double-conversion-fast-dtoa.cpp (32210B)
1 // © 2018 and later: Unicode, Inc. and others. 2 // License & terms of use: http://www.unicode.org/copyright.html 3 // 4 // From the double-conversion library. Original license: 5 // 6 // Copyright 2012 the V8 project authors. All rights reserved. 7 // Redistribution and use in source and binary forms, with or without 8 // modification, are permitted provided that the following conditions are 9 // met: 10 // 11 // * Redistributions of source code must retain the above copyright 12 // notice, this list of conditions and the following disclaimer. 13 // * Redistributions in binary form must reproduce the above 14 // copyright notice, this list of conditions and the following 15 // disclaimer in the documentation and/or other materials provided 16 // with the distribution. 17 // * Neither the name of Google Inc. nor the names of its 18 // contributors may be used to endorse or promote products derived 19 // from this software without specific prior written permission. 20 // 21 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 22 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 23 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 24 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT 25 // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 26 // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT 27 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 28 // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 29 // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 30 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 31 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 32 33 // ICU PATCH: ifdef around UCONFIG_NO_FORMATTING 34 #include "unicode/utypes.h" 35 #if !UCONFIG_NO_FORMATTING 36 37 // ICU PATCH: Customize header file paths for ICU. 38 39 #include "double-conversion-fast-dtoa.h" 40 41 #include "double-conversion-cached-powers.h" 42 #include "double-conversion-diy-fp.h" 43 #include "double-conversion-ieee.h" 44 45 // ICU PATCH: Wrap in ICU namespace 46 U_NAMESPACE_BEGIN 47 48 namespace double_conversion { 49 50 // The minimal and maximal target exponent define the range of w's binary 51 // exponent, where 'w' is the result of multiplying the input by a cached power 52 // of ten. 53 // 54 // A different range might be chosen on a different platform, to optimize digit 55 // generation, but a smaller range requires more powers of ten to be cached. 56 static const int kMinimalTargetExponent = -60; 57 static const int kMaximalTargetExponent = -32; 58 59 60 // Adjusts the last digit of the generated number, and screens out generated 61 // solutions that may be inaccurate. A solution may be inaccurate if it is 62 // outside the safe interval, or if we cannot prove that it is closer to the 63 // input than a neighboring representation of the same length. 64 // 65 // Input: * buffer containing the digits of too_high / 10^kappa 66 // * the buffer's length 67 // * distance_too_high_w == (too_high - w).f() * unit 68 // * unsafe_interval == (too_high - too_low).f() * unit 69 // * rest = (too_high - buffer * 10^kappa).f() * unit 70 // * ten_kappa = 10^kappa * unit 71 // * unit = the common multiplier 72 // Output: returns true if the buffer is guaranteed to contain the closest 73 // representable number to the input. 74 // Modifies the generated digits in the buffer to approach (round towards) w. 75 static bool RoundWeed(Vector<char> buffer, 76 int length, 77 uint64_t distance_too_high_w, 78 uint64_t unsafe_interval, 79 uint64_t rest, 80 uint64_t ten_kappa, 81 uint64_t unit) { 82 uint64_t small_distance = distance_too_high_w - unit; 83 uint64_t big_distance = distance_too_high_w + unit; 84 // Let w_low = too_high - big_distance, and 85 // w_high = too_high - small_distance. 86 // Note: w_low < w < w_high 87 // 88 // The real w (* unit) must lie somewhere inside the interval 89 // ]w_low; w_high[ (often written as "(w_low; w_high)") 90 91 // Basically the buffer currently contains a number in the unsafe interval 92 // ]too_low; too_high[ with too_low < w < too_high 93 // 94 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 95 // ^v 1 unit ^ ^ ^ ^ 96 // boundary_high --------------------- . . . . 97 // ^v 1 unit . . . . 98 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . 99 // . . ^ . . 