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 <cmath> 6 7 #include "include/v8stdint.h" 8 #include "src/checks.h" 9 #include "src/utils.h" 10 11 #include "src/double.h" 12 #include "src/fixed-dtoa.h" 13 14 namespace v8 { 15 namespace internal { 16 17 // Represents a 128bit type. This class should be replaced by a native type on 18 // platforms that support 128bit integers. 19 class UInt128 { 20 public: 21 UInt128() : high_bits_(0), low_bits_(0) { } 22 UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { } 23 24 void Multiply(uint32_t multiplicand) { 25 uint64_t accumulator; 26 27 accumulator = (low_bits_ & kMask32) * multiplicand; 28 uint32_t part = static_cast<uint32_t>(accumulator & kMask32); 29 accumulator >>= 32; 30 accumulator = accumulator + (low_bits_ >> 32) * multiplicand; 31 low_bits_ = (accumulator << 32) + part; 32 accumulator >>= 32; 33 accumulator = accumulator + (high_bits_ & kMask32) * multiplicand; 34 part = static_cast<uint32_t>(accumulator & kMask32); 35 accumulator >>= 32; 36 accumulator = accumulator + (high_bits_ >> 32) * multiplicand; 37 high_bits_ = (accumulator << 32) + part; 38 ASSERT((accumulator >> 32) == 0); 39 } 40 41 void Shift(int shift_amount) { 42 ASSERT(-64 <= shift_amount && shift_amount <= 64); 43 if (shift_amount == 0) { 44 return; 45 } else if (shift_amount == -64) { 46 high_bits_ = low_bits_; 47 low_bits_ = 0; 48 } else if (shift_amount == 64) { 49 low_bits_ = high_bits_; 50 high_bits_ = 0; 51 } else if (shift_amount <= 0) { 52 high_bits_ <<= -shift_amount; 53 high_bits_ += low_bits_ >> (64 + shift_amount); 54 low_bits_ <<= -shift_amount; 55 } else { 56 low_bits_ >>= shift_amount; 57 low_bits_ += high_bits_ << (64 - shift_amount); 58 high_bits_ >>= shift_amount; 59 } 60 } 61 62 // Modifies *this to *this MOD (2^power). 63 // Returns *this DIV (2^power). 64 int DivModPowerOf2(int power) { 65 if (power >= 64) { 66 int result = static_cast<int>(high_bits_ >> (power - 64)); 67 high_bits_ -= static_cast<uint64_t>(result) << (power - 64); 68 return result; 69 } else { 70 uint64_t part_low = low_bits_ >> power; 71 uint64_t part_high = high_bits_ << (64 - power); 72 int result = static_cast<int>(part_low + part_high); 73 high_bits_ = 0; 74 low_bits_ -= part_low << power; 75 return result; 76 } 77 } 78 79 bool IsZero() const { 80 return high_bits_ == 0 && low_bits_ == 0; 81 } 82 83 int BitAt(int position) { 84 if (position >= 64) { 85 return static_cast<int>(high_bits_ >> (position - 64)) & 1; 86 } else { 87 return static_cast<int>(low_bits_ >> position) & 1; 88 } 89 } 90 91 private: 92 static const uint64_t kMask32 = 0xFFFFFFFF; 93 // Value == (high_bits_ << 64) + low_bits_ 94 uint64_t high_bits_; 95 uint64_t low_bits_; 96 }; 97 98 99 static const int kDoubleSignificandSize = 53; // Includes the hidden bit. 100 101 102 static void FillDigits32FixedLength(uint32_t number, int requested_length, 103 Vector<char> buffer, int* length) { 104 for (int i = requested_length - 1; i >= 0; --i) { 105 buffer[(*length) + i] = '0' + number % 10; 106 number /= 10; 107 } 108 *length += requested_length; 109 } 110 111 112 static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) { 113 int number_length = 0; 114 // We fill the digits in reverse order and exchange them afterwards. 115 while (number != 0) { 116 int digit = number % 10; 117 number /= 10; 118 buffer[(*length) + number_length] = '0' + digit; 119 number_length++; 120 } 121 // Exchange the digits. 122 int i = *length; 123 int j = *length + number_length - 1; 124 while (i < j) { 125 char tmp = buffer[i]; 126 buffer[i] = buffer[j]; 127 buffer[j] = tmp; 128 i++; 129 j--; 130 } 131 *length += number_length; 132 } 133 134 135 static void FillDigits64FixedLength(uint64_t number, int requested_length, 136 Vector<char> buffer, int* length) { 137 const uint32_t kTen7 = 10000000; 138 // For efficiency cut the number into 3 uint32_t parts, and print those. 139 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 140 number /= kTen7; 141 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 142 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 143 144 FillDigits32FixedLength(part0, 3, buffer, length); 145 FillDigits32FixedLength(part1, 7, buffer, length); 146 FillDigits32FixedLength(part2, 7, buffer, length); 147 } 148 149 150 static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) { 151 const uint32_t kTen7 = 10000000; 152 // For efficiency cut the number into 3 uint32_t parts, and print those. 