimal-point 217 // (indicating that the point is now one digit to the right). 218 if (buffer[0] == '0' + 10) { 219 buffer[0] = '1'; 220 (*decimal_point)++; 221 } 222 } 223 224 225 // The given fractionals number represents a fixed-point number with binary 226 // point at bit (-exponent). 227 // Preconditions: 228 // -128 <= exponent <= 0. 229 // 0 <= fractionals * 2^exponent < 1 230 // The buffer holds the result. 231 // The function will round its result. During the rounding-process digits not 232 // generated by this function might be updated, and the decimal-point variable 233 // might be updated. If this function generates the digits 99 and the buffer 234 // already contained "199" (thus yielding a buffer of "19999") then a 235 // rounding-up will change the contents of the buffer to "20000". 236 static void FillFractionals(uint64_t fractionals, int exponent, 237 int fractional_count, Vector<char> buffer, 238 int* length, int* decimal_point) { 239 ASSERT(-128 <= exponent && exponent <= 0); 240 // 'fractionals' is a fixed-point number, with binary point at bit 241 // (-exponent). Inside the function the non-converted remainder of fractionals 242 // is a fixed-point number, with binary point at bit 'point'. 243 if (-exponent <= 64) { 244 // One 64 bit number is sufficient. 245 ASSERT(fractionals >> 56 == 0); 246 int point = -exponent; 247 for (int i = 0; i < fractional_count; ++i) { 248 if (fractionals == 0) break; 249 // Instead of multiplying by 10 we multiply by 5 and adjust the point 250 // location. This way the fractionals variable will not overflow. 251 // Invariant at the beginning of the loop: fractionals < 2^point. 252 // Initially we have: point <= 64 and fractionals < 2^56 253 // After each iteration the point is decremented by one. 254 // Note that 5^3 = 125 < 128 = 2^7. 255 // Therefore three iterations of this loop will not overflow fractionals 256 // (even without the subtraction at the end of the loop body). At this 257 // time point will satisfy point <= 61 and therefore fractionals < 2^point 258 // and any further multiplication of fractionals by 5 will not overflow. 259 fractionals *= 5; 260 point--; 261 int digit = static_cast<int>(fractionals >> point); 262 buffer[*length] = '0' + digit; 263 (*length)++; 264 fractionals -= static_cast<uint64_t>(digit) << point; 265 } 266 // If the first bit after the point is set we have to round up. 267 if (((fractionals >> (point - 1)) & 1) == 1) { 268 RoundUp(buffer, length, decimal_point); 269 } 270 } else { // We need 128 bits. 271 ASSERT(64 < -exponent && -exponent <= 128); 272 UInt128 fractionals128 = UInt128(fractionals, 0); 273 fractionals128.Shift(-exponent - 64); 274 int point = 128; 275 for (int i = 0; i < fractional_count; ++i) { 276 if (fractionals128.IsZero()) break; 277 // As before: instead of multiplying by 10 we multiply by 5 and adjust the 278 // point location. 279 // This multiplication will not overflow for the same reasons as before. 280 fractionals128.Multiply(5); 281 point--; 282 int digit = fractionals128.DivModPowerOf2(point); 283 buffer[*length] = '0' + digit; 284 (*length)++; 285 } 286 if (fractionals128.BitAt(point - 1) == 1) { 287 RoundUp(buffer, length, decimal_point); 288 } 289 } 290 } 291 292 293 // Removes leading and trailing zeros. 294 // If leading zeros are removed then the decimal point position is adjusted. 295 static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) { 296 while (*length > 0 && buffer[(*length) - 1] == '0') { 297 (*length)--; 298 } 299 int first_non_zero = 0; 300 while (first_non_zero < *length && buffer[first_non_zero] == '0') { 301 first_non_zero++; 302 } 303 if (first_non_zero != 0) { 304 for (int i = first_non_zero; i < *length; ++i) { 305 buffer[i - first_non_zero] = buffer[i]; 306 } 307 *length -= first_non_zero; 308 *decimal_point -= first_non_zero; 309 } 310 } 311 312 313 bool FastFixedDtoa(double v, 314 int fractional_count, 315 Vector<char> buffer, 316 int* length, 317 int* decimal_point) { 318 const uint32_t kMaxUInt32 = 0xFFFFFFFF; 319 uint64_t significand = Double(v).Significand(); 320 int exponent = Double(v).Exponent(); 321 // v = significand * 2^exponent (with significand a 53bit integer). 