100 // . big_distance . . . 101 // . . . . rest 102 // small_distance . . . . 103 // v . . . . 104 // w_high - - - - - - - - - - - - - - - - - - . . . . 105 // ^v 1 unit . . . . 106 // w ---------------------------------------- . . . . 107 // ^v 1 unit v . . . 108 // w_low - - - - - - - - - - - - - - - - - - - - - . . . 109 // . . v 110 // buffer --------------------------------------------------+-------+-------- 111 // . . 112 // safe_interval . 113 // v . 114 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . 115 // ^v 1 unit . 116 // boundary_low ------------------------- unsafe_interval 117 // ^v 1 unit v 118 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 119 // 120 // 121 // Note that the value of buffer could lie anywhere inside the range too_low 122 // to too_high. 123 // 124 // boundary_low, boundary_high and w are approximations of the real boundaries 125 // and v (the input number). They are guaranteed to be precise up to one unit. 126 // In fact the error is guaranteed to be strictly less than one unit. 127 // 128 // Anything that lies outside the unsafe interval is guaranteed not to round 129 // to v when read again. 130 // Anything that lies inside the safe interval is guaranteed to round to v 131 // when read again. 132 // If the number inside the buffer lies inside the unsafe interval but not 133 // inside the safe interval then we simply do not know and bail out (returning 134 // false). 135 // 136 // Similarly we have to take into account the imprecision of 'w' when finding 137 // the closest representation of 'w'. If we have two potential 138 // representations, and one is closer to both w_low and w_high, then we know 139 // it is closer to the actual value v. 140 // 141 // By generating the digits of too_high we got the largest (closest to 142 // too_high) buffer that is still in the unsafe interval. In the case where 143 // w_high < buffer < too_high we try to decrement the buffer. 144 // This way the buffer approaches (rounds towards) w. 145 // There are 3 conditions that stop the decrementation process: 146 // 1) the buffer is already below w_high 147 // 2) decrementing the buffer would make it leave the unsafe interval 148 // 3) decrementing the buffer would yield a number below w_high and farther 149 // away than the current number. In other words: 150 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high 151 // Instead of using the buffer directly we use its distance to too_high. 152 // Conceptually rest ~= too_high - buffer 153 // We need to do the following tests in this order to avoid over- and 154 // underflows. 155 DOUBLE_CONVERSION_ASSERT(rest <= unsafe_interval); 156 while (rest < small_distance && // Negated condition 1 157 unsafe_interval - rest >= ten_kappa && // Negated condition 2 158 (rest + ten_kappa < small_distance || // buffer{-1} > w_high 159 small_distance - rest >= rest + ten_kappa - small_distance)) { 160 buffer[length - 1]--; 161 rest += ten_kappa; 162 } 163 164 // We have approached w+ as much as possible. We now test if approaching w- 165 // would require changing the buffer. If yes, then we have two possible 166 // representations close to w, but we cannot decide which one is closer. 167 if (rest < big_distance && 168 unsafe_interval - rest >= ten_kappa && 169 (rest + ten_kappa < big_distance || 170 big_distance - rest > rest + ten_kappa - big_distance)) { 171 return false; 172 } 173 174 // Weeding test. 175 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] 176 // Since too_low = too_high - unsafe_interval this is equivalent to 177 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] 178 // Conceptually we have: rest ~= too_high - buffer 179 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); 180 } 181 182 183 // Rounds the buffer upwards if the result is closer to v by possibly adding 184 // 1 to the buffer. If the precision of the calculation is not sufficient to 185 // round correctly, return false. 186 // The rounding might shift the whole buffer in which case the kappa is 187 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. 188 // 189 // If 2*rest > ten_kappa then the buffer needs to be round up. 190 // rest can have an error of +/- 1 unit. This function accounts for the 191 // imprecision and returns false, if the rounding direction cannot be 192 // unambiguously determined. 