153 uint32_t part2 = static_cast<uint32_t>(number % kTen7); 154 number /= kTen7; 155 uint32_t part1 = static_cast<uint32_t>(number % kTen7); 156 uint32_t part0 = static_cast<uint32_t>(number / kTen7); 157 158 if (part0 != 0) { 159 FillDigits32(part0, buffer, length); 160 FillDigits32FixedLength(part1, 7, buffer, length); 161 FillDigits32FixedLength(part2, 7, buffer, length); 162 } else if (part1 != 0) { 163 FillDigits32(part1, buffer, length); 164 FillDigits32FixedLength(part2, 7, buffer, length); 165 } else { 166 FillDigits32(part2, buffer, length); 167 } 168 } 169 170 171 static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) { 172 // An empty buffer represents 0. 173 if (*length == 0) { 174 buffer[0] = '1'; 175 *decimal_point = 1; 176 *length = 1; 177 return; 178 } 179 // Round the last digit until we either have a digit that was not '9' or until 180 // we reached the first digit. 181 buffer[(*length) - 1]++; 182 for (int i = (*length) - 1; i > 0; --i) { 183 if (buffer[i] != '0' + 10) { 184 return; 185 } 186 buffer[i] = '0'; 187 buffer[i - 1]++; 188 } 189 // If the first digit is now '0' + 10, we would need to set it to '0' and add 190 // a '1' in front. However we reach the first digit only if all following 191 // digits had been '9' before rounding up. Now all trailing digits are '0' and 192 // we simply switch the first digit to '1' and update the decimal-point 193 // (indicating that the point is now one digit to the right). 194 if (buffer[0] == '0' + 10) { 195 buffer[0] = '1'; 196 (*decimal_point)++; 197 } 198 } 199 200 201 // The given fractionals number represents a fixed-point number with binary 202 // point at bit (-exponent). 203 // Preconditions: 204 // -128 <= exponent <= 0. 205 // 0 <= fractionals * 2^exponent < 1 206 // The buffer holds the result. 207 // The function will round its result. During the rounding-process digits not 208 // generated by this function might be updated, and the decimal-point variable 209 // might be updated. If this function generates the digits 99 and the buffer 210 // already contained "199" (thus yielding a buffer of "19999") then a 211 // rounding-up will change the contents of the buffer to "20000". 212 static void FillFractionals(uint64_t fractionals, int exponent, 213 int fractional_count, Vector<char> buffer, 214 int* length, int* decimal_point) { 215 ASSERT(-128 <= exponent && exponent <= 0); 216 // 'fractionals' is a fixed-point number, with binary point at bit 217 // (-exponent). Inside the function the non-converted remainder of fractionals 218 // is a fixed-point number, with binary point at bit 'point'. 219 if (-exponent <= 64) { 220 // One 64 bit number is sufficient. 221 ASSERT(fractionals >> 56 == 0); 222 int point = -exponent; 223 for (int i = 0; i < fractional_count; ++i) { 224 if (fractionals == 0) break; 225 // Instead of multiplying by 10 we multiply by 5 and adjust the point 226 // location. This way the fractionals variable will not overflow. 227 // Invariant at the beginning of the loop: fractionals < 2^point. 228 // Initially we have: point <= 64 and fractionals < 2^56 229 // After each iteration the point is decremented by one. 230 // Note that 5^3 = 125 < 128 = 2^7. 231 // Therefore three iterations of this loop will not overflow fractionals 232 // (even without the subtraction at the end of the loop body). At this 233 // time point will satisfy point <= 61 and therefore fractionals < 2^point 234 // and any further multiplication of fractionals by 5 will not overflow. 235 fractionals *= 5; 236 point--; 237 int digit = static_cast<int>(fractionals >> point); 238 buffer[*length] = '0' + digit; 239 (*length)++; 240 fractionals -= static_cast<uint64_t>(digit) << point; 241 } 242 // If the first bit after the point is set we have to round up. 243 if (((fractionals >> (point - 1)) & 1) == 1) { 244 RoundUp(buffer, length, decimal_point); 245 } 246 } else { // We need 128 bits. 247 ASSERT(64 < -exponent && -exponent <= 128); 248 UInt128 fractionals128 = UInt128(fractionals, 0); 249 fractionals128.Shift(-exponent - 64); 250 int point = 128; 251 for (int i = 0; i < fractional_count; ++i) { 252 if (fractionals128.IsZero()) break; 253 // As before: instead of multiplying by 10 we multiply by 5 and adjust the 254 // point location. 255 // This multiplication will not overflow for the same reasons as before. 