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| 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 "fast-dtoa.h" |
| 29 |
| 30 #include "cached-powers.h" |
| 31 #include "diy-fp.h" |
| 32 #include "double.h" |
| 33 |
| 34 #include <stdio.h> |
| 35 |
| 36 namespace double_conversion { |
| 37 |
| 38 // The minimal and maximal target exponent define the range of w's binary |
| 39 // exponent, where 'w' is the result of multiplying the input by a cached power |
| 40 // of ten. |
| 41 // |
| 42 // A different range might be chosen on a different platform, to optimize digit |
| 43 // generation, but a smaller range requires more powers of ten to be cached. |
| 44 static const int kMinimalTargetExponent = -60; |
| 45 static const int kMaximalTargetExponent = -32; |
| 46 |
| 47 |
| 48 // Adjusts the last digit of the generated number, and screens out generated |
| 49 // solutions that may be inaccurate. A solution may be inaccurate if it is |
| 50 // outside the safe interval, or if we cannot prove that it is closer to the |
| 51 // input than a neighboring representation of the same length. |
| 52 // |
| 53 // Input: * buffer containing the digits of too_high / 10^kappa |
| 54 // * the buffer's length |
| 55 // * distance_too_high_w == (too_high - w).f() * unit |
| 56 // * unsafe_interval == (too_high - too_low).f() * unit |
| 57 // * rest = (too_high - buffer * 10^kappa).f() * unit |
| 58 // * ten_kappa = 10^kappa * unit |
| 59 // * unit = the common multiplier |
| 60 // Output: returns true if the buffer is guaranteed to contain the closest |
| 61 // representable number to the input. |
| 62 // Modifies the generated digits in the buffer to approach (round towards) w. |
| 63 static bool RoundWeed(Vector<char> buffer, |
| 64 int length, |
| 65 uint64_t distance_too_high_w, |
| 66 uint64_t unsafe_interval, |
| 67 uint64_t rest, |
| 68 uint64_t ten_kappa, |
| 69 uint64_t unit) { |
| 70 uint64_t small_distance = distance_too_high_w - unit; |
| 71 uint64_t big_distance = distance_too_high_w + unit; |
| 72 // Let w_low = too_high - big_distance, and |
| 73 // w_high = too_high - small_distance. |
| 74 // Note: w_low < w < w_high |
| 75 // |
| 76 // The real w (* unit) must lie somewhere inside the interval |
| 77 // ]w_low; w_high[ (often written as "(w_low; w_high)") |
| 78 |
| 79 // Basically the buffer currently contains a number in the unsafe interval |
| 80 // ]too_low; too_high[ with too_low < w < too_high |
| 81 // |
| 82 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
| 83 // ^v 1 unit ^ ^ ^ ^ |
| 84 // boundary_high --------------------- . . . . |
| 85 // ^v 1 unit . . . . |
| 86 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . |
| 87 // . . ^ . . |
| 88 // . big_distance . . . |
| 89 // . . . . rest |
| 90 // small_distance . . . . |
| 91 // v . . . . |
| 92 // w_high - - - - - - - - - - - - - - - - - - . . . . |
| 93 // ^v 1 unit . . . . |
| 94 // w ---------------------------------------- . . . . |
| 95 // ^v 1 unit v . . . |
| 96 // w_low - - - - - - - - - - - - - - - - - - - - - . . . |
| 97 // . . v |
| 98 // buffer --------------------------------------------------+-------+-------- |
| 99 // . . |
| 100 // safe_interval . |
| 101 // v . |
| 102 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . |
| 103 // ^v 1 unit . |
| 104 // boundary_low ------------------------- unsafe_interval |
| 105 // ^v 1 unit v |
| 106 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
| 107 // |
| 108 // |
| 109 // Note that the value of buffer could lie anywhere inside the range too_low |
| 110 // to too_high. |
| 111 // |
| 112 // boundary_low, boundary_high and w are approximations of the real boundaries |
| 113 // and v (the input number). They are guaranteed to be precise up to one unit. |
| 114 // In fact the error is guaranteed to be strictly less than one unit. |
| 115 // |
| 116 // Anything that lies outside the unsafe interval is guaranteed not to round |
| 117 // to v when read again. |
| 118 // Anything that lies inside the safe interval is guaranteed to round to v |
| 119 // when read again. |
| 120 // If the number inside the buffer lies inside the unsafe interval but not |
| 121 // inside the safe interval then we simply do not know and bail out (returning |
| 122 // false). |
| 123 // |