193 // 194 // Precondition: rest < ten_kappa. 195 static bool RoundWeedCounted(Vector<char> buffer, 196 int length, 197 uint64_t rest, 198 uint64_t ten_kappa, 199 uint64_t unit, 200 int* kappa) { 201 DOUBLE_CONVERSION_ASSERT(rest < ten_kappa); 202 // The following tests are done in a specific order to avoid overflows. They 203 // will work correctly with any uint64 values of rest < ten_kappa and unit. 204 // 205 // If the unit is too big, then we don't know which way to round. For example 206 // a unit of 50 means that the real number lies within rest +/- 50. If 207 // 10^kappa == 40 then there is no way to tell which way to round. 208 if (unit >= ten_kappa) return false; 209 // Even if unit is just half the size of 10^kappa we are already completely 210 // lost. (And after the previous test we know that the expression will not 211 // over/underflow.) 212 if (ten_kappa - unit <= unit) return false; 213 // If 2 * (rest + unit) <= 10^kappa we can safely round down. 214 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) { 215 return true; 216 } 217 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. 218 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) { 219 // Increment the last digit recursively until we find a non '9' digit. 220 buffer[length - 1]++; 221 for (int i = length - 1; i > 0; --i) { 222 if (buffer[i] != '0' + 10) break; 223 buffer[i] = '0'; 224 buffer[i - 1]++; 225 } 226 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the 227 // exception of the first digit all digits are now '0'. Simply switch the 228 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and 229 // the power (the kappa) is increased. 230 if (buffer[0] == '0' + 10) { 231 buffer[0] = '1'; 232 (*kappa) += 1; 233 } 234 return true; 235 } 236 return false; 237 } 238 239 // Returns the biggest power of ten that is less than or equal to the given 240 // number. We furthermore receive the maximum number of bits 'number' has. 241 // 242 // Returns power == 10^(exponent_plus_one-1) such that 243 // power <= number < power * 10. 244 // If number_bits == 0 then 0^(0-1) is returned. 245 // The number of bits must be <= 32. 246 // Precondition: number < (1 << (number_bits + 1)). 247 248 // Inspired by the method for finding an integer log base 10 from here: 249 // http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10 250 static unsigned int const kSmallPowersOfTen[] = 251 {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 252 1000000000}; 253 254 static void BiggestPowerTen(uint32_t number, 255 int number_bits, 256 uint32_t* power, 257 int* exponent_plus_one) { 258 DOUBLE_CONVERSION_ASSERT(number < (1u << (number_bits + 1))); 259 // 1233/4096 is approximately 1/lg(10). 260 int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12); 261 // We increment to skip over the first entry in the kPowersOf10 table. 262 // Note: kPowersOf10[i] == 10^(i-1). 263 exponent_plus_one_guess++; 264 // We don't have any guarantees that 2^number_bits <= number. 265 if (number < kSmallPowersOfTen[exponent_plus_one_guess]) { 266 exponent_plus_one_guess--; 267 } 268 *power = kSmallPowersOfTen[exponent_plus_one_guess]; 269 *exponent_plus_one = exponent_plus_one_guess; 270 } 271 272 // Generates the digits of input number w. 273 // w is a floating-point number (DiyFp), consisting of a significand and an 274 // exponent. Its exponent is bounded by kMinimalTargetExponent and 275 // kMaximalTargetExponent. 276 // Hence -60 <= w.e() <= -32. 277 // 278 // Returns false if it fails, in which case the generated digits in the buffer 279 // should not be used. 280 // Preconditions: 281 // * low, w and high are correct up to 1 ulp (unit in the last place). That 282 // is, their error must be less than a unit of their last digits. 283 // * low.e() == w.e() == high.e() 284 // * low < w < high, and taking into account their error: low~ <= high~ 285 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 286 // Postconditions: returns false if procedure fails. 287 // otherwise: 288 // * buffer is not null-terminated, but len contains the number of digits. 289 // * buffer contains the shortest possible decimal digit-sequence 290 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the 291 // correct values of low and high (without their error). 