256 fractionals128.Multiply(5); 257 point--; 258 int digit = fractionals128.DivModPowerOf2(point); 259 buffer[*length] = '0' + digit; 260 (*length)++; 261 } 262 if (fractionals128.BitAt(point - 1) == 1) { 263 RoundUp(buffer, length, decimal_point); 264 } 265 } 266 } 267 268 269 // Removes leading and trailing zeros. 270 // If leading zeros are removed then the decimal point position is adjusted. 271 static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) { 272 while (*length > 0 && buffer[(*length) - 1] == '0') { 273 (*length)--; 274 } 275 int first_non_zero = 0; 276 while (first_non_zero < *length && buffer[first_non_zero] == '0') { 277 first_non_zero++; 278 } 279 if (first_non_zero != 0) { 280 for (int i = first_non_zero; i < *length; ++i) { 281 buffer[i - first_non_zero] = buffer[i]; 282 } 283 *length -= first_non_zero; 284 *decimal_point -= first_non_zero; 285 } 286 } 287 288 289 bool FastFixedDtoa(double v, 290 int fractional_count, 291 Vector<char> buffer, 292 int* length, 293 int* decimal_point) { 294 const uint32_t kMaxUInt32 = 0xFFFFFFFF; 295 uint64_t significand = Double(v).Significand(); 296 int exponent = Double(v).Exponent(); 297 // v = significand * 2^exponent (with significand a 53bit integer). 298 // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we 299 // don't know how to compute the representation. 2^73 ~= 9.5*10^21. 300 // If necessary this limit could probably be increased, but we don't need 301 // more. 302 if (exponent > 20) return false; 303 if (fractional_count > 20) return false; 304 *length = 0; 305 // At most kDoubleSignificandSize bits of the significand are non-zero. 306 // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero 307 // bits: 0..11*..0xxx..53*..xx 308 if (exponent + kDoubleSignificandSize > 64) { 309 // The exponent must be > 11. 310 // 311 // We know that v = significand * 2^exponent. 312 // And the exponent > 11. 313 // We simplify the task by dividing v by 10^17. 314 // The quotient delivers the first digits, and the remainder fits into a 64 315 // bit number. 316 // Dividing by 10^17 is equivalent to dividing by 5^17*2^17. 317 const uint64_t kFive17 = V8_2PART_UINT64_C(0xB1, A2BC2EC5); // 5^17 318 uint64_t divisor = kFive17; 319 int divisor_power = 17; 320 uint64_t dividend = significand; 321 uint32_t quotient; 322 uint64_t remainder; 323 // Let v = f * 2^e with f == significand and e == exponent. 324 // Then need q (quotient) and r (remainder) as follows: 325 // v = q * 10^17 + r 326 // f * 2^e = q * 10^17 + r 327 // f * 2^e = q * 5^17 * 2^17 + r 328 // If e > 17 then 329 // f * 2^(e-17) = q * 5^17 + r/2^17 330 // else 331 // f = q * 5^17 * 2^(17-e) + r/2^e 332 if (exponent > divisor_power) { 333 // We only allow exponents of up to 20 and therefore (17 - e) <= 3 334 dividend <<= exponent - divisor_power; 335 quotient = static_cast<uint32_t>(dividend / divisor); 336 remainder = (dividend % divisor) << divisor_power; 337 } else { 338 divisor <<= divisor_power - exponent; 339 quotient = static_cast<uint32_t>(dividend / divisor); 340 remainder = (dividend % divisor) << exponent; 341 } 342 FillDigits32(quotient, buffer, length); 343 FillDigits64FixedLength(remainder, divisor_power, buffer, length); 344 *decimal_point = *length; 345 } else if (exponent >= 0) { 346 // 0 <= exponent <= 11 347 significand <<= exponent; 348 FillDigits64(significand, buffer, length); 349 *decimal_point = *length; 350 } else if (exponent > -kDoubleSignificandSize) { 351 // We have to cut the number. 352 uint64_t integrals = significand >> -exponent; 353 uint64_t fractionals = significand - (integrals << -exponent); 354 if (integrals > kMaxUInt32) { 355 FillDigits64(integrals, buffer, length); 356 } else { 357 FillDigits32(static_cast<uint32_t>(integrals), buffer, length); 358 } 359 *decimal_point = *length; 360 FillFractionals(fractionals, exponent, fractional_count, 361 buffer, length, decimal_point); 362 } else if (exponent < -128) { 363 // This configuration (with at most 20 digits) means that all digits must be 364 // 0. 365 ASSERT(fractional_count <= 20); 366 buffer[0] = '\0'; 367 *length = 0; 368 *decimal_point = -fractional_count; 369 } else { 370 *decimal_point = 0; 371 FillFractionals(significand, exponent, fractional_count, 372 buffer, length, decimal_point); 373 } 374 TrimZeros(buffer, length, decimal_point); 375 buffer[*length] = '\0'; 376 if ((*length) == 0) { 377 // The string is empty and the decimal_point thus has no importance. Mimick 378 // Gay's dtoa and and set it to -fractional_count. 379 *decimal_point = -fractional_count; 380 } 381 return true; 382 } 383 384 } } // namespace v8::internal 385