| 124 // Similarly we have to take into account the imprecision of 'w' when finding |
| 125 // the closest representation of 'w'. If we have two potential |
| 126 // representations, and one is closer to both w_low and w_high, then we know |
| 127 // it is closer to the actual value v. |
| 128 // |
| 129 // By generating the digits of too_high we got the largest (closest to |
| 130 // too_high) buffer that is still in the unsafe interval. In the case where |
| 131 // w_high < buffer < too_high we try to decrement the buffer. |
| 132 // This way the buffer approaches (rounds towards) w. |
| 133 // There are 3 conditions that stop the decrementation process: |
| 134 // 1) the buffer is already below w_high |
| 135 // 2) decrementing the buffer would make it leave the unsafe interval |
| 136 // 3) decrementing the buffer would yield a number below w_high and farther |
| 137 // away than the current number. In other words: |
| 138 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high |
| 139 // Instead of using the buffer directly we use its distance to too_high. |
| 140 // Conceptually rest ~= too_high - buffer |
| 141 // We need to do the following tests in this order to avoid over- and |
| 142 // underflows. |
| 143 ASSERT(rest <= unsafe_interval); |
| 144 while (rest < small_distance && // Negated condition 1 |
| 145 unsafe_interval - rest >= ten_kappa && // Negated condition 2 |
| 146 (rest + ten_kappa < small_distance || // buffer{-1} > w_high |
| 147 small_distance - rest >= rest + ten_kappa - small_distance)) { |
| 148 buffer[length - 1]--; |
| 149 rest += ten_kappa; |
| 150 } |
| 151 |
| 152 // We have approached w+ as much as possible. We now test if approaching w- |
| 153 // would require changing the buffer. If yes, then we have two possible |
| 154 // representations close to w, but we cannot decide which one is closer. |
| 155 if (rest < big_distance && |
| 156 unsafe_interval - rest >= ten_kappa && |
| 157 (rest + ten_kappa < big_distance || |
| 158 big_distance - rest > rest + ten_kappa - big_distance)) { |
| 159 return false; |
| 160 } |
| 161 |
| 162 // Weeding test. |
| 163 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] |
| 164 // Since too_low = too_high - unsafe_interval this is equivalent to |
| 165 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] |
| 166 // Conceptually we have: rest ~= too_high - buffer |
| 167 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); |
| 168 } |
| 169 |
| 170 |
| 171 // Rounds the buffer upwards if the result is closer to v by possibly adding |
| 172 // 1 to the buffer. If the precision of the calculation is not sufficient to |
| 173 // round correctly, return false. |
| 174 // The rounding might shift the whole buffer in which case the kappa is |
| 175 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. |
| 176 // |
| 177 // If 2*rest > ten_kappa then the buffer needs to be round up. |
| 178 // rest can have an error of +/- 1 unit. This function accounts for the |
| 179 // imprecision and returns false, if the rounding direction cannot be |
| 180 // unambiguously determined. |
| 181 // |
| 182 // Precondition: rest < ten_kappa. |
| 183 static bool RoundWeedCounted(Vector<char> buffer, |
| 184 int length, |
| 185 uint64_t rest, |
| 186 uint64_t ten_kappa, |
| 187 uint64_t unit, |
| 188 int* kappa) { |
| 189 ASSERT(rest < ten_kappa); |
| 190 // The following tests are done in a specific order to avoid overflows. They |
| 191 // will work correctly with any uint64 values of rest < ten_kappa and unit. |
| 192 // |
| 193 // If the unit is too big, then we don't know which way to round. For example |
| 194 // a unit of 50 means that the real number lies within rest +/- 50. If |
| 195 // 10^kappa == 40 then there is no way to tell which way to round. |
| 196 if (unit >= ten_kappa) return false; |
| 197 // Even if unit is just half the size of 10^kappa we are already completely |
| 198 // lost. (And after the previous test we know that the expression will not |
| 199 // over/underflow.) |
| 200 if (ten_kappa - unit <= unit) return false; |
| 201 // If 2 * (rest + unit) <= 10^kappa we can safely round down. |
| 202 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) { |
| 203 return true; |
| 204 } |
| 205 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. |