292 // * if more than one decimal representation gives the minimal number of 293 // decimal digits then the one closest to W (where W is the correct value 294 // of w) is chosen. 295 // Remark: this procedure takes into account the imprecision of its input 296 // numbers. If the precision is not enough to guarantee all the postconditions 297 // then false is returned. This usually happens rarely (~0.5%). 298 // 299 // Say, for the sake of example, that 300 // w.e() == -48, and w.f() == 0x1234567890abcdef 301 // w's value can be computed by w.f() * 2^w.e() 302 // We can obtain w's integral digits by simply shifting w.f() by -w.e(). 303 // -> w's integral part is 0x1234 304 // w's fractional part is therefore 0x567890abcdef. 305 // Printing w's integral part is easy (simply print 0x1234 in decimal). 306 // In order to print its fraction we repeatedly multiply the fraction by 10 and 307 // get each digit. Example the first digit after the point would be computed by 308 // (0x567890abcdef * 10) >> 48. -> 3 309 // The whole thing becomes slightly more complicated because we want to stop 310 // once we have enough digits. That is, once the digits inside the buffer 311 // represent 'w' we can stop. Everything inside the interval low - high 312 // represents w. However we have to pay attention to low, high and w's 313 // imprecision. 314 static bool DigitGen(DiyFp low, 315 DiyFp w, 316 DiyFp high, 317 Vector<char> buffer, 318 int* length, 319 int* kappa) { 320 DOUBLE_CONVERSION_ASSERT(low.e() == w.e() && w.e() == high.e()); 321 DOUBLE_CONVERSION_ASSERT(low.f() + 1 <= high.f() - 1); 322 DOUBLE_CONVERSION_ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 323 // low, w and high are imprecise, but by less than one ulp (unit in the last 324 // place). 325 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that 326 // the new numbers are outside of the interval we want the final 327 // representation to lie in. 328 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield 329 // numbers that are certain to lie in the interval. We will use this fact 330 // later on. 331 // We will now start by generating the digits within the uncertain 332 // interval. Later we will weed out representations that lie outside the safe 333 // interval and thus _might_ lie outside the correct interval. 334 uint64_t unit = 1; 335 DiyFp too_low = DiyFp(low.f() - unit, low.e()); 336 DiyFp too_high = DiyFp(high.f() + unit, high.e()); 337 // too_low and too_high are guaranteed to lie outside the interval we want the 338 // generated number in. 339 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low); 340 // We now cut the input number into two parts: the integral digits and the 341 // fractionals. We will not write any decimal separator though, but adapt 342 // kappa instead. 343 // Reminder: we are currently computing the digits (stored inside the buffer) 344 // such that: too_low < buffer * 10^kappa < too_high 345 // We use too_high for the digit_generation and stop as soon as possible. 346 // If we stop early we effectively round down. 347 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 348 // Division by one is a shift. 349 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); 350 // Modulo by one is an and. 351 uint64_t fractionals = too_high.f() & (one.f() - 1); 352 uint32_t divisor; 353 int divisor_exponent_plus_one; 354 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 355 &divisor, &divisor_exponent_plus_one); 356 *kappa = divisor_exponent_plus_one; 357 *length = 0; 358 // Loop invariant: buffer = too_high / 10^kappa (integer division) 359 // The invariant holds for the first iteration: kappa has been initialized 360 // with the divisor exponent + 1. And the divisor is the biggest power of ten 361 // that is smaller than integrals. 362 while (*kappa > 0) { 363 int digit = integrals / divisor; 364 DOUBLE_CONVERSION_ASSERT(digit <= 9); 365 buffer[*length] = static_cast<char>('0' + digit); 366 (*length)++; 367 integrals %= divisor; 368 (*kappa)--; 369 // Note that kappa now equals the exponent of the divisor and that the 370 // invariant thus holds again. 371 uint64_t rest = 372 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 373 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) 374 // Reminder: unsafe_interval.e() == one.e() 375 if (rest < unsafe_interval.f()) { 376 // Rounding down (by not emitting the remaining digits) yields a number 377 // that lies within the unsafe interval. 