322 // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we 323 // don't know how to compute the representation. 2^73 ~= 9.5*10^21. 324 // If necessary this limit could probably be increased, but we don't need 325 // more. 326 if (exponent > 20) return false; 327 if (fractional_count > 20) return false; 328 *length = 0; 329 // At most kDoubleSignificandSize bits of the significand are non-zero. 330 // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero 331 // bits: 0..11*..0xxx..53*..xx 332 if (exponent + kDoubleSignificandSize > 64) { 333 // The exponent must be > 11. 334 // 335 // We know that v = significand * 2^exponent. 336 // And the exponent > 11. 337 // We simplify the task by dividing v by 10^17. 338 // The quotient delivers the first digits, and the remainder fits into a 64 339 // bit number. 340 // Dividing by 10^17 is equivalent to dividing by 5^17*2^17. 341 const uint64_t kFive17 = UINT64_2PART_C(0xB1, A2BC2EC5); // 5^17 342 uint64_t divisor = kFive17; 343 int divisor_power = 17; 344 uint64_t dividend = significand; 345 uint32_t quotient; 346 uint64_t remainder; 347 // Let v = f * 2^e with f == significand and e == exponent. 348 // Then need q (quotient) and r (remainder) as follows: 349 // v = q * 10^17 + r 350 // f * 2^e = q * 10^17 + r 351 // f * 2^e = q * 5^17 * 2^17 + r 352 // If e > 17 then 353 // f * 2^(e-17) = q * 5^17 + r/2^17 354 // else 355 // f = q * 5^17 * 2^(17-e) + r/2^e 356 if (exponent > divisor_power) { 357 // We only allow exponents of up to 20 and therefore (17 - e) <= 3 358 dividend <<= exponent - divisor_power; 359 quotient = static_cast<uint32_t>(dividend / divisor); 360 remainder = (dividend % divisor) << divisor_power; 361 } else { 362 divisor <<= divisor_power - exponent; 363 quotient = static_cast<uint32_t>(dividend / divisor); 364 remainder = (dividend % divisor) << exponent; 365 } 366 FillDigits32(quotient, buffer, length); 367 FillDigits64FixedLength(remainder, divisor_power, buffer, length); 368 *decimal_point = *length; 369 } else if (exponent >= 0) { 370 // 0 <= exponent <= 11 371 significand <<= exponent; 372 FillDigits64(significand, buffer, length); 373 *decimal_point = *length; 374 } else if (exponent > -kDoubleSignificandSize) { 375 // We have to cut the number. 376 uint64_t integrals = significand >> -exponent; 377 uint64_t fractionals = significand - (integrals << -exponent); 378 if (integrals > kMaxUInt32) { 379 FillDigits64(integrals, buffer, length); 380 } else { 381 FillDigits32(static_cast<uint32_t>(integrals), buffer, length); 382 } 383 *decimal_point = *length; 384 FillFractionals(fractionals, exponent, fractional_count, 385 buffer, length, decimal_point); 386 } else if (exponent < -128) { 387 // This configuration (with at most 20 digits) means that all digits must be 388 // 0. 389 ASSERT(fractional_count <= 20); 390 buffer[0] = '\0'; 391 *length = 0; 392 *decimal_point = -fractional_count; 393 } else { 394 *decimal_point = 0; 395 FillFractionals(significand, exponent, fractional_count, 396 buffer, length, decimal_point); 397 } 398 TrimZeros(buffer, length, decimal_point); 399 buffer[*length] = '\0'; 400 if ((*length) == 0) { 401 // The string is empty and the decimal_point thus has no importance. Mimick 402 // Gay's dtoa and and set it to -fractional_count. 403 *decimal_point = -fractional_count; 404 } 405 return true; 406 } 407 408 } // namespace double_conversion 409 410 } // namespace WTF 411