| 206 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) { |
| 207 // Increment the last digit recursively until we find a non '9' digit. |
| 208 buffer[length - 1]++; |
| 209 for (int i = length - 1; i > 0; --i) { |
| 210 if (buffer[i] != '0' + 10) break; |
| 211 buffer[i] = '0'; |
| 212 buffer[i - 1]++; |
| 213 } |
| 214 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the |
| 215 // exception of the first digit all digits are now '0'. Simply switch the |
| 216 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and |
| 217 // the power (the kappa) is increased. |
| 218 if (buffer[0] == '0' + 10) { |
| 219 buffer[0] = '1'; |
| 220 (*kappa) += 1; |
| 221 } |
| 222 return true; |
| 223 } |
| 224 return false; |
| 225 } |
| 226 |
| 227 // Returns the biggest power of ten that is less than or equal to the given |
| 228 // number. We furthermore receive the maximum number of bits 'number' has. |
| 229 // |
| 230 // Returns power == 10^(exponent_plus_one-1) such that |
| 231 // power <= number < power * 10. |
| 232 // If number_bits == 0 then 0^(0-1) is returned. |
| 233 // The number of bits must be <= 32. |
| 234 // Precondition: number < (1 << (number_bits + 1)). |
| 235 |
| 236 // Inspired by the method for finding an integer log base 10 from here: |
| 237 // http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10 |
| 238 static unsigned int const kSmallPowersOfTen[] = |
| 239 {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, |
| 240 1000000000}; |
| 241 |
| 242 static void BiggestPowerTen(uint32_t number, |
| 243 int number_bits, |
| 244 uint32_t* power, |
| 245 int* exponent_plus_one) { |
| 246 ASSERT(number < (1 << (number_bits + 1))); |
| 247 // 1233/4096 is approximately 1/lg(10). |
| 248 int exponent_plus_one_guess = ((number_bits + 1) * 1233 >> 12); |
| 249 // We increment to skip over the first entry in the kPowersOf10 table. |
| 250 // Note: kPowersOf10[i] == 10^(i-1). |
| 251 exponent_plus_one_guess++; |
| 252 // We don't have any guarantees that 2^number_bits <= number. |
| 253 // TODO(floitsch): can we change the 'while' into an 'if'? We definitely see |
| 254 // number < (2^number_bits - 1), but I haven't encountered |
| 255 // number < (2^number_bits - 2) yet. |
| 256 while (number < kSmallPowersOfTen[exponent_plus_one_guess]) { |
| 257 exponent_plus_one_guess--; |
| 258 } |
| 259 *power = kSmallPowersOfTen[exponent_plus_one_guess]; |
| 260 *exponent_plus_one = exponent_plus_one_guess; |
| 261 } |
| 262 |
| 263 // Generates the digits of input number w. |
| 264 // w is a floating-point number (DiyFp), consisting of a significand and an |
| 265 // exponent. Its exponent is bounded by kMinimalTargetExponent and |
| 266 // kMaximalTargetExponent. |
| 267 // Hence -60 <= w.e() <= -32. |
| 268 // |
| 269 // Returns false if it fails, in which case the generated digits in the buffer |
| 270 // should not be used. |
| 271 // Preconditions: |
| 272 // * low, w and high are correct up to 1 ulp (unit in the last place). That |
| 273 // is, their error must be less than a unit of their last digits. |
| 274 // * low.e() == w.e() == high.e() |
| 275 // * low < w < high, and taking into account their error: low~ <= high~ |
| 276 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent |
| 277 // Postconditions: returns false if procedure fails. |
| 278 // otherwise: |
| 279 // * buffer is not null-terminated, but len contains the number of digits. |
| 280 // * buffer contains the shortest possible decimal digit-sequence |
| 281 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the |
| 282 // correct values of low and high (without their error). |
| 283 // * if more than one decimal representation gives the minimal number of |
| 284 // decimal digits then the one closest to W (where W is the correct value |
| 285 // of w) is chosen. |
| 286 // Remark: this procedure takes into account the imprecision of its input |
| 287 // numbers. If the precision is not enough to guarantee all the postconditions |
| 288 // then false is returned. This usually happens rarely (~0.5%). |
| 289 // |
| 290 // Say, for the sake of example, that |
| 291 // w.e() == -48, and w.f() == 0x1234567890abcdef |
| 292 // w's value can be computed by w.f() * 2^w.e() |
| 293 // We can obtain w's integral digits by simply shifting w.f() by -w.e(). |