378 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), 379 unsafe_interval.f(), rest, 380 static_cast<uint64_t>(divisor) << -one.e(), unit); 381 } 382 divisor /= 10; 383 } 384 385 // The integrals have been generated. We are at the point of the decimal 386 // separator. In the following loop we simply multiply the remaining digits by 387 // 10 and divide by one. We just need to pay attention to multiply associated 388 // data (like the interval or 'unit'), too. 389 // Note that the multiplication by 10 does not overflow, because w.e >= -60 390 // and thus one.e >= -60. 391 DOUBLE_CONVERSION_ASSERT(one.e() >= -60); 392 DOUBLE_CONVERSION_ASSERT(fractionals < one.f()); 393 DOUBLE_CONVERSION_ASSERT(DOUBLE_CONVERSION_UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); 394 for (;;) { 395 fractionals *= 10; 396 unit *= 10; 397 unsafe_interval.set_f(unsafe_interval.f() * 10); 398 // Integer division by one. 399 int digit = static_cast<int>(fractionals >> -one.e()); 400 DOUBLE_CONVERSION_ASSERT(digit <= 9); 401 buffer[*length] = static_cast<char>('0' + digit); 402 (*length)++; 403 fractionals &= one.f() - 1; // Modulo by one. 404 (*kappa)--; 405 if (fractionals < unsafe_interval.f()) { 406 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit, 407 unsafe_interval.f(), fractionals, one.f(), unit); 408 } 409 } 410 } 411 412 413 414 // Generates (at most) requested_digits digits of input number w. 415 // w is a floating-point number (DiyFp), consisting of a significand and an 416 // exponent. Its exponent is bounded by kMinimalTargetExponent and 417 // kMaximalTargetExponent. 418 // Hence -60 <= w.e() <= -32. 419 // 420 // Returns false if it fails, in which case the generated digits in the buffer 421 // should not be used. 422 // Preconditions: 423 // * w is correct up to 1 ulp (unit in the last place). That 424 // is, its error must be strictly less than a unit of its last digit. 425 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 426 // 427 // Postconditions: returns false if procedure fails. 428 // otherwise: 429 // * buffer is not null-terminated, but length contains the number of 430 // digits. 431 // * the representation in buffer is the most precise representation of 432 // requested_digits digits. 433 // * buffer contains at most requested_digits digits of w. If there are less 434 // than requested_digits digits then some trailing '0's have been removed. 435 // * kappa is such that 436 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. 437 // 438 // Remark: This procedure takes into account the imprecision of its input 439 // numbers. If the precision is not enough to guarantee all the postconditions 440 // then false is returned. This usually happens rarely, but the failure-rate 441 // increases with higher requested_digits. 442 static bool DigitGenCounted(DiyFp w, 443 int requested_digits, 444 Vector<char> buffer, 445 int* length, 446 int* kappa) { 447 DOUBLE_CONVERSION_ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 448 DOUBLE_CONVERSION_ASSERT(kMinimalTargetExponent >= -60); 449 DOUBLE_CONVERSION_ASSERT(kMaximalTargetExponent <= -32); 450 // w is assumed to have an error less than 1 unit. Whenever w is scaled we 451 // also scale its error. 452 uint64_t w_error = 1; 453 // We cut the input number into two parts: the integral digits and the 454 // fractional digits. We don't emit any decimal separator, but adapt kappa 455 // instead. Example: instead of writing "1.2" we put "12" into the buffer and 456 // increase kappa by 1. 457 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 458 // Division by one is a shift. 459 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); 460 // Modulo by one is an and. 461 uint64_t fractionals = w.f() & (one.f() - 1); 462 uint32_t divisor; 463 int divisor_exponent_plus_one; 464 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 465 &divisor, &divisor_exponent_plus_one); 466 *kappa = divisor_exponent_plus_one; 467 *length = 0; 468 469 // Loop invariant: buffer = w / 10^kappa (integer division) 470 // The invariant holds for the first iteration: kappa has been initialized 471 // with the divisor exponent + 1. And the divisor is the biggest power of ten 472 // that is smaller than 'integrals'. 473 while (*kappa > 0) { 474 int digit = integrals / divisor; 475 DOUBLE_CONVERSION_ASSERT(digit <= 9); 476 buffer[*length] = static_cast<char>('0' + digit); 477 (*length)++; 478 requested_digits--; 479 integrals %= divisor; 480 (*kappa)--; 481 // Note that kappa now equals the exponent of the divisor and that the 482 // invariant thus holds again. 483 if (requested_digits == 0) break; 484 divisor /= 10; 485 } 486 487 if (requested_digits == 0) { 488 uint64_t rest = 489 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 490 return RoundWeedCounted(buffer, *length, rest, 491 static_cast<uint64_t>(divisor) << -one.e(), w_error, 492 kappa); 493 } 494 495 // The integrals have been generated. We are at the point of the decimal 496 // separator. In the following loop we simply multiply the remaining digits by 497 // 10 and divide by one. We just need to pay attention to multiply associated 498 // data (the 'unit'), too. 499 // Note that the multiplication by 10 does not overflow, because w.e >= -60 500 // and thus one.e >= -60. 501 DOUBLE_CONVERSION_ASSERT(one.e() >= -60); 502 DOUBLE_CONVERSION_ASSERT(fractionals < one.f()); 503 DOUBLE_CONVERSION_ASSERT(DOUBLE_CONVERSION_UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); 504 while (requested_digits > 0 && fractionals > w_error) { 505 fractionals *= 10; 506 w_error *= 10; 507 // Integer division by one. 508 int digit = static_cast<int>(fractionals >> -one.e()); 509 DOUBLE_CONVERSION_ASSERT(digit <= 9); 510 buffer[*length] = static_cast<char>('0' + digit); 511 (*length)++; 512 requested_digits--; 513 fractionals &= one.f() - 1; // Modulo by one. 514 (*kappa)--; 515 } 516 if (requested_digits != 0) return false; 517 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, 518 kappa); 519 } 520 521 522 // Provides a decimal representation of v. 523 // Returns true if it succeeds, otherwise the result cannot be trusted. 524 // There will be *length digits inside the buffer (not null-terminated). 525 // If the function returns true then 526 // v == (double) (buffer * 10^decimal_exponent). 527 // The digits in the buffer are the shortest representation possible: no 528 // 0.09999999999999999 instead of 0.1. The shorter representation will even be 529 // chosen even if the longer one would be closer to v. 530 // The last digit will be closest to the actual v. That is, even if several 531 // digits might correctly yield 'v' when read again, the closest will be 532 // computed. 533 static bool Grisu3(double v, 534 FastDtoaMode mode, 535 Vector<char> buffer, 536 int* length, 537 int* decimal_exponent) { 538 DiyFp w = Double(v).AsNormalizedDiyFp(); 539 // boundary_minus and boundary_plus are the boundaries between v and its 540 // closest floating-point neighbors. Any number strictly between 541 // boundary_minus and boundary_plus will round to v when convert to a double. 542 // Grisu3 will never output representations that lie exactly on a boundary. 543 DiyFp boundary_minus, boundary_plus; 544 if (mode == FAST_DTOA_SHORTEST) { 545 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); 546 } else { 547 DOUBLE_CONVERSION_ASSERT(mode == FAST_DTOA_SHORTEST_SINGLE); 548 float single_v = static_cast<float>(v); 549 Single(single_v).NormalizedBoundaries(&boundary_minus, &boundary_plus); 550 } 551 DOUBLE_CONVERSION_ASSERT(boundary_plus.e() == w.e()); 552 DiyFp ten_mk; // Cached power of ten: 10^-k 553 int mk; // -k 554 int ten_mk_minimal_binary_exponent = 555 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 556 int ten_mk_maximal_binary_exponent = 557 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 558 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 559 ten_mk_minimal_binary_exponent, 560 ten_mk_maximal_binary_exponent, 561 &ten_mk, &mk); 562 DOUBLE_CONVERSION_ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 563 DiyFp::kSignificandSize) && 564 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 565 DiyFp::kSignificandSize)); 566 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 567 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 568 569 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 570 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 571 // off by a small amount. 