| 294 // -> w's integral part is 0x1234 |
| 295 // w's fractional part is therefore 0x567890abcdef. |
| 296 // Printing w's integral part is easy (simply print 0x1234 in decimal). |
| 297 // In order to print its fraction we repeatedly multiply the fraction by 10 and |
| 298 // get each digit. Example the first digit after the point would be computed by |
| 299 // (0x567890abcdef * 10) >> 48. -> 3 |
| 300 // The whole thing becomes slightly more complicated because we want to stop |
| 301 // once we have enough digits. That is, once the digits inside the buffer |
| 302 // represent 'w' we can stop. Everything inside the interval low - high |
| 303 // represents w. However we have to pay attention to low, high and w's |
| 304 // imprecision. |
| 305 static bool DigitGen(DiyFp low, |
| 306 DiyFp w, |
| 307 DiyFp high, |
| 308 Vector<char> buffer, |
| 309 int* length, |
| 310 int* kappa) { |
| 311 ASSERT(low.e() == w.e() && w.e() == high.e()); |
| 312 ASSERT(low.f() + 1 <= high.f() - 1); |
| 313 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); |
| 314 // low, w and high are imprecise, but by less than one ulp (unit in the last |
| 315 // place). |
| 316 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that |
| 317 // the new numbers are outside of the interval we want the final |
| 318 // representation to lie in. |
| 319 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield |
| 320 // numbers that are certain to lie in the interval. We will use this fact |
| 321 // later on. |
| 322 // We will now start by generating the digits within the uncertain |
| 323 // interval. Later we will weed out representations that lie outside the safe |
| 324 // interval and thus _might_ lie outside the correct interval. |
| 325 uint64_t unit = 1; |
| 326 DiyFp too_low = DiyFp(low.f() - unit, low.e()); |
| 327 DiyFp too_high = DiyFp(high.f() + unit, high.e()); |
| 328 // too_low and too_high are guaranteed to lie outside the interval we want the |
| 329 // generated number in. |
| 330 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low); |
| 331 // We now cut the input number into two parts: the integral digits and the |
| 332 // fractionals. We will not write any decimal separator though, but adapt |
| 333 // kappa instead. |
| 334 // Reminder: we are currently computing the digits (stored inside the buffer) |
| 335 // such that: too_low < buffer * 10^kappa < too_high |
| 336 // We use too_high for the digit_generation and stop as soon as possible. |
| 337 // If we stop early we effectively round down. |
| 338 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); |
| 339 // Division by one is a shift. |
| 340 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); |
| 341 // Modulo by one is an and. |
| 342 uint64_t fractionals = too_high.f() & (one.f() - 1); |
| 343 uint32_t divisor; |
| 344 int divisor_exponent_plus_one; |
| 345 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), |
| 346 &divisor, &divisor_exponent_plus_one); |
| 347 *kappa = divisor_exponent_plus_one; |
| 348 *length = 0; |
| 349 // Loop invariant: buffer = too_high / 10^kappa (integer division) |
| 350 // The invariant holds for the first iteration: kappa has been initialized |
| 351 // with the divisor exponent + 1. And the divisor is the biggest power of ten |
| 352 // that is smaller than integrals. |
| 353 while (*kappa > 0) { |
| 354 int digit = integrals / divisor; |
| 355 buffer[*length] = '0' + digit; |
| 356 (*length)++; |
| 357 integrals %= divisor; |
| 358 (*kappa)--; |
| 359 // Note that kappa now equals the exponent of the divisor and that the |
| 360 // invariant thus holds again. |
| 361 uint64_t rest = |
| 362 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; |
| 363 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) |
| 364 // Reminder: unsafe_interval.e() == one.e() |
| 365 if (rest < unsafe_interval.f()) { |
| 366 // Rounding down (by not emitting the remaining digits) yields a number |
| 367 // that lies within the unsafe interval. |
| 368 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), |
| 369 unsafe_interval.f(), rest, |
| 370 static_cast<uint64_t>(divisor) << -one.e(), unit); |
| 371 } |
| 372 divisor /= 10; |
| 373 } |
| 374 |
| 375 // The integrals have been generated. We are at the point of the decimal |
| 376 // separator. In the following loop we simply multiply the remaining digits by |
| 377 // 10 and divide by one. We just need to pay attention to multiply associated |
| 378 // data (like the interval or 'unit'), too. |
| 379 // Note that the multiplication by 10 does not overflow, because w.e >= -60 |
| 380 // and thus one.e >= -60. |
| 381 ASSERT(one.e() >= -60); |
| 382 ASSERT(fractionals < one.f()); |
| 383 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); |
| 384 while (true) { |
| 385 fractionals *= 10; |
| 386 unit *= 10; |
| 387 unsafe_interval.set_f(unsafe_interval.f() * 10); |
| 388 // Integer division by one. |
| 389 int digit = static_cast<int>(fractionals >> -one.e()); |
| 390 buffer[*length] = '0' + digit; |
| 391 (*length)++; |
| 392 fractionals &= one.f() - 1; // Modulo by one. |
| 393 (*kappa)--; |
| 394 if (fractionals < unsafe_interval.f()) { |
| 395 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit, |
| 396 unsafe_interval.f(), fractionals, one.f(), unit); |
| 397 } |
| 398 } |
| 399 } |
| 400 |
| 401 |
| 402 |
| 403 // Generates (at most) requested_digits digits of input number w. |
| 404 // w is a floating-point number (DiyFp), consisting of a significand and an |
| 405 // exponent. Its exponent is bounded by kMinimalTargetExponent and |
| 406 // kMaximalTargetExponent. |
| 407 // Hence -60 <= w.e() <= -32. |
| 408 // |
| 409 // Returns false if it fails, in which case the generated digits in the buffer |
| 410 // should not be used. |
| 411 // Preconditions: |
| 412 // * w is correct up to 1 ulp (unit in the last place). That |
| 413 // is, its error must be strictly less than a unit of its last digit. |
| 414 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent |
| 415 // |
| 416 // Postconditions: returns false if procedure fails. |
| 417 // otherwise: |
| 418 // * buffer is not null-terminated, but length contains the number of |
| 419 // digits. |
| 420 // * the representation in buffer is the most precise representation of |
| 421 // requested_digits digits. |
| 422 // * buffer contains at most requested_digits digits of w. If there are less |
| 423 // than requested_digits digits then some trailing '0's have been removed. |
| 424 // * kappa is such that |
| 425 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. |
| 426 // |
| 427 // Remark: This procedure takes into account the imprecision of its input |
| 428 // numbers. If the precision is not enough to guarantee all the postconditions |
| 429 // then false is returned. This usually happens rarely, but the failure-rate |
| 430 // increases with higher requested_digits. |
| 431 static bool DigitGenCounted(DiyFp w, |
| 432 int requested_digits, |
| 433 Vector<char> buffer, |
| 434 int* length, |
| 435 int* kappa) { |
| 436 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); |
| 437 ASSERT(kMinimalTargetExponent >= -60); |
| 438 ASSERT(kMaximalTargetExponent <= -32); |
| 439 // w is assumed to have an error less than 1 unit. Whenever w is scaled we |
| 440 // also scale its error. |
| 441 uint64_t w_error = 1; |
| 442 // We cut the input number into two parts: the integral digits and the |
| 443 // fractional digits. We don't emit any decimal separator, but adapt kappa |
| 444 // instead. Example: instead of writing "1.2" we put "12" into the buffer and |
| 445 // increase kappa by 1. |
| 446 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); |
| 447 // Division by one is a shift. |
| 448 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); |
| 449 // Modulo by one is an and. |
| 450 uint64_t fractionals = w.f() & (one.f() - 1); |
| 451 uint32_t divisor; |
| 452 int divisor_exponent_plus_one; |
| 453 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), |
| 454 &divisor, &divisor_exponent_plus_one); |
| 455 *kappa = divisor_exponent_plus_one; |
| 456 *length = 0; |
| 457 |
| 458 // Loop invariant: buffer = w / 10^kappa (integer division) |
| 459 // The invariant holds for the first iteration: kappa has been initialized |
| 460 // with the divisor exponent + 1. And the divisor is the biggest power of ten |
| 461 // that is smaller than 'integrals'. |
| 462 while (*kappa > 0) { |
| 463 int digit = integrals / divisor; |
| 464 buffer[*length] = '0' + digit; |
| 465 (*length)++; |
| 466 requested_digits--; |
| 467 integrals %= divisor; |
| 468 (*kappa)--; |
| 469 // Note that kappa now equals the exponent of the divisor and that the |
| 470 // invariant thus holds again. |
| 471 if (requested_digits == 0) break; |
| 472 divisor /= 10; |
| 473 } |
| 474 |
| 475 if (requested_digits == 0) { |
| 476 uint64_t rest = |
| 477 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; |
| 478 return RoundWeedCounted(buffer, *length, rest, |
| 479 static_cast<uint64_t>(divisor) << -one.e(), w_error, |
| 480 kappa); |
| 481 } |
| 482 |
| 483 // The integrals have been generated. We are at the point of the decimal |
| 484 // separator. In the following loop we simply multiply the remaining digits by |
| 485 // 10 and divide by one. We just need to pay attention to multiply associated |
| 486 // data (the 'unit'), too. |
| 487 // Note that the multiplication by 10 does not overflow, because w.e >= -60 |
| 488 // and thus one.e >= -60. |
| 489 ASSERT(one.e() >= -60); |
| 490 ASSERT(fractionals < one.f()); |
| 491 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); |
| 492 while (requested_digits > 0 && fractionals > w_error) { |
| 493 fractionals *= 10; |
| 494 w_error *= 10; |
| 495 // Integer division by one. |
| 496 int digit = static_cast<int>(fractionals >> -one.e()); |
| 497 buffer[*length] = '0' + digit; |
| 498 (*length)++; |
| 499 requested_digits--; |
| 500 fractionals &= one.f() - 1; // Modulo by one. |
| 501 (*kappa)--; |
| 502 } |
| 503 if (requested_digits != 0) return false; |
| 504 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, |
| 505 kappa); |
| 506 } |
| 507 |
| 508 |
| 509 // Provides a decimal representation of v. |
| 510 // Returns true if it succeeds, otherwise the result cannot be trusted. |
| 511 // There will be *length digits inside the buffer (not null-terminated). |
| 512 // If the function returns true then |
| 513 // v == (double) (buffer * 10^decimal_exponent). |
| 514 // The digits in the buffer are the shortest representation possible: no |
| 515 // 0.09999999999999999 instead of 0.1. The shorter representation will even be |
| 516 // chosen even if the longer one would be closer to v. |
| 517 // The last digit will be closest to the actual v. That is, even if several |
| 518 // digits might correctly yield 'v' when read again, the closest will be |
| 519 // computed. |
| 520 static bool Grisu3(double v, |
| 521 Vector<char> buffer, |
| 522 int* length, |
| 523 int* decimal_exponent) { |
| 524 DiyFp w = Double(v).AsNormalizedDiyFp(); |
| 525 // boundary_minus and boundary_plus are the boundaries between v and its |
| 526 // closest floating-point neighbors. Any number strictly between |
| 527 // boundary_minus and boundary_plus will round to v when convert to a double. |
| 528 // Grisu3 will never output representations that lie exactly on a boundary. |
| 529 DiyFp boundary_minus, boundary_plus; |
| 530 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); |
| 531 ASSERT(boundary_plus.e() == w.e()); |
| 532 DiyFp ten_mk; // Cached power of ten: 10^-k |
| 533 int mk; // -k |
| 534 int ten_mk_minimal_binary_exponent = |
| 535 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 536 int ten_mk_maximal_binary_exponent = |
| 537 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 538 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( |
| 539 ten_mk_minimal_binary_exponent, |
| 540 ten_mk_maximal_binary_exponent, |
| 541 &ten_mk, &mk); |
| 542 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + |
| 543 DiyFp::kSignificandSize) && |
| 544 (kMaximalTargetExponent >= w.e() + ten_mk.e() + |
| 545 DiyFp::kSignificandSize)); |
| 546 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a |
| 547 // 64 bit significand and ten_mk is thus only precise up to 64 bits. |
| 548 |
| 549 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated |
| 550 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now |
| 551 // off by a small amount. |
| 552 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. |
| 553 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then |
| 554 // (f-1) * 2^e < w*10^k < (f+1) * 2^e |
| 555 DiyFp scaled_w = DiyFp::Times(w, ten_mk); |
| 556 ASSERT(scaled_w.e() == |
| 557 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); |