572 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 573 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 574 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 575 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 576 DOUBLE_CONVERSION_ASSERT(scaled_w.e() == 577 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); 578 // In theory it would be possible to avoid some recomputations by computing 579 // the difference between w and boundary_minus/plus (a power of 2) and to 580 // compute scaled_boundary_minus/plus by subtracting/adding from 581 // scaled_w. However the code becomes much less readable and the speed 582 // enhancements are not terrific. 583 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); 584 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); 585 586 // DigitGen will generate the digits of scaled_w. Therefore we have 587 // v == (double) (scaled_w * 10^-mk). 588 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an 589 // integer than it will be updated. For instance if scaled_w == 1.23 then 590 // the buffer will be filled with "123" and the decimal_exponent will be 591 // decreased by 2. 592 int kappa; 593 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, 594 buffer, length, &kappa); 595 *decimal_exponent = -mk + kappa; 596 return result; 597 } 598 599 600 // The "counted" version of grisu3 (see above) only generates requested_digits 601 // number of digits. This version does not generate the shortest representation, 602 // and with enough requested digits 0.1 will at some point print as 0.9999999... 603 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and 604 // therefore the rounding strategy for halfway cases is irrelevant. 605 static bool Grisu3Counted(double v, 606 int requested_digits, 607 Vector<char> buffer, 608 int* length, 609 int* decimal_exponent) { 610 DiyFp w = Double(v).AsNormalizedDiyFp(); 611 DiyFp ten_mk; // Cached power of ten: 10^-k 612 int mk; // -k 613 int ten_mk_minimal_binary_exponent = 614 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 615 int ten_mk_maximal_binary_exponent = 616 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 617 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 618 ten_mk_minimal_binary_exponent, 619 ten_mk_maximal_binary_exponent, 620 &ten_mk, &mk); 621 DOUBLE_CONVERSION_ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 622 DiyFp::kSignificandSize) && 623 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 624 DiyFp::kSignificandSize)); 625 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 626 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 627 628 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 629 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 630 // off by a small amount. 631 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 632 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 633 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 634 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 635 636 // We now have (double) (scaled_w * 10^-mk). 637 // DigitGen will generate the first requested_digits digits of scaled_w and 638 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It 639 // will not always be exactly the same since DigitGenCounted only produces a 640 // limited number of digits.) 641 int kappa; 642 bool result = DigitGenCounted(scaled_w, requested_digits, 643 buffer, length, &kappa); 644 *decimal_exponent = -mk + kappa; 645 return result; 646 } 647 648 649 bool FastDtoa(double v, 650 FastDtoaMode mode, 651 int requested_digits, 652 Vector<char> buffer, 653 int* length, 654 int* decimal_point) { 655 DOUBLE_CONVERSION_ASSERT(v > 0); 656 DOUBLE_CONVERSION_ASSERT(!Double(v).IsSpecial()); 657 658 bool result = false; 659 int decimal_exponent = 0; 660 switch (mode) { 661 case FAST_DTOA_SHORTEST: 662 case FAST_DTOA_SHORTEST_SINGLE: 663 result = Grisu3(v, mode, buffer, length, &decimal_exponent); 664 break; 665 case FAST_DTOA_PRECISION: 666 result = Grisu3Counted(v, requested_digits, 667 buffer, length, &decimal_exponent); 668 break; 669 default: 670 DOUBLE_CONVERSION_UNREACHABLE(); 671 } 672 if (result) { 673 *decimal_point = *length + decimal_exponent; 674 buffer[*length] = '\0'; 675 } 676 return result; 677 } 678 679 } // namespace double_conversion 680 681 // ICU PATCH: Close ICU namespace 682 U_NAMESPACE_END 683 #endif // ICU PATCH: close #if !UCONFIG_NO_FORMATTING