| 558 // In theory it would be possible to avoid some recomputations by computing |
| 559 // the difference between w and boundary_minus/plus (a power of 2) and to |
| 560 // compute scaled_boundary_minus/plus by subtracting/adding from |
| 561 // scaled_w. However the code becomes much less readable and the speed |
| 562 // enhancements are not terriffic. |
| 563 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); |
| 564 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); |
| 565 |
| 566 // DigitGen will generate the digits of scaled_w. Therefore we have |
| 567 // v == (double) (scaled_w * 10^-mk). |
| 568 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an |
| 569 // integer than it will be updated. For instance if scaled_w == 1.23 then |
| 570 // the buffer will be filled with "123" und the decimal_exponent will be |
| 571 // decreased by 2. |
| 572 int kappa; |
| 573 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, |
| 574 buffer, length, &kappa); |
| 575 *decimal_exponent = -mk + kappa; |
| 576 return result; |
| 577 } |
| 578 |
| 579 |
| 580 // The "counted" version of grisu3 (see above) only generates requested_digits |
| 581 // number of digits. This version does not generate the shortest representation, |
| 582 // and with enough requested digits 0.1 will at some point print as 0.9999999... |
| 583 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and |
| 584 // therefore the rounding strategy for halfway cases is irrelevant. |
| 585 static bool Grisu3Counted(double v, |
| 586 int requested_digits, |
| 587 Vector<char> buffer, |
| 588 int* length, |
| 589 int* decimal_exponent) { |
| 590 DiyFp w = Double(v).AsNormalizedDiyFp(); |
| 591 DiyFp ten_mk; // Cached power of ten: 10^-k |
| 592 int mk; // -k |
| 593 int ten_mk_minimal_binary_exponent = |
| 594 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 595 int ten_mk_maximal_binary_exponent = |
| 596 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 597 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( |
| 598 ten_mk_minimal_binary_exponent, |
| 599 ten_mk_maximal_binary_exponent, |
| 600 &ten_mk, &mk); |
| 601 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + |
| 602 DiyFp::kSignificandSize) && |
| 603 (kMaximalTargetExponent >= w.e() + ten_mk.e() + |
| 604 DiyFp::kSignificandSize)); |
| 605 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a |
| 606 // 64 bit significand and ten_mk is thus only precise up to 64 bits. |
| 607 |
| 608 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated |
| 609 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now |
| 610 // off by a small amount. |
| 611 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. |
| 612 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then |
| 613 // (f-1) * 2^e < w*10^k < (f+1) * 2^e |
| 614 DiyFp scaled_w = DiyFp::Times(w, ten_mk); |
| 615 |
| 616 // We now have (double) (scaled_w * 10^-mk). |
| 617 // DigitGen will generate the first requested_digits digits of scaled_w and |
| 618 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It |
| 619 // will not always be exactly the same since DigitGenCounted only produces a |
| 620 // limited number of digits.) |
| 621 int kappa; |
| 622 bool result = DigitGenCounted(scaled_w, requested_digits, |
| 623 buffer, length, &kappa); |
| 624 *decimal_exponent = -mk + kappa; |
| 625 return result; |
| 626 } |
| 627 |
| 628 |
| 629 bool FastDtoa(double v, |
| 630 FastDtoaMode mode, |
| 631 int requested_digits, |
| 632 Vector<char> buffer, |
| 633 int* length, |
| 634 int* decimal_point) { |
| 635 ASSERT(v > 0); |
| 636 ASSERT(!Double(v).IsSpecial()); |
| 637 |
| 638 bool result = false; |
| 639 int decimal_exponent = 0; |
| 640 switch (mode) { |
| 641 case FAST_DTOA_SHORTEST: |
| 642 result = Grisu3(v, buffer, length, &decimal_exponent); |
| 643 break; |
| 644 case FAST_DTOA_PRECISION: |
| 645 result = Grisu3Counted(v, requested_digits, |
| 646 buffer, length, &decimal_exponent); |
| 647 break; |
| 648 default: |
| 649 UNREACHABLE(); |
| 650 } |
| 651 if (result) { |
| 652 *decimal_point = *length + decimal_exponent; |
| 653 buffer[*length] = '\0'; |
| 654 } |
| 655 return result; |
| 656 } |
| 657 |
| 658 } // namespace